diff options
| author | Eric Andersen <andersen@codepoet.org> | 2001-11-22 14:04:29 +0000 |
|---|---|---|
| committer | Eric Andersen <andersen@codepoet.org> | 2001-11-22 14:04:29 +0000 |
| commit | 7ce331c01ce6eb7b3f5c715a38a24359da9c6ee2 (patch) | |
| tree | 3a7e8476e868ae15f4da1b7ce26b2db6f434468c /libm/float | |
| parent | c117dd5fb183afb1a4790a6f6110d88704be6bf8 (diff) | |
Totally rework the math library, this time based on the MacOs X
math library (which is itself based on the math lib from FreeBSD).
-Erik
Diffstat (limited to 'libm/float')
82 files changed, 0 insertions, 20647 deletions
diff --git a/libm/float/Makefile b/libm/float/Makefile deleted file mode 100644 index 80f7aa1ff..000000000 --- a/libm/float/Makefile +++ /dev/null @@ -1,59 +0,0 @@ -# Makefile for uClibc's math library -# Copyright (C) 2001 by Lineo, inc. -# -# This math library is derived primarily from the Cephes Math Library, -# copyright by Stephen L. Moshier <moshier@world.std.com> -# -# This program is free software; you can redistribute it and/or modify it under -# the terms of the GNU Library General Public License as published by the Free -# Software Foundation; either version 2 of the License, or (at your option) any -# later version. -# -# This program is distributed in the hope that it will be useful, but WITHOUT -# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS -# FOR A PARTICULAR PURPOSE. See the GNU Library General Public License for more -# details. -# -# You should have received a copy of the GNU Library General Public License -# along with this program; if not, write to the Free Software Foundation, Inc., -# 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA -# - -TOPDIR=../../ -include $(TOPDIR)Rules.mak - -LIBM=../libm.a -TARGET_CC= $(TOPDIR)/extra/gcc-uClibc/$(TARGET_ARCH)-uclibc-gcc - -CSRC= acoshf.c airyf.c asinf.c asinhf.c atanf.c \ - atanhf.c bdtrf.c betaf.c cbrtf.c chbevlf.c chdtrf.c \ - clogf.c cmplxf.c constf.c coshf.c dawsnf.c ellief.c \ - ellikf.c ellpef.c ellpkf.c ellpjf.c expf.c exp2f.c \ - exp10f.c expnf.c facf.c fdtrf.c floorf.c fresnlf.c \ - gammaf.c gdtrf.c hypergf.c hyp2f1f.c igamf.c igamif.c \ - incbetf.c incbif.c i0f.c i1f.c ivf.c j0f.c j1f.c \ - jnf.c jvf.c k0f.c k1f.c knf.c logf.c log2f.c \ - log10f.c nbdtrf.c ndtrf.c ndtrif.c pdtrf.c polynf.c \ - powif.c powf.c psif.c rgammaf.c shichif.c sicif.c \ - sindgf.c sinf.c sinhf.c spencef.c sqrtf.c stdtrf.c \ - struvef.c tandgf.c tanf.c tanhf.c ynf.c zetaf.c \ - zetacf.c polevlf.c setprec.c mtherr.c -COBJS=$(patsubst %.c,%.o, $(CSRC)) -OBJS=$(COBJS) - -all: $(OBJS) $(LIBM) - -$(LIBM): ar-target - -ar-target: $(OBJS) - $(AR) $(ARFLAGS) $(LIBM) $(OBJS) - -$(COBJS): %.o : %.c - $(TARGET_CC) $(TARGET_CFLAGS) -c $< -o $@ - $(STRIPTOOL) -x -R .note -R .comment $*.o - -$(OBJ): Makefile - -clean: - rm -f *.[oa] *~ core - diff --git a/libm/float/README.txt b/libm/float/README.txt deleted file mode 100644 index 30a10b083..000000000 --- a/libm/float/README.txt +++ /dev/null @@ -1,4721 +0,0 @@ -/* acoshf.c - * - * Inverse hyperbolic cosine - * - * - * - * SYNOPSIS: - * - * float x, y, acoshf(); - * - * y = acoshf( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic cosine of argument. - * - * If 1 <= x < 1.5, a polynomial approximation - * - * sqrt(z) * P(z) - * - * where z = x-1, is used. Otherwise, - * - * acosh(x) = log( x + sqrt( (x-1)(x+1) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 1,3 100000 1.8e-7 3.9e-8 - * IEEE 1,2000 100000 3.0e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * acoshf domain |x| < 1 0.0 - * - */ - -/* airy.c - * - * Airy function - * - * - * - * SYNOPSIS: - * - * float x, ai, aip, bi, bip; - * int airyf(); - * - * airyf( x, _&ai, _&aip, _&bi, _&bip ); - * - * - * - * DESCRIPTION: - * - * Solution of the differential equation - * - * y"(x) = xy. - * - * The function returns the two independent solutions Ai, Bi - * and their first derivatives Ai'(x), Bi'(x). - * - * Evaluation is by power series summation for small x, - * by rational minimax approximations for large x. - * - * - * - * ACCURACY: - * Error criterion is absolute when function <= 1, relative - * when function > 1, except * denotes relative error criterion. - * For large negative x, the absolute error increases as x^1.5. - * For large positive x, the relative error increases as x^1.5. - * - * Arithmetic domain function # trials peak rms - * IEEE -10, 0 Ai 50000 7.0e-7 1.2e-7 - * IEEE 0, 10 Ai 50000 9.9e-6* 6.8e-7* - * IEEE -10, 0 Ai' 50000 2.4e-6 3.5e-7 - * IEEE 0, 10 Ai' 50000 8.7e-6* 6.2e-7* - * IEEE -10, 10 Bi 100000 2.2e-6 2.6e-7 - * IEEE -10, 10 Bi' 50000 2.2e-6 3.5e-7 - * - */ - -/* asinf.c - * - * Inverse circular sine - * - * - * - * SYNOPSIS: - * - * float x, y, asinf(); - * - * y = asinf( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between -pi/2 and +pi/2 whose sine is x. - * - * A polynomial of the form x + x**3 P(x**2) - * is used for |x| in the interval [0, 0.5]. If |x| > 0.5 it is - * transformed by the identity - * - * asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -1, 1 100000 2.5e-7 5.0e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * asinf domain |x| > 1 0.0 - * - */ - /* acosf() - * - * Inverse circular cosine - * - * - * - * SYNOPSIS: - * - * float x, y, acosf(); - * - * y = acosf( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between -pi/2 and +pi/2 whose cosine - * is x. - * - * Analytically, acos(x) = pi/2 - asin(x). However if |x| is - * near 1, there is cancellation error in subtracting asin(x) - * from pi/2. Hence if x < -0.5, - * - * acos(x) = pi - 2.0 * asin( sqrt((1+x)/2) ); - * - * or if x > +0.5, - * - * acos(x) = 2.0 * asin( sqrt((1-x)/2) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -1, 1 100000 1.4e-7 4.2e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * acosf domain |x| > 1 0.0 - */ - -/* asinhf.c - * - * Inverse hyperbolic sine - * - * - * - * SYNOPSIS: - * - * float x, y, asinhf(); - * - * y = asinhf( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic sine of argument. - * - * If |x| < 0.5, the function is approximated by a rational - * form x + x**3 P(x)/Q(x). Otherwise, - * - * asinh(x) = log( x + sqrt(1 + x*x) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -3,3 100000 2.4e-7 4.1e-8 - * - */ - -/* atanf.c - * - * Inverse circular tangent - * (arctangent) - * - * - * - * SYNOPSIS: - * - * float x, y, atanf(); - * - * y = atanf( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between -pi/2 and +pi/2 whose tangent - * is x. - * - * Range reduction is from four intervals into the interval - * from zero to tan( pi/8 ). A polynomial approximates - * the function in this basic interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10, 10 100000 1.9e-7 4.1e-8 - * - */ - /* atan2f() - * - * Quadrant correct inverse circular tangent - * - * - * - * SYNOPSIS: - * - * float x, y, z, atan2f(); - * - * z = atan2f( y, x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle whose tangent is y/x. - * Define compile time symbol ANSIC = 1 for ANSI standard, - * range -PI < z <= +PI, args (y,x); else ANSIC = 0 for range - * 0 to 2PI, args (x,y). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10, 10 100000 1.9e-7 4.1e-8 - * See atan.c. - * - */ - -/* atanhf.c - * - * Inverse hyperbolic tangent - * - * - * - * SYNOPSIS: - * - * float x, y, atanhf(); - * - * y = atanhf( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic tangent of argument in the range - * MINLOGF to MAXLOGF. - * - * If |x| < 0.5, a polynomial approximation is used. - * Otherwise, - * atanh(x) = 0.5 * log( (1+x)/(1-x) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -1,1 100000 1.4e-7 3.1e-8 - * - */ - -/* bdtrf.c - * - * Binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, bdtrf(); - * - * y = bdtrf( k, n, p ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms 0 through k of the Binomial - * probability density: - * - * k - * -- ( n ) j n-j - * > ( ) p (1-p) - * -- ( j ) - * j=0 - * - * The terms are not summed directly; instead the incomplete - * beta integral is employed, according to the formula - * - * y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * - * - * ACCURACY: - * - * Relative error (p varies from 0 to 1): - * arithmetic domain # trials peak rms - * IEEE 0,100 2000 6.9e-5 1.1e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * bdtrf domain k < 0 0.0 - * n < k - * x < 0, x > 1 - * - */ - /* bdtrcf() - * - * Complemented binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, bdtrcf(); - * - * y = bdtrcf( k, n, p ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms k+1 through n of the Binomial - * probability density: - * - * n - * -- ( n ) j n-j - * > ( ) p (1-p) - * -- ( j ) - * j=k+1 - * - * The terms are not summed directly; instead the incomplete - * beta integral is employed, according to the formula - * - * y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * - * - * ACCURACY: - * - * Relative error (p varies from 0 to 1): - * arithmetic domain # trials peak rms - * IEEE 0,100 2000 6.0e-5 1.2e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * bdtrcf domain x<0, x>1, n<k 0.0 - */ - /* bdtrif() - * - * Inverse binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, bdtrif(); - * - * p = bdtrf( k, n, y ); - * - * - * - * DESCRIPTION: - * - * Finds the event probability p such that the sum of the - * terms 0 through k of the Binomial probability density - * is equal to the given cumulative probability y. - * - * This is accomplished using the inverse beta integral - * function and the relation - * - * 1 - p = incbi( n-k, k+1, y ). - * - * - * - * - * ACCURACY: - * - * Relative error (p varies from 0 to 1): - * arithmetic domain # trials peak rms - * IEEE 0,100 2000 3.5e-5 3.3e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * bdtrif domain k < 0, n <= k 0.0 - * x < 0, x > 1 - * - */ - -/* betaf.c - * - * Beta function - * - * - * - * SYNOPSIS: - * - * float a, b, y, betaf(); - * - * y = betaf( a, b ); - * - * - * - * DESCRIPTION: - * - * - - - * | (a) | (b) - * beta( a, b ) = -----------. - * - - * | (a+b) - * - * For large arguments the logarithm of the function is - * evaluated using lgam(), then exponentiated. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 10000 4.0e-5 6.0e-6 - * IEEE -20,0 10000 4.9e-3 5.4e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * betaf overflow log(beta) > MAXLOG 0.0 - * a or b <0 integer 0.0 - * - */ - -/* cbrtf.c - * - * Cube root - * - * - * - * SYNOPSIS: - * - * float x, y, cbrtf(); - * - * y = cbrtf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the cube root of the argument, which may be negative. - * - * Range reduction involves determining the power of 2 of - * the argument. A polynomial of degree 2 applied to the - * mantissa, and multiplication by the cube root of 1, 2, or 4 - * approximates the root to within about 0.1%. Then Newton's - * iteration is used to converge to an accurate result. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,1e38 100000 7.6e-8 2.7e-8 - * - */ - -/* chbevlf.c - * - * Evaluate Chebyshev series - * - * - * - * SYNOPSIS: - * - * int N; - * float x, y, coef[N], chebevlf(); - * - * y = chbevlf( x, coef, N ); - * - * - * - * DESCRIPTION: - * - * Evaluates the series - * - * N-1 - * - ' - * y = > coef[i] T (x/2) - * - i - * i=0 - * - * of Chebyshev polynomials Ti at argument x/2. - * - * Coefficients are stored in reverse order, i.e. the zero - * order term is last in the array. Note N is the number of - * coefficients, not the order. - * - * If coefficients are for the interval a to b, x must - * have been transformed to x -> 2(2x - b - a)/(b-a) before - * entering the routine. This maps x from (a, b) to (-1, 1), - * over which the Chebyshev polynomials are defined. - * - * If the coefficients are for the inverted interval, in - * which (a, b) is mapped to (1/b, 1/a), the transformation - * required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity, - * this becomes x -> 4a/x - 1. - * - * - * - * SPEED: - * - * Taking advantage of the recurrence properties of the - * Chebyshev polynomials, the routine requires one more - * addition per loop than evaluating a nested polynomial of - * the same degree. - * - */ - -/* chdtrf.c - * - * Chi-square distribution - * - * - * - * SYNOPSIS: - * - * float df, x, y, chdtrf(); - * - * y = chdtrf( df, x ); - * - * - * - * DESCRIPTION: - * - * Returns the area under the left hand tail (from 0 to x) - * of the Chi square probability density function with - * v degrees of freedom. - * - * - * inf. - * - - * 1 | | v/2-1 -t/2 - * P( x | v ) = ----------- | t e dt - * v/2 - | | - * 2 | (v/2) - - * x - * - * where x is the Chi-square variable. - * - * The incomplete gamma integral is used, according to the - * formula - * - * y = chdtr( v, x ) = igam( v/2.0, x/2.0 ). - * - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 3.2e-5 5.0e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * chdtrf domain x < 0 or v < 1 0.0 - */ - /* chdtrcf() - * - * Complemented Chi-square distribution - * - * - * - * SYNOPSIS: - * - * float v, x, y, chdtrcf(); - * - * y = chdtrcf( v, x ); - * - * - * - * DESCRIPTION: - * - * Returns the area under the right hand tail (from x to - * infinity) of the Chi square probability density function - * with v degrees of freedom: - * - * - * inf. - * - - * 1 | | v/2-1 -t/2 - * P( x | v ) = ----------- | t e dt - * v/2 - | | - * 2 | (v/2) - - * x - * - * where x is the Chi-square variable. - * - * The incomplete gamma integral is used, according to the - * formula - * - * y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ). - * - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 2.7e-5 3.2e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * chdtrc domain x < 0 or v < 1 0.0 - */ - /* chdtrif() - * - * Inverse of complemented Chi-square distribution - * - * - * - * SYNOPSIS: - * - * float df, x, y, chdtrif(); - * - * x = chdtrif( df, y ); - * - * - * - * - * DESCRIPTION: - * - * Finds the Chi-square argument x such that the integral - * from x to infinity of the Chi-square density is equal - * to the given cumulative probability y. - * - * This is accomplished using the inverse gamma integral - * function and the relation - * - * x/2 = igami( df/2, y ); - * - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 10000 2.2e-5 8.5e-7 - * - * ERROR MESSAGES: - * - * message condition value returned - * chdtri domain y < 0 or y > 1 0.0 - * v < 1 - * - */ - -/* clogf.c - * - * Complex natural logarithm - * - * - * - * SYNOPSIS: - * - * void clogf(); - * cmplxf z, w; - * - * clogf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * Returns complex logarithm to the base e (2.718...) of - * the complex argument x. - * - * If z = x + iy, r = sqrt( x**2 + y**2 ), - * then - * w = log(r) + i arctan(y/x). - * - * The arctangent ranges from -PI to +PI. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.9e-6 6.2e-8 - * - * Larger relative error can be observed for z near 1 +i0. - * In IEEE arithmetic the peak absolute error is 3.1e-7. - * - */ - /* cexpf() - * - * Complex exponential function - * - * - * - * SYNOPSIS: - * - * void cexpf(); - * cmplxf z, w; - * - * cexpf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * Returns the exponential of the complex argument z - * into the complex result w. - * - * If - * z = x + iy, - * r = exp(x), - * - * then - * - * w = r cos y + i r sin y. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.4e-7 4.5e-8 - * - */ - /* csinf() - * - * Complex circular sine - * - * - * - * SYNOPSIS: - * - * void csinf(); - * cmplxf z, w; - * - * csinf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * w = sin x cosh y + i cos x sinh y. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.9e-7 5.5e-8 - * - */ - /* ccosf() - * - * Complex circular cosine - * - * - * - * SYNOPSIS: - * - * void ccosf(); - * cmplxf z, w; - * - * ccosf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * w = cos x cosh y - i sin x sinh y. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.8e-7 5.5e-8 - */ - /* ctanf() - * - * Complex circular tangent - * - * - * - * SYNOPSIS: - * - * void ctanf(); - * cmplxf z, w; - * - * ctanf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * sin 2x + i sinh 2y - * w = --------------------. - * cos 2x + cosh 2y - * - * On the real axis the denominator is zero at odd multiples - * of PI/2. The denominator is evaluated by its Taylor - * series near these points. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 3.3e-7 5.1e-8 - */ - /* ccotf() - * - * Complex circular cotangent - * - * - * - * SYNOPSIS: - * - * void ccotf(); - * cmplxf z, w; - * - * ccotf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * sin 2x - i sinh 2y - * w = --------------------. - * cosh 2y - cos 2x - * - * On the real axis, the denominator has zeros at even - * multiples of PI/2. Near these points it is evaluated - * by a Taylor series. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 3.6e-7 5.7e-8 - * Also tested by ctan * ccot = 1 + i0. - */ - /* casinf() - * - * Complex circular arc sine - * - * - * - * SYNOPSIS: - * - * void casinf(); - * cmplxf z, w; - * - * casinf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * Inverse complex sine: - * - * 2 - * w = -i clog( iz + csqrt( 1 - z ) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.1e-5 1.5e-6 - * Larger relative error can be observed for z near zero. - * - */ - /* cacosf() - * - * Complex circular arc cosine - * - * - * - * SYNOPSIS: - * - * void cacosf(); - * cmplxf z, w; - * - * cacosf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * - * w = arccos z = PI/2 - arcsin z. - * - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 9.2e-6 1.2e-6 - * - */ - /* catan() - * - * Complex circular arc tangent - * - * - * - * SYNOPSIS: - * - * void catan(); - * cmplxf z, w; - * - * catan( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * 1 ( 2x ) - * Re w = - arctan(-----------) + k PI - * 2 ( 2 2) - * (1 - x - y ) - * - * ( 2 2) - * 1 (x + (y+1) ) - * Im w = - log(------------) - * 4 ( 2 2) - * (x + (y-1) ) - * - * Where k is an arbitrary integer. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 2.3e-6 5.2e-8 - * - */ - -/* cmplxf.c - * - * Complex number arithmetic - * - * - * - * SYNOPSIS: - * - * typedef struct { - * float r; real part - * float i; imaginary part - * }cmplxf; - * - * cmplxf *a, *b, *c; - * - * caddf( a, b, c ); c = b + a - * csubf( a, b, c ); c = b - a - * cmulf( a, b, c ); c = b * a - * cdivf( a, b, c ); c = b / a - * cnegf( c ); c = -c - * cmovf( b, c ); c = b - * - * - * - * DESCRIPTION: - * - * Addition: - * c.r = b.r + a.r - * c.i = b.i + a.i - * - * Subtraction: - * c.r = b.r - a.r - * c.i = b.i - a.i - * - * Multiplication: - * c.r = b.r * a.r - b.i * a.i - * c.i = b.r * a.i + b.i * a.r - * - * Division: - * d = a.r * a.r + a.i * a.i - * c.r = (b.r * a.r + b.i * a.i)/d - * c.i = (b.i * a.r - b.r * a.i)/d - * ACCURACY: - * - * In DEC arithmetic, the test (1/z) * z = 1 had peak relative - * error 3.1e-17, rms 1.2e-17. The test (y/z) * (z/y) = 1 had - * peak relative error 8.3e-17, rms 2.1e-17. - * - * Tests in the rectangle {-10,+10}: - * Relative error: - * arithmetic function # trials peak rms - * IEEE cadd 30000 5.9e-8 2.6e-8 - * IEEE csub 30000 6.0e-8 2.6e-8 - * IEEE cmul 30000 1.1e-7 3.7e-8 - * IEEE cdiv 30000 2.1e-7 5.7e-8 - */ - -/* cabsf() - * - * Complex absolute value - * - * - * - * SYNOPSIS: - * - * float cabsf(); - * cmplxf z; - * float a; - * - * a = cabsf( &z ); - * - * - * - * DESCRIPTION: - * - * - * If z = x + iy - * - * then - * - * a = sqrt( x**2 + y**2 ). - * - * Overflow and underflow are avoided by testing the magnitudes - * of x and y before squaring. If either is outside half of - * the floating point full scale range, both are rescaled. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.2e-7 3.4e-8 - */ - /* csqrtf() - * - * Complex square root - * - * - * - * SYNOPSIS: - * - * void csqrtf(); - * cmplxf z, w; - * - * csqrtf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * - * If z = x + iy, r = |z|, then - * - * 1/2 - * Im w = [ (r - x)/2 ] , - * - * Re w = y / 2 Im w. - * - * - * Note that -w is also a square root of z. The solution - * reported is always in the upper half plane. - * - * Because of the potential for cancellation error in r - x, - * the result is sharpened by doing a Heron iteration - * (see sqrt.c) in complex arithmetic. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 100000 1.8e-7 4.2e-8 - * - */ - -/* coshf.c - * - * Hyperbolic cosine - * - * - * - * SYNOPSIS: - * - * float x, y, coshf(); - * - * y = coshf( x ); - * - * - * - * DESCRIPTION: - * - * Returns hyperbolic cosine of argument in the range MINLOGF to - * MAXLOGF. - * - * cosh(x) = ( exp(x) + exp(-x) )/2. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-MAXLOGF 100000 1.2e-7 2.8e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * coshf overflow |x| > MAXLOGF MAXNUMF - * - * - */ - -/* dawsnf.c - * - * Dawson's Integral - * - * - * - * SYNOPSIS: - * - * float x, y, dawsnf(); - * - * y = dawsnf( x ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * x - * - - * 2 | | 2 - * dawsn(x) = exp( -x ) | exp( t ) dt - * | | - * - - * 0 - * - * Three different rational approximations are employed, for - * the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,10 50000 4.4e-7 6.3e-8 - * - * - */ - -/* ellief.c - * - * Incomplete elliptic integral of the second kind - * - * - * - * SYNOPSIS: - * - * float phi, m, y, ellief(); - * - * y = ellief( phi, m ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * phi - * - - * | | - * | 2 - * E(phi\m) = | sqrt( 1 - m sin t ) dt - * | - * | | - * - - * 0 - * - * of amplitude phi and modulus m, using the arithmetic - - * geometric mean algorithm. - * - * - * - * ACCURACY: - * - * Tested at random arguments with phi in [0, 2] and m in - * [0, 1]. - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,2 10000 4.5e-7 7.4e-8 - * - * - */ - -/* ellikf.c - * - * Incomplete elliptic integral of the first kind - * - * - * - * SYNOPSIS: - * - * float phi, m, y, ellikf(); - * - * y = ellikf( phi, m ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * - * phi - * - - * | | - * | dt - * F(phi\m) = | ------------------ - * | 2 - * | | sqrt( 1 - m sin t ) - * - - * 0 - * - * of amplitude phi and modulus m, using the arithmetic - - * geometric mean algorithm. - * - * - * - * - * ACCURACY: - * - * Tested at random points with phi in [0, 2] and m in - * [0, 1]. - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,2 10000 2.9e-7 5.8e-8 - * - * - */ - -/* ellpef.c - * - * Complete elliptic integral of the second kind - * - * - * - * SYNOPSIS: - * - * float m1, y, ellpef(); - * - * y = ellpef( m1 ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * pi/2 - * - - * | | 2 - * E(m) = | sqrt( 1 - m sin t ) dt - * | | - * - - * 0 - * - * Where m = 1 - m1, using the approximation - * - * P(x) - x log x Q(x). - * - * Though there are no singularities, the argument m1 is used - * rather than m for compatibility with ellpk(). - * - * E(1) = 1; E(0) = pi/2. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 1 30000 1.1e-7 3.9e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * ellpef domain x<0, x>1 0.0 - * - */ - -/* ellpjf.c - * - * Jacobian Elliptic Functions - * - * - * - * SYNOPSIS: - * - * float u, m, sn, cn, dn, phi; - * int ellpj(); - * - * ellpj( u, m, _&sn, _&cn, _&dn, _&phi ); - * - * - * - * DESCRIPTION: - * - * - * Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m), - * and dn(u|m) of parameter m between 0 and 1, and real - * argument u. - * - * These functions are periodic, with quarter-period on the - * real axis equal to the complete elliptic integral - * ellpk(1.0-m). - * - * Relation to incomplete elliptic integral: - * If u = ellik(phi,m), then sn(u|m) = sin(phi), - * and cn(u|m) = cos(phi). Phi is called the amplitude of u. - * - * Computation is by means of the arithmetic-geometric mean - * algorithm, except when m is within 1e-9 of 0 or 1. In the - * latter case with m close to 1, the approximation applies - * only for phi < pi/2. - * - * ACCURACY: - * - * Tested at random points with u between 0 and 10, m between - * 0 and 1. - * - * Absolute error (* = relative error): - * arithmetic function # trials peak rms - * IEEE sn 10000 1.7e-6 2.2e-7 - * IEEE cn 10000 1.6e-6 2.2e-7 - * IEEE dn 10000 1.4e-3 1.9e-5 - * IEEE phi 10000 3.9e-7* 6.7e-8* - * - * Peak error observed in consistency check using addition - * theorem for sn(u+v) was 4e-16 (absolute). Also tested by - * the above relation to the incomplete elliptic integral. - * Accuracy deteriorates when u is large. - * - */ - -/* ellpkf.c - * - * Complete elliptic integral of the first kind - * - * - * - * SYNOPSIS: - * - * float m1, y, ellpkf(); - * - * y = ellpkf( m1 ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * - * pi/2 - * - - * | | - * | dt - * K(m) = | ------------------ - * | 2 - * | | sqrt( 1 - m sin t ) - * - - * 0 - * - * where m = 1 - m1, using the approximation - * - * P(x) - log x Q(x). - * - * The argument m1 is used rather than m so that the logarithmic - * singularity at m = 1 will be shifted to the origin; this - * preserves maximum accuracy. - * - * K(0) = pi/2. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,1 30000 1.3e-7 3.4e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * ellpkf domain x<0, x>1 0.0 - * - */ - -/* exp10f.c - * - * Base 10 exponential function - * (Common antilogarithm) - * - * - * - * SYNOPSIS: - * - * float x, y, exp10f(); - * - * y = exp10f( x ); - * - * - * - * DESCRIPTION: - * - * Returns 10 raised to the x power. - * - * Range reduction is accomplished by expressing the argument - * as 10**x = 2**n 10**f, with |f| < 0.5 log10(2). - * A polynomial approximates 10**f. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -38,+38 100000 9.8e-8 2.8e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * exp10 underflow x < -MAXL10 0.0 - * exp10 overflow x > MAXL10 MAXNUM - * - * IEEE single arithmetic: MAXL10 = 38.230809449325611792. - * - */ - -/* exp2f.c - * - * Base 2 exponential function - * - * - * - * SYNOPSIS: - * - * float x, y, exp2f(); - * - * y = exp2f( x ); - * - * - * - * DESCRIPTION: - * - * Returns 2 raised to the x power. - * - * Range reduction is accomplished by separating the argument - * into an integer k and fraction f such that - * x k f - * 2 = 2 2. - * - * A polynomial approximates 2**x in the basic range [-0.5, 0.5]. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -127,+127 100000 1.7e-7 2.8e-8 - * - * - * See exp.c for comments on error amplification. - * - * - * ERROR MESSAGES: - * - * message condition value returned - * exp underflow x < -MAXL2 0.0 - * exp overflow x > MAXL2 MAXNUMF - * - * For IEEE arithmetic, MAXL2 = 127. - */ - -/* expf.c - * - * Exponential function - * - * - * - * SYNOPSIS: - * - * float x, y, expf(); - * - * y = expf( x ); - * - * - * - * DESCRIPTION: - * - * Returns e (2.71828...) raised to the x power. - * - * Range reduction is accomplished by separating the argument - * into an integer k and fraction f such that - * - * x k f - * e = 2 e. - * - * A polynomial is used to approximate exp(f) - * in the basic range [-0.5, 0.5]. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +- MAXLOG 100000 1.7e-7 2.8e-8 - * - * - * Error amplification in the exponential function can be - * a serious matter. The error propagation involves - * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ), - * which shows that a 1 lsb error in representing X produces - * a relative error of X times 1 lsb in the function. - * While the routine gives an accurate result for arguments - * that are exactly represented by a double precision - * computer number, the result contains amplified roundoff - * error for large arguments not exactly represented. - * - * - * ERROR MESSAGES: - * - * message condition value returned - * expf underflow x < MINLOGF 0.0 - * expf overflow x > MAXLOGF MAXNUMF - * - */ - -/* expnf.c - * - * Exponential integral En - * - * - * - * SYNOPSIS: - * - * int n; - * float x, y, expnf(); - * - * y = expnf( n, x ); - * - * - * - * DESCRIPTION: - * - * Evaluates the exponential integral - * - * inf. - * - - * | | -xt - * | e - * E (x) = | ---- dt. - * n | n - * | | t - * - - * 1 - * - * - * Both n and x must be nonnegative. - * - * The routine employs either a power series, a continued - * fraction, or an asymptotic formula depending on the - * relative values of n and x. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 10000 5.6e-7 1.2e-7 - * - */ - -/* facf.c - * - * Factorial function - * - * - * - * SYNOPSIS: - * - * float y, facf(); - * int i; - * - * y = facf( i ); - * - * - * - * DESCRIPTION: - * - * Returns factorial of i = 1 * 2 * 3 * ... * i. - * fac(0) = 1.0. - * - * Due to machine arithmetic bounds the largest value of - * i accepted is 33 in single precision arithmetic. - * Greater values, or negative ones, - * produce an error message and return MAXNUM. - * - * - * - * ACCURACY: - * - * For i < 34 the values are simply tabulated, and have - * full machine accuracy. - * - */ - -/* fdtrf.c - * - * F distribution - * - * - * - * SYNOPSIS: - * - * int df1, df2; - * float x, y, fdtrf(); - * - * y = fdtrf( df1, df2, x ); - * - * - * - * DESCRIPTION: - * - * Returns the area from zero to x under the F density - * function (also known as Snedcor's density or the - * variance ratio density). This is the density - * of x = (u1/df1)/(u2/df2), where u1 and u2 are random - * variables having Chi square distributions with df1 - * and df2 degrees of freedom, respectively. - * - * The incomplete beta integral is used, according to the - * formula - * - * P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ). - * - * - * The arguments a and b are greater than zero, and x - * x is nonnegative. - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 2.2e-5 1.1e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * fdtrf domain a<0, b<0, x<0 0.0 - * - */ - /* fdtrcf() - * - * Complemented F distribution - * - * - * - * SYNOPSIS: - * - * int df1, df2; - * float x, y, fdtrcf(); - * - * y = fdtrcf( df1, df2, x ); - * - * - * - * DESCRIPTION: - * - * Returns the area from x to infinity under the F density - * function (also known as Snedcor's density or the - * variance ratio density). - * - * - * inf. - * - - * 1 | | a-1 b-1 - * 1-P(x) = ------ | t (1-t) dt - * B(a,b) | | - * - - * x - * - * (See fdtr.c.) - * - * The incomplete beta integral is used, according to the - * formula - * - * P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 7.3e-5 1.2e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * fdtrcf domain a<0, b<0, x<0 0.0 - * - */ - /* fdtrif() - * - * Inverse of complemented F distribution - * - * - * - * SYNOPSIS: - * - * float df1, df2, x, y, fdtrif(); - * - * x = fdtrif( df1, df2, y ); - * - * - * - * - * DESCRIPTION: - * - * Finds the F density argument x such that the integral - * from x to infinity of the F density is equal to the - * given probability y. - * - * This is accomplished using the inverse beta integral - * function and the relations - * - * z = incbi( df2/2, df1/2, y ) - * x = df2 (1-z) / (df1 z). - * - * Note: the following relations hold for the inverse of - * the uncomplemented F distribution: - * - * z = incbi( df1/2, df2/2, y ) - * x = df2 z / (df1 (1-z)). - * - * - * - * ACCURACY: - * - * arithmetic domain # trials peak rms - * Absolute error: - * IEEE 0,100 5000 4.0e-5 3.2e-6 - * Relative error: - * IEEE 0,100 5000 1.2e-3 1.8e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * fdtrif domain y <= 0 or y > 1 0.0 - * v < 1 - * - */ - -/* ceilf() - * floorf() - * frexpf() - * ldexpf() - * - * Single precision floating point numeric utilities - * - * - * - * SYNOPSIS: - * - * float x, y; - * float ceilf(), floorf(), frexpf(), ldexpf(); - * int expnt, n; - * - * y = floorf(x); - * y = ceilf(x); - * y = frexpf( x, &expnt ); - * y = ldexpf( x, n ); - * - * - * - * DESCRIPTION: - * - * All four routines return a single precision floating point - * result. - * - * sfloor() returns the largest integer less than or equal to x. - * It truncates toward minus infinity. - * - * sceil() returns the smallest integer greater than or equal - * to x. It truncates toward plus infinity. - * - * sfrexp() extracts the exponent from x. It returns an integer - * power of two to expnt and the significand between 0.5 and 1 - * to y. Thus x = y * 2**expn. - * - * sldexp() multiplies x by 2**n. - * - * These functions are part of the standard C run time library - * for many but not all C compilers. The ones supplied are - * written in C for either DEC or IEEE arithmetic. They should - * be used only if your compiler library does not already have - * them. - * - * The IEEE versions assume that denormal numbers are implemented - * in the arithmetic. Some modifications will be required if - * the arithmetic has abrupt rather than gradual underflow. - */ - -/* fresnlf.c - * - * Fresnel integral - * - * - * - * SYNOPSIS: - * - * float x, S, C; - * void fresnlf(); - * - * fresnlf( x, _&S, _&C ); - * - * - * DESCRIPTION: - * - * Evaluates the Fresnel integrals - * - * x - * - - * | | - * C(x) = | cos(pi/2 t**2) dt, - * | | - * - - * 0 - * - * x - * - - * | | - * S(x) = | sin(pi/2 t**2) dt. - * | | - * - - * 0 - * - * - * The integrals are evaluated by power series for small x. - * For x >= 1 auxiliary functions f(x) and g(x) are employed - * such that - * - * C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 ) - * S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 ) - * - * - * - * ACCURACY: - * - * Relative error. - * - * Arithmetic function domain # trials peak rms - * IEEE S(x) 0, 10 30000 1.1e-6 1.9e-7 - * IEEE C(x) 0, 10 30000 1.1e-6 2.0e-7 - */ - -/* gammaf.c - * - * Gamma function - * - * - * - * SYNOPSIS: - * - * float x, y, gammaf(); - * extern int sgngamf; - * - * y = gammaf( x ); - * - * - * - * DESCRIPTION: - * - * Returns gamma function of the argument. The result is - * correctly signed, and the sign (+1 or -1) is also - * returned in a global (extern) variable named sgngamf. - * This same variable is also filled in by the logarithmic - * gamma function lgam(). - * - * Arguments between 0 and 10 are reduced by recurrence and the - * function is approximated by a polynomial function covering - * the interval (2,3). Large arguments are handled by Stirling's - * formula. Negative arguments are made positive using - * a reflection formula. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,-33 100,000 5.7e-7 1.0e-7 - * IEEE -33,0 100,000 6.1e-7 1.2e-7 - * - * - */ -/* lgamf() - * - * Natural logarithm of gamma function - * - * - * - * SYNOPSIS: - * - * float x, y, lgamf(); - * extern int sgngamf; - * - * y = lgamf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base e (2.718...) logarithm of the absolute - * value of the gamma function of the argument. - * The sign (+1 or -1) of the gamma function is returned in a - * global (extern) variable named sgngamf. - * - * For arguments greater than 6.5, the logarithm of the gamma - * function is approximated by the logarithmic version of - * Stirling's formula. Arguments between 0 and +6.5 are reduced by - * by recurrence to the interval [.75,1.25] or [1.5,2.5] of a rational - * approximation. The cosecant reflection formula is employed for - * arguments less than zero. - * - * Arguments greater than MAXLGM = 2.035093e36 return MAXNUM and an - * error message. - * - * - * - * ACCURACY: - * - * - * - * arithmetic domain # trials peak rms - * IEEE -100,+100 500,000 7.4e-7 6.8e-8 - * The error criterion was relative when the function magnitude - * was greater than one but absolute when it was less than one. - * The routine has low relative error for positive arguments. - * - * The following test used the relative error criterion. - * IEEE -2, +3 100000 4.0e-7 5.6e-8 - * - */ - -/* gdtrf.c - * - * Gamma distribution function - * - * - * - * SYNOPSIS: - * - * float a, b, x, y, gdtrf(); - * - * y = gdtrf( a, b, x ); - * - * - * - * DESCRIPTION: - * - * Returns the integral from zero to x of the gamma probability - * density function: - * - * - * x - * b - - * a | | b-1 -at - * y = ----- | t e dt - * - | | - * | (b) - - * 0 - * - * The incomplete gamma integral is used, according to the - * relation - * - * y = igam( b, ax ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 5.8e-5 3.0e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * gdtrf domain x < 0 0.0 - * - */ - /* gdtrcf.c - * - * Complemented gamma distribution function - * - * - * - * SYNOPSIS: - * - * float a, b, x, y, gdtrcf(); - * - * y = gdtrcf( a, b, x ); - * - * - * - * DESCRIPTION: - * - * Returns the integral from x to infinity of the gamma - * probability density function: - * - * - * inf. - * b - - * a | | b-1 -at - * y = ----- | t e dt - * - | | - * | (b) - - * x - * - * The incomplete gamma integral is used, according to the - * relation - * - * y = igamc( b, ax ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 9.1e-5 1.5e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * gdtrcf domain x < 0 0.0 - * - */ - -/* hyp2f1f.c - * - * Gauss hypergeometric function F - * 2 1 - * - * - * SYNOPSIS: - * - * float a, b, c, x, y, hyp2f1f(); - * - * y = hyp2f1f( a, b, c, x ); - * - * - * DESCRIPTION: - * - * - * hyp2f1( a, b, c, x ) = F ( a, b; c; x ) - * 2 1 - * - * inf. - * - a(a+1)...(a+k) b(b+1)...(b+k) k+1 - * = 1 + > ----------------------------- x . - * - c(c+1)...(c+k) (k+1)! - * k = 0 - * - * Cases addressed are - * Tests and escapes for negative integer a, b, or c - * Linear transformation if c - a or c - b negative integer - * Special case c = a or c = b - * Linear transformation for x near +1 - * Transformation for x < -0.5 - * Psi function expansion if x > 0.5 and c - a - b integer - * Conditionally, a recurrence on c to make c-a-b > 0 - * - * |x| > 1 is rejected. - * - * The parameters a, b, c are considered to be integer - * valued if they are within 1.0e-6 of the nearest integer. - * - * ACCURACY: - * - * Relative error (-1 < x < 1): - * arithmetic domain # trials peak rms - * IEEE 0,3 30000 5.8e-4 4.3e-6 - */ - -/* hypergf.c - * - * Confluent hypergeometric function - * - * - * - * SYNOPSIS: - * - * float a, b, x, y, hypergf(); - * - * y = hypergf( a, b, x ); - * - * - * - * DESCRIPTION: - * - * Computes the confluent hypergeometric function - * - * 1 2 - * a x a(a+1) x - * F ( a,b;x ) = 1 + ---- + --------- + ... - * 1 1 b 1! b(b+1) 2! - * - * Many higher transcendental functions are special cases of - * this power series. - * - * As is evident from the formula, b must not be a negative - * integer or zero unless a is an integer with 0 >= a > b. - * - * The routine attempts both a direct summation of the series - * and an asymptotic expansion. In each case error due to - * roundoff, cancellation, and nonconvergence is estimated. - * The result with smaller estimated error is returned. - * - * - * - * ACCURACY: - * - * Tested at random points (a, b, x), all three variables - * ranging from 0 to 30. - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,5 10000 6.6e-7 1.3e-7 - * IEEE 0,30 30000 1.1e-5 6.5e-7 - * - * Larger errors can be observed when b is near a negative - * integer or zero. Certain combinations of arguments yield - * serious cancellation error in the power series summation - * and also are not in the region of near convergence of the - * asymptotic series. An error message is printed if the - * self-estimated relative error is greater than 1.0e-3. - * - */ - -/* i0f.c - * - * Modified Bessel function of order zero - * - * - * - * SYNOPSIS: - * - * float x, y, i0(); - * - * y = i0f( x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of order zero of the - * argument. - * - * The function is defined as i0(x) = j0( ix ). - * - * The range is partitioned into the two intervals [0,8] and - * (8, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 100000 4.0e-7 7.9e-8 - * - */ - /* i0ef.c - * - * Modified Bessel function of order zero, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * float x, y, i0ef(); - * - * y = i0ef( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of order zero of the argument. - * - * The function is defined as i0e(x) = exp(-|x|) j0( ix ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 100000 3.7e-7 7.0e-8 - * See i0f(). - * - */ - -/* i1f.c - * - * Modified Bessel function of order one - * - * - * - * SYNOPSIS: - * - * float x, y, i1f(); - * - * y = i1f( x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of order one of the - * argument. - * - * The function is defined as i1(x) = -i j1( ix ). - * - * The range is partitioned into the two intervals [0,8] and - * (8, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 100000 1.5e-6 1.6e-7 - * - * - */ - /* i1ef.c - * - * Modified Bessel function of order one, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * float x, y, i1ef(); - * - * y = i1ef( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of order one of the argument. - * - * The function is defined as i1(x) = -i exp(-|x|) j1( ix ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 1.5e-6 1.5e-7 - * See i1(). - * - */ - -/* igamf.c - * - * Incomplete gamma integral - * - * - * - * SYNOPSIS: - * - * float a, x, y, igamf(); - * - * y = igamf( a, x ); - * - * - * - * DESCRIPTION: - * - * The function is defined by - * - * x - * - - * 1 | | -t a-1 - * igam(a,x) = ----- | e t dt. - * - | | - * | (a) - - * 0 - * - * - * In this implementation both arguments must be positive. - * The integral is evaluated by either a power series or - * continued fraction expansion, depending on the relative - * values of a and x. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 20000 7.8e-6 5.9e-7 - * - */ - /* igamcf() - * - * Complemented incomplete gamma integral - * - * - * - * SYNOPSIS: - * - * float a, x, y, igamcf(); - * - * y = igamcf( a, x ); - * - * - * - * DESCRIPTION: - * - * The function is defined by - * - * - * igamc(a,x) = 1 - igam(a,x) - * - * inf. - * - - * 1 | | -t a-1 - * = ----- | e t dt. - * - | | - * | (a) - - * x - * - * - * In this implementation both arguments must be positive. - * The integral is evaluated by either a power series or - * continued fraction expansion, depending on the relative - * values of a and x. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 30000 7.8e-6 5.9e-7 - * - */ - -/* igamif() - * - * Inverse of complemented imcomplete gamma integral - * - * - * - * SYNOPSIS: - * - * float a, x, y, igamif(); - * - * x = igamif( a, y ); - * - * - * - * DESCRIPTION: - * - * Given y, the function finds x such that - * - * igamc( a, x ) = y. - * - * Starting with the approximate value - * - * 3 - * x = a t - * - * where - * - * t = 1 - d - ndtri(y) sqrt(d) - * - * and - * - * d = 1/9a, - * - * the routine performs up to 10 Newton iterations to find the - * root of igamc(a,x) - y = 0. - * - * - * ACCURACY: - * - * Tested for a ranging from 0 to 100 and x from 0 to 1. - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 1.0e-5 1.5e-6 - * - */ - -/* incbetf.c - * - * Incomplete beta integral - * - * - * SYNOPSIS: - * - * float a, b, x, y, incbetf(); - * - * y = incbetf( a, b, x ); - * - * - * DESCRIPTION: - * - * Returns incomplete beta integral of the arguments, evaluated - * from zero to x. The function is defined as - * - * x - * - - - * | (a+b) | | a-1 b-1 - * ----------- | t (1-t) dt. - * - - | | - * | (a) | (b) - - * 0 - * - * The domain of definition is 0 <= x <= 1. In this - * implementation a and b are restricted to positive values. - * The integral from x to 1 may be obtained by the symmetry - * relation - * - * 1 - incbet( a, b, x ) = incbet( b, a, 1-x ). - * - * The integral is evaluated by a continued fraction expansion. - * If a < 1, the function calls itself recursively after a - * transformation to increase a to a+1. - * - * ACCURACY: - * - * Tested at random points (a,b,x) with a and b in the indicated - * interval and x between 0 and 1. - * - * arithmetic domain # trials peak rms - * Relative error: - * IEEE 0,30 10000 3.7e-5 5.1e-6 - * IEEE 0,100 10000 1.7e-4 2.5e-5 - * The useful domain for relative error is limited by underflow - * of the single precision exponential function. - * Absolute error: - * IEEE 0,30 100000 2.2e-5 9.6e-7 - * IEEE 0,100 10000 6.5e-5 3.7e-6 - * - * Larger errors may occur for extreme ratios of a and b. - * - * ERROR MESSAGES: - * message condition value returned - * incbetf domain x<0, x>1 0.0 - */ - -/* incbif() - * - * Inverse of imcomplete beta integral - * - * - * - * SYNOPSIS: - * - * float a, b, x, y, incbif(); - * - * x = incbif( a, b, y ); - * - * - * - * DESCRIPTION: - * - * Given y, the function finds x such that - * - * incbet( a, b, x ) = y. - * - * the routine performs up to 10 Newton iterations to find the - * root of incbet(a,b,x) - y = 0. - * - * - * ACCURACY: - * - * Relative error: - * x a,b - * arithmetic domain domain # trials peak rms - * IEEE 0,1 0,100 5000 2.8e-4 8.3e-6 - * - * Overflow and larger errors may occur for one of a or b near zero - * and the other large. - */ - -/* ivf.c - * - * Modified Bessel function of noninteger order - * - * - * - * SYNOPSIS: - * - * float v, x, y, ivf(); - * - * y = ivf( v, x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of order v of the - * argument. If x is negative, v must be integer valued. - * - * The function is defined as Iv(x) = Jv( ix ). It is - * here computed in terms of the confluent hypergeometric - * function, according to the formula - * - * v -x - * Iv(x) = (x/2) e hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1) - * - * If v is a negative integer, then v is replaced by -v. - * - * - * ACCURACY: - * - * Tested at random points (v, x), with v between 0 and - * 30, x between 0 and 28. - * arithmetic domain # trials peak rms - * Relative error: - * IEEE 0,15 3000 4.7e-6 5.4e-7 - * Absolute error (relative when function > 1) - * IEEE 0,30 5000 8.5e-6 1.3e-6 - * - * Accuracy is diminished if v is near a negative integer. - * The useful domain for relative error is limited by overflow - * of the single precision exponential function. - * - * See also hyperg.c. - * - */ - -/* j0f.c - * - * Bessel function of order zero - * - * - * - * SYNOPSIS: - * - * float x, y, j0f(); - * - * y = j0f( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order zero of the argument. - * - * The domain is divided into the intervals [0, 2] and - * (2, infinity). In the first interval the following polynomial - * approximation is used: - * - * - * 2 2 2 - * (w - r ) (w - r ) (w - r ) P(w) - * 1 2 3 - * - * 2 - * where w = x and the three r's are zeros of the function. - * - * In the second interval, the modulus and phase are approximated - * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) - * and Phase(x) = x + 1/x R(1/x^2) - pi/4. The function is - * - * j0(x) = Modulus(x) cos( Phase(x) ). - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic domain # trials peak rms - * IEEE 0, 2 100000 1.3e-7 3.6e-8 - * IEEE 2, 32 100000 1.9e-7 5.4e-8 - * - */ - /* y0f.c - * - * Bessel function of the second kind, order zero - * - * - * - * SYNOPSIS: - * - * float x, y, y0f(); - * - * y = y0f( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of the second kind, of order - * zero, of the argument. - * - * The domain is divided into the intervals [0, 2] and - * (2, infinity). In the first interval a rational approximation - * R(x) is employed to compute - * - * 2 2 2 - * y0(x) = (w - r ) (w - r ) (w - r ) R(x) + 2/pi ln(x) j0(x). - * 1 2 3 - * - * Thus a call to j0() is required. The three zeros are removed - * from R(x) to improve its numerical stability. - * - * In the second interval, the modulus and phase are approximated - * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) - * and Phase(x) = x + 1/x S(1/x^2) - pi/4. Then the function is - * - * y0(x) = Modulus(x) sin( Phase(x) ). - * - * - * - * - * ACCURACY: - * - * Absolute error, when y0(x) < 1; else relative error: - * - * arithmetic domain # trials peak rms - * IEEE 0, 2 100000 2.4e-7 3.4e-8 - * IEEE 2, 32 100000 1.8e-7 5.3e-8 - * - */ - -/* j1f.c - * - * Bessel function of order one - * - * - * - * SYNOPSIS: - * - * float x, y, j1f(); - * - * y = j1f( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order one of the argument. - * - * The domain is divided into the intervals [0, 2] and - * (2, infinity). In the first interval a polynomial approximation - * 2 - * (w - r ) x P(w) - * 1 - * 2 - * is used, where w = x and r is the first zero of the function. - * - * In the second interval, the modulus and phase are approximated - * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) - * and Phase(x) = x + 1/x R(1/x^2) - 3pi/4. The function is - * - * j0(x) = Modulus(x) cos( Phase(x) ). - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic domain # trials peak rms - * IEEE 0, 2 100000 1.2e-7 2.5e-8 - * IEEE 2, 32 100000 2.0e-7 5.3e-8 - * - * - */ - /* y1.c - * - * Bessel function of second kind of order one - * - * - * - * SYNOPSIS: - * - * double x, y, y1(); - * - * y = y1( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of the second kind of order one - * of the argument. - * - * The domain is divided into the intervals [0, 2] and - * (2, infinity). In the first interval a rational approximation - * R(x) is employed to compute - * - * 2 - * y0(x) = (w - r ) x R(x^2) + 2/pi (ln(x) j1(x) - 1/x) . - * 1 - * - * Thus a call to j1() is required. - * - * In the second interval, the modulus and phase are approximated - * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) - * and Phase(x) = x + 1/x S(1/x^2) - 3pi/4. Then the function is - * - * y0(x) = Modulus(x) sin( Phase(x) ). - * - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic domain # trials peak rms - * IEEE 0, 2 100000 2.2e-7 4.6e-8 - * IEEE 2, 32 100000 1.9e-7 5.3e-8 - * - * (error criterion relative when |y1| > 1). - * - */ - -/* jnf.c - * - * Bessel function of integer order - * - * - * - * SYNOPSIS: - * - * int n; - * float x, y, jnf(); - * - * y = jnf( n, x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order n, where n is a - * (possibly negative) integer. - * - * The ratio of jn(x) to j0(x) is computed by backward - * recurrence. First the ratio jn/jn-1 is found by a - * continued fraction expansion. Then the recurrence - * relating successive orders is applied until j0 or j1 is - * reached. - * - * If n = 0 or 1 the routine for j0 or j1 is called - * directly. - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic range # trials peak rms - * IEEE 0, 15 30000 3.6e-7 3.6e-8 - * - * - * Not suitable for large n or x. Use jvf() instead. - * - */ - -/* jvf.c - * - * Bessel function of noninteger order - * - * - * - * SYNOPSIS: - * - * float v, x, y, jvf(); - * - * y = jvf( v, x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order v of the argument, - * where v is real. Negative x is allowed if v is an integer. - * - * Several expansions are included: the ascending power - * series, the Hankel expansion, and two transitional - * expansions for large v. If v is not too large, it - * is reduced by recurrence to a region of best accuracy. - * - * The single precision routine accepts negative v, but with - * reduced accuracy. - * - * - * - * ACCURACY: - * Results for integer v are indicated by *. - * Error criterion is absolute, except relative when |jv()| > 1. - * - * arithmetic domain # trials peak rms - * v x - * IEEE 0,125 0,125 30000 2.0e-6 2.0e-7 - * IEEE -17,0 0,125 30000 1.1e-5 4.0e-7 - * IEEE -100,0 0,125 3000 1.5e-4 7.8e-6 - */ - -/* k0f.c - * - * Modified Bessel function, third kind, order zero - * - * - * - * SYNOPSIS: - * - * float x, y, k0f(); - * - * y = k0f( x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of the third kind - * of order zero of the argument. - * - * The range is partitioned into the two intervals [0,8] and - * (8, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Tested at 2000 random points between 0 and 8. Peak absolute - * error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15. - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 7.8e-7 8.5e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * K0 domain x <= 0 MAXNUM - * - */ - /* k0ef() - * - * Modified Bessel function, third kind, order zero, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * float x, y, k0ef(); - * - * y = k0ef( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of the third kind of order zero of the argument. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 8.1e-7 7.8e-8 - * See k0(). - * - */ - -/* k1f.c - * - * Modified Bessel function, third kind, order one - * - * - * - * SYNOPSIS: - * - * float x, y, k1f(); - * - * y = k1f( x ); - * - * - * - * DESCRIPTION: - * - * Computes the modified Bessel function of the third kind - * of order one of the argument. - * - * The range is partitioned into the two intervals [0,2] and - * (2, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 4.6e-7 7.6e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * k1 domain x <= 0 MAXNUM - * - */ - /* k1ef.c - * - * Modified Bessel function, third kind, order one, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * float x, y, k1ef(); - * - * y = k1ef( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of the third kind of order one of the argument: - * - * k1e(x) = exp(x) * k1(x). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 4.9e-7 6.7e-8 - * See k1(). - * - */ - -/* knf.c - * - * Modified Bessel function, third kind, integer order - * - * - * - * SYNOPSIS: - * - * float x, y, knf(); - * int n; - * - * y = knf( n, x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of the third kind - * of order n of the argument. - * - * The range is partitioned into the two intervals [0,9.55] and - * (9.55, infinity). An ascending power series is used in the - * low range, and an asymptotic expansion in the high range. - * - * - * - * ACCURACY: - * - * Absolute error, relative when function > 1: - * arithmetic domain # trials peak rms - * IEEE 0,30 10000 2.0e-4 3.8e-6 - * - * Error is high only near the crossover point x = 9.55 - * between the two expansions used. - */ - -/* log10f.c - * - * Common logarithm - * - * - * - * SYNOPSIS: - * - * float x, y, log10f(); - * - * y = log10f( x ); - * - * - * - * DESCRIPTION: - * - * Returns logarithm to the base 10 of x. - * - * The argument is separated into its exponent and fractional - * parts. The logarithm of the fraction is approximated by - * - * log(1+x) = x - 0.5 x**2 + x**3 P(x). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0.5, 2.0 100000 1.3e-7 3.4e-8 - * IEEE 0, MAXNUMF 100000 1.3e-7 2.6e-8 - * - * In the tests over the interval [0, MAXNUM], the logarithms - * of the random arguments were uniformly distributed over - * [-MAXL10, MAXL10]. - * - * ERROR MESSAGES: - * - * log10f singularity: x = 0; returns -MAXL10 - * log10f domain: x < 0; returns -MAXL10 - * MAXL10 = 38.230809449325611792 - */ - -/* log2f.c - * - * Base 2 logarithm - * - * - * - * SYNOPSIS: - * - * float x, y, log2f(); - * - * y = log2f( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base 2 logarithm of x. - * - * The argument is separated into its exponent and fractional - * parts. If the exponent is between -1 and +1, the base e - * logarithm of the fraction is approximated by - * - * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). - * - * Otherwise, setting z = 2(x-1)/x+1), - * - * log(x) = z + z**3 P(z)/Q(z). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE exp(+-88) 100000 1.1e-7 2.4e-8 - * IEEE 0.5, 2.0 100000 1.1e-7 3.0e-8 - * - * In the tests over the interval [exp(+-88)], the logarithms - * of the random arguments were uniformly distributed. - * - * ERROR MESSAGES: - * - * log singularity: x = 0; returns MINLOGF/log(2) - * log domain: x < 0; returns MINLOGF/log(2) - */ - -/* logf.c - * - * Natural logarithm - * - * - * - * SYNOPSIS: - * - * float x, y, logf(); - * - * y = logf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base e (2.718...) logarithm of x. - * - * The argument is separated into its exponent and fractional - * parts. If the exponent is between -1 and +1, the logarithm - * of the fraction is approximated by - * - * log(1+x) = x - 0.5 x**2 + x**3 P(x) - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0.5, 2.0 100000 7.6e-8 2.7e-8 - * IEEE 1, MAXNUMF 100000 2.6e-8 - * - * In the tests over the interval [1, MAXNUM], the logarithms - * of the random arguments were uniformly distributed over - * [0, MAXLOGF]. - * - * ERROR MESSAGES: - * - * logf singularity: x = 0; returns MINLOG - * logf domain: x < 0; returns MINLOG - */ - -/* mtherr.c - * - * Library common error handling routine - * - * - * - * SYNOPSIS: - * - * char *fctnam; - * int code; - * void mtherr(); - * - * mtherr( fctnam, code ); - * - * - * - * DESCRIPTION: - * - * This routine may be called to report one of the following - * error conditions (in the include file math.h). - * - * Mnemonic Value Significance - * - * DOMAIN 1 argument domain error - * SING 2 function singularity - * OVERFLOW 3 overflow range error - * UNDERFLOW 4 underflow range error - * TLOSS 5 total loss of precision - * PLOSS 6 partial loss of precision - * EDOM 33 Unix domain error code - * ERANGE 34 Unix range error code - * - * The default version of the file prints the function name, - * passed to it by the pointer fctnam, followed by the - * error condition. The display is directed to the standard - * output device. The routine then returns to the calling - * program. Users may wish to modify the program to abort by - * calling exit() under severe error conditions such as domain - * errors. - * - * Since all error conditions pass control to this function, - * the display may be easily changed, eliminated, or directed - * to an error logging device. - * - * SEE ALSO: - * - * math.h - * - */ - -/* nbdtrf.c - * - * Negative binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, nbdtrf(); - * - * y = nbdtrf( k, n, p ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms 0 through k of the negative - * binomial distribution: - * - * k - * -- ( n+j-1 ) n j - * > ( ) p (1-p) - * -- ( j ) - * j=0 - * - * In a sequence of Bernoulli trials, this is the probability - * that k or fewer failures precede the nth success. - * - * The terms are not computed individually; instead the incomplete - * beta integral is employed, according to the formula - * - * y = nbdtr( k, n, p ) = incbet( n, k+1, p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 1.5e-4 1.9e-5 - * - */ - /* nbdtrcf.c - * - * Complemented negative binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, nbdtrcf(); - * - * y = nbdtrcf( k, n, p ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms k+1 to infinity of the negative - * binomial distribution: - * - * inf - * -- ( n+j-1 ) n j - * > ( ) p (1-p) - * -- ( j ) - * j=k+1 - * - * The terms are not computed individually; instead the incomplete - * beta integral is employed, according to the formula - * - * y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 1.4e-4 2.0e-5 - * - */ - -/* ndtrf.c - * - * Normal distribution function - * - * - * - * SYNOPSIS: - * - * float x, y, ndtrf(); - * - * y = ndtrf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the area under the Gaussian probability density - * function, integrated from minus infinity to x: - * - * x - * - - * 1 | | 2 - * ndtr(x) = --------- | exp( - t /2 ) dt - * sqrt(2pi) | | - * - - * -inf. - * - * = ( 1 + erf(z) ) / 2 - * = erfc(z) / 2 - * - * where z = x/sqrt(2). Computation is via the functions - * erf and erfc. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -13,0 50000 1.5e-5 2.6e-6 - * - * - * ERROR MESSAGES: - * - * See erfcf(). - * - */ - /* erff.c - * - * Error function - * - * - * - * SYNOPSIS: - * - * float x, y, erff(); - * - * y = erff( x ); - * - * - * - * DESCRIPTION: - * - * The integral is - * - * x - * - - * 2 | | 2 - * erf(x) = -------- | exp( - t ) dt. - * sqrt(pi) | | - * - - * 0 - * - * The magnitude of x is limited to 9.231948545 for DEC - * arithmetic; 1 or -1 is returned outside this range. - * - * For 0 <= |x| < 1, erf(x) = x * P(x**2); otherwise - * erf(x) = 1 - erfc(x). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -9.3,9.3 50000 1.7e-7 2.8e-8 - * - */ - /* erfcf.c - * - * Complementary error function - * - * - * - * SYNOPSIS: - * - * float x, y, erfcf(); - * - * y = erfcf( x ); - * - * - * - * DESCRIPTION: - * - * - * 1 - erf(x) = - * - * inf. - * - - * 2 | | 2 - * erfc(x) = -------- | exp( - t ) dt - * sqrt(pi) | | - * - - * x - * - * - * For small x, erfc(x) = 1 - erf(x); otherwise polynomial - * approximations 1/x P(1/x**2) are computed. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -9.3,9.3 50000 3.9e-6 7.2e-7 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * erfcf underflow x**2 > MAXLOGF 0.0 - * - * - */ - -/* ndtrif.c - * - * Inverse of Normal distribution function - * - * - * - * SYNOPSIS: - * - * float x, y, ndtrif(); - * - * x = ndtrif( y ); - * - * - * - * DESCRIPTION: - * - * Returns the argument, x, for which the area under the - * Gaussian probability density function (integrated from - * minus infinity to x) is equal to y. - * - * - * For small arguments 0 < y < exp(-2), the program computes - * z = sqrt( -2.0 * log(y) ); then the approximation is - * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z). - * There are two rational functions P/Q, one for 0 < y < exp(-32) - * and the other for y up to exp(-2). For larger arguments, - * w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 1e-38, 1 30000 3.6e-7 5.0e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * ndtrif domain x <= 0 -MAXNUM - * ndtrif domain x >= 1 MAXNUM - * - */ - -/* pdtrf.c - * - * Poisson distribution - * - * - * - * SYNOPSIS: - * - * int k; - * float m, y, pdtrf(); - * - * y = pdtrf( k, m ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the first k terms of the Poisson - * distribution: - * - * k j - * -- -m m - * > e -- - * -- j! - * j=0 - * - * The terms are not summed directly; instead the incomplete - * gamma integral is employed, according to the relation - * - * y = pdtr( k, m ) = igamc( k+1, m ). - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 6.9e-5 8.0e-6 - * - */ - /* pdtrcf() - * - * Complemented poisson distribution - * - * - * - * SYNOPSIS: - * - * int k; - * float m, y, pdtrcf(); - * - * y = pdtrcf( k, m ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms k+1 to infinity of the Poisson - * distribution: - * - * inf. j - * -- -m m - * > e -- - * -- j! - * j=k+1 - * - * The terms are not summed directly; instead the incomplete - * gamma integral is employed, according to the formula - * - * y = pdtrc( k, m ) = igam( k+1, m ). - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 8.4e-5 1.2e-5 - * - */ - /* pdtrif() - * - * Inverse Poisson distribution - * - * - * - * SYNOPSIS: - * - * int k; - * float m, y, pdtrf(); - * - * m = pdtrif( k, y ); - * - * - * - * - * DESCRIPTION: - * - * Finds the Poisson variable x such that the integral - * from 0 to x of the Poisson density is equal to the - * given probability y. - * - * This is accomplished using the inverse gamma integral - * function and the relation - * - * m = igami( k+1, y ). - * - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 8.7e-6 1.4e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * pdtri domain y < 0 or y >= 1 0.0 - * k < 0 - * - */ - -/* polevlf.c - * p1evlf.c - * - * Evaluate polynomial - * - * - * - * SYNOPSIS: - * - * int N; - * float x, y, coef[N+1], polevlf[]; - * - * y = polevlf( x, coef, N ); - * - * - * - * DESCRIPTION: - * - * Evaluates polynomial of degree N: - * - * 2 N - * y = C + C x + C x +...+ C x - * 0 1 2 N - * - * Coefficients are stored in reverse order: - * - * coef[0] = C , ..., coef[N] = C . - * N 0 - * - * The function p1evl() assumes that coef[N] = 1.0 and is - * omitted from the array. Its calling arguments are - * otherwise the same as polevl(). - * - * - * SPEED: - * - * In the interest of speed, there are no checks for out - * of bounds arithmetic. This routine is used by most of - * the functions in the library. Depending on available - * equipment features, the user may wish to rewrite the - * program in microcode or assembly language. - * - */ - -/* polynf.c - * polyrf.c - * Arithmetic operations on polynomials - * - * In the following descriptions a, b, c are polynomials of degree - * na, nb, nc respectively. The degree of a polynomial cannot - * exceed a run-time value MAXPOLF. An operation that attempts - * to use or generate a polynomial of higher degree may produce a - * result that suffers truncation at degree MAXPOL. The value of - * MAXPOL is set by calling the function - * - * polinif( maxpol ); - * - * where maxpol is the desired maximum degree. This must be - * done prior to calling any of the other functions in this module. - * Memory for internal temporary polynomial storage is allocated - * by polinif(). - * - * Each polynomial is represented by an array containing its - * coefficients, together with a separately declared integer equal - * to the degree of the polynomial. The coefficients appear in - * ascending order; that is, - * - * 2 na - * a(x) = a[0] + a[1] * x + a[2] * x + ... + a[na] * x . - * - * - * - * sum = poleva( a, na, x ); Evaluate polynomial a(t) at t = x. - * polprtf( a, na, D ); Print the coefficients of a to D digits. - * polclrf( a, na ); Set a identically equal to zero, up to a[na]. - * polmovf( a, na, b ); Set b = a. - * poladdf( a, na, b, nb, c ); c = b + a, nc = max(na,nb) - * polsubf( a, na, b, nb, c ); c = b - a, nc = max(na,nb) - * polmulf( a, na, b, nb, c ); c = b * a, nc = na+nb - * - * - * Division: - * - * i = poldivf( a, na, b, nb, c ); c = b / a, nc = MAXPOL - * - * returns i = the degree of the first nonzero coefficient of a. - * The computed quotient c must be divided by x^i. An error message - * is printed if a is identically zero. - * - * - * Change of variables: - * If a and b are polynomials, and t = a(x), then - * c(t) = b(a(x)) - * is a polynomial found by substituting a(x) for t. The - * subroutine call for this is - * - * polsbtf( a, na, b, nb, c ); - * - * - * Notes: - * poldivf() is an integer routine; polevaf() is float. - * Any of the arguments a, b, c may refer to the same array. - * - */ - -/* powf.c - * - * Power function - * - * - * - * SYNOPSIS: - * - * float x, y, z, powf(); - * - * z = powf( x, y ); - * - * - * - * DESCRIPTION: - * - * Computes x raised to the yth power. Analytically, - * - * x**y = exp( y log(x) ). - * - * Following Cody and Waite, this program uses a lookup table - * of 2**-i/16 and pseudo extended precision arithmetic to - * obtain an extra three bits of accuracy in both the logarithm - * and the exponential. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,10 100,000 1.4e-7 3.6e-8 - * 1/10 < x < 10, x uniformly distributed. - * -10 < y < 10, y uniformly distributed. - * - * - * ERROR MESSAGES: - * - * message condition value returned - * powf overflow x**y > MAXNUMF MAXNUMF - * powf underflow x**y < 1/MAXNUMF 0.0 - * powf domain x<0 and y noninteger 0.0 - * - */ - -/* powif.c - * - * Real raised to integer power - * - * - * - * SYNOPSIS: - * - * float x, y, powif(); - * int n; - * - * y = powif( x, n ); - * - * - * - * DESCRIPTION: - * - * Returns argument x raised to the nth power. - * The routine efficiently decomposes n as a sum of powers of - * two. The desired power is a product of two-to-the-kth - * powers of x. Thus to compute the 32767 power of x requires - * 28 multiplications instead of 32767 multiplications. - * - * - * - * ACCURACY: - * - * - * Relative error: - * arithmetic x domain n domain # trials peak rms - * IEEE .04,26 -26,26 100000 1.1e-6 2.0e-7 - * IEEE 1,2 -128,128 100000 1.1e-5 1.0e-6 - * - * Returns MAXNUMF on overflow, zero on underflow. - * - */ - -/* psif.c - * - * Psi (digamma) function - * - * - * SYNOPSIS: - * - * float x, y, psif(); - * - * y = psif( x ); - * - * - * DESCRIPTION: - * - * d - - * psi(x) = -- ln | (x) - * dx - * - * is the logarithmic derivative of the gamma function. - * For integer x, - * n-1 - * - - * psi(n) = -EUL + > 1/k. - * - - * k=1 - * - * This formula is used for 0 < n <= 10. If x is negative, it - * is transformed to a positive argument by the reflection - * formula psi(1-x) = psi(x) + pi cot(pi x). - * For general positive x, the argument is made greater than 10 - * using the recurrence psi(x+1) = psi(x) + 1/x. - * Then the following asymptotic expansion is applied: - * - * inf. B - * - 2k - * psi(x) = log(x) - 1/2x - > ------- - * - 2k - * k=1 2k x - * - * where the B2k are Bernoulli numbers. - * - * ACCURACY: - * Absolute error, relative when |psi| > 1 : - * arithmetic domain # trials peak rms - * IEEE -33,0 30000 8.2e-7 1.2e-7 - * IEEE 0,33 100000 7.3e-7 7.7e-8 - * - * ERROR MESSAGES: - * message condition value returned - * psi singularity x integer <=0 MAXNUMF - */ - -/* rgammaf.c - * - * Reciprocal gamma function - * - * - * - * SYNOPSIS: - * - * float x, y, rgammaf(); - * - * y = rgammaf( x ); - * - * - * - * DESCRIPTION: - * - * Returns one divided by the gamma function of the argument. - * - * The function is approximated by a Chebyshev expansion in - * the interval [0,1]. Range reduction is by recurrence - * for arguments between -34.034 and +34.84425627277176174. - * 1/MAXNUMF is returned for positive arguments outside this - * range. - * - * The reciprocal gamma function has no singularities, - * but overflow and underflow may occur for large arguments. - * These conditions return either MAXNUMF or 1/MAXNUMF with - * appropriate sign. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -34,+34 100000 8.9e-7 1.1e-7 - */ - -/* shichif.c - * - * Hyperbolic sine and cosine integrals - * - * - * - * SYNOPSIS: - * - * float x, Chi, Shi; - * - * shichi( x, &Chi, &Shi ); - * - * - * DESCRIPTION: - * - * Approximates the integrals - * - * x - * - - * | | cosh t - 1 - * Chi(x) = eul + ln x + | ----------- dt, - * | | t - * - - * 0 - * - * x - * - - * | | sinh t - * Shi(x) = | ------ dt - * | | t - * - - * 0 - * - * where eul = 0.57721566490153286061 is Euler's constant. - * The integrals are evaluated by power series for x < 8 - * and by Chebyshev expansions for x between 8 and 88. - * For large x, both functions approach exp(x)/2x. - * Arguments greater than 88 in magnitude return MAXNUM. - * - * - * ACCURACY: - * - * Test interval 0 to 88. - * Relative error: - * arithmetic function # trials peak rms - * IEEE Shi 20000 3.5e-7 7.0e-8 - * Absolute error, except relative when |Chi| > 1: - * IEEE Chi 20000 3.8e-7 7.6e-8 - */ - -/* sicif.c - * - * Sine and cosine integrals - * - * - * - * SYNOPSIS: - * - * float x, Ci, Si; - * - * sicif( x, &Si, &Ci ); - * - * - * DESCRIPTION: - * - * Evaluates the integrals - * - * x - * - - * | cos t - 1 - * Ci(x) = eul + ln x + | --------- dt, - * | t - * - - * 0 - * x - * - - * | sin t - * Si(x) = | ----- dt - * | t - * - - * 0 - * - * where eul = 0.57721566490153286061 is Euler's constant. - * The integrals are approximated by rational functions. - * For x > 8 auxiliary functions f(x) and g(x) are employed - * such that - * - * Ci(x) = f(x) sin(x) - g(x) cos(x) - * Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x) - * - * - * ACCURACY: - * Test interval = [0,50]. - * Absolute error, except relative when > 1: - * arithmetic function # trials peak rms - * IEEE Si 30000 2.1e-7 4.3e-8 - * IEEE Ci 30000 3.9e-7 2.2e-8 - */ - -/* sindgf.c - * - * Circular sine of angle in degrees - * - * - * - * SYNOPSIS: - * - * float x, y, sindgf(); - * - * y = sindgf( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of 45 degrees. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the sine is approximated by - * x + x**3 P(x**2). - * Between pi/4 and pi/2 the cosine is represented as - * 1 - x**2 Q(x**2). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-3600 100,000 1.2e-7 3.0e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * sin total loss x > 2^24 0.0 - * - */ - -/* cosdgf.c - * - * Circular cosine of angle in degrees - * - * - * - * SYNOPSIS: - * - * float x, y, cosdgf(); - * - * y = cosdgf( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of 45 degrees. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the cosine is approximated by - * 1 - x**2 Q(x**2). - * Between pi/4 and pi/2 the sine is represented as - * x + x**3 P(x**2). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8 - */ - -/* sinf.c - * - * Circular sine - * - * - * - * SYNOPSIS: - * - * float x, y, sinf(); - * - * y = sinf( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of pi/4. The reduction - * error is nearly eliminated by contriving an extended precision - * modular arithmetic. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the sine is approximated by - * x + x**3 P(x**2). - * Between pi/4 and pi/2 the cosine is represented as - * 1 - x**2 Q(x**2). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -4096,+4096 100,000 1.2e-7 3.0e-8 - * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * sin total loss x > 2^24 0.0 - * - * Partial loss of accuracy begins to occur at x = 2^13 - * = 8192. Results may be meaningless for x >= 2^24 - * The routine as implemented flags a TLOSS error - * for x >= 2^24 and returns 0.0. - */ - -/* cosf.c - * - * Circular cosine - * - * - * - * SYNOPSIS: - * - * float x, y, cosf(); - * - * y = cosf( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of pi/4. The reduction - * error is nearly eliminated by contriving an extended precision - * modular arithmetic. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the cosine is approximated by - * 1 - x**2 Q(x**2). - * Between pi/4 and pi/2 the sine is represented as - * x + x**3 P(x**2). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8 - */ - -/* sinhf.c - * - * Hyperbolic sine - * - * - * - * SYNOPSIS: - * - * float x, y, sinhf(); - * - * y = sinhf( x ); - * - * - * - * DESCRIPTION: - * - * Returns hyperbolic sine of argument in the range MINLOGF to - * MAXLOGF. - * - * The range is partitioned into two segments. If |x| <= 1, a - * polynomial approximation is used. - * Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-MAXLOG 100000 1.1e-7 2.9e-8 - * - */ - -/* spencef.c - * - * Dilogarithm - * - * - * - * SYNOPSIS: - * - * float x, y, spencef(); - * - * y = spencef( x ); - * - * - * - * DESCRIPTION: - * - * Computes the integral - * - * x - * - - * | | log t - * spence(x) = - | ----- dt - * | | t - 1 - * - - * 1 - * - * for x >= 0. A rational approximation gives the integral in - * the interval (0.5, 1.5). Transformation formulas for 1/x - * and 1-x are employed outside the basic expansion range. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,4 30000 4.4e-7 6.3e-8 - * - * - */ - -/* sqrtf.c - * - * Square root - * - * - * - * SYNOPSIS: - * - * float x, y, sqrtf(); - * - * y = sqrtf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the square root of x. - * - * Range reduction involves isolating the power of two of the - * argument and using a polynomial approximation to obtain - * a rough value for the square root. Then Heron's iteration - * is used three times to converge to an accurate value. - * - * - * - * ACCURACY: - * - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,1.e38 100000 8.7e-8 2.9e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * sqrtf domain x < 0 0.0 - * - */ - -/* stdtrf.c - * - * Student's t distribution - * - * - * - * SYNOPSIS: - * - * float t, stdtrf(); - * short k; - * - * y = stdtrf( k, t ); - * - * - * DESCRIPTION: - * - * Computes the integral from minus infinity to t of the Student - * t distribution with integer k > 0 degrees of freedom: - * - * t - * - - * | | - * - | 2 -(k+1)/2 - * | ( (k+1)/2 ) | ( x ) - * ---------------------- | ( 1 + --- ) dx - * - | ( k ) - * sqrt( k pi ) | ( k/2 ) | - * | | - * - - * -inf. - * - * Relation to incomplete beta integral: - * - * 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z ) - * where - * z = k/(k + t**2). - * - * For t < -1, this is the method of computation. For higher t, - * a direct method is derived from integration by parts. - * Since the function is symmetric about t=0, the area under the - * right tail of the density is found by calling the function - * with -t instead of t. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +/- 100 5000 2.3e-5 2.9e-6 - */ - -/* struvef.c - * - * Struve function - * - * - * - * SYNOPSIS: - * - * float v, x, y, struvef(); - * - * y = struvef( v, x ); - * - * - * - * DESCRIPTION: - * - * Computes the Struve function Hv(x) of order v, argument x. - * Negative x is rejected unless v is an integer. - * - * This module also contains the hypergeometric functions 1F2 - * and 3F0 and a routine for the Bessel function Yv(x) with - * noninteger v. - * - * - * - * ACCURACY: - * - * v varies from 0 to 10. - * Absolute error (relative error when |Hv(x)| > 1): - * arithmetic domain # trials peak rms - * IEEE -10,10 100000 9.0e-5 4.0e-6 - * - */ - -/* tandgf.c - * - * Circular tangent of angle in degrees - * - * - * - * SYNOPSIS: - * - * float x, y, tandgf(); - * - * y = tandgf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the circular tangent of the radian argument x. - * - * Range reduction is into intervals of 45 degrees. - * - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-2^24 50000 2.4e-7 4.8e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * tanf total loss x > 2^24 0.0 - * - */ - /* cotdgf.c - * - * Circular cotangent of angle in degrees - * - * - * - * SYNOPSIS: - * - * float x, y, cotdgf(); - * - * y = cotdgf( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of 45 degrees. - * A common routine computes either the tangent or cotangent. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-2^24 50000 2.4e-7 4.8e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * cot total loss x > 2^24 0.0 - * cot singularity x = 0 MAXNUMF - * - */ - -/* tanf.c - * - * Circular tangent - * - * - * - * SYNOPSIS: - * - * float x, y, tanf(); - * - * y = tanf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the circular tangent of the radian argument x. - * - * Range reduction is modulo pi/4. A polynomial approximation - * is employed in the basic interval [0, pi/4]. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-4096 100000 3.3e-7 4.5e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * tanf total loss x > 2^24 0.0 - * - */ - /* cotf.c - * - * Circular cotangent - * - * - * - * SYNOPSIS: - * - * float x, y, cotf(); - * - * y = cotf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the circular cotangent of the radian argument x. - * A common routine computes either the tangent or cotangent. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-4096 100000 3.0e-7 4.5e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * cot total loss x > 2^24 0.0 - * cot singularity x = 0 MAXNUMF - * - */ - -/* tanhf.c - * - * Hyperbolic tangent - * - * - * - * SYNOPSIS: - * - * float x, y, tanhf(); - * - * y = tanhf( x ); - * - * - * - * DESCRIPTION: - * - * Returns hyperbolic tangent of argument in the range MINLOG to - * MAXLOG. - * - * A polynomial approximation is used for |x| < 0.625. - * Otherwise, - * - * tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -2,2 100000 1.3e-7 2.6e-8 - * - */ - -/* ynf.c - * - * Bessel function of second kind of integer order - * - * - * - * SYNOPSIS: - * - * float x, y, ynf(); - * int n; - * - * y = ynf( n, x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order n, where n is a - * (possibly negative) integer. - * - * The function is evaluated by forward recurrence on - * n, starting with values computed by the routines - * y0() and y1(). - * - * If n = 0 or 1 the routine for y0 or y1 is called - * directly. - * - * - * - * ACCURACY: - * - * - * Absolute error, except relative when y > 1: - * - * arithmetic domain # trials peak rms - * IEEE 0, 30 10000 2.3e-6 3.4e-7 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * yn singularity x = 0 MAXNUMF - * yn overflow MAXNUMF - * - * Spot checked against tables for x, n between 0 and 100. - * - */ - - /* zetacf.c - * - * Riemann zeta function - * - * - * - * SYNOPSIS: - * - * float x, y, zetacf(); - * - * y = zetacf( x ); - * - * - * - * DESCRIPTION: - * - * - * - * inf. - * - -x - * zetac(x) = > k , x > 1, - * - - * k=2 - * - * is related to the Riemann zeta function by - * - * Riemann zeta(x) = zetac(x) + 1. - * - * Extension of the function definition for x < 1 is implemented. - * Zero is returned for x > log2(MAXNUM). - * - * An overflow error may occur for large negative x, due to the - * gamma function in the reflection formula. - * - * ACCURACY: - * - * Tabulated values have full machine accuracy. - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 1,50 30000 5.5e-7 7.5e-8 - * - * - */ - -/* zetaf.c - * - * Riemann zeta function of two arguments - * - * - * - * SYNOPSIS: - * - * float x, q, y, zetaf(); - * - * y = zetaf( x, q ); - * - * - * - * DESCRIPTION: - * - * - * - * inf. - * - -x - * zeta(x,q) = > (k+q) - * - - * k=0 - * - * where x > 1 and q is not a negative integer or zero. - * The Euler-Maclaurin summation formula is used to obtain - * the expansion - * - * n - * - -x - * zeta(x,q) = > (k+q) - * - - * k=1 - * - * 1-x inf. B x(x+1)...(x+2j) - * (n+q) 1 - 2j - * + --------- - ------- + > -------------------- - * x-1 x - x+2j+1 - * 2(n+q) j=1 (2j)! (n+q) - * - * where the B2j are Bernoulli numbers. Note that (see zetac.c) - * zeta(x,1) = zetac(x) + 1. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,25 10000 6.9e-7 1.0e-7 - * - * Large arguments may produce underflow in powf(), in which - * case the results are inaccurate. - * - * REFERENCE: - * - * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, - * Series, and Products, p. 1073; Academic Press, 1980. - * - */ diff --git a/libm/float/acoshf.c b/libm/float/acoshf.c deleted file mode 100644 index c45206125..000000000 --- a/libm/float/acoshf.c +++ /dev/null @@ -1,97 +0,0 @@ -/* acoshf.c - * - * Inverse hyperbolic cosine - * - * - * - * SYNOPSIS: - * - * float x, y, acoshf(); - * - * y = acoshf( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic cosine of argument. - * - * If 1 <= x < 1.5, a polynomial approximation - * - * sqrt(z) * P(z) - * - * where z = x-1, is used. Otherwise, - * - * acosh(x) = log( x + sqrt( (x-1)(x+1) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 1,3 100000 1.8e-7 3.9e-8 - * IEEE 1,2000 100000 3.0e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * acoshf domain |x| < 1 0.0 - * - */ - -/* acosh.c */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision inverse hyperbolic cosine - * test interval: [1.0, 1.5] - * trials: 10000 - * peak relative error: 1.7e-7 - * rms relative error: 5.0e-8 - * - * Copyright (C) 1989 by Stephen L. Moshier. All rights reserved. - */ -#include <math.h> -extern float LOGE2F; - -float sqrtf( float ); -float logf( float ); - -float acoshf( float xx ) -{ -float x, z; - -x = xx; -if( x < 1.0 ) - { - mtherr( "acoshf", DOMAIN ); - return(0.0); - } - -if( x > 1500.0 ) - return( logf(x) + LOGE2F ); - -z = x - 1.0; - -if( z < 0.5 ) - { - z = - (((( 1.7596881071E-3 * z - - 7.5272886713E-3) * z - + 2.6454905019E-2) * z - - 1.1784741703E-1) * z - + 1.4142135263E0) * sqrtf( z ); - } -else - { - z = sqrtf( z*(x+1.0) ); - z = logf(x + z); - } -return( z ); -} diff --git a/libm/float/airyf.c b/libm/float/airyf.c deleted file mode 100644 index a84a5c861..000000000 --- a/libm/float/airyf.c +++ /dev/null @@ -1,377 +0,0 @@ -/* airy.c - * - * Airy function - * - * - * - * SYNOPSIS: - * - * float x, ai, aip, bi, bip; - * int airyf(); - * - * airyf( x, _&ai, _&aip, _&bi, _&bip ); - * - * - * - * DESCRIPTION: - * - * Solution of the differential equation - * - * y"(x) = xy. - * - * The function returns the two independent solutions Ai, Bi - * and their first derivatives Ai'(x), Bi'(x). - * - * Evaluation is by power series summation for small x, - * by rational minimax approximations for large x. - * - * - * - * ACCURACY: - * Error criterion is absolute when function <= 1, relative - * when function > 1, except * denotes relative error criterion. - * For large negative x, the absolute error increases as x^1.5. - * For large positive x, the relative error increases as x^1.5. - * - * Arithmetic domain function # trials peak rms - * IEEE -10, 0 Ai 50000 7.0e-7 1.2e-7 - * IEEE 0, 10 Ai 50000 9.9e-6* 6.8e-7* - * IEEE -10, 0 Ai' 50000 2.4e-6 3.5e-7 - * IEEE 0, 10 Ai' 50000 8.7e-6* 6.2e-7* - * IEEE -10, 10 Bi 100000 2.2e-6 2.6e-7 - * IEEE -10, 10 Bi' 50000 2.2e-6 3.5e-7 - * - */ - /* airy.c */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -static float c1 = 0.35502805388781723926; -static float c2 = 0.258819403792806798405; -static float sqrt3 = 1.732050807568877293527; -static float sqpii = 5.64189583547756286948E-1; -extern float PIF; - -extern float MAXNUMF, MACHEPF; -#define MAXAIRY 25.77 - -/* Note, these expansions are for double precision accuracy; - * they have not yet been redesigned for single precision. - */ -static float AN[8] = { - 3.46538101525629032477e-1, - 1.20075952739645805542e1, - 7.62796053615234516538e1, - 1.68089224934630576269e2, - 1.59756391350164413639e2, - 7.05360906840444183113e1, - 1.40264691163389668864e1, - 9.99999999999999995305e-1, -}; -static float AD[8] = { - 5.67594532638770212846e-1, - 1.47562562584847203173e1, - 8.45138970141474626562e1, - 1.77318088145400459522e2, - 1.64234692871529701831e2, - 7.14778400825575695274e1, - 1.40959135607834029598e1, - 1.00000000000000000470e0, -}; - - -static float APN[8] = { - 6.13759184814035759225e-1, - 1.47454670787755323881e1, - 8.20584123476060982430e1, - 1.71184781360976385540e2, - 1.59317847137141783523e2, - 6.99778599330103016170e1, - 1.39470856980481566958e1, - 1.00000000000000000550e0, -}; -static float APD[8] = { - 3.34203677749736953049e-1, - 1.11810297306158156705e1, - 7.11727352147859965283e1, - 1.58778084372838313640e2, - 1.53206427475809220834e2, - 6.86752304592780337944e1, - 1.38498634758259442477e1, - 9.99999999999999994502e-1, -}; - -static float BN16[5] = { --2.53240795869364152689e-1, - 5.75285167332467384228e-1, --3.29907036873225371650e-1, - 6.44404068948199951727e-2, --3.82519546641336734394e-3, -}; -static float BD16[5] = { -/* 1.00000000000000000000e0,*/ --7.15685095054035237902e0, - 1.06039580715664694291e1, --5.23246636471251500874e0, - 9.57395864378383833152e-1, --5.50828147163549611107e-2, -}; - -static float BPPN[5] = { - 4.65461162774651610328e-1, --1.08992173800493920734e0, - 6.38800117371827987759e-1, --1.26844349553102907034e-1, - 7.62487844342109852105e-3, -}; -static float BPPD[5] = { -/* 1.00000000000000000000e0,*/ --8.70622787633159124240e0, - 1.38993162704553213172e1, --7.14116144616431159572e0, - 1.34008595960680518666e0, --7.84273211323341930448e-2, -}; - -static float AFN[9] = { --1.31696323418331795333e-1, --6.26456544431912369773e-1, --6.93158036036933542233e-1, --2.79779981545119124951e-1, --4.91900132609500318020e-2, --4.06265923594885404393e-3, --1.59276496239262096340e-4, --2.77649108155232920844e-6, --1.67787698489114633780e-8, -}; -static float AFD[9] = { -/* 1.00000000000000000000e0,*/ - 1.33560420706553243746e1, - 3.26825032795224613948e1, - 2.67367040941499554804e1, - 9.18707402907259625840e0, - 1.47529146771666414581e0, - 1.15687173795188044134e-1, - 4.40291641615211203805e-3, - 7.54720348287414296618e-5, - 4.51850092970580378464e-7, -}; - -static float AGN[11] = { - 1.97339932091685679179e-2, - 3.91103029615688277255e-1, - 1.06579897599595591108e0, - 9.39169229816650230044e-1, - 3.51465656105547619242e-1, - 6.33888919628925490927e-2, - 5.85804113048388458567e-3, - 2.82851600836737019778e-4, - 6.98793669997260967291e-6, - 8.11789239554389293311e-8, - 3.41551784765923618484e-10, -}; -static float AGD[10] = { -/* 1.00000000000000000000e0,*/ - 9.30892908077441974853e0, - 1.98352928718312140417e1, - 1.55646628932864612953e1, - 5.47686069422975497931e0, - 9.54293611618961883998e-1, - 8.64580826352392193095e-2, - 4.12656523824222607191e-3, - 1.01259085116509135510e-4, - 1.17166733214413521882e-6, - 4.91834570062930015649e-9, -}; - -static float APFN[9] = { - 1.85365624022535566142e-1, - 8.86712188052584095637e-1, - 9.87391981747398547272e-1, - 4.01241082318003734092e-1, - 7.10304926289631174579e-2, - 5.90618657995661810071e-3, - 2.33051409401776799569e-4, - 4.08718778289035454598e-6, - 2.48379932900442457853e-8, -}; -static float APFD[9] = { -/* 1.00000000000000000000e0,*/ - 1.47345854687502542552e1, - 3.75423933435489594466e1, - 3.14657751203046424330e1, - 1.09969125207298778536e1, - 1.78885054766999417817e0, - 1.41733275753662636873e-1, - 5.44066067017226003627e-3, - 9.39421290654511171663e-5, - 5.65978713036027009243e-7, -}; - -static float APGN[11] = { --3.55615429033082288335e-2, --6.37311518129435504426e-1, --1.70856738884312371053e0, --1.50221872117316635393e0, --5.63606665822102676611e-1, --1.02101031120216891789e-1, --9.48396695961445269093e-3, --4.60325307486780994357e-4, --1.14300836484517375919e-5, --1.33415518685547420648e-7, --5.63803833958893494476e-10, -}; -static float APGD[11] = { -/* 1.00000000000000000000e0,*/ - 9.85865801696130355144e0, - 2.16401867356585941885e1, - 1.73130776389749389525e1, - 6.17872175280828766327e0, - 1.08848694396321495475e0, - 9.95005543440888479402e-2, - 4.78468199683886610842e-3, - 1.18159633322838625562e-4, - 1.37480673554219441465e-6, - 5.79912514929147598821e-9, -}; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -float polevlf(float, float *, int); -float p1evlf(float, float *, int); -float sinf(float), cosf(float), expf(float), sqrtf(float); - -int airyf( float xx, float *ai, float *aip, float *bi, float *bip ) -{ -float x, z, zz, t, f, g, uf, ug, k, zeta, theta; -int domflg; - -x = xx; -domflg = 0; -if( x > MAXAIRY ) - { - *ai = 0; - *aip = 0; - *bi = MAXNUMF; - *bip = MAXNUMF; - return(-1); - } - -if( x < -2.09 ) - { - domflg = 15; - t = sqrtf(-x); - zeta = -2.0 * x * t / 3.0; - t = sqrtf(t); - k = sqpii / t; - z = 1.0/zeta; - zz = z * z; - uf = 1.0 + zz * polevlf( zz, AFN, 8 ) / p1evlf( zz, AFD, 9 ); - ug = z * polevlf( zz, AGN, 10 ) / p1evlf( zz, AGD, 10 ); - theta = zeta + 0.25 * PIF; - f = sinf( theta ); - g = cosf( theta ); - *ai = k * (f * uf - g * ug); - *bi = k * (g * uf + f * ug); - uf = 1.0 + zz * polevlf( zz, APFN, 8 ) / p1evlf( zz, APFD, 9 ); - ug = z * polevlf( zz, APGN, 10 ) / p1evlf( zz, APGD, 10 ); - k = sqpii * t; - *aip = -k * (g * uf + f * ug); - *bip = k * (f * uf - g * ug); - return(0); - } - -if( x >= 2.09 ) /* cbrt(9) */ - { - domflg = 5; - t = sqrtf(x); - zeta = 2.0 * x * t / 3.0; - g = expf( zeta ); - t = sqrtf(t); - k = 2.0 * t * g; - z = 1.0/zeta; - f = polevlf( z, AN, 7 ) / polevlf( z, AD, 7 ); - *ai = sqpii * f / k; - k = -0.5 * sqpii * t / g; - f = polevlf( z, APN, 7 ) / polevlf( z, APD, 7 ); - *aip = f * k; - - if( x > 8.3203353 ) /* zeta > 16 */ - { - f = z * polevlf( z, BN16, 4 ) / p1evlf( z, BD16, 5 ); - k = sqpii * g; - *bi = k * (1.0 + f) / t; - f = z * polevlf( z, BPPN, 4 ) / p1evlf( z, BPPD, 5 ); - *bip = k * t * (1.0 + f); - return(0); - } - } - -f = 1.0; -g = x; -t = 1.0; -uf = 1.0; -ug = x; -k = 1.0; -z = x * x * x; -while( t > MACHEPF ) - { - uf *= z; - k += 1.0; - uf /=k; - ug *= z; - k += 1.0; - ug /=k; - uf /=k; - f += uf; - k += 1.0; - ug /=k; - g += ug; - t = fabsf(uf/f); - } -uf = c1 * f; -ug = c2 * g; -if( (domflg & 1) == 0 ) - *ai = uf - ug; -if( (domflg & 2) == 0 ) - *bi = sqrt3 * (uf + ug); - -/* the deriviative of ai */ -k = 4.0; -uf = x * x/2.0; -ug = z/3.0; -f = uf; -g = 1.0 + ug; -uf /= 3.0; -t = 1.0; - -while( t > MACHEPF ) - { - uf *= z; - ug /=k; - k += 1.0; - ug *= z; - uf /=k; - f += uf; - k += 1.0; - ug /=k; - uf /=k; - g += ug; - k += 1.0; - t = fabsf(ug/g); - } - -uf = c1 * f; -ug = c2 * g; -if( (domflg & 4) == 0 ) - *aip = uf - ug; -if( (domflg & 8) == 0 ) - *bip = sqrt3 * (uf + ug); -return(0); -} diff --git a/libm/float/asinf.c b/libm/float/asinf.c deleted file mode 100644 index c96d75acb..000000000 --- a/libm/float/asinf.c +++ /dev/null @@ -1,186 +0,0 @@ -/* asinf.c - * - * Inverse circular sine - * - * - * - * SYNOPSIS: - * - * float x, y, asinf(); - * - * y = asinf( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between -pi/2 and +pi/2 whose sine is x. - * - * A polynomial of the form x + x**3 P(x**2) - * is used for |x| in the interval [0, 0.5]. If |x| > 0.5 it is - * transformed by the identity - * - * asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -1, 1 100000 2.5e-7 5.0e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * asinf domain |x| > 1 0.0 - * - */ - /* acosf() - * - * Inverse circular cosine - * - * - * - * SYNOPSIS: - * - * float x, y, acosf(); - * - * y = acosf( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between -pi/2 and +pi/2 whose cosine - * is x. - * - * Analytically, acos(x) = pi/2 - asin(x). However if |x| is - * near 1, there is cancellation error in subtracting asin(x) - * from pi/2. Hence if x < -0.5, - * - * acos(x) = pi - 2.0 * asin( sqrt((1+x)/2) ); - * - * or if x > +0.5, - * - * acos(x) = 2.0 * asin( sqrt((1-x)/2) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -1, 1 100000 1.4e-7 4.2e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * acosf domain |x| > 1 0.0 - */ - -/* asin.c */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision circular arcsine - * test interval: [-0.5, +0.5] - * trials: 10000 - * peak relative error: 6.7e-8 - * rms relative error: 2.5e-8 - */ -#include <math.h> -extern float PIF, PIO2F; - -float sqrtf( float ); - -float asinf( float xx ) -{ -float a, x, z; -int sign, flag; - -x = xx; - -if( x > 0 ) - { - sign = 1; - a = x; - } -else - { - sign = -1; - a = -x; - } - -if( a > 1.0 ) - { - mtherr( "asinf", DOMAIN ); - return( 0.0 ); - } - -if( a < 1.0e-4 ) - { - z = a; - goto done; - } - -if( a > 0.5 ) - { - z = 0.5 * (1.0 - a); - x = sqrtf( z ); - flag = 1; - } -else - { - x = a; - z = x * x; - flag = 0; - } - -z = -(((( 4.2163199048E-2 * z - + 2.4181311049E-2) * z - + 4.5470025998E-2) * z - + 7.4953002686E-2) * z - + 1.6666752422E-1) * z * x - + x; - -if( flag != 0 ) - { - z = z + z; - z = PIO2F - z; - } -done: -if( sign < 0 ) - z = -z; -return( z ); -} - - - - -float acosf( float x ) -{ - -if( x < -1.0 ) - goto domerr; - -if( x < -0.5) - return( PIF - 2.0 * asinf( sqrtf(0.5*(1.0+x)) ) ); - -if( x > 1.0 ) - { -domerr: mtherr( "acosf", DOMAIN ); - return( 0.0 ); - } - -if( x > 0.5 ) - return( 2.0 * asinf( sqrtf(0.5*(1.0-x) ) ) ); - -return( PIO2F - asinf(x) ); -} - diff --git a/libm/float/asinhf.c b/libm/float/asinhf.c deleted file mode 100644 index d3fbe10a7..000000000 --- a/libm/float/asinhf.c +++ /dev/null @@ -1,88 +0,0 @@ -/* asinhf.c - * - * Inverse hyperbolic sine - * - * - * - * SYNOPSIS: - * - * float x, y, asinhf(); - * - * y = asinhf( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic sine of argument. - * - * If |x| < 0.5, the function is approximated by a rational - * form x + x**3 P(x)/Q(x). Otherwise, - * - * asinh(x) = log( x + sqrt(1 + x*x) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -3,3 100000 2.4e-7 4.1e-8 - * - */ - -/* asinh.c */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision inverse hyperbolic sine - * test interval: [-0.5, +0.5] - * trials: 10000 - * peak relative error: 8.8e-8 - * rms relative error: 3.2e-8 - */ -#include <math.h> -extern float LOGE2F; - -float logf( float ); -float sqrtf( float ); - -float asinhf( float xx ) -{ -float x, z; - -if( xx < 0 ) - x = -xx; -else - x = xx; - -if( x > 1500.0 ) - { - z = logf(x) + LOGE2F; - goto done; - } -z = x * x; -if( x < 0.5 ) - { - z = - ((( 2.0122003309E-2 * z - - 4.2699340972E-2) * z - + 7.4847586088E-2) * z - - 1.6666288134E-1) * z * x - + x; - } -else - { - z = sqrtf( z + 1.0 ); - z = logf( x + z ); - } -done: -if( xx < 0 ) - z = -z; -return( z ); -} - diff --git a/libm/float/atanf.c b/libm/float/atanf.c deleted file mode 100644 index 321e3be39..000000000 --- a/libm/float/atanf.c +++ /dev/null @@ -1,190 +0,0 @@ -/* atanf.c - * - * Inverse circular tangent - * (arctangent) - * - * - * - * SYNOPSIS: - * - * float x, y, atanf(); - * - * y = atanf( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between -pi/2 and +pi/2 whose tangent - * is x. - * - * Range reduction is from four intervals into the interval - * from zero to tan( pi/8 ). A polynomial approximates - * the function in this basic interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10, 10 100000 1.9e-7 4.1e-8 - * - */ - /* atan2f() - * - * Quadrant correct inverse circular tangent - * - * - * - * SYNOPSIS: - * - * float x, y, z, atan2f(); - * - * z = atan2f( y, x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle whose tangent is y/x. - * Define compile time symbol ANSIC = 1 for ANSI standard, - * range -PI < z <= +PI, args (y,x); else ANSIC = 0 for range - * 0 to 2PI, args (x,y). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10, 10 100000 1.9e-7 4.1e-8 - * See atan.c. - * - */ - -/* atan.c */ - - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision circular arcsine - * test interval: [-tan(pi/8), +tan(pi/8)] - * trials: 10000 - * peak relative error: 7.7e-8 - * rms relative error: 2.9e-8 - */ -#include <math.h> -extern float PIF, PIO2F, PIO4F; - -float atanf( float xx ) -{ -float x, y, z; -int sign; - -x = xx; - -/* make argument positive and save the sign */ -if( xx < 0.0 ) - { - sign = -1; - x = -xx; - } -else - { - sign = 1; - x = xx; - } -/* range reduction */ -if( x > 2.414213562373095 ) /* tan 3pi/8 */ - { - y = PIO2F; - x = -( 1.0/x ); - } - -else if( x > 0.4142135623730950 ) /* tan pi/8 */ - { - y = PIO4F; - x = (x-1.0)/(x+1.0); - } -else - y = 0.0; - -z = x * x; -y += -((( 8.05374449538e-2 * z - - 1.38776856032E-1) * z - + 1.99777106478E-1) * z - - 3.33329491539E-1) * z * x - + x; - -if( sign < 0 ) - y = -y; - -return( y ); -} - - - - -float atan2f( float y, float x ) -{ -float z, w; -int code; - - -code = 0; - -if( x < 0.0 ) - code = 2; -if( y < 0.0 ) - code |= 1; - -if( x == 0.0 ) - { - if( code & 1 ) - { -#if ANSIC - return( -PIO2F ); -#else - return( 3.0*PIO2F ); -#endif - } - if( y == 0.0 ) - return( 0.0 ); - return( PIO2F ); - } - -if( y == 0.0 ) - { - if( code & 2 ) - return( PIF ); - return( 0.0 ); - } - - -switch( code ) - { - default: -#if ANSIC - case 0: - case 1: w = 0.0; break; - case 2: w = PIF; break; - case 3: w = -PIF; break; -#else - case 0: w = 0.0; break; - case 1: w = 2.0 * PIF; break; - case 2: - case 3: w = PIF; break; -#endif - } - -z = atanf( y/x ); - -return( w + z ); -} - diff --git a/libm/float/atanhf.c b/libm/float/atanhf.c deleted file mode 100644 index dfadad09e..000000000 --- a/libm/float/atanhf.c +++ /dev/null @@ -1,92 +0,0 @@ -/* atanhf.c - * - * Inverse hyperbolic tangent - * - * - * - * SYNOPSIS: - * - * float x, y, atanhf(); - * - * y = atanhf( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic tangent of argument in the range - * MINLOGF to MAXLOGF. - * - * If |x| < 0.5, a polynomial approximation is used. - * Otherwise, - * atanh(x) = 0.5 * log( (1+x)/(1-x) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -1,1 100000 1.4e-7 3.1e-8 - * - */ - -/* atanh.c */ - - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright (C) 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision inverse hyperbolic tangent - * test interval: [-0.5, +0.5] - * trials: 10000 - * peak relative error: 8.2e-8 - * rms relative error: 3.0e-8 - */ -#include <math.h> -extern float MAXNUMF; - -float logf( float ); - -float atanhf( float xx ) -{ -float x, z; - -x = xx; -if( x < 0 ) - z = -x; -else - z = x; -if( z >= 1.0 ) - { - if( x == 1.0 ) - return( MAXNUMF ); - if( x == -1.0 ) - return( -MAXNUMF ); - mtherr( "atanhl", DOMAIN ); - return( MAXNUMF ); - } - -if( z < 1.0e-4 ) - return(x); - -if( z < 0.5 ) - { - z = x * x; - z = - (((( 1.81740078349E-1 * z - + 8.24370301058E-2) * z - + 1.46691431730E-1) * z - + 1.99782164500E-1) * z - + 3.33337300303E-1) * z * x - + x; - } -else - { - z = 0.5 * logf( (1.0+x)/(1.0-x) ); - } -return( z ); -} diff --git a/libm/float/bdtrf.c b/libm/float/bdtrf.c deleted file mode 100644 index e063f1c77..000000000 --- a/libm/float/bdtrf.c +++ /dev/null @@ -1,247 +0,0 @@ -/* bdtrf.c - * - * Binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, bdtrf(); - * - * y = bdtrf( k, n, p ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms 0 through k of the Binomial - * probability density: - * - * k - * -- ( n ) j n-j - * > ( ) p (1-p) - * -- ( j ) - * j=0 - * - * The terms are not summed directly; instead the incomplete - * beta integral is employed, according to the formula - * - * y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * - * - * ACCURACY: - * - * Relative error (p varies from 0 to 1): - * arithmetic domain # trials peak rms - * IEEE 0,100 2000 6.9e-5 1.1e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * bdtrf domain k < 0 0.0 - * n < k - * x < 0, x > 1 - * - */ - /* bdtrcf() - * - * Complemented binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, bdtrcf(); - * - * y = bdtrcf( k, n, p ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms k+1 through n of the Binomial - * probability density: - * - * n - * -- ( n ) j n-j - * > ( ) p (1-p) - * -- ( j ) - * j=k+1 - * - * The terms are not summed directly; instead the incomplete - * beta integral is employed, according to the formula - * - * y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * - * - * ACCURACY: - * - * Relative error (p varies from 0 to 1): - * arithmetic domain # trials peak rms - * IEEE 0,100 2000 6.0e-5 1.2e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * bdtrcf domain x<0, x>1, n<k 0.0 - */ - /* bdtrif() - * - * Inverse binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, bdtrif(); - * - * p = bdtrf( k, n, y ); - * - * - * - * DESCRIPTION: - * - * Finds the event probability p such that the sum of the - * terms 0 through k of the Binomial probability density - * is equal to the given cumulative probability y. - * - * This is accomplished using the inverse beta integral - * function and the relation - * - * 1 - p = incbi( n-k, k+1, y ). - * - * - * - * - * ACCURACY: - * - * Relative error (p varies from 0 to 1): - * arithmetic domain # trials peak rms - * IEEE 0,100 2000 3.5e-5 3.3e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * bdtrif domain k < 0, n <= k 0.0 - * x < 0, x > 1 - * - */ - -/* bdtr() */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -#ifdef ANSIC -float incbetf(float, float, float), powf(float, float); -float incbif( float, float, float ); -#else -float incbetf(), powf(), incbif(); -#endif - -float bdtrcf( int k, int n, float pp ) -{ -float p, dk, dn; - -p = pp; -if( (p < 0.0) || (p > 1.0) ) - goto domerr; -if( k < 0 ) - return( 1.0 ); - -if( n < k ) - { -domerr: - mtherr( "bdtrcf", DOMAIN ); - return( 0.0 ); - } - -if( k == n ) - return( 0.0 ); -dn = n - k; -if( k == 0 ) - { - dk = 1.0 - powf( 1.0-p, dn ); - } -else - { - dk = k + 1; - dk = incbetf( dk, dn, p ); - } -return( dk ); -} - - - -float bdtrf( int k, int n, float pp ) -{ -float p, dk, dn; - -p = pp; -if( (p < 0.0) || (p > 1.0) ) - goto domerr; -if( (k < 0) || (n < k) ) - { -domerr: - mtherr( "bdtrf", DOMAIN ); - return( 0.0 ); - } - -if( k == n ) - return( 1.0 ); - -dn = n - k; -if( k == 0 ) - { - dk = powf( 1.0-p, dn ); - } -else - { - dk = k + 1; - dk = incbetf( dn, dk, 1.0 - p ); - } -return( dk ); -} - - -float bdtrif( int k, int n, float yy ) -{ -float y, dk, dn, p; - -y = yy; -if( (y < 0.0) || (y > 1.0) ) - goto domerr; -if( (k < 0) || (n <= k) ) - { -domerr: - mtherr( "bdtrif", DOMAIN ); - return( 0.0 ); - } - -dn = n - k; -if( k == 0 ) - { - p = 1.0 - powf( y, 1.0/dn ); - } -else - { - dk = k + 1; - p = 1.0 - incbif( dn, dk, y ); - } -return( p ); -} diff --git a/libm/float/betaf.c b/libm/float/betaf.c deleted file mode 100644 index 7a1963191..000000000 --- a/libm/float/betaf.c +++ /dev/null @@ -1,122 +0,0 @@ -/* betaf.c - * - * Beta function - * - * - * - * SYNOPSIS: - * - * float a, b, y, betaf(); - * - * y = betaf( a, b ); - * - * - * - * DESCRIPTION: - * - * - - - * | (a) | (b) - * beta( a, b ) = -----------. - * - - * | (a+b) - * - * For large arguments the logarithm of the function is - * evaluated using lgam(), then exponentiated. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 10000 4.0e-5 6.0e-6 - * IEEE -20,0 10000 4.9e-3 5.4e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * betaf overflow log(beta) > MAXLOG 0.0 - * a or b <0 integer 0.0 - * - */ - -/* beta.c */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#define MAXGAM 34.84425627277176174 - - -extern float MAXLOGF, MAXNUMF; -extern int sgngamf; - -#ifdef ANSIC -float gammaf(float), lgamf(float), expf(float), floorf(float); -#else -float gammaf(), lgamf(), expf(), floorf(); -#endif - -float betaf( float aa, float bb ) -{ -float a, b, y; -int sign; - -sign = 1; -a = aa; -b = bb; -if( a <= 0.0 ) - { - if( a == floorf(a) ) - goto over; - } -if( b <= 0.0 ) - { - if( b == floorf(b) ) - goto over; - } - - -y = a + b; -if( fabsf(y) > MAXGAM ) - { - y = lgamf(y); - sign *= sgngamf; /* keep track of the sign */ - y = lgamf(b) - y; - sign *= sgngamf; - y = lgamf(a) + y; - sign *= sgngamf; - if( y > MAXLOGF ) - { -over: - mtherr( "betaf", OVERFLOW ); - return( sign * MAXNUMF ); - } - return( sign * expf(y) ); - } - -y = gammaf(y); -if( y == 0.0 ) - goto over; - -if( a > b ) - { - y = gammaf(a)/y; - y *= gammaf(b); - } -else - { - y = gammaf(b)/y; - y *= gammaf(a); - } - -return(y); -} diff --git a/libm/float/cbrtf.c b/libm/float/cbrtf.c deleted file mode 100644 index ca9b433d9..000000000 --- a/libm/float/cbrtf.c +++ /dev/null @@ -1,119 +0,0 @@ -/* cbrtf.c - * - * Cube root - * - * - * - * SYNOPSIS: - * - * float x, y, cbrtf(); - * - * y = cbrtf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the cube root of the argument, which may be negative. - * - * Range reduction involves determining the power of 2 of - * the argument. A polynomial of degree 2 applied to the - * mantissa, and multiplication by the cube root of 1, 2, or 4 - * approximates the root to within about 0.1%. Then Newton's - * iteration is used to converge to an accurate result. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,1e38 100000 7.6e-8 2.7e-8 - * - */ - /* cbrt.c */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> - -static float CBRT2 = 1.25992104989487316477; -static float CBRT4 = 1.58740105196819947475; - - -float frexpf(float, int *), ldexpf(float, int); - -float cbrtf( float xx ) -{ -int e, rem, sign; -float x, z; - -x = xx; -if( x == 0 ) - return( 0.0 ); -if( x > 0 ) - sign = 1; -else - { - sign = -1; - x = -x; - } - -z = x; -/* extract power of 2, leaving - * mantissa between 0.5 and 1 - */ -x = frexpf( x, &e ); - -/* Approximate cube root of number between .5 and 1, - * peak relative error = 9.2e-6 - */ -x = (((-0.13466110473359520655053 * x - + 0.54664601366395524503440 ) * x - - 0.95438224771509446525043 ) * x - + 1.1399983354717293273738 ) * x - + 0.40238979564544752126924; - -/* exponent divided by 3 */ -if( e >= 0 ) - { - rem = e; - e /= 3; - rem -= 3*e; - if( rem == 1 ) - x *= CBRT2; - else if( rem == 2 ) - x *= CBRT4; - } - - -/* argument less than 1 */ - -else - { - e = -e; - rem = e; - e /= 3; - rem -= 3*e; - if( rem == 1 ) - x /= CBRT2; - else if( rem == 2 ) - x /= CBRT4; - e = -e; - } - -/* multiply by power of 2 */ -x = ldexpf( x, e ); - -/* Newton iteration */ -x -= ( x - (z/(x*x)) ) * 0.333333333333; - -if( sign < 0 ) - x = -x; -return(x); -} diff --git a/libm/float/chbevlf.c b/libm/float/chbevlf.c deleted file mode 100644 index 343d00a22..000000000 --- a/libm/float/chbevlf.c +++ /dev/null @@ -1,86 +0,0 @@ -/* chbevlf.c - * - * Evaluate Chebyshev series - * - * - * - * SYNOPSIS: - * - * int N; - * float x, y, coef[N], chebevlf(); - * - * y = chbevlf( x, coef, N ); - * - * - * - * DESCRIPTION: - * - * Evaluates the series - * - * N-1 - * - ' - * y = > coef[i] T (x/2) - * - i - * i=0 - * - * of Chebyshev polynomials Ti at argument x/2. - * - * Coefficients are stored in reverse order, i.e. the zero - * order term is last in the array. Note N is the number of - * coefficients, not the order. - * - * If coefficients are for the interval a to b, x must - * have been transformed to x -> 2(2x - b - a)/(b-a) before - * entering the routine. This maps x from (a, b) to (-1, 1), - * over which the Chebyshev polynomials are defined. - * - * If the coefficients are for the inverted interval, in - * which (a, b) is mapped to (1/b, 1/a), the transformation - * required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity, - * this becomes x -> 4a/x - 1. - * - * - * - * SPEED: - * - * Taking advantage of the recurrence properties of the - * Chebyshev polynomials, the routine requires one more - * addition per loop than evaluating a nested polynomial of - * the same degree. - * - */ - /* chbevl.c */ - -/* -Cephes Math Library Release 2.0: April, 1987 -Copyright 1985, 1987 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#ifdef ANSIC -float chbevlf( float x, float *array, int n ) -#else -float chbevlf( x, array, n ) -float x; -float *array; -int n; -#endif -{ -float b0, b1, b2, *p; -int i; - -p = array; -b0 = *p++; -b1 = 0.0; -i = n - 1; - -do - { - b2 = b1; - b1 = b0; - b0 = x * b1 - b2 + *p++; - } -while( --i ); - -return( 0.5*(b0-b2) ); -} diff --git a/libm/float/chdtrf.c b/libm/float/chdtrf.c deleted file mode 100644 index 53bd3d961..000000000 --- a/libm/float/chdtrf.c +++ /dev/null @@ -1,210 +0,0 @@ -/* chdtrf.c - * - * Chi-square distribution - * - * - * - * SYNOPSIS: - * - * float df, x, y, chdtrf(); - * - * y = chdtrf( df, x ); - * - * - * - * DESCRIPTION: - * - * Returns the area under the left hand tail (from 0 to x) - * of the Chi square probability density function with - * v degrees of freedom. - * - * - * inf. - * - - * 1 | | v/2-1 -t/2 - * P( x | v ) = ----------- | t e dt - * v/2 - | | - * 2 | (v/2) - - * x - * - * where x is the Chi-square variable. - * - * The incomplete gamma integral is used, according to the - * formula - * - * y = chdtr( v, x ) = igam( v/2.0, x/2.0 ). - * - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 3.2e-5 5.0e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * chdtrf domain x < 0 or v < 1 0.0 - */ - /* chdtrcf() - * - * Complemented Chi-square distribution - * - * - * - * SYNOPSIS: - * - * float v, x, y, chdtrcf(); - * - * y = chdtrcf( v, x ); - * - * - * - * DESCRIPTION: - * - * Returns the area under the right hand tail (from x to - * infinity) of the Chi square probability density function - * with v degrees of freedom: - * - * - * inf. - * - - * 1 | | v/2-1 -t/2 - * P( x | v ) = ----------- | t e dt - * v/2 - | | - * 2 | (v/2) - - * x - * - * where x is the Chi-square variable. - * - * The incomplete gamma integral is used, according to the - * formula - * - * y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ). - * - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 2.7e-5 3.2e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * chdtrc domain x < 0 or v < 1 0.0 - */ - /* chdtrif() - * - * Inverse of complemented Chi-square distribution - * - * - * - * SYNOPSIS: - * - * float df, x, y, chdtrif(); - * - * x = chdtrif( df, y ); - * - * - * - * - * DESCRIPTION: - * - * Finds the Chi-square argument x such that the integral - * from x to infinity of the Chi-square density is equal - * to the given cumulative probability y. - * - * This is accomplished using the inverse gamma integral - * function and the relation - * - * x/2 = igami( df/2, y ); - * - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 10000 2.2e-5 8.5e-7 - * - * ERROR MESSAGES: - * - * message condition value returned - * chdtri domain y < 0 or y > 1 0.0 - * v < 1 - * - */ - -/* chdtr() */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -#ifdef ANSIC -float igamcf(float, float), igamf(float, float), igamif(float, float); -#else -float igamcf(), igamf(), igamif(); -#endif - -float chdtrcf(float dff, float xx) -{ -float df, x; - -df = dff; -x = xx; - -if( (x < 0.0) || (df < 1.0) ) - { - mtherr( "chdtrcf", DOMAIN ); - return(0.0); - } -return( igamcf( 0.5*df, 0.5*x ) ); -} - - -float chdtrf(float dff, float xx) -{ -float df, x; - -df = dff; -x = xx; -if( (x < 0.0) || (df < 1.0) ) - { - mtherr( "chdtrf", DOMAIN ); - return(0.0); - } -return( igamf( 0.5*df, 0.5*x ) ); -} - - -float chdtrif( float dff, float yy ) -{ -float y, df, x; - -y = yy; -df = dff; -if( (y < 0.0) || (y > 1.0) || (df < 1.0) ) - { - mtherr( "chdtrif", DOMAIN ); - return(0.0); - } - -x = igamif( 0.5 * df, y ); -return( 2.0 * x ); -} diff --git a/libm/float/clogf.c b/libm/float/clogf.c deleted file mode 100644 index 5f4944eba..000000000 --- a/libm/float/clogf.c +++ /dev/null @@ -1,669 +0,0 @@ -/* clogf.c - * - * Complex natural logarithm - * - * - * - * SYNOPSIS: - * - * void clogf(); - * cmplxf z, w; - * - * clogf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * Returns complex logarithm to the base e (2.718...) of - * the complex argument x. - * - * If z = x + iy, r = sqrt( x**2 + y**2 ), - * then - * w = log(r) + i arctan(y/x). - * - * The arctangent ranges from -PI to +PI. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.9e-6 6.2e-8 - * - * Larger relative error can be observed for z near 1 +i0. - * In IEEE arithmetic the peak absolute error is 3.1e-7. - * - */ - -#include <math.h> -extern float MAXNUMF, MACHEPF, PIF, PIO2F; -#ifdef ANSIC -float cabsf(cmplxf *), sqrtf(float), logf(float), atan2f(float, float); -float expf(float), sinf(float), cosf(float); -float coshf(float), sinhf(float), asinf(float); -float ctansf(cmplxf *), redupif(float); -void cchshf( float, float *, float * ); -void caddf( cmplxf *, cmplxf *, cmplxf * ); -void csqrtf( cmplxf *, cmplxf * ); -#else -float cabsf(), sqrtf(), logf(), atan2f(); -float expf(), sinf(), cosf(); -float coshf(), sinhf(), asinf(); -float ctansf(), redupif(); -void cchshf(), csqrtf(), caddf(); -#endif - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -void clogf( z, w ) -register cmplxf *z, *w; -{ -float p, rr; - -/*rr = sqrtf( z->r * z->r + z->i * z->i );*/ -rr = cabsf(z); -p = logf(rr); -#if ANSIC -rr = atan2f( z->i, z->r ); -#else -rr = atan2f( z->r, z->i ); -if( rr > PIF ) - rr -= PIF + PIF; -#endif -w->i = rr; -w->r = p; -} - /* cexpf() - * - * Complex exponential function - * - * - * - * SYNOPSIS: - * - * void cexpf(); - * cmplxf z, w; - * - * cexpf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * Returns the exponential of the complex argument z - * into the complex result w. - * - * If - * z = x + iy, - * r = exp(x), - * - * then - * - * w = r cos y + i r sin y. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.4e-7 4.5e-8 - * - */ - -void cexpf( z, w ) -register cmplxf *z, *w; -{ -float r; - -r = expf( z->r ); -w->r = r * cosf( z->i ); -w->i = r * sinf( z->i ); -} - /* csinf() - * - * Complex circular sine - * - * - * - * SYNOPSIS: - * - * void csinf(); - * cmplxf z, w; - * - * csinf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * w = sin x cosh y + i cos x sinh y. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.9e-7 5.5e-8 - * - */ - -void csinf( z, w ) -register cmplxf *z, *w; -{ -float ch, sh; - -cchshf( z->i, &ch, &sh ); -w->r = sinf( z->r ) * ch; -w->i = cosf( z->r ) * sh; -} - - - -/* calculate cosh and sinh */ - -void cchshf( float xx, float *c, float *s ) -{ -float x, e, ei; - -x = xx; -if( fabsf(x) <= 0.5f ) - { - *c = coshf(x); - *s = sinhf(x); - } -else - { - e = expf(x); - ei = 0.5f/e; - e = 0.5f * e; - *s = e - ei; - *c = e + ei; - } -} - - /* ccosf() - * - * Complex circular cosine - * - * - * - * SYNOPSIS: - * - * void ccosf(); - * cmplxf z, w; - * - * ccosf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * w = cos x cosh y - i sin x sinh y. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.8e-7 5.5e-8 - */ - -void ccosf( z, w ) -register cmplxf *z, *w; -{ -float ch, sh; - -cchshf( z->i, &ch, &sh ); -w->r = cosf( z->r ) * ch; -w->i = -sinf( z->r ) * sh; -} - /* ctanf() - * - * Complex circular tangent - * - * - * - * SYNOPSIS: - * - * void ctanf(); - * cmplxf z, w; - * - * ctanf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * sin 2x + i sinh 2y - * w = --------------------. - * cos 2x + cosh 2y - * - * On the real axis the denominator is zero at odd multiples - * of PI/2. The denominator is evaluated by its Taylor - * series near these points. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 3.3e-7 5.1e-8 - */ - -void ctanf( z, w ) -register cmplxf *z, *w; -{ -float d; - -d = cosf( 2.0f * z->r ) + coshf( 2.0f * z->i ); - -if( fabsf(d) < 0.25f ) - d = ctansf(z); - -if( d == 0.0f ) - { - mtherr( "ctanf", OVERFLOW ); - w->r = MAXNUMF; - w->i = MAXNUMF; - return; - } - -w->r = sinf( 2.0f * z->r ) / d; -w->i = sinhf( 2.0f * z->i ) / d; -} - /* ccotf() - * - * Complex circular cotangent - * - * - * - * SYNOPSIS: - * - * void ccotf(); - * cmplxf z, w; - * - * ccotf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * - * sin 2x - i sinh 2y - * w = --------------------. - * cosh 2y - cos 2x - * - * On the real axis, the denominator has zeros at even - * multiples of PI/2. Near these points it is evaluated - * by a Taylor series. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 3.6e-7 5.7e-8 - * Also tested by ctan * ccot = 1 + i0. - */ - -void ccotf( z, w ) -register cmplxf *z, *w; -{ -float d; - - -d = coshf(2.0f * z->i) - cosf(2.0f * z->r); - -if( fabsf(d) < 0.25f ) - d = ctansf(z); - -if( d == 0.0f ) - { - mtherr( "ccotf", OVERFLOW ); - w->r = MAXNUMF; - w->i = MAXNUMF; - return; - } - -d = 1.0f/d; -w->r = sinf( 2.0f * z->r ) * d; -w->i = -sinhf( 2.0f * z->i ) * d; -} - -/* Program to subtract nearest integer multiple of PI */ -/* extended precision value of PI: */ - -static float DP1 = 3.140625; -static float DP2 = 9.67502593994140625E-4; -static float DP3 = 1.509957990978376432E-7; - - -float redupif(float xx) -{ -float x, t; -long i; - -x = xx; -t = x/PIF; -if( t >= 0.0f ) - t += 0.5f; -else - t -= 0.5f; - -i = t; /* the multiple */ -t = i; -t = ((x - t * DP1) - t * DP2) - t * DP3; -return(t); -} - -/* Taylor series expansion for cosh(2y) - cos(2x) */ - -float ctansf(z) -cmplxf *z; -{ -float f, x, x2, y, y2, rn, t, d; - -x = fabsf( 2.0f * z->r ); -y = fabsf( 2.0f * z->i ); - -x = redupif(x); - -x = x * x; -y = y * y; -x2 = 1.0f; -y2 = 1.0f; -f = 1.0f; -rn = 0.0f; -d = 0.0f; -do - { - rn += 1.0f; - f *= rn; - rn += 1.0f; - f *= rn; - x2 *= x; - y2 *= y; - t = y2 + x2; - t /= f; - d += t; - - rn += 1.0f; - f *= rn; - rn += 1.0f; - f *= rn; - x2 *= x; - y2 *= y; - t = y2 - x2; - t /= f; - d += t; - } -while( fabsf(t/d) > MACHEPF ); -return(d); -} - /* casinf() - * - * Complex circular arc sine - * - * - * - * SYNOPSIS: - * - * void casinf(); - * cmplxf z, w; - * - * casinf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * Inverse complex sine: - * - * 2 - * w = -i clog( iz + csqrt( 1 - z ) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.1e-5 1.5e-6 - * Larger relative error can be observed for z near zero. - * - */ - -void casinf( z, w ) -cmplxf *z, *w; -{ -float x, y; -static cmplxf ca, ct, zz, z2; -/* -float cn, n; -static float a, b, s, t, u, v, y2; -static cmplxf sum; -*/ - -x = z->r; -y = z->i; - -if( y == 0.0f ) - { - if( fabsf(x) > 1.0f ) - { - w->r = PIO2F; - w->i = 0.0f; - mtherr( "casinf", DOMAIN ); - } - else - { - w->r = asinf(x); - w->i = 0.0f; - } - return; - } - -/* Power series expansion */ -/* -b = cabsf(z); -if( b < 0.125 ) -{ -z2.r = (x - y) * (x + y); -z2.i = 2.0 * x * y; - -cn = 1.0; -n = 1.0; -ca.r = x; -ca.i = y; -sum.r = x; -sum.i = y; -do - { - ct.r = z2.r * ca.r - z2.i * ca.i; - ct.i = z2.r * ca.i + z2.i * ca.r; - ca.r = ct.r; - ca.i = ct.i; - - cn *= n; - n += 1.0; - cn /= n; - n += 1.0; - b = cn/n; - - ct.r *= b; - ct.i *= b; - sum.r += ct.r; - sum.i += ct.i; - b = fabsf(ct.r) + fabsf(ct.i); - } -while( b > MACHEPF ); -w->r = sum.r; -w->i = sum.i; -return; -} -*/ - - -ca.r = x; -ca.i = y; - -ct.r = -ca.i; /* iz */ -ct.i = ca.r; - - /* sqrt( 1 - z*z) */ -/* cmul( &ca, &ca, &zz ) */ -zz.r = (ca.r - ca.i) * (ca.r + ca.i); /*x * x - y * y */ -zz.i = 2.0f * ca.r * ca.i; - -zz.r = 1.0f - zz.r; -zz.i = -zz.i; -csqrtf( &zz, &z2 ); - -caddf( &z2, &ct, &zz ); -clogf( &zz, &zz ); -w->r = zz.i; /* mult by 1/i = -i */ -w->i = -zz.r; -return; -} - /* cacosf() - * - * Complex circular arc cosine - * - * - * - * SYNOPSIS: - * - * void cacosf(); - * cmplxf z, w; - * - * cacosf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * - * w = arccos z = PI/2 - arcsin z. - * - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 9.2e-6 1.2e-6 - * - */ - -void cacosf( z, w ) -cmplxf *z, *w; -{ - -casinf( z, w ); -w->r = PIO2F - w->r; -w->i = -w->i; -} - /* catan() - * - * Complex circular arc tangent - * - * - * - * SYNOPSIS: - * - * void catan(); - * cmplxf z, w; - * - * catan( &z, &w ); - * - * - * - * DESCRIPTION: - * - * If - * z = x + iy, - * - * then - * 1 ( 2x ) - * Re w = - arctan(-----------) + k PI - * 2 ( 2 2) - * (1 - x - y ) - * - * ( 2 2) - * 1 (x + (y+1) ) - * Im w = - log(------------) - * 4 ( 2 2) - * (x + (y-1) ) - * - * Where k is an arbitrary integer. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 2.3e-6 5.2e-8 - * - */ - -void catanf( z, w ) -cmplxf *z, *w; -{ -float a, t, x, x2, y; - -x = z->r; -y = z->i; - -if( (x == 0.0f) && (y > 1.0f) ) - goto ovrf; - -x2 = x * x; -a = 1.0f - x2 - (y * y); -if( a == 0.0f ) - goto ovrf; - -#if ANSIC -t = 0.5f * atan2f( 2.0f * x, a ); -#else -t = 0.5f * atan2f( a, 2.0f * x ); -#endif -w->r = redupif( t ); - -t = y - 1.0f; -a = x2 + (t * t); -if( a == 0.0f ) - goto ovrf; - -t = y + 1.0f; -a = (x2 + (t * t))/a; -w->i = 0.25f*logf(a); -return; - -ovrf: -mtherr( "catanf", OVERFLOW ); -w->r = MAXNUMF; -w->i = MAXNUMF; -} diff --git a/libm/float/cmplxf.c b/libm/float/cmplxf.c deleted file mode 100644 index 949b94e3d..000000000 --- a/libm/float/cmplxf.c +++ /dev/null @@ -1,407 +0,0 @@ -/* cmplxf.c - * - * Complex number arithmetic - * - * - * - * SYNOPSIS: - * - * typedef struct { - * float r; real part - * float i; imaginary part - * }cmplxf; - * - * cmplxf *a, *b, *c; - * - * caddf( a, b, c ); c = b + a - * csubf( a, b, c ); c = b - a - * cmulf( a, b, c ); c = b * a - * cdivf( a, b, c ); c = b / a - * cnegf( c ); c = -c - * cmovf( b, c ); c = b - * - * - * - * DESCRIPTION: - * - * Addition: - * c.r = b.r + a.r - * c.i = b.i + a.i - * - * Subtraction: - * c.r = b.r - a.r - * c.i = b.i - a.i - * - * Multiplication: - * c.r = b.r * a.r - b.i * a.i - * c.i = b.r * a.i + b.i * a.r - * - * Division: - * d = a.r * a.r + a.i * a.i - * c.r = (b.r * a.r + b.i * a.i)/d - * c.i = (b.i * a.r - b.r * a.i)/d - * ACCURACY: - * - * In DEC arithmetic, the test (1/z) * z = 1 had peak relative - * error 3.1e-17, rms 1.2e-17. The test (y/z) * (z/y) = 1 had - * peak relative error 8.3e-17, rms 2.1e-17. - * - * Tests in the rectangle {-10,+10}: - * Relative error: - * arithmetic function # trials peak rms - * IEEE cadd 30000 5.9e-8 2.6e-8 - * IEEE csub 30000 6.0e-8 2.6e-8 - * IEEE cmul 30000 1.1e-7 3.7e-8 - * IEEE cdiv 30000 2.1e-7 5.7e-8 - */ - /* cmplx.c - * complex number arithmetic - */ - - -/* -Cephes Math Library Release 2.1: December, 1988 -Copyright 1984, 1987, 1988 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> -extern float MAXNUMF, MACHEPF, PIF, PIO2F; -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) -#ifdef ANSIC -float sqrtf(float), frexpf(float, int *); -float ldexpf(float, int); -float cabsf(cmplxf *), atan2f(float, float), cosf(float), sinf(float); -#else -float sqrtf(), frexpf(), ldexpf(); -float cabsf(), atan2f(), cosf(), sinf(); -#endif -/* -typedef struct - { - float r; - float i; - }cmplxf; -*/ -cmplxf czerof = {0.0, 0.0}; -extern cmplxf czerof; -cmplxf conef = {1.0, 0.0}; -extern cmplxf conef; - -/* c = b + a */ - -void caddf( a, b, c ) -register cmplxf *a, *b; -cmplxf *c; -{ - -c->r = b->r + a->r; -c->i = b->i + a->i; -} - - -/* c = b - a */ - -void csubf( a, b, c ) -register cmplxf *a, *b; -cmplxf *c; -{ - -c->r = b->r - a->r; -c->i = b->i - a->i; -} - -/* c = b * a */ - -void cmulf( a, b, c ) -register cmplxf *a, *b; -cmplxf *c; -{ -register float y; - -y = b->r * a->r - b->i * a->i; -c->i = b->r * a->i + b->i * a->r; -c->r = y; -} - - - -/* c = b / a */ - -void cdivf( a, b, c ) -register cmplxf *a, *b; -cmplxf *c; -{ -float y, p, q, w; - - -y = a->r * a->r + a->i * a->i; -p = b->r * a->r + b->i * a->i; -q = b->i * a->r - b->r * a->i; - -if( y < 1.0f ) - { - w = MAXNUMF * y; - if( (fabsf(p) > w) || (fabsf(q) > w) || (y == 0.0f) ) - { - c->r = MAXNUMF; - c->i = MAXNUMF; - mtherr( "cdivf", OVERFLOW ); - return; - } - } -c->r = p/y; -c->i = q/y; -} - - -/* b = a */ - -void cmovf( a, b ) -register short *a, *b; -{ -int i; - - -i = 8; -do - *b++ = *a++; -while( --i ); -} - - -void cnegf( a ) -register cmplxf *a; -{ - -a->r = -a->r; -a->i = -a->i; -} - -/* cabsf() - * - * Complex absolute value - * - * - * - * SYNOPSIS: - * - * float cabsf(); - * cmplxf z; - * float a; - * - * a = cabsf( &z ); - * - * - * - * DESCRIPTION: - * - * - * If z = x + iy - * - * then - * - * a = sqrt( x**2 + y**2 ). - * - * Overflow and underflow are avoided by testing the magnitudes - * of x and y before squaring. If either is outside half of - * the floating point full scale range, both are rescaled. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 30000 1.2e-7 3.4e-8 - */ - - -/* -Cephes Math Library Release 2.1: January, 1989 -Copyright 1984, 1987, 1989 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -/* -typedef struct - { - float r; - float i; - }cmplxf; -*/ -/* square root of max and min numbers */ -#define SMAX 1.3043817825332782216E+19 -#define SMIN 7.6664670834168704053E-20 -#define PREC 12 -#define MAXEXPF 128 - - -#define SMAXT (2.0f * SMAX) -#define SMINT (0.5f * SMIN) - -float cabsf( z ) -register cmplxf *z; -{ -float x, y, b, re, im; -int ex, ey, e; - -re = fabsf( z->r ); -im = fabsf( z->i ); - -if( re == 0.0f ) - { - return( im ); - } -if( im == 0.0f ) - { - return( re ); - } - -/* Get the exponents of the numbers */ -x = frexpf( re, &ex ); -y = frexpf( im, &ey ); - -/* Check if one number is tiny compared to the other */ -e = ex - ey; -if( e > PREC ) - return( re ); -if( e < -PREC ) - return( im ); - -/* Find approximate exponent e of the geometric mean. */ -e = (ex + ey) >> 1; - -/* Rescale so mean is about 1 */ -x = ldexpf( re, -e ); -y = ldexpf( im, -e ); - -/* Hypotenuse of the right triangle */ -b = sqrtf( x * x + y * y ); - -/* Compute the exponent of the answer. */ -y = frexpf( b, &ey ); -ey = e + ey; - -/* Check it for overflow and underflow. */ -if( ey > MAXEXPF ) - { - mtherr( "cabsf", OVERFLOW ); - return( MAXNUMF ); - } -if( ey < -MAXEXPF ) - return(0.0f); - -/* Undo the scaling */ -b = ldexpf( b, e ); -return( b ); -} - /* csqrtf() - * - * Complex square root - * - * - * - * SYNOPSIS: - * - * void csqrtf(); - * cmplxf z, w; - * - * csqrtf( &z, &w ); - * - * - * - * DESCRIPTION: - * - * - * If z = x + iy, r = |z|, then - * - * 1/2 - * Im w = [ (r - x)/2 ] , - * - * Re w = y / 2 Im w. - * - * - * Note that -w is also a square root of z. The solution - * reported is always in the upper half plane. - * - * Because of the potential for cancellation error in r - x, - * the result is sharpened by doing a Heron iteration - * (see sqrt.c) in complex arithmetic. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,+10 100000 1.8e-7 4.2e-8 - * - */ - - -void csqrtf( z, w ) -cmplxf *z, *w; -{ -cmplxf q, s; -float x, y, r, t; - -x = z->r; -y = z->i; - -if( y == 0.0f ) - { - if( x < 0.0f ) - { - w->r = 0.0f; - w->i = sqrtf(-x); - return; - } - else - { - w->r = sqrtf(x); - w->i = 0.0f; - return; - } - } - -if( x == 0.0f ) - { - r = fabsf(y); - r = sqrtf(0.5f*r); - if( y > 0 ) - w->r = r; - else - w->r = -r; - w->i = r; - return; - } - -/* Approximate sqrt(x^2+y^2) - x = y^2/2x - y^4/24x^3 + ... . - * The relative error in the first term is approximately y^2/12x^2 . - */ -if( (fabsf(y) < fabsf(0.015f*x)) - && (x > 0) ) - { - t = 0.25f*y*(y/x); - } -else - { - r = cabsf(z); - t = 0.5f*(r - x); - } - -r = sqrtf(t); -q.i = r; -q.r = 0.5f*y/r; - -/* Heron iteration in complex arithmetic: - * q = (q + z/q)/2 - */ -cdivf( &q, z, &s ); -caddf( &q, &s, w ); -w->r *= 0.5f; -w->i *= 0.5f; -} - diff --git a/libm/float/constf.c b/libm/float/constf.c deleted file mode 100644 index bf6b6f657..000000000 --- a/libm/float/constf.c +++ /dev/null @@ -1,20 +0,0 @@ - -#ifdef DEC -/* MAXNUMF = 2^127 * (1 - 2^-24) */ -float MAXNUMF = 1.7014117331926442990585209174225846272e38; -float MAXLOGF = 88.02969187150841; -float MINLOGF = -88.7228391116729996; /* log(2^-128) */ -#else -/* MAXNUMF = 2^128 * (1 - 2^-24) */ -float MAXNUMF = 3.4028234663852885981170418348451692544e38; -float MAXLOGF = 88.72283905206835; -float MINLOGF = -103.278929903431851103; /* log(2^-149) */ -#endif - -float LOG2EF = 1.44269504088896341; -float LOGE2F = 0.693147180559945309; -float SQRTHF = 0.707106781186547524; -float PIF = 3.141592653589793238; -float PIO2F = 1.5707963267948966192; -float PIO4F = 0.7853981633974483096; -float MACHEPF = 5.9604644775390625E-8; diff --git a/libm/float/coshf.c b/libm/float/coshf.c deleted file mode 100644 index 2b44fdeb3..000000000 --- a/libm/float/coshf.c +++ /dev/null @@ -1,67 +0,0 @@ -/* coshf.c - * - * Hyperbolic cosine - * - * - * - * SYNOPSIS: - * - * float x, y, coshf(); - * - * y = coshf( x ); - * - * - * - * DESCRIPTION: - * - * Returns hyperbolic cosine of argument in the range MINLOGF to - * MAXLOGF. - * - * cosh(x) = ( exp(x) + exp(-x) )/2. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-MAXLOGF 100000 1.2e-7 2.8e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * coshf overflow |x| > MAXLOGF MAXNUMF - * - * - */ - -/* cosh.c */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1985, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> -extern float MAXLOGF, MAXNUMF; - -float expf(float); - -float coshf(float xx) -{ -float x, y; - -x = xx; -if( x < 0 ) - x = -x; -if( x > MAXLOGF ) - { - mtherr( "coshf", OVERFLOW ); - return( MAXNUMF ); - } -y = expf(x); -y = y + 1.0/y; -return( 0.5*y ); -} diff --git a/libm/float/dawsnf.c b/libm/float/dawsnf.c deleted file mode 100644 index d00607719..000000000 --- a/libm/float/dawsnf.c +++ /dev/null @@ -1,168 +0,0 @@ -/* dawsnf.c - * - * Dawson's Integral - * - * - * - * SYNOPSIS: - * - * float x, y, dawsnf(); - * - * y = dawsnf( x ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * x - * - - * 2 | | 2 - * dawsn(x) = exp( -x ) | exp( t ) dt - * | | - * - - * 0 - * - * Three different rational approximations are employed, for - * the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,10 50000 4.4e-7 6.3e-8 - * - * - */ - -/* dawsn.c */ - - -/* -Cephes Math Library Release 2.1: January, 1989 -Copyright 1984, 1987, 1989 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> -/* Dawson's integral, interval 0 to 3.25 */ -static float AN[10] = { - 1.13681498971755972054E-11, - 8.49262267667473811108E-10, - 1.94434204175553054283E-8, - 9.53151741254484363489E-7, - 3.07828309874913200438E-6, - 3.52513368520288738649E-4, --8.50149846724410912031E-4, - 4.22618223005546594270E-2, --9.17480371773452345351E-2, - 9.99999999999999994612E-1, -}; -static float AD[11] = { - 2.40372073066762605484E-11, - 1.48864681368493396752E-9, - 5.21265281010541664570E-8, - 1.27258478273186970203E-6, - 2.32490249820789513991E-5, - 3.25524741826057911661E-4, - 3.48805814657162590916E-3, - 2.79448531198828973716E-2, - 1.58874241960120565368E-1, - 5.74918629489320327824E-1, - 1.00000000000000000539E0, -}; - -/* interval 3.25 to 6.25 */ -static float BN[11] = { - 5.08955156417900903354E-1, --2.44754418142697847934E-1, - 9.41512335303534411857E-2, --2.18711255142039025206E-2, - 3.66207612329569181322E-3, --4.23209114460388756528E-4, - 3.59641304793896631888E-5, --2.14640351719968974225E-6, - 9.10010780076391431042E-8, --2.40274520828250956942E-9, - 3.59233385440928410398E-11, -}; -static float BD[10] = { -/* 1.00000000000000000000E0,*/ --6.31839869873368190192E-1, - 2.36706788228248691528E-1, --5.31806367003223277662E-2, - 8.48041718586295374409E-3, --9.47996768486665330168E-4, - 7.81025592944552338085E-5, --4.55875153252442634831E-6, - 1.89100358111421846170E-7, --4.91324691331920606875E-9, - 7.18466403235734541950E-11, -}; - -/* 6.25 to infinity */ -static float CN[5] = { --5.90592860534773254987E-1, - 6.29235242724368800674E-1, --1.72858975380388136411E-1, - 1.64837047825189632310E-2, --4.86827613020462700845E-4, -}; -static float CD[5] = { -/* 1.00000000000000000000E0,*/ --2.69820057197544900361E0, - 1.73270799045947845857E0, --3.93708582281939493482E-1, - 3.44278924041233391079E-2, --9.73655226040941223894E-4, -}; - - -extern float PIF, MACHEPF; -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) -#ifdef ANSIC -float polevlf(float, float *, int); -float p1evlf(float, float *, int); -#else -float polevlf(), p1evlf(); -#endif - -float dawsnf( float xxx ) -{ -float xx, x, y; -int sign; - -xx = xxx; -sign = 1; -if( xx < 0.0 ) - { - sign = -1; - xx = -xx; - } - -if( xx < 3.25 ) - { - x = xx*xx; - y = xx * polevlf( x, AN, 9 )/polevlf( x, AD, 10 ); - return( sign * y ); - } - - -x = 1.0/(xx*xx); - -if( xx < 6.25 ) - { - y = 1.0/xx + x * polevlf( x, BN, 10) / (p1evlf( x, BD, 10) * xx); - return( sign * 0.5 * y ); - } - - -if( xx > 1.0e9 ) - return( (sign * 0.5)/xx ); - -/* 6.25 to infinity */ -y = 1.0/xx + x * polevlf( x, CN, 4) / (p1evlf( x, CD, 5) * xx); -return( sign * 0.5 * y ); -} diff --git a/libm/float/ellief.c b/libm/float/ellief.c deleted file mode 100644 index 5c3f822df..000000000 --- a/libm/float/ellief.c +++ /dev/null @@ -1,115 +0,0 @@ -/* ellief.c - * - * Incomplete elliptic integral of the second kind - * - * - * - * SYNOPSIS: - * - * float phi, m, y, ellief(); - * - * y = ellief( phi, m ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * phi - * - - * | | - * | 2 - * E(phi\m) = | sqrt( 1 - m sin t ) dt - * | - * | | - * - - * 0 - * - * of amplitude phi and modulus m, using the arithmetic - - * geometric mean algorithm. - * - * - * - * ACCURACY: - * - * Tested at random arguments with phi in [0, 2] and m in - * [0, 1]. - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,2 10000 4.5e-7 7.4e-8 - * - * - */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Incomplete elliptic integral of second kind */ - -#include <math.h> - -extern float PIF, PIO2F, MACHEPF; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float sqrtf(float), logf(float), sinf(float), tanf(float), atanf(float); -float ellpef(float), ellpkf(float); -#else -float sqrtf(), logf(), sinf(), tanf(), atanf(); -float ellpef(), ellpkf(); -#endif - - -float ellief( float phia, float ma ) -{ -float phi, m, a, b, c, e, temp; -float lphi, t; -int d, mod; - -phi = phia; -m = ma; -if( m == 0.0 ) - return( phi ); -if( m == 1.0 ) - return( sinf(phi) ); -lphi = phi; -if( lphi < 0.0 ) - lphi = -lphi; -a = 1.0; -b = 1.0 - m; -b = sqrtf(b); -c = sqrtf(m); -d = 1; -e = 0.0; -t = tanf( lphi ); -mod = (lphi + PIO2F)/PIF; - -while( fabsf(c/a) > MACHEPF ) - { - temp = b/a; - lphi = lphi + atanf(t*temp) + mod * PIF; - mod = (lphi + PIO2F)/PIF; - t = t * ( 1.0 + temp )/( 1.0 - temp * t * t ); - c = 0.5 * ( a - b ); - temp = sqrtf( a * b ); - a = 0.5 * ( a + b ); - b = temp; - d += d; - e += c * sinf(lphi); - } - -b = 1.0 - m; -temp = ellpef(b)/ellpkf(b); -temp *= (atanf(t) + mod * PIF)/(d * a); -temp += e; -if( phi < 0.0 ) - temp = -temp; -return( temp ); -} diff --git a/libm/float/ellikf.c b/libm/float/ellikf.c deleted file mode 100644 index 8ec890926..000000000 --- a/libm/float/ellikf.c +++ /dev/null @@ -1,113 +0,0 @@ -/* ellikf.c - * - * Incomplete elliptic integral of the first kind - * - * - * - * SYNOPSIS: - * - * float phi, m, y, ellikf(); - * - * y = ellikf( phi, m ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * - * phi - * - - * | | - * | dt - * F(phi\m) = | ------------------ - * | 2 - * | | sqrt( 1 - m sin t ) - * - - * 0 - * - * of amplitude phi and modulus m, using the arithmetic - - * geometric mean algorithm. - * - * - * - * - * ACCURACY: - * - * Tested at random points with phi in [0, 2] and m in - * [0, 1]. - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,2 10000 2.9e-7 5.8e-8 - * - * - */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Incomplete elliptic integral of first kind */ - -#include <math.h> -extern float PIF, PIO2F, MACHEPF; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float sqrtf(float), logf(float), sinf(float), tanf(float), atanf(float); -#else -float sqrtf(), logf(), sinf(), tanf(), atanf(); -#endif - - -float ellikf( float phia, float ma ) -{ -float phi, m, a, b, c, temp; -float t; -int d, mod, sign; - -phi = phia; -m = ma; -if( m == 0.0 ) - return( phi ); -if( phi < 0.0 ) - { - phi = -phi; - sign = -1; - } -else - sign = 0; -a = 1.0; -b = 1.0 - m; -if( b == 0.0 ) - return( logf( tanf( 0.5*(PIO2F + phi) ) ) ); -b = sqrtf(b); -c = sqrtf(m); -d = 1; -t = tanf( phi ); -mod = (phi + PIO2F)/PIF; - -while( fabsf(c/a) > MACHEPF ) - { - temp = b/a; - phi = phi + atanf(t*temp) + mod * PIF; - mod = (phi + PIO2F)/PIF; - t = t * ( 1.0 + temp )/( 1.0 - temp * t * t ); - c = ( a - b )/2.0; - temp = sqrtf( a * b ); - a = ( a + b )/2.0; - b = temp; - d += d; - } - -temp = (atanf(t) + mod * PIF)/(d * a); -if( sign < 0 ) - temp = -temp; -return( temp ); -} diff --git a/libm/float/ellpef.c b/libm/float/ellpef.c deleted file mode 100644 index 645bc55ba..000000000 --- a/libm/float/ellpef.c +++ /dev/null @@ -1,105 +0,0 @@ -/* ellpef.c - * - * Complete elliptic integral of the second kind - * - * - * - * SYNOPSIS: - * - * float m1, y, ellpef(); - * - * y = ellpef( m1 ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * pi/2 - * - - * | | 2 - * E(m) = | sqrt( 1 - m sin t ) dt - * | | - * - - * 0 - * - * Where m = 1 - m1, using the approximation - * - * P(x) - x log x Q(x). - * - * Though there are no singularities, the argument m1 is used - * rather than m for compatibility with ellpk(). - * - * E(1) = 1; E(0) = pi/2. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 1 30000 1.1e-7 3.9e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * ellpef domain x<0, x>1 0.0 - * - */ - -/* ellpe.c */ - -/* Elliptic integral of second kind */ - -/* -Cephes Math Library, Release 2.1: February, 1989 -Copyright 1984, 1987, 1989 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - - -static float P[] = { - 1.53552577301013293365E-4, - 2.50888492163602060990E-3, - 8.68786816565889628429E-3, - 1.07350949056076193403E-2, - 7.77395492516787092951E-3, - 7.58395289413514708519E-3, - 1.15688436810574127319E-2, - 2.18317996015557253103E-2, - 5.68051945617860553470E-2, - 4.43147180560990850618E-1, - 1.00000000000000000299E0 -}; -static float Q[] = { - 3.27954898576485872656E-5, - 1.00962792679356715133E-3, - 6.50609489976927491433E-3, - 1.68862163993311317300E-2, - 2.61769742454493659583E-2, - 3.34833904888224918614E-2, - 4.27180926518931511717E-2, - 5.85936634471101055642E-2, - 9.37499997197644278445E-2, - 2.49999999999888314361E-1 -}; - -float polevlf(float, float *, int), logf(float); -float ellpef( float xx) -{ -float x; - -x = xx; -if( (x <= 0.0) || (x > 1.0) ) - { - if( x == 0.0 ) - return( 1.0 ); - mtherr( "ellpef", DOMAIN ); - return( 0.0 ); - } -return( polevlf(x,P,10) - logf(x) * (x * polevlf(x,Q,9)) ); -} diff --git a/libm/float/ellpjf.c b/libm/float/ellpjf.c deleted file mode 100644 index 552f5ffe4..000000000 --- a/libm/float/ellpjf.c +++ /dev/null @@ -1,161 +0,0 @@ -/* ellpjf.c - * - * Jacobian Elliptic Functions - * - * - * - * SYNOPSIS: - * - * float u, m, sn, cn, dn, phi; - * int ellpj(); - * - * ellpj( u, m, _&sn, _&cn, _&dn, _&phi ); - * - * - * - * DESCRIPTION: - * - * - * Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m), - * and dn(u|m) of parameter m between 0 and 1, and real - * argument u. - * - * These functions are periodic, with quarter-period on the - * real axis equal to the complete elliptic integral - * ellpk(1.0-m). - * - * Relation to incomplete elliptic integral: - * If u = ellik(phi,m), then sn(u|m) = sin(phi), - * and cn(u|m) = cos(phi). Phi is called the amplitude of u. - * - * Computation is by means of the arithmetic-geometric mean - * algorithm, except when m is within 1e-9 of 0 or 1. In the - * latter case with m close to 1, the approximation applies - * only for phi < pi/2. - * - * ACCURACY: - * - * Tested at random points with u between 0 and 10, m between - * 0 and 1. - * - * Absolute error (* = relative error): - * arithmetic function # trials peak rms - * IEEE sn 10000 1.7e-6 2.2e-7 - * IEEE cn 10000 1.6e-6 2.2e-7 - * IEEE dn 10000 1.4e-3 1.9e-5 - * IEEE phi 10000 3.9e-7* 6.7e-8* - * - * Peak error observed in consistency check using addition - * theorem for sn(u+v) was 4e-16 (absolute). Also tested by - * the above relation to the incomplete elliptic integral. - * Accuracy deteriorates when u is large. - * - */ - -/* ellpj.c */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> -extern float PIO2F, MACHEPF; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float sqrtf(float), sinf(float), cosf(float), asinf(float), tanhf(float); -float sinhf(float), coshf(float), atanf(float), expf(float); -#else -float sqrtf(), sinf(), cosf(), asinf(), tanhf(); -float sinhf(), coshf(), atanf(), expf(); -#endif - -int ellpjf( float uu, float mm, - float *sn, float *cn, float *dn, float *ph ) -{ -float u, m, ai, b, phi, t, twon; -float a[10], c[10]; -int i; - -u = uu; -m = mm; -/* Check for special cases */ - -if( m < 0.0 || m > 1.0 ) - { - mtherr( "ellpjf", DOMAIN ); - return(-1); - } -if( m < 1.0e-5 ) - { - t = sinf(u); - b = cosf(u); - ai = 0.25 * m * (u - t*b); - *sn = t - ai*b; - *cn = b + ai*t; - *ph = u - ai; - *dn = 1.0 - 0.5*m*t*t; - return(0); - } - -if( m >= 0.99999 ) - { - ai = 0.25 * (1.0-m); - b = coshf(u); - t = tanhf(u); - phi = 1.0/b; - twon = b * sinhf(u); - *sn = t + ai * (twon - u)/(b*b); - *ph = 2.0*atanf(expf(u)) - PIO2F + ai*(twon - u)/b; - ai *= t * phi; - *cn = phi - ai * (twon - u); - *dn = phi + ai * (twon + u); - return(0); - } - - -/* A. G. M. scale */ -a[0] = 1.0; -b = sqrtf(1.0 - m); -c[0] = sqrtf(m); -twon = 1.0; -i = 0; - -while( fabsf( (c[i]/a[i]) ) > MACHEPF ) - { - if( i > 8 ) - { -/* mtherr( "ellpjf", OVERFLOW );*/ - break; - } - ai = a[i]; - ++i; - c[i] = 0.5 * ( ai - b ); - t = sqrtf( ai * b ); - a[i] = 0.5 * ( ai + b ); - b = t; - twon += twon; - } - - -/* backward recurrence */ -phi = twon * a[i] * u; -do - { - t = c[i] * sinf(phi) / a[i]; - b = phi; - phi = 0.5 * (asinf(t) + phi); - } -while( --i ); - -*sn = sinf(phi); -t = cosf(phi); -*cn = t; -*dn = t/cosf(phi-b); -*ph = phi; -return(0); -} diff --git a/libm/float/ellpkf.c b/libm/float/ellpkf.c deleted file mode 100644 index 2cc13d90a..000000000 --- a/libm/float/ellpkf.c +++ /dev/null @@ -1,128 +0,0 @@ -/* ellpkf.c - * - * Complete elliptic integral of the first kind - * - * - * - * SYNOPSIS: - * - * float m1, y, ellpkf(); - * - * y = ellpkf( m1 ); - * - * - * - * DESCRIPTION: - * - * Approximates the integral - * - * - * - * pi/2 - * - - * | | - * | dt - * K(m) = | ------------------ - * | 2 - * | | sqrt( 1 - m sin t ) - * - - * 0 - * - * where m = 1 - m1, using the approximation - * - * P(x) - log x Q(x). - * - * The argument m1 is used rather than m so that the logarithmic - * singularity at m = 1 will be shifted to the origin; this - * preserves maximum accuracy. - * - * K(0) = pi/2. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,1 30000 1.3e-7 3.4e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * ellpkf domain x<0, x>1 0.0 - * - */ - -/* ellpk.c */ - - -/* -Cephes Math Library, Release 2.0: April, 1987 -Copyright 1984, 1987 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -static float P[] = -{ - 1.37982864606273237150E-4, - 2.28025724005875567385E-3, - 7.97404013220415179367E-3, - 9.85821379021226008714E-3, - 6.87489687449949877925E-3, - 6.18901033637687613229E-3, - 8.79078273952743772254E-3, - 1.49380448916805252718E-2, - 3.08851465246711995998E-2, - 9.65735902811690126535E-2, - 1.38629436111989062502E0 -}; - -static float Q[] = -{ - 2.94078955048598507511E-5, - 9.14184723865917226571E-4, - 5.94058303753167793257E-3, - 1.54850516649762399335E-2, - 2.39089602715924892727E-2, - 3.01204715227604046988E-2, - 3.73774314173823228969E-2, - 4.88280347570998239232E-2, - 7.03124996963957469739E-2, - 1.24999999999870820058E-1, - 4.99999999999999999821E-1 -}; -static float C1 = 1.3862943611198906188E0; /* log(4) */ - -extern float MACHEPF, MAXNUMF; - -float polevlf(float, float *, int); -float p1evlf(float, float *, int); -float logf(float); -float ellpkf(float xx) -{ -float x; - -x = xx; -if( (x < 0.0) || (x > 1.0) ) - { - mtherr( "ellpkf", DOMAIN ); - return( 0.0 ); - } - -if( x > MACHEPF ) - { - return( polevlf(x,P,10) - logf(x) * polevlf(x,Q,10) ); - } -else - { - if( x == 0.0 ) - { - mtherr( "ellpkf", SING ); - return( MAXNUMF ); - } - else - { - return( C1 - 0.5 * logf(x) ); - } - } -} diff --git a/libm/float/exp10f.c b/libm/float/exp10f.c deleted file mode 100644 index c7c62c567..000000000 --- a/libm/float/exp10f.c +++ /dev/null @@ -1,115 +0,0 @@ -/* exp10f.c - * - * Base 10 exponential function - * (Common antilogarithm) - * - * - * - * SYNOPSIS: - * - * float x, y, exp10f(); - * - * y = exp10f( x ); - * - * - * - * DESCRIPTION: - * - * Returns 10 raised to the x power. - * - * Range reduction is accomplished by expressing the argument - * as 10**x = 2**n 10**f, with |f| < 0.5 log10(2). - * A polynomial approximates 10**f. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -38,+38 100000 9.8e-8 2.8e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * exp10 underflow x < -MAXL10 0.0 - * exp10 overflow x > MAXL10 MAXNUM - * - * IEEE single arithmetic: MAXL10 = 38.230809449325611792. - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> - -static float P[] = { - 2.063216740311022E-001, - 5.420251702225484E-001, - 1.171292686296281E+000, - 2.034649854009453E+000, - 2.650948748208892E+000, - 2.302585167056758E+000 -}; - -/*static float LOG102 = 3.01029995663981195214e-1;*/ -static float LOG210 = 3.32192809488736234787e0; -static float LG102A = 3.00781250000000000000E-1; -static float LG102B = 2.48745663981195213739E-4; -static float MAXL10 = 38.230809449325611792; - - - - -extern float MAXNUMF; - -float floorf(float), ldexpf(float, int), polevlf(float, float *, int); - -float exp10f(float xx) -{ -float x, px, qx; -short n; - -x = xx; -if( x > MAXL10 ) - { - mtherr( "exp10f", OVERFLOW ); - return( MAXNUMF ); - } - -if( x < -MAXL10 ) /* Would like to use MINLOG but can't */ - { - mtherr( "exp10f", UNDERFLOW ); - return(0.0); - } - -/* The following is necessary because range reduction blows up: */ -if( x == 0 ) - return(1.0); - -/* Express 10**x = 10**g 2**n - * = 10**g 10**( n log10(2) ) - * = 10**( g + n log10(2) ) - */ -px = x * LOG210; -qx = floorf( px + 0.5 ); -n = qx; -x -= qx * LG102A; -x -= qx * LG102B; - -/* rational approximation for exponential - * of the fractional part: - * 10**x - 1 = 2x P(x**2)/( Q(x**2) - P(x**2) ) - */ -px = 1.0 + x * polevlf( x, P, 5 ); - -/* multiply by power of 2 */ -x = ldexpf( px, n ); - -return(x); -} diff --git a/libm/float/exp2f.c b/libm/float/exp2f.c deleted file mode 100644 index 0de21decd..000000000 --- a/libm/float/exp2f.c +++ /dev/null @@ -1,116 +0,0 @@ -/* exp2f.c - * - * Base 2 exponential function - * - * - * - * SYNOPSIS: - * - * float x, y, exp2f(); - * - * y = exp2f( x ); - * - * - * - * DESCRIPTION: - * - * Returns 2 raised to the x power. - * - * Range reduction is accomplished by separating the argument - * into an integer k and fraction f such that - * x k f - * 2 = 2 2. - * - * A polynomial approximates 2**x in the basic range [-0.5, 0.5]. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -127,+127 100000 1.7e-7 2.8e-8 - * - * - * See exp.c for comments on error amplification. - * - * - * ERROR MESSAGES: - * - * message condition value returned - * exp underflow x < -MAXL2 0.0 - * exp overflow x > MAXL2 MAXNUMF - * - * For IEEE arithmetic, MAXL2 = 127. - */ - - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - - -#include <math.h> -static char fname[] = {"exp2f"}; - -static float P[] = { - 1.535336188319500E-004, - 1.339887440266574E-003, - 9.618437357674640E-003, - 5.550332471162809E-002, - 2.402264791363012E-001, - 6.931472028550421E-001 -}; -#define MAXL2 127.0 -#define MINL2 -127.0 - - - -extern float MAXNUMF; - -float polevlf(float, float *, int), floorf(float), ldexpf(float, int); - -float exp2f( float xx ) -{ -float x, px; -int i0; - -x = xx; -if( x > MAXL2) - { - mtherr( fname, OVERFLOW ); - return( MAXNUMF ); - } - -if( x < MINL2 ) - { - mtherr( fname, UNDERFLOW ); - return(0.0); - } - -/* The following is necessary because range reduction blows up: */ -if( x == 0 ) - return(1.0); - -/* separate into integer and fractional parts */ -px = floorf(x); -i0 = px; -x = x - px; - -if( x > 0.5 ) - { - i0 += 1; - x -= 1.0; - } - -/* rational approximation - * exp2(x) = 1.0 + xP(x) - */ -px = 1.0 + x * polevlf( x, P, 5 ); - -/* scale by power of 2 */ -px = ldexpf( px, i0 ); -return(px); -} diff --git a/libm/float/expf.c b/libm/float/expf.c deleted file mode 100644 index 073678b99..000000000 --- a/libm/float/expf.c +++ /dev/null @@ -1,122 +0,0 @@ -/* expf.c - * - * Exponential function - * - * - * - * SYNOPSIS: - * - * float x, y, expf(); - * - * y = expf( x ); - * - * - * - * DESCRIPTION: - * - * Returns e (2.71828...) raised to the x power. - * - * Range reduction is accomplished by separating the argument - * into an integer k and fraction f such that - * - * x k f - * e = 2 e. - * - * A polynomial is used to approximate exp(f) - * in the basic range [-0.5, 0.5]. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +- MAXLOG 100000 1.7e-7 2.8e-8 - * - * - * Error amplification in the exponential function can be - * a serious matter. The error propagation involves - * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ), - * which shows that a 1 lsb error in representing X produces - * a relative error of X times 1 lsb in the function. - * While the routine gives an accurate result for arguments - * that are exactly represented by a double precision - * computer number, the result contains amplified roundoff - * error for large arguments not exactly represented. - * - * - * ERROR MESSAGES: - * - * message condition value returned - * expf underflow x < MINLOGF 0.0 - * expf overflow x > MAXLOGF MAXNUMF - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1989 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision exponential function. - * test interval: [-0.5, +0.5] - * trials: 80000 - * peak relative error: 7.6e-8 - * rms relative error: 2.8e-8 - */ -#include <math.h> -extern float LOG2EF, MAXLOGF, MINLOGF, MAXNUMF; - -static float C1 = 0.693359375; -static float C2 = -2.12194440e-4; - - - -float floorf( float ), ldexpf( float, int ); - -float expf( float xx ) -{ -float x, z; -int n; - -x = xx; - - -if( x > MAXLOGF) - { - mtherr( "expf", OVERFLOW ); - return( MAXNUMF ); - } - -if( x < MINLOGF ) - { - mtherr( "expf", UNDERFLOW ); - return(0.0); - } - -/* Express e**x = e**g 2**n - * = e**g e**( n loge(2) ) - * = e**( g + n loge(2) ) - */ -z = floorf( LOG2EF * x + 0.5 ); /* floor() truncates toward -infinity. */ -x -= z * C1; -x -= z * C2; -n = z; - -z = x * x; -/* Theoretical peak relative error in [-0.5, +0.5] is 4.2e-9. */ -z = -((((( 1.9875691500E-4 * x - + 1.3981999507E-3) * x - + 8.3334519073E-3) * x - + 4.1665795894E-2) * x - + 1.6666665459E-1) * x - + 5.0000001201E-1) * z - + x - + 1.0; - -/* multiply by power of 2 */ -x = ldexpf( z, n ); - -return( x ); -} diff --git a/libm/float/expnf.c b/libm/float/expnf.c deleted file mode 100644 index ebf0ccb3e..000000000 --- a/libm/float/expnf.c +++ /dev/null @@ -1,207 +0,0 @@ -/* expnf.c - * - * Exponential integral En - * - * - * - * SYNOPSIS: - * - * int n; - * float x, y, expnf(); - * - * y = expnf( n, x ); - * - * - * - * DESCRIPTION: - * - * Evaluates the exponential integral - * - * inf. - * - - * | | -xt - * | e - * E (x) = | ---- dt. - * n | n - * | | t - * - - * 1 - * - * - * Both n and x must be nonnegative. - * - * The routine employs either a power series, a continued - * fraction, or an asymptotic formula depending on the - * relative values of n and x. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 10000 5.6e-7 1.2e-7 - * - */ - -/* expn.c */ - -/* Cephes Math Library Release 2.2: July, 1992 - * Copyright 1985, 1992 by Stephen L. Moshier - * Direct inquiries to 30 Frost Street, Cambridge, MA 02140 */ - -#include <math.h> - -#define EUL 0.57721566490153286060 -#define BIG 16777216. -extern float MAXNUMF, MACHEPF, MAXLOGF; -#ifdef ANSIC -float powf(float, float), gammaf(float), logf(float), expf(float); -#else -float powf(), gammaf(), logf(), expf(); -#endif -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - - -float expnf( int n, float xx ) -{ -float x, ans, r, t, yk, xk; -float pk, pkm1, pkm2, qk, qkm1, qkm2; -float psi, z; -int i, k; -static float big = BIG; - - -x = xx; -if( n < 0 ) - goto domerr; - -if( x < 0 ) - { -domerr: mtherr( "expnf", DOMAIN ); - return( MAXNUMF ); - } - -if( x > MAXLOGF ) - return( 0.0 ); - -if( x == 0.0 ) - { - if( n < 2 ) - { - mtherr( "expnf", SING ); - return( MAXNUMF ); - } - else - return( 1.0/(n-1.0) ); - } - -if( n == 0 ) - return( expf(-x)/x ); - -/* expn.c */ -/* Expansion for large n */ - -if( n > 5000 ) - { - xk = x + n; - yk = 1.0 / (xk * xk); - t = n; - ans = yk * t * (6.0 * x * x - 8.0 * t * x + t * t); - ans = yk * (ans + t * (t - 2.0 * x)); - ans = yk * (ans + t); - ans = (ans + 1.0) * expf( -x ) / xk; - goto done; - } - -if( x > 1.0 ) - goto cfrac; - -/* expn.c */ - -/* Power series expansion */ - -psi = -EUL - logf(x); -for( i=1; i<n; i++ ) - psi = psi + 1.0/i; - -z = -x; -xk = 0.0; -yk = 1.0; -pk = 1.0 - n; -if( n == 1 ) - ans = 0.0; -else - ans = 1.0/pk; -do - { - xk += 1.0; - yk *= z/xk; - pk += 1.0; - if( pk != 0.0 ) - { - ans += yk/pk; - } - if( ans != 0.0 ) - t = fabsf(yk/ans); - else - t = 1.0; - } -while( t > MACHEPF ); -k = xk; -t = n; -r = n - 1; -ans = (powf(z, r) * psi / gammaf(t)) - ans; -goto done; - -/* expn.c */ -/* continued fraction */ -cfrac: -k = 1; -pkm2 = 1.0; -qkm2 = x; -pkm1 = 1.0; -qkm1 = x + n; -ans = pkm1/qkm1; - -do - { - k += 1; - if( k & 1 ) - { - yk = 1.0; - xk = n + (k-1)/2; - } - else - { - yk = x; - xk = k/2; - } - pk = pkm1 * yk + pkm2 * xk; - qk = qkm1 * yk + qkm2 * xk; - if( qk != 0 ) - { - r = pk/qk; - t = fabsf( (ans - r)/r ); - ans = r; - } - else - t = 1.0; - pkm2 = pkm1; - pkm1 = pk; - qkm2 = qkm1; - qkm1 = qk; -if( fabsf(pk) > big ) - { - pkm2 *= MACHEPF; - pkm1 *= MACHEPF; - qkm2 *= MACHEPF; - qkm1 *= MACHEPF; - } - } -while( t > MACHEPF ); - -ans *= expf( -x ); - -done: -return( ans ); -} - diff --git a/libm/float/facf.c b/libm/float/facf.c deleted file mode 100644 index c69738897..000000000 --- a/libm/float/facf.c +++ /dev/null @@ -1,106 +0,0 @@ -/* facf.c - * - * Factorial function - * - * - * - * SYNOPSIS: - * - * float y, facf(); - * int i; - * - * y = facf( i ); - * - * - * - * DESCRIPTION: - * - * Returns factorial of i = 1 * 2 * 3 * ... * i. - * fac(0) = 1.0. - * - * Due to machine arithmetic bounds the largest value of - * i accepted is 33 in single precision arithmetic. - * Greater values, or negative ones, - * produce an error message and return MAXNUM. - * - * - * - * ACCURACY: - * - * For i < 34 the values are simply tabulated, and have - * full machine accuracy. - * - */ - -/* -Cephes Math Library Release 2.0: April, 1987 -Copyright 1984, 1987 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -/* Factorials of integers from 0 through 33 */ -static float factbl[] = { - 1.00000000000000000000E0, - 1.00000000000000000000E0, - 2.00000000000000000000E0, - 6.00000000000000000000E0, - 2.40000000000000000000E1, - 1.20000000000000000000E2, - 7.20000000000000000000E2, - 5.04000000000000000000E3, - 4.03200000000000000000E4, - 3.62880000000000000000E5, - 3.62880000000000000000E6, - 3.99168000000000000000E7, - 4.79001600000000000000E8, - 6.22702080000000000000E9, - 8.71782912000000000000E10, - 1.30767436800000000000E12, - 2.09227898880000000000E13, - 3.55687428096000000000E14, - 6.40237370572800000000E15, - 1.21645100408832000000E17, - 2.43290200817664000000E18, - 5.10909421717094400000E19, - 1.12400072777760768000E21, - 2.58520167388849766400E22, - 6.20448401733239439360E23, - 1.55112100433309859840E25, - 4.03291461126605635584E26, - 1.0888869450418352160768E28, - 3.04888344611713860501504E29, - 8.841761993739701954543616E30, - 2.6525285981219105863630848E32, - 8.22283865417792281772556288E33, - 2.6313083693369353016721801216E35, - 8.68331761881188649551819440128E36 -}; -#define MAXFACF 33 - -extern float MAXNUMF; - -#ifdef ANSIC -float facf( int i ) -#else -float facf(i) -int i; -#endif -{ - -if( i < 0 ) - { - mtherr( "facf", SING ); - return( MAXNUMF ); - } - -if( i > MAXFACF ) - { - mtherr( "facf", OVERFLOW ); - return( MAXNUMF ); - } - -/* Get answer from table for small i. */ -return( factbl[i] ); -} diff --git a/libm/float/fdtrf.c b/libm/float/fdtrf.c deleted file mode 100644 index 5fdc6d81d..000000000 --- a/libm/float/fdtrf.c +++ /dev/null @@ -1,214 +0,0 @@ -/* fdtrf.c - * - * F distribution - * - * - * - * SYNOPSIS: - * - * int df1, df2; - * float x, y, fdtrf(); - * - * y = fdtrf( df1, df2, x ); - * - * - * - * DESCRIPTION: - * - * Returns the area from zero to x under the F density - * function (also known as Snedcor's density or the - * variance ratio density). This is the density - * of x = (u1/df1)/(u2/df2), where u1 and u2 are random - * variables having Chi square distributions with df1 - * and df2 degrees of freedom, respectively. - * - * The incomplete beta integral is used, according to the - * formula - * - * P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ). - * - * - * The arguments a and b are greater than zero, and x - * x is nonnegative. - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 2.2e-5 1.1e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * fdtrf domain a<0, b<0, x<0 0.0 - * - */ - /* fdtrcf() - * - * Complemented F distribution - * - * - * - * SYNOPSIS: - * - * int df1, df2; - * float x, y, fdtrcf(); - * - * y = fdtrcf( df1, df2, x ); - * - * - * - * DESCRIPTION: - * - * Returns the area from x to infinity under the F density - * function (also known as Snedcor's density or the - * variance ratio density). - * - * - * inf. - * - - * 1 | | a-1 b-1 - * 1-P(x) = ------ | t (1-t) dt - * B(a,b) | | - * - - * x - * - * (See fdtr.c.) - * - * The incomplete beta integral is used, according to the - * formula - * - * P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 7.3e-5 1.2e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * fdtrcf domain a<0, b<0, x<0 0.0 - * - */ - /* fdtrif() - * - * Inverse of complemented F distribution - * - * - * - * SYNOPSIS: - * - * float df1, df2, x, y, fdtrif(); - * - * x = fdtrif( df1, df2, y ); - * - * - * - * - * DESCRIPTION: - * - * Finds the F density argument x such that the integral - * from x to infinity of the F density is equal to the - * given probability y. - * - * This is accomplished using the inverse beta integral - * function and the relations - * - * z = incbi( df2/2, df1/2, y ) - * x = df2 (1-z) / (df1 z). - * - * Note: the following relations hold for the inverse of - * the uncomplemented F distribution: - * - * z = incbi( df1/2, df2/2, y ) - * x = df2 z / (df1 (1-z)). - * - * - * - * ACCURACY: - * - * arithmetic domain # trials peak rms - * Absolute error: - * IEEE 0,100 5000 4.0e-5 3.2e-6 - * Relative error: - * IEEE 0,100 5000 1.2e-3 1.8e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * fdtrif domain y <= 0 or y > 1 0.0 - * v < 1 - * - */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> - -#ifdef ANSIC -float incbetf(float, float, float); -float incbif(float, float, float); -#else -float incbetf(), incbif(); -#endif - -float fdtrcf( int ia, int ib, float xx ) -{ -float x, a, b, w; - -x = xx; -if( (ia < 1) || (ib < 1) || (x < 0.0) ) - { - mtherr( "fdtrcf", DOMAIN ); - return( 0.0 ); - } -a = ia; -b = ib; -w = b / (b + a * x); -return( incbetf( 0.5*b, 0.5*a, w ) ); -} - - - -float fdtrf( int ia, int ib, int xx ) -{ -float x, a, b, w; - -x = xx; -if( (ia < 1) || (ib < 1) || (x < 0.0) ) - { - mtherr( "fdtrf", DOMAIN ); - return( 0.0 ); - } -a = ia; -b = ib; -w = a * x; -w = w / (b + w); -return( incbetf( 0.5*a, 0.5*b, w) ); -} - - -float fdtrif( int ia, int ib, float yy ) -{ -float y, a, b, w, x; - -y = yy; -if( (ia < 1) || (ib < 1) || (y <= 0.0) || (y > 1.0) ) - { - mtherr( "fdtrif", DOMAIN ); - return( 0.0 ); - } -a = ia; -b = ib; -w = incbif( 0.5*b, 0.5*a, y ); -x = (b - b*w)/(a*w); -return(x); -} diff --git a/libm/float/floorf.c b/libm/float/floorf.c deleted file mode 100644 index 7a2f3530d..000000000 --- a/libm/float/floorf.c +++ /dev/null @@ -1,526 +0,0 @@ -/* ceilf() - * floorf() - * frexpf() - * ldexpf() - * signbitf() - * isnanf() - * isfinitef() - * - * Single precision floating point numeric utilities - * - * - * - * SYNOPSIS: - * - * float x, y; - * float ceilf(), floorf(), frexpf(), ldexpf(); - * int signbit(), isnan(), isfinite(); - * int expnt, n; - * - * y = floorf(x); - * y = ceilf(x); - * y = frexpf( x, &expnt ); - * y = ldexpf( x, n ); - * n = signbit(x); - * n = isnan(x); - * n = isfinite(x); - * - * - * - * DESCRIPTION: - * - * All four routines return a single precision floating point - * result. - * - * sfloor() returns the largest integer less than or equal to x. - * It truncates toward minus infinity. - * - * sceil() returns the smallest integer greater than or equal - * to x. It truncates toward plus infinity. - * - * sfrexp() extracts the exponent from x. It returns an integer - * power of two to expnt and the significand between 0.5 and 1 - * to y. Thus x = y * 2**expn. - * - * ldexpf() multiplies x by 2**n. - * - * signbit(x) returns 1 if the sign bit of x is 1, else 0. - * - * These functions are part of the standard C run time library - * for many but not all C compilers. The ones supplied are - * written in C for either DEC or IEEE arithmetic. They should - * be used only if your compiler library does not already have - * them. - * - * The IEEE versions assume that denormal numbers are implemented - * in the arithmetic. Some modifications will be required if - * the arithmetic has abrupt rather than gradual underflow. - */ - - -/* -Cephes Math Library Release 2.1: December, 1988 -Copyright 1984, 1987, 1988 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> -#ifdef DEC -#undef DENORMAL -#define DENORMAL 0 -#endif - -#ifdef UNK -#undef UNK -#if BIGENDIAN -#define MIEEE 1 -#else -#define IBMPC 1 -#endif -/* -char *unkmsg = "ceil(), floor(), frexp(), ldexp() must be rewritten!\n"; -*/ -#endif - -#define EXPMSK 0x807f -#define MEXP 255 -#define NBITS 24 - - -extern float MAXNUMF; /* (2^24 - 1) * 2^103 */ -#ifdef ANSIC -float floorf(float); -#else -float floorf(); -#endif - -float ceilf( float x ) -{ -float y; - -#ifdef UNK -printf( "%s\n", unkmsg ); -return(0.0); -#endif - -y = floorf( (float )x ); -if( y < x ) - y += 1.0; -return(y); -} - - - - -/* Bit clearing masks: */ - -static unsigned short bmask[] = { -0xffff, -0xfffe, -0xfffc, -0xfff8, -0xfff0, -0xffe0, -0xffc0, -0xff80, -0xff00, -0xfe00, -0xfc00, -0xf800, -0xf000, -0xe000, -0xc000, -0x8000, -0x0000, -}; - - - -float floorf( float x ) -{ -unsigned short *p; -union - { - float y; - unsigned short i[2]; - } u; -int e; - -#ifdef UNK -printf( "%s\n", unkmsg ); -return(0.0); -#endif - -u.y = x; -/* find the exponent (power of 2) */ -#ifdef DEC -p = &u.i[0]; -e = (( *p >> 7) & 0377) - 0201; -p += 3; -#endif - -#ifdef IBMPC -p = &u.i[1]; -e = (( *p >> 7) & 0xff) - 0x7f; -p -= 1; -#endif - -#ifdef MIEEE -p = &u.i[0]; -e = (( *p >> 7) & 0xff) - 0x7f; -p += 1; -#endif - -if( e < 0 ) - { - if( u.y < 0 ) - return( -1.0 ); - else - return( 0.0 ); - } - -e = (NBITS -1) - e; -/* clean out 16 bits at a time */ -while( e >= 16 ) - { -#ifdef IBMPC - *p++ = 0; -#endif - -#ifdef DEC - *p-- = 0; -#endif - -#ifdef MIEEE - *p-- = 0; -#endif - e -= 16; - } - -/* clear the remaining bits */ -if( e > 0 ) - *p &= bmask[e]; - -if( (x < 0) && (u.y != x) ) - u.y -= 1.0; - -return(u.y); -} - - - -float frexpf( float x, int *pw2 ) -{ -union - { - float y; - unsigned short i[2]; - } u; -int i, k; -short *q; - -u.y = x; - -#ifdef UNK -printf( "%s\n", unkmsg ); -return(0.0); -#endif - -#ifdef IBMPC -q = &u.i[1]; -#endif - -#ifdef DEC -q = &u.i[0]; -#endif - -#ifdef MIEEE -q = &u.i[0]; -#endif - -/* find the exponent (power of 2) */ - -i = ( *q >> 7) & 0xff; -if( i == 0 ) - { - if( u.y == 0.0 ) - { - *pw2 = 0; - return(0.0); - } -/* Number is denormal or zero */ -#if DENORMAL -/* Handle denormal number. */ - do - { - u.y *= 2.0; - i -= 1; - k = ( *q >> 7) & 0xff; - } - while( k == 0 ); - i = i + k; -#else - *pw2 = 0; - return( 0.0 ); -#endif /* DENORMAL */ - } -i -= 0x7e; -*pw2 = i; -*q &= 0x807f; /* strip all exponent bits */ -*q |= 0x3f00; /* mantissa between 0.5 and 1 */ -return( u.y ); -} - - - - - -float ldexpf( float x, int pw2 ) -{ -union - { - float y; - unsigned short i[2]; - } u; -short *q; -int e; - -#ifdef UNK -printf( "%s\n", unkmsg ); -return(0.0); -#endif - -u.y = x; -#ifdef DEC -q = &u.i[0]; -#endif - -#ifdef IBMPC -q = &u.i[1]; -#endif -#ifdef MIEEE -q = &u.i[0]; -#endif -while( (e = ( *q >> 7) & 0xff) == 0 ) - { - if( u.y == (float )0.0 ) - { - return( 0.0 ); - } -/* Input is denormal. */ - if( pw2 > 0 ) - { - u.y *= 2.0; - pw2 -= 1; - } - if( pw2 < 0 ) - { - if( pw2 < -24 ) - return( 0.0 ); - u.y *= 0.5; - pw2 += 1; - } - if( pw2 == 0 ) - return(u.y); - } - -e += pw2; - -/* Handle overflow */ -if( e > MEXP ) - { - return( MAXNUMF ); - } - -*q &= 0x807f; - -/* Handle denormalized results */ -if( e < 1 ) - { -#if DENORMAL - if( e < -24 ) - return( 0.0 ); - *q |= 0x80; /* Set LSB of exponent. */ - /* For denormals, significant bits may be lost even - when dividing by 2. Construct 2^-(1-e) so the result - is obtained with only one multiplication. */ - u.y *= ldexpf(1.0f, e - 1); - return(u.y); -#else - return( 0.0 ); -#endif - } -*q |= (e & 0xff) << 7; -return(u.y); -} - - -/* Return 1 if the sign bit of x is 1, else 0. */ - -int signbitf(x) -float x; -{ -union - { - float f; - short s[4]; - int i; - } u; - -u.f = x; - -if( sizeof(int) == 4 ) - { -#ifdef IBMPC - return( u.i < 0 ); -#endif -#ifdef DEC - return( u.s[1] < 0 ); -#endif -#ifdef MIEEE - return( u.i < 0 ); -#endif - } -else - { -#ifdef IBMPC - return( u.s[1] < 0 ); -#endif -#ifdef DEC - return( u.s[1] < 0 ); -#endif -#ifdef MIEEE - return( u.s[0] < 0 ); -#endif - } -} - - -/* Return 1 if x is a number that is Not a Number, else return 0. */ - -int isnanf(x) -float x; -{ -#ifdef NANS -union - { - float f; - unsigned short s[2]; - unsigned int i; - } u; - -u.f = x; - -if( sizeof(int) == 4 ) - { -#ifdef IBMPC - if( ((u.i & 0x7f800000) == 0x7f800000) - && ((u.i & 0x007fffff) != 0) ) - return 1; -#endif -#ifdef DEC - if( (u.s[1] & 0x7f80) == 0) - { - if( (u.s[1] | u.s[0]) != 0 ) - return(1); - } -#endif -#ifdef MIEEE - if( ((u.i & 0x7f800000) == 0x7f800000) - && ((u.i & 0x007fffff) != 0) ) - return 1; -#endif - return(0); - } -else - { /* size int not 4 */ -#ifdef IBMPC - if( (u.s[1] & 0x7f80) == 0x7f80) - { - if( ((u.s[1] & 0x007f) | u.s[0]) != 0 ) - return(1); - } -#endif -#ifdef DEC - if( (u.s[1] & 0x7f80) == 0) - { - if( (u.s[1] | u.s[0]) != 0 ) - return(1); - } -#endif -#ifdef MIEEE - if( (u.s[0] & 0x7f80) == 0x7f80) - { - if( ((u.s[0] & 0x000f) | u.s[1]) != 0 ) - return(1); - } -#endif - return(0); - } /* size int not 4 */ - -#else -/* No NANS. */ -return(0); -#endif -} - - -/* Return 1 if x is not infinite and is not a NaN. */ - -int isfinitef(x) -float x; -{ -#ifdef INFINITIES -union - { - float f; - unsigned short s[2]; - unsigned int i; - } u; - -u.f = x; - -if( sizeof(int) == 4 ) - { -#ifdef IBMPC - if( (u.i & 0x7f800000) != 0x7f800000) - return 1; -#endif -#ifdef DEC - if( (u.s[1] & 0x7f80) == 0) - { - if( (u.s[1] | u.s[0]) != 0 ) - return(1); - } -#endif -#ifdef MIEEE - if( (u.i & 0x7f800000) != 0x7f800000) - return 1; -#endif - return(0); - } -else - { -#ifdef IBMPC - if( (u.s[1] & 0x7f80) != 0x7f80) - return 1; -#endif -#ifdef DEC - if( (u.s[1] & 0x7f80) == 0) - { - if( (u.s[1] | u.s[0]) != 0 ) - return(1); - } -#endif -#ifdef MIEEE - if( (u.s[0] & 0x7f80) != 0x7f80) - return 1; -#endif - return(0); - } -#else -/* No INFINITY. */ -return(1); -#endif -} diff --git a/libm/float/fresnlf.c b/libm/float/fresnlf.c deleted file mode 100644 index d6ae773b1..000000000 --- a/libm/float/fresnlf.c +++ /dev/null @@ -1,173 +0,0 @@ -/* fresnlf.c - * - * Fresnel integral - * - * - * - * SYNOPSIS: - * - * float x, S, C; - * void fresnlf(); - * - * fresnlf( x, _&S, _&C ); - * - * - * DESCRIPTION: - * - * Evaluates the Fresnel integrals - * - * x - * - - * | | - * C(x) = | cos(pi/2 t**2) dt, - * | | - * - - * 0 - * - * x - * - - * | | - * S(x) = | sin(pi/2 t**2) dt. - * | | - * - - * 0 - * - * - * The integrals are evaluated by power series for small x. - * For x >= 1 auxiliary functions f(x) and g(x) are employed - * such that - * - * C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 ) - * S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 ) - * - * - * - * ACCURACY: - * - * Relative error. - * - * Arithmetic function domain # trials peak rms - * IEEE S(x) 0, 10 30000 1.1e-6 1.9e-7 - * IEEE C(x) 0, 10 30000 1.1e-6 2.0e-7 - */ - -/* -Cephes Math Library Release 2.1: January, 1989 -Copyright 1984, 1987, 1989 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -/* S(x) for small x */ -static float sn[7] = { - 1.647629463788700E-009, --1.522754752581096E-007, - 8.424748808502400E-006, --3.120693124703272E-004, - 7.244727626597022E-003, --9.228055941124598E-002, - 5.235987735681432E-001 -}; - -/* C(x) for small x */ -static float cn[7] = { - 1.416802502367354E-008, --1.157231412229871E-006, - 5.387223446683264E-005, --1.604381798862293E-003, - 2.818489036795073E-002, --2.467398198317899E-001, - 9.999999760004487E-001 -}; - - -/* Auxiliary function f(x) */ -static float fn[8] = { --1.903009855649792E+012, - 1.355942388050252E+011, --4.158143148511033E+009, - 7.343848463587323E+007, --8.732356681548485E+005, - 8.560515466275470E+003, --1.032877601091159E+002, - 2.999401847870011E+000 -}; - -/* Auxiliary function g(x) */ -static float gn[8] = { --1.860843997624650E+011, - 1.278350673393208E+010, --3.779387713202229E+008, - 6.492611570598858E+006, --7.787789623358162E+004, - 8.602931494734327E+002, --1.493439396592284E+001, - 9.999841934744914E-001 -}; - - -extern float PIF, PIO2F; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float polevlf( float, float *, int ); -float cosf(float), sinf(float); -#else -float polevlf(), cosf(), sinf(); -#endif - -void fresnlf( float xxa, float *ssa, float *cca ) -{ -float f, g, cc, ss, c, s, t, u, x, x2; - -x = xxa; -x = fabsf(x); -x2 = x * x; -if( x2 < 2.5625 ) - { - t = x2 * x2; - ss = x * x2 * polevlf( t, sn, 6); - cc = x * polevlf( t, cn, 6); - goto done; - } - -if( x > 36974.0 ) - { - cc = 0.5; - ss = 0.5; - goto done; - } - - -/* Asymptotic power series auxiliary functions - * for large argument - */ - x2 = x * x; - t = PIF * x2; - u = 1.0/(t * t); - t = 1.0/t; - f = 1.0 - u * polevlf( u, fn, 7); - g = t * polevlf( u, gn, 7); - - t = PIO2F * x2; - c = cosf(t); - s = sinf(t); - t = PIF * x; - cc = 0.5 + (f * s - g * c)/t; - ss = 0.5 - (f * c + g * s)/t; - -done: -if( xxa < 0.0 ) - { - cc = -cc; - ss = -ss; - } - -*cca = cc; -*ssa = ss; -#if !ANSIC -return 0; -#endif -} diff --git a/libm/float/gammaf.c b/libm/float/gammaf.c deleted file mode 100644 index e8c4694c4..000000000 --- a/libm/float/gammaf.c +++ /dev/null @@ -1,423 +0,0 @@ -/* gammaf.c - * - * Gamma function - * - * - * - * SYNOPSIS: - * - * float x, y, gammaf(); - * extern int sgngamf; - * - * y = gammaf( x ); - * - * - * - * DESCRIPTION: - * - * Returns gamma function of the argument. The result is - * correctly signed, and the sign (+1 or -1) is also - * returned in a global (extern) variable named sgngamf. - * This same variable is also filled in by the logarithmic - * gamma function lgam(). - * - * Arguments between 0 and 10 are reduced by recurrence and the - * function is approximated by a polynomial function covering - * the interval (2,3). Large arguments are handled by Stirling's - * formula. Negative arguments are made positive using - * a reflection formula. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,-33 100,000 5.7e-7 1.0e-7 - * IEEE -33,0 100,000 6.1e-7 1.2e-7 - * - * - */ -/* lgamf() - * - * Natural logarithm of gamma function - * - * - * - * SYNOPSIS: - * - * float x, y, lgamf(); - * extern int sgngamf; - * - * y = lgamf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base e (2.718...) logarithm of the absolute - * value of the gamma function of the argument. - * The sign (+1 or -1) of the gamma function is returned in a - * global (extern) variable named sgngamf. - * - * For arguments greater than 6.5, the logarithm of the gamma - * function is approximated by the logarithmic version of - * Stirling's formula. Arguments between 0 and +6.5 are reduced by - * by recurrence to the interval [.75,1.25] or [1.5,2.5] of a rational - * approximation. The cosecant reflection formula is employed for - * arguments less than zero. - * - * Arguments greater than MAXLGM = 2.035093e36 return MAXNUM and an - * error message. - * - * - * - * ACCURACY: - * - * - * - * arithmetic domain # trials peak rms - * IEEE -100,+100 500,000 7.4e-7 6.8e-8 - * The error criterion was relative when the function magnitude - * was greater than one but absolute when it was less than one. - * The routine has low relative error for positive arguments. - * - * The following test used the relative error criterion. - * IEEE -2, +3 100000 4.0e-7 5.6e-8 - * - */ - -/* gamma.c */ -/* gamma function */ - -/* -Cephes Math Library Release 2.7: July, 1998 -Copyright 1984, 1987, 1989, 1992, 1998 by Stephen L. Moshier -*/ - - -#include <math.h> - -/* define MAXGAM 34.84425627277176174 */ - -/* Stirling's formula for the gamma function - * gamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) ( 1 + 1/x P(1/x) ) - * .028 < 1/x < .1 - * relative error < 1.9e-11 - */ -static float STIR[] = { --2.705194986674176E-003, - 3.473255786154910E-003, - 8.333331788340907E-002, -}; -static float MAXSTIR = 26.77; -static float SQTPIF = 2.50662827463100050242; /* sqrt( 2 pi ) */ - -int sgngamf = 0; -extern int sgngamf; -extern float MAXLOGF, MAXNUMF, PIF; - -#ifdef ANSIC -float expf(float); -float logf(float); -float powf( float, float ); -float sinf(float); -float gammaf(float); -float floorf(float); -static float stirf(float); -float polevlf( float, float *, int ); -float p1evlf( float, float *, int ); -#else -float expf(), logf(), powf(), sinf(), floorf(); -float polevlf(), p1evlf(); -static float stirf(); -#endif - -/* Gamma function computed by Stirling's formula, - * sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x)) - * The polynomial STIR is valid for 33 <= x <= 172. - */ -static float stirf( float xx ) -{ -float x, y, w, v; - -x = xx; -w = 1.0/x; -w = 1.0 + w * polevlf( w, STIR, 2 ); -y = expf( -x ); -if( x > MAXSTIR ) - { /* Avoid overflow in pow() */ - v = powf( x, 0.5 * x - 0.25 ); - y *= v; - y *= v; - } -else - { - y = powf( x, x - 0.5 ) * y; - } -y = SQTPIF * y * w; -return( y ); -} - - -/* gamma(x+2), 0 < x < 1 */ -static float P[] = { - 1.536830450601906E-003, - 5.397581592950993E-003, - 4.130370201859976E-003, - 7.232307985516519E-002, - 8.203960091619193E-002, - 4.117857447645796E-001, - 4.227867745131584E-001, - 9.999999822945073E-001, -}; - -float gammaf( float xx ) -{ -float p, q, x, z, nz; -int i, direction, negative; - -x = xx; -sgngamf = 1; -negative = 0; -nz = 0.0; -if( x < 0.0 ) - { - negative = 1; - q = -x; - p = floorf(q); - if( p == q ) - goto goverf; - i = p; - if( (i & 1) == 0 ) - sgngamf = -1; - nz = q - p; - if( nz > 0.5 ) - { - p += 1.0; - nz = q - p; - } - nz = q * sinf( PIF * nz ); - if( nz == 0.0 ) - { -goverf: - mtherr( "gamma", OVERFLOW ); - return( sgngamf * MAXNUMF); - } - if( nz < 0 ) - nz = -nz; - x = q; - } -if( x >= 10.0 ) - { - z = stirf(x); - } -if( x < 2.0 ) - direction = 1; -else - direction = 0; -z = 1.0; -while( x >= 3.0 ) - { - x -= 1.0; - z *= x; - } -/* -while( x < 0.0 ) - { - if( x > -1.E-4 ) - goto small; - z *=x; - x += 1.0; - } -*/ -while( x < 2.0 ) - { - if( x < 1.e-4 ) - goto small; - z *=x; - x += 1.0; - } - -if( direction ) - z = 1.0/z; - -if( x == 2.0 ) - return(z); - -x -= 2.0; -p = z * polevlf( x, P, 7 ); - -gdone: - -if( negative ) - { - p = sgngamf * PIF/(nz * p ); - } -return(p); - -small: -if( x == 0.0 ) - { - mtherr( "gamma", SING ); - return( MAXNUMF ); - } -else - { - p = z / ((1.0 + 0.5772156649015329 * x) * x); - goto gdone; - } -} - - - - -/* log gamma(x+2), -.5 < x < .5 */ -static float B[] = { - 6.055172732649237E-004, --1.311620815545743E-003, - 2.863437556468661E-003, --7.366775108654962E-003, - 2.058355474821512E-002, --6.735323259371034E-002, - 3.224669577325661E-001, - 4.227843421859038E-001 -}; - -/* log gamma(x+1), -.25 < x < .25 */ -static float C[] = { - 1.369488127325832E-001, --1.590086327657347E-001, - 1.692415923504637E-001, --2.067882815621965E-001, - 2.705806208275915E-001, --4.006931650563372E-001, - 8.224670749082976E-001, --5.772156501719101E-001 -}; - -/* log( sqrt( 2*pi ) ) */ -static float LS2PI = 0.91893853320467274178; -#define MAXLGM 2.035093e36 -static float PIINV = 0.318309886183790671538; - -/* Logarithm of gamma function */ - - -float lgamf( float xx ) -{ -float p, q, w, z, x; -float nx, tx; -int i, direction; - -sgngamf = 1; - -x = xx; -if( x < 0.0 ) - { - q = -x; - w = lgamf(q); /* note this modifies sgngam! */ - p = floorf(q); - if( p == q ) - goto loverf; - i = p; - if( (i & 1) == 0 ) - sgngamf = -1; - else - sgngamf = 1; - z = q - p; - if( z > 0.5 ) - { - p += 1.0; - z = p - q; - } - z = q * sinf( PIF * z ); - if( z == 0.0 ) - goto loverf; - z = -logf( PIINV*z ) - w; - return( z ); - } - -if( x < 6.5 ) - { - direction = 0; - z = 1.0; - tx = x; - nx = 0.0; - if( x >= 1.5 ) - { - while( tx > 2.5 ) - { - nx -= 1.0; - tx = x + nx; - z *=tx; - } - x += nx - 2.0; -iv1r5: - p = x * polevlf( x, B, 7 ); - goto cont; - } - if( x >= 1.25 ) - { - z *= x; - x -= 1.0; /* x + 1 - 2 */ - direction = 1; - goto iv1r5; - } - if( x >= 0.75 ) - { - x -= 1.0; - p = x * polevlf( x, C, 7 ); - q = 0.0; - goto contz; - } - while( tx < 1.5 ) - { - if( tx == 0.0 ) - goto loverf; - z *=tx; - nx += 1.0; - tx = x + nx; - } - direction = 1; - x += nx - 2.0; - p = x * polevlf( x, B, 7 ); - -cont: - if( z < 0.0 ) - { - sgngamf = -1; - z = -z; - } - else - { - sgngamf = 1; - } - q = logf(z); - if( direction ) - q = -q; -contz: - return( p + q ); - } - -if( x > MAXLGM ) - { -loverf: - mtherr( "lgamf", OVERFLOW ); - return( sgngamf * MAXNUMF ); - } - -/* Note, though an asymptotic formula could be used for x >= 3, - * there is cancellation error in the following if x < 6.5. */ -q = LS2PI - x; -q += ( x - 0.5 ) * logf(x); - -if( x <= 1.0e4 ) - { - z = 1.0/x; - p = z * z; - q += (( 6.789774945028216E-004 * p - - 2.769887652139868E-003 ) * p - + 8.333316229807355E-002 ) * z; - } -return( q ); -} diff --git a/libm/float/gdtrf.c b/libm/float/gdtrf.c deleted file mode 100644 index e7e02026b..000000000 --- a/libm/float/gdtrf.c +++ /dev/null @@ -1,144 +0,0 @@ -/* gdtrf.c - * - * Gamma distribution function - * - * - * - * SYNOPSIS: - * - * float a, b, x, y, gdtrf(); - * - * y = gdtrf( a, b, x ); - * - * - * - * DESCRIPTION: - * - * Returns the integral from zero to x of the gamma probability - * density function: - * - * - * x - * b - - * a | | b-1 -at - * y = ----- | t e dt - * - | | - * | (b) - - * 0 - * - * The incomplete gamma integral is used, according to the - * relation - * - * y = igam( b, ax ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 5.8e-5 3.0e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * gdtrf domain x < 0 0.0 - * - */ - /* gdtrcf.c - * - * Complemented gamma distribution function - * - * - * - * SYNOPSIS: - * - * float a, b, x, y, gdtrcf(); - * - * y = gdtrcf( a, b, x ); - * - * - * - * DESCRIPTION: - * - * Returns the integral from x to infinity of the gamma - * probability density function: - * - * - * inf. - * b - - * a | | b-1 -at - * y = ----- | t e dt - * - | | - * | (b) - - * x - * - * The incomplete gamma integral is used, according to the - * relation - * - * y = igamc( b, ax ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 9.1e-5 1.5e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * gdtrcf domain x < 0 0.0 - * - */ - -/* gdtr() */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> -#ifdef ANSIC -float igamf(float, float), igamcf(float, float); -#else -float igamf(), igamcf(); -#endif - - - -float gdtrf( float aa, float bb, float xx ) -{ -float a, b, x; - -a = aa; -b = bb; -x = xx; - - -if( x < 0.0 ) - { - mtherr( "gdtrf", DOMAIN ); - return( 0.0 ); - } -return( igamf( b, a * x ) ); -} - - - -float gdtrcf( float aa, float bb, float xx ) -{ -float a, b, x; - -a = aa; -b = bb; -x = xx; -if( x < 0.0 ) - { - mtherr( "gdtrcf", DOMAIN ); - return( 0.0 ); - } -return( igamcf( b, a * x ) ); -} diff --git a/libm/float/hyp2f1f.c b/libm/float/hyp2f1f.c deleted file mode 100644 index 01fe54928..000000000 --- a/libm/float/hyp2f1f.c +++ /dev/null @@ -1,442 +0,0 @@ -/* hyp2f1f.c - * - * Gauss hypergeometric function F - * 2 1 - * - * - * SYNOPSIS: - * - * float a, b, c, x, y, hyp2f1f(); - * - * y = hyp2f1f( a, b, c, x ); - * - * - * DESCRIPTION: - * - * - * hyp2f1( a, b, c, x ) = F ( a, b; c; x ) - * 2 1 - * - * inf. - * - a(a+1)...(a+k) b(b+1)...(b+k) k+1 - * = 1 + > ----------------------------- x . - * - c(c+1)...(c+k) (k+1)! - * k = 0 - * - * Cases addressed are - * Tests and escapes for negative integer a, b, or c - * Linear transformation if c - a or c - b negative integer - * Special case c = a or c = b - * Linear transformation for x near +1 - * Transformation for x < -0.5 - * Psi function expansion if x > 0.5 and c - a - b integer - * Conditionally, a recurrence on c to make c-a-b > 0 - * - * |x| > 1 is rejected. - * - * The parameters a, b, c are considered to be integer - * valued if they are within 1.0e-6 of the nearest integer. - * - * ACCURACY: - * - * Relative error (-1 < x < 1): - * arithmetic domain # trials peak rms - * IEEE 0,3 30000 5.8e-4 4.3e-6 - */ - -/* hyp2f1 */ - - -/* -Cephes Math Library Release 2.2: November, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> - -#define EPS 1.0e-5 -#define EPS2 1.0e-5 -#define ETHRESH 1.0e-5 - -extern float MAXNUMF, MACHEPF; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float powf(float, float); -static float hys2f1f(float, float, float, float, float *); -static float hyt2f1f(float, float, float, float, float *); -float gammaf(float), logf(float), expf(float), psif(float); -float floorf(float); -#else -float powf(), gammaf(), logf(), expf(), psif(); -float floorf(); -static float hyt2f1f(), hys2f1f(); -#endif - -#define roundf(x) (floorf((x)+(float )0.5)) - - - - -float hyp2f1f( float aa, float bb, float cc, float xx ) -{ -float a, b, c, x; -float d, d1, d2, e; -float p, q, r, s, y, ax; -float ia, ib, ic, id, err; -int flag, i, aid; - -a = aa; -b = bb; -c = cc; -x = xx; -err = 0.0; -ax = fabsf(x); -s = 1.0 - x; -flag = 0; -ia = roundf(a); /* nearest integer to a */ -ib = roundf(b); - -if( a <= 0 ) - { - if( fabsf(a-ia) < EPS ) /* a is a negative integer */ - flag |= 1; - } - -if( b <= 0 ) - { - if( fabsf(b-ib) < EPS ) /* b is a negative integer */ - flag |= 2; - } - -if( ax < 1.0 ) - { - if( fabsf(b-c) < EPS ) /* b = c */ - { - y = powf( s, -a ); /* s to the -a power */ - goto hypdon; - } - if( fabsf(a-c) < EPS ) /* a = c */ - { - y = powf( s, -b ); /* s to the -b power */ - goto hypdon; - } - } - - - -if( c <= 0.0 ) - { - ic = roundf(c); /* nearest integer to c */ - if( fabsf(c-ic) < EPS ) /* c is a negative integer */ - { - /* check if termination before explosion */ - if( (flag & 1) && (ia > ic) ) - goto hypok; - if( (flag & 2) && (ib > ic) ) - goto hypok; - goto hypdiv; - } - } - -if( flag ) /* function is a polynomial */ - goto hypok; - -if( ax > 1.0 ) /* series diverges */ - goto hypdiv; - -p = c - a; -ia = roundf(p); -if( (ia <= 0.0) && (fabsf(p-ia) < EPS) ) /* negative int c - a */ - flag |= 4; - -r = c - b; -ib = roundf(r); /* nearest integer to r */ -if( (ib <= 0.0) && (fabsf(r-ib) < EPS) ) /* negative int c - b */ - flag |= 8; - -d = c - a - b; -id = roundf(d); /* nearest integer to d */ -q = fabsf(d-id); - -if( fabsf(ax-1.0) < EPS ) /* |x| == 1.0 */ - { - if( x > 0.0 ) - { - if( flag & 12 ) /* negative int c-a or c-b */ - { - if( d >= 0.0 ) - goto hypf; - else - goto hypdiv; - } - if( d <= 0.0 ) - goto hypdiv; - y = gammaf(c)*gammaf(d)/(gammaf(p)*gammaf(r)); - goto hypdon; - } - - if( d <= -1.0 ) - goto hypdiv; - } - -/* Conditionally make d > 0 by recurrence on c - * AMS55 #15.2.27 - */ -if( d < 0.0 ) - { -/* Try the power series first */ - y = hyt2f1f( a, b, c, x, &err ); - if( err < ETHRESH ) - goto hypdon; -/* Apply the recurrence if power series fails */ - err = 0.0; - aid = 2 - id; - e = c + aid; - d2 = hyp2f1f(a,b,e,x); - d1 = hyp2f1f(a,b,e+1.0,x); - q = a + b + 1.0; - for( i=0; i<aid; i++ ) - { - r = e - 1.0; - y = (e*(r-(2.0*e-q)*x)*d2 + (e-a)*(e-b)*x*d1)/(e*r*s); - e = r; - d1 = d2; - d2 = y; - } - goto hypdon; - } - - -if( flag & 12 ) - goto hypf; /* negative integer c-a or c-b */ - -hypok: -y = hyt2f1f( a, b, c, x, &err ); - -hypdon: -if( err > ETHRESH ) - { - mtherr( "hyp2f1", PLOSS ); -/* printf( "Estimated err = %.2e\n", err );*/ - } -return(y); - -/* The transformation for c-a or c-b negative integer - * AMS55 #15.3.3 - */ -hypf: -y = powf( s, d ) * hys2f1f( c-a, c-b, c, x, &err ); -goto hypdon; - -/* The alarm exit */ -hypdiv: -mtherr( "hyp2f1f", OVERFLOW ); -return( MAXNUMF ); -} - - - - -/* Apply transformations for |x| near 1 - * then call the power series - */ -static float hyt2f1f( float aa, float bb, float cc, float xx, float *loss ) -{ -float a, b, c, x; -float p, q, r, s, t, y, d, err, err1; -float ax, id, d1, d2, e, y1; -int i, aid; - -a = aa; -b = bb; -c = cc; -x = xx; -err = 0.0; -s = 1.0 - x; -if( x < -0.5 ) - { - if( b > a ) - y = powf( s, -a ) * hys2f1f( a, c-b, c, -x/s, &err ); - - else - y = powf( s, -b ) * hys2f1f( c-a, b, c, -x/s, &err ); - - goto done; - } - - - -d = c - a - b; -id = roundf(d); /* nearest integer to d */ - -if( x > 0.8 ) -{ - -if( fabsf(d-id) > EPS2 ) /* test for integer c-a-b */ - { -/* Try the power series first */ - y = hys2f1f( a, b, c, x, &err ); - if( err < ETHRESH ) - goto done; -/* If power series fails, then apply AMS55 #15.3.6 */ - q = hys2f1f( a, b, 1.0-d, s, &err ); - q *= gammaf(d) /(gammaf(c-a) * gammaf(c-b)); - r = powf(s,d) * hys2f1f( c-a, c-b, d+1.0, s, &err1 ); - r *= gammaf(-d)/(gammaf(a) * gammaf(b)); - y = q + r; - - q = fabsf(q); /* estimate cancellation error */ - r = fabsf(r); - if( q > r ) - r = q; - err += err1 + (MACHEPF*r)/y; - - y *= gammaf(c); - goto done; - } -else - { -/* Psi function expansion, AMS55 #15.3.10, #15.3.11, #15.3.12 */ - if( id >= 0.0 ) - { - e = d; - d1 = d; - d2 = 0.0; - aid = id; - } - else - { - e = -d; - d1 = 0.0; - d2 = d; - aid = -id; - } - - ax = logf(s); - - /* sum for t = 0 */ - y = psif(1.0) + psif(1.0+e) - psif(a+d1) - psif(b+d1) - ax; - y /= gammaf(e+1.0); - - p = (a+d1) * (b+d1) * s / gammaf(e+2.0); /* Poch for t=1 */ - t = 1.0; - do - { - r = psif(1.0+t) + psif(1.0+t+e) - psif(a+t+d1) - - psif(b+t+d1) - ax; - q = p * r; - y += q; - p *= s * (a+t+d1) / (t+1.0); - p *= (b+t+d1) / (t+1.0+e); - t += 1.0; - } - while( fabsf(q/y) > EPS ); - - - if( id == 0.0 ) - { - y *= gammaf(c)/(gammaf(a)*gammaf(b)); - goto psidon; - } - - y1 = 1.0; - - if( aid == 1 ) - goto nosum; - - t = 0.0; - p = 1.0; - for( i=1; i<aid; i++ ) - { - r = 1.0-e+t; - p *= s * (a+t+d2) * (b+t+d2) / r; - t += 1.0; - p /= t; - y1 += p; - } - - -nosum: - p = gammaf(c); - y1 *= gammaf(e) * p / (gammaf(a+d1) * gammaf(b+d1)); - y *= p / (gammaf(a+d2) * gammaf(b+d2)); - if( (aid & 1) != 0 ) - y = -y; - - q = powf( s, id ); /* s to the id power */ - if( id > 0.0 ) - y *= q; - else - y1 *= q; - - y += y1; -psidon: - goto done; - } -} - - -/* Use defining power series if no special cases */ -y = hys2f1f( a, b, c, x, &err ); - -done: -*loss = err; -return(y); -} - - - - - -/* Defining power series expansion of Gauss hypergeometric function */ - -static float hys2f1f( float aa, float bb, float cc, float xx, float *loss ) -{ -int i; -float a, b, c, x; -float f, g, h, k, m, s, u, umax; - - -a = aa; -b = bb; -c = cc; -x = xx; -i = 0; -umax = 0.0; -f = a; -g = b; -h = c; -k = 0.0; -s = 1.0; -u = 1.0; - -do - { - if( fabsf(h) < EPS ) - return( MAXNUMF ); - m = k + 1.0; - u = u * ((f+k) * (g+k) * x / ((h+k) * m)); - s += u; - k = fabsf(u); /* remember largest term summed */ - if( k > umax ) - umax = k; - k = m; - if( ++i > 10000 ) /* should never happen */ - { - *loss = 1.0; - return(s); - } - } -while( fabsf(u/s) > MACHEPF ); - -/* return estimated relative error */ -*loss = (MACHEPF*umax)/fabsf(s) + (MACHEPF*i); - -return(s); -} - - diff --git a/libm/float/hypergf.c b/libm/float/hypergf.c deleted file mode 100644 index 60d0eb4c5..000000000 --- a/libm/float/hypergf.c +++ /dev/null @@ -1,384 +0,0 @@ -/* hypergf.c - * - * Confluent hypergeometric function - * - * - * - * SYNOPSIS: - * - * float a, b, x, y, hypergf(); - * - * y = hypergf( a, b, x ); - * - * - * - * DESCRIPTION: - * - * Computes the confluent hypergeometric function - * - * 1 2 - * a x a(a+1) x - * F ( a,b;x ) = 1 + ---- + --------- + ... - * 1 1 b 1! b(b+1) 2! - * - * Many higher transcendental functions are special cases of - * this power series. - * - * As is evident from the formula, b must not be a negative - * integer or zero unless a is an integer with 0 >= a > b. - * - * The routine attempts both a direct summation of the series - * and an asymptotic expansion. In each case error due to - * roundoff, cancellation, and nonconvergence is estimated. - * The result with smaller estimated error is returned. - * - * - * - * ACCURACY: - * - * Tested at random points (a, b, x), all three variables - * ranging from 0 to 30. - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,5 10000 6.6e-7 1.3e-7 - * IEEE 0,30 30000 1.1e-5 6.5e-7 - * - * Larger errors can be observed when b is near a negative - * integer or zero. Certain combinations of arguments yield - * serious cancellation error in the power series summation - * and also are not in the region of near convergence of the - * asymptotic series. An error message is printed if the - * self-estimated relative error is greater than 1.0e-3. - * - */ - -/* hyperg.c */ - - -/* -Cephes Math Library Release 2.1: November, 1988 -Copyright 1984, 1987, 1988 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -extern float MAXNUMF, MACHEPF; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float expf(float); -float hyp2f0f(float, float, float, int, float *); -static float hy1f1af(float, float, float, float *); -static float hy1f1pf(float, float, float, float *); -float logf(float), gammaf(float), lgamf(float); -#else -float expf(), hyp2f0f(); -float logf(), gammaf(), lgamf(); -static float hy1f1pf(), hy1f1af(); -#endif - -float hypergf( float aa, float bb, float xx ) -{ -float a, b, x, asum, psum, acanc, pcanc, temp; - - -a = aa; -b = bb; -x = xx; -/* See if a Kummer transformation will help */ -temp = b - a; -if( fabsf(temp) < 0.001 * fabsf(a) ) - return( expf(x) * hypergf( temp, b, -x ) ); - -psum = hy1f1pf( a, b, x, &pcanc ); -if( pcanc < 1.0e-6 ) - goto done; - - -/* try asymptotic series */ - -asum = hy1f1af( a, b, x, &acanc ); - - -/* Pick the result with less estimated error */ - -if( acanc < pcanc ) - { - pcanc = acanc; - psum = asum; - } - -done: -if( pcanc > 1.0e-3 ) - mtherr( "hyperg", PLOSS ); - -return( psum ); -} - - - - -/* Power series summation for confluent hypergeometric function */ - - -static float hy1f1pf( float aa, float bb, float xx, float *err ) -{ -float a, b, x, n, a0, sum, t, u, temp; -float an, bn, maxt, pcanc; - -a = aa; -b = bb; -x = xx; -/* set up for power series summation */ -an = a; -bn = b; -a0 = 1.0; -sum = 1.0; -n = 1.0; -t = 1.0; -maxt = 0.0; - - -while( t > MACHEPF ) - { - if( bn == 0 ) /* check bn first since if both */ - { - mtherr( "hypergf", SING ); - return( MAXNUMF ); /* an and bn are zero it is */ - } - if( an == 0 ) /* a singularity */ - return( sum ); - if( n > 200 ) - goto pdone; - u = x * ( an / (bn * n) ); - - /* check for blowup */ - temp = fabsf(u); - if( (temp > 1.0 ) && (maxt > (MAXNUMF/temp)) ) - { - pcanc = 1.0; /* estimate 100% error */ - goto blowup; - } - - a0 *= u; - sum += a0; - t = fabsf(a0); - if( t > maxt ) - maxt = t; -/* - if( (maxt/fabsf(sum)) > 1.0e17 ) - { - pcanc = 1.0; - goto blowup; - } -*/ - an += 1.0; - bn += 1.0; - n += 1.0; - } - -pdone: - -/* estimate error due to roundoff and cancellation */ -if( sum != 0.0 ) - maxt /= fabsf(sum); -maxt *= MACHEPF; /* this way avoids multiply overflow */ -pcanc = fabsf( MACHEPF * n + maxt ); - -blowup: - -*err = pcanc; - -return( sum ); -} - - -/* hy1f1a() */ -/* asymptotic formula for hypergeometric function: - * - * ( -a - * -- ( |z| - * | (b) ( -------- 2f0( a, 1+a-b, -1/x ) - * ( -- - * ( | (b-a) - * - * - * x a-b ) - * e |x| ) - * + -------- 2f0( b-a, 1-a, 1/x ) ) - * -- ) - * | (a) ) - */ - -static float hy1f1af( float aa, float bb, float xx, float *err ) -{ -float a, b, x, h1, h2, t, u, temp, acanc, asum, err1, err2; - -a = aa; -b = bb; -x = xx; -if( x == 0 ) - { - acanc = 1.0; - asum = MAXNUMF; - goto adone; - } -temp = logf( fabsf(x) ); -t = x + temp * (a-b); -u = -temp * a; - -if( b > 0 ) - { - temp = lgamf(b); - t += temp; - u += temp; - } - -h1 = hyp2f0f( a, a-b+1, -1.0/x, 1, &err1 ); - -temp = expf(u) / gammaf(b-a); -h1 *= temp; -err1 *= temp; - -h2 = hyp2f0f( b-a, 1.0-a, 1.0/x, 2, &err2 ); - -if( a < 0 ) - temp = expf(t) / gammaf(a); -else - temp = expf( t - lgamf(a) ); - -h2 *= temp; -err2 *= temp; - -if( x < 0.0 ) - asum = h1; -else - asum = h2; - -acanc = fabsf(err1) + fabsf(err2); - - -if( b < 0 ) - { - temp = gammaf(b); - asum *= temp; - acanc *= fabsf(temp); - } - - -if( asum != 0.0 ) - acanc /= fabsf(asum); - -acanc *= 30.0; /* fudge factor, since error of asymptotic formula - * often seems this much larger than advertised */ - -adone: - - -*err = acanc; -return( asum ); -} - -/* hyp2f0() */ - -float hyp2f0f(float aa, float bb, float xx, int type, float *err) -{ -float a, b, x, a0, alast, t, tlast, maxt; -float n, an, bn, u, sum, temp; - -a = aa; -b = bb; -x = xx; -an = a; -bn = b; -a0 = 1.0; -alast = 1.0; -sum = 0.0; -n = 1.0; -t = 1.0; -tlast = 1.0e9; -maxt = 0.0; - -do - { - if( an == 0 ) - goto pdone; - if( bn == 0 ) - goto pdone; - - u = an * (bn * x / n); - - /* check for blowup */ - temp = fabsf(u); - if( (temp > 1.0 ) && (maxt > (MAXNUMF/temp)) ) - goto error; - - a0 *= u; - t = fabsf(a0); - - /* terminating condition for asymptotic series */ - if( t > tlast ) - goto ndone; - - tlast = t; - sum += alast; /* the sum is one term behind */ - alast = a0; - - if( n > 200 ) - goto ndone; - - an += 1.0; - bn += 1.0; - n += 1.0; - if( t > maxt ) - maxt = t; - } -while( t > MACHEPF ); - - -pdone: /* series converged! */ - -/* estimate error due to roundoff and cancellation */ -*err = fabsf( MACHEPF * (n + maxt) ); - -alast = a0; -goto done; - -ndone: /* series did not converge */ - -/* The following "Converging factors" are supposed to improve accuracy, - * but do not actually seem to accomplish very much. */ - -n -= 1.0; -x = 1.0/x; - -switch( type ) /* "type" given as subroutine argument */ -{ -case 1: - alast *= ( 0.5 + (0.125 + 0.25*b - 0.5*a + 0.25*x - 0.25*n)/x ); - break; - -case 2: - alast *= 2.0/3.0 - b + 2.0*a + x - n; - break; - -default: - ; -} - -/* estimate error due to roundoff, cancellation, and nonconvergence */ -*err = MACHEPF * (n + maxt) + fabsf( a0 ); - - -done: -sum += alast; -return( sum ); - -/* series blew up: */ -error: -*err = MAXNUMF; -mtherr( "hypergf", TLOSS ); -return( sum ); -} diff --git a/libm/float/i0f.c b/libm/float/i0f.c deleted file mode 100644 index bb62cf60a..000000000 --- a/libm/float/i0f.c +++ /dev/null @@ -1,160 +0,0 @@ -/* i0f.c - * - * Modified Bessel function of order zero - * - * - * - * SYNOPSIS: - * - * float x, y, i0(); - * - * y = i0f( x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of order zero of the - * argument. - * - * The function is defined as i0(x) = j0( ix ). - * - * The range is partitioned into the two intervals [0,8] and - * (8, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 100000 4.0e-7 7.9e-8 - * - */ - /* i0ef.c - * - * Modified Bessel function of order zero, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * float x, y, i0ef(); - * - * y = i0ef( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of order zero of the argument. - * - * The function is defined as i0e(x) = exp(-|x|) j0( ix ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 100000 3.7e-7 7.0e-8 - * See i0f(). - * - */ - -/* i0.c */ - - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -/* Chebyshev coefficients for exp(-x) I0(x) - * in the interval [0,8]. - * - * lim(x->0){ exp(-x) I0(x) } = 1. - */ - -static float A[] = -{ --1.30002500998624804212E-8f, - 6.04699502254191894932E-8f, --2.67079385394061173391E-7f, - 1.11738753912010371815E-6f, --4.41673835845875056359E-6f, - 1.64484480707288970893E-5f, --5.75419501008210370398E-5f, - 1.88502885095841655729E-4f, --5.76375574538582365885E-4f, - 1.63947561694133579842E-3f, --4.32430999505057594430E-3f, - 1.05464603945949983183E-2f, --2.37374148058994688156E-2f, - 4.93052842396707084878E-2f, --9.49010970480476444210E-2f, - 1.71620901522208775349E-1f, --3.04682672343198398683E-1f, - 6.76795274409476084995E-1f -}; - - -/* Chebyshev coefficients for exp(-x) sqrt(x) I0(x) - * in the inverted interval [8,infinity]. - * - * lim(x->inf){ exp(-x) sqrt(x) I0(x) } = 1/sqrt(2pi). - */ - -static float B[] = -{ - 3.39623202570838634515E-9f, - 2.26666899049817806459E-8f, - 2.04891858946906374183E-7f, - 2.89137052083475648297E-6f, - 6.88975834691682398426E-5f, - 3.36911647825569408990E-3f, - 8.04490411014108831608E-1f -}; - - -float chbevlf(float, float *, int), expf(float), sqrtf(float); - -float i0f( float x ) -{ -float y; - -if( x < 0 ) - x = -x; -if( x <= 8.0f ) - { - y = 0.5f*x - 2.0f; - return( expf(x) * chbevlf( y, A, 18 ) ); - } - -return( expf(x) * chbevlf( 32.0f/x - 2.0f, B, 7 ) / sqrtf(x) ); -} - - - -float chbevlf(float, float *, int), expf(float), sqrtf(float); - -float i0ef( float x ) -{ -float y; - -if( x < 0 ) - x = -x; -if( x <= 8.0f ) - { - y = 0.5f*x - 2.0f; - return( chbevlf( y, A, 18 ) ); - } - -return( chbevlf( 32.0f/x - 2.0f, B, 7 ) / sqrtf(x) ); -} diff --git a/libm/float/i1f.c b/libm/float/i1f.c deleted file mode 100644 index e9741e1da..000000000 --- a/libm/float/i1f.c +++ /dev/null @@ -1,177 +0,0 @@ -/* i1f.c - * - * Modified Bessel function of order one - * - * - * - * SYNOPSIS: - * - * float x, y, i1f(); - * - * y = i1f( x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of order one of the - * argument. - * - * The function is defined as i1(x) = -i j1( ix ). - * - * The range is partitioned into the two intervals [0,8] and - * (8, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 100000 1.5e-6 1.6e-7 - * - * - */ - /* i1ef.c - * - * Modified Bessel function of order one, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * float x, y, i1ef(); - * - * y = i1ef( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of order one of the argument. - * - * The function is defined as i1(x) = -i exp(-|x|) j1( ix ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 1.5e-6 1.5e-7 - * See i1(). - * - */ - -/* i1.c 2 */ - - -/* -Cephes Math Library Release 2.0: March, 1987 -Copyright 1985, 1987 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -/* Chebyshev coefficients for exp(-x) I1(x) / x - * in the interval [0,8]. - * - * lim(x->0){ exp(-x) I1(x) / x } = 1/2. - */ - -static float A[] = -{ - 9.38153738649577178388E-9f, --4.44505912879632808065E-8f, - 2.00329475355213526229E-7f, --8.56872026469545474066E-7f, - 3.47025130813767847674E-6f, --1.32731636560394358279E-5f, - 4.78156510755005422638E-5f, --1.61760815825896745588E-4f, - 5.12285956168575772895E-4f, --1.51357245063125314899E-3f, - 4.15642294431288815669E-3f, --1.05640848946261981558E-2f, - 2.47264490306265168283E-2f, --5.29459812080949914269E-2f, - 1.02643658689847095384E-1f, --1.76416518357834055153E-1f, - 2.52587186443633654823E-1f -}; - - -/* Chebyshev coefficients for exp(-x) sqrt(x) I1(x) - * in the inverted interval [8,infinity]. - * - * lim(x->inf){ exp(-x) sqrt(x) I1(x) } = 1/sqrt(2pi). - */ - -static float B[] = -{ --3.83538038596423702205E-9f, --2.63146884688951950684E-8f, --2.51223623787020892529E-7f, --3.88256480887769039346E-6f, --1.10588938762623716291E-4f, --9.76109749136146840777E-3f, - 7.78576235018280120474E-1f -}; - -/* i1.c */ - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float chbevlf(float, float *, int); -float expf(float), sqrtf(float); -#else -float chbevlf(), expf(), sqrtf(); -#endif - - -float i1f(float xx) -{ -float x, y, z; - -x = xx; -z = fabsf(x); -if( z <= 8.0f ) - { - y = 0.5f*z - 2.0f; - z = chbevlf( y, A, 17 ) * z * expf(z); - } -else - { - z = expf(z) * chbevlf( 32.0f/z - 2.0f, B, 7 ) / sqrtf(z); - } -if( x < 0.0f ) - z = -z; -return( z ); -} - -/* i1e() */ - -float i1ef( float xx ) -{ -float x, y, z; - -x = xx; -z = fabsf(x); -if( z <= 8.0f ) - { - y = 0.5f*z - 2.0f; - z = chbevlf( y, A, 17 ) * z; - } -else - { - z = chbevlf( 32.0f/z - 2.0f, B, 7 ) / sqrtf(z); - } -if( x < 0.0f ) - z = -z; -return( z ); -} diff --git a/libm/float/igamf.c b/libm/float/igamf.c deleted file mode 100644 index c54225df4..000000000 --- a/libm/float/igamf.c +++ /dev/null @@ -1,223 +0,0 @@ -/* igamf.c - * - * Incomplete gamma integral - * - * - * - * SYNOPSIS: - * - * float a, x, y, igamf(); - * - * y = igamf( a, x ); - * - * - * - * DESCRIPTION: - * - * The function is defined by - * - * x - * - - * 1 | | -t a-1 - * igam(a,x) = ----- | e t dt. - * - | | - * | (a) - - * 0 - * - * - * In this implementation both arguments must be positive. - * The integral is evaluated by either a power series or - * continued fraction expansion, depending on the relative - * values of a and x. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 20000 7.8e-6 5.9e-7 - * - */ - /* igamcf() - * - * Complemented incomplete gamma integral - * - * - * - * SYNOPSIS: - * - * float a, x, y, igamcf(); - * - * y = igamcf( a, x ); - * - * - * - * DESCRIPTION: - * - * The function is defined by - * - * - * igamc(a,x) = 1 - igam(a,x) - * - * inf. - * - - * 1 | | -t a-1 - * = ----- | e t dt. - * - | | - * | (a) - - * x - * - * - * In this implementation both arguments must be positive. - * The integral is evaluated by either a power series or - * continued fraction expansion, depending on the relative - * values of a and x. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,30 30000 7.8e-6 5.9e-7 - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1985, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -/* BIG = 1/MACHEPF */ -#define BIG 16777216. - -extern float MACHEPF, MAXLOGF; - -#ifdef ANSIC -float lgamf(float), expf(float), logf(float), igamf(float, float); -#else -float lgamf(), expf(), logf(), igamf(); -#endif - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - - - -float igamcf( float aa, float xx ) -{ -float a, x, ans, c, yc, ax, y, z; -float pk, pkm1, pkm2, qk, qkm1, qkm2; -float r, t; -static float big = BIG; - -a = aa; -x = xx; -if( (x <= 0) || ( a <= 0) ) - return( 1.0 ); - -if( (x < 1.0) || (x < a) ) - return( 1.0 - igamf(a,x) ); - -ax = a * logf(x) - x - lgamf(a); -if( ax < -MAXLOGF ) - { - mtherr( "igamcf", UNDERFLOW ); - return( 0.0 ); - } -ax = expf(ax); - -/* continued fraction */ -y = 1.0 - a; -z = x + y + 1.0; -c = 0.0; -pkm2 = 1.0; -qkm2 = x; -pkm1 = x + 1.0; -qkm1 = z * x; -ans = pkm1/qkm1; - -do - { - c += 1.0; - y += 1.0; - z += 2.0; - yc = y * c; - pk = pkm1 * z - pkm2 * yc; - qk = qkm1 * z - qkm2 * yc; - if( qk != 0 ) - { - r = pk/qk; - t = fabsf( (ans - r)/r ); - ans = r; - } - else - t = 1.0; - pkm2 = pkm1; - pkm1 = pk; - qkm2 = qkm1; - qkm1 = qk; - if( fabsf(pk) > big ) - { - pkm2 *= MACHEPF; - pkm1 *= MACHEPF; - qkm2 *= MACHEPF; - qkm1 *= MACHEPF; - } - } -while( t > MACHEPF ); - -return( ans * ax ); -} - - - -/* left tail of incomplete gamma function: - * - * inf. k - * a -x - x - * x e > ---------- - * - - - * k=0 | (a+k+1) - * - */ - -float igamf( float aa, float xx ) -{ -float a, x, ans, ax, c, r; - -a = aa; -x = xx; -if( (x <= 0) || ( a <= 0) ) - return( 0.0 ); - -if( (x > 1.0) && (x > a ) ) - return( 1.0 - igamcf(a,x) ); - -/* Compute x**a * exp(-x) / gamma(a) */ -ax = a * logf(x) - x - lgamf(a); -if( ax < -MAXLOGF ) - { - mtherr( "igamf", UNDERFLOW ); - return( 0.0 ); - } -ax = expf(ax); - -/* power series */ -r = a; -c = 1.0; -ans = 1.0; - -do - { - r += 1.0; - c *= x/r; - ans += c; - } -while( c/ans > MACHEPF ); - -return( ans * ax/a ); -} diff --git a/libm/float/igamif.c b/libm/float/igamif.c deleted file mode 100644 index 5a33b4982..000000000 --- a/libm/float/igamif.c +++ /dev/null @@ -1,112 +0,0 @@ -/* igamif() - * - * Inverse of complemented imcomplete gamma integral - * - * - * - * SYNOPSIS: - * - * float a, x, y, igamif(); - * - * x = igamif( a, y ); - * - * - * - * DESCRIPTION: - * - * Given y, the function finds x such that - * - * igamc( a, x ) = y. - * - * Starting with the approximate value - * - * 3 - * x = a t - * - * where - * - * t = 1 - d - ndtri(y) sqrt(d) - * - * and - * - * d = 1/9a, - * - * the routine performs up to 10 Newton iterations to find the - * root of igamc(a,x) - y = 0. - * - * - * ACCURACY: - * - * Tested for a ranging from 0 to 100 and x from 0 to 1. - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 1.0e-5 1.5e-6 - * - */ - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -extern float MACHEPF, MAXLOGF; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float igamcf(float, float); -float ndtrif(float), expf(float), logf(float), sqrtf(float), lgamf(float); -#else -float igamcf(); -float ndtrif(), expf(), logf(), sqrtf(), lgamf(); -#endif - - -float igamif( float aa, float yy0 ) -{ -float a, y0, d, y, x0, lgm; -int i; - -a = aa; -y0 = yy0; -/* approximation to inverse function */ -d = 1.0/(9.0*a); -y = ( 1.0 - d - ndtrif(y0) * sqrtf(d) ); -x0 = a * y * y * y; - -lgm = lgamf(a); - -for( i=0; i<10; i++ ) - { - if( x0 <= 0.0 ) - { - mtherr( "igamif", UNDERFLOW ); - return(0.0); - } - y = igamcf(a,x0); -/* compute the derivative of the function at this point */ - d = (a - 1.0) * logf(x0) - x0 - lgm; - if( d < -MAXLOGF ) - { - mtherr( "igamif", UNDERFLOW ); - goto done; - } - d = -expf(d); -/* compute the step to the next approximation of x */ - if( d == 0.0 ) - goto done; - d = (y - y0)/d; - x0 = x0 - d; - if( i < 3 ) - continue; - if( fabsf(d/x0) < (2.0 * MACHEPF) ) - goto done; - } - -done: -return( x0 ); -} diff --git a/libm/float/incbetf.c b/libm/float/incbetf.c deleted file mode 100644 index fed9aae4b..000000000 --- a/libm/float/incbetf.c +++ /dev/null @@ -1,424 +0,0 @@ -/* incbetf.c - * - * Incomplete beta integral - * - * - * SYNOPSIS: - * - * float a, b, x, y, incbetf(); - * - * y = incbetf( a, b, x ); - * - * - * DESCRIPTION: - * - * Returns incomplete beta integral of the arguments, evaluated - * from zero to x. The function is defined as - * - * x - * - - - * | (a+b) | | a-1 b-1 - * ----------- | t (1-t) dt. - * - - | | - * | (a) | (b) - - * 0 - * - * The domain of definition is 0 <= x <= 1. In this - * implementation a and b are restricted to positive values. - * The integral from x to 1 may be obtained by the symmetry - * relation - * - * 1 - incbet( a, b, x ) = incbet( b, a, 1-x ). - * - * The integral is evaluated by a continued fraction expansion. - * If a < 1, the function calls itself recursively after a - * transformation to increase a to a+1. - * - * ACCURACY: - * - * Tested at random points (a,b,x) with a and b in the indicated - * interval and x between 0 and 1. - * - * arithmetic domain # trials peak rms - * Relative error: - * IEEE 0,30 10000 3.7e-5 5.1e-6 - * IEEE 0,100 10000 1.7e-4 2.5e-5 - * The useful domain for relative error is limited by underflow - * of the single precision exponential function. - * Absolute error: - * IEEE 0,30 100000 2.2e-5 9.6e-7 - * IEEE 0,100 10000 6.5e-5 3.7e-6 - * - * Larger errors may occur for extreme ratios of a and b. - * - * ERROR MESSAGES: - * message condition value returned - * incbetf domain x<0, x>1 0.0 - */ - - -/* -Cephes Math Library, Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -#ifdef ANSIC -float lgamf(float), expf(float), logf(float); -static float incbdf(float, float, float); -static float incbcff(float, float, float); -float incbpsf(float, float, float); -#else -float lgamf(), expf(), logf(); -float incbpsf(); -static float incbcff(), incbdf(); -#endif - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -/* BIG = 1/MACHEPF */ -#define BIG 16777216. -extern float MACHEPF, MAXLOGF; -#define MINLOGF (-MAXLOGF) - -float incbetf( float aaa, float bbb, float xxx ) -{ -float aa, bb, xx, ans, a, b, t, x, onemx; -int flag; - -aa = aaa; -bb = bbb; -xx = xxx; -if( (xx <= 0.0) || ( xx >= 1.0) ) - { - if( xx == 0.0 ) - return(0.0); - if( xx == 1.0 ) - return( 1.0 ); - mtherr( "incbetf", DOMAIN ); - return( 0.0 ); - } - -onemx = 1.0 - xx; - - -/* transformation for small aa */ - -if( aa <= 1.0 ) - { - ans = incbetf( aa+1.0, bb, xx ); - t = aa*logf(xx) + bb*logf( 1.0-xx ) - + lgamf(aa+bb) - lgamf(aa+1.0) - lgamf(bb); - if( t > MINLOGF ) - ans += expf(t); - return( ans ); - } - - -/* see if x is greater than the mean */ - -if( xx > (aa/(aa+bb)) ) - { - flag = 1; - a = bb; - b = aa; - t = xx; - x = onemx; - } -else - { - flag = 0; - a = aa; - b = bb; - t = onemx; - x = xx; - } - -/* transformation for small aa */ -/* -if( a <= 1.0 ) - { - ans = a*logf(x) + b*logf( onemx ) - + lgamf(a+b) - lgamf(a+1.0) - lgamf(b); - t = incbetf( a+1.0, b, x ); - if( ans > MINLOGF ) - t += expf(ans); - goto bdone; - } -*/ -/* Choose expansion for optimal convergence */ - - -if( b > 10.0 ) - { -if( fabsf(b*x/a) < 0.3 ) - { - t = incbpsf( a, b, x ); - goto bdone; - } - } - -ans = x * (a+b-2.0)/(a-1.0); -if( ans < 1.0 ) - { - ans = incbcff( a, b, x ); - t = b * logf( t ); - } -else - { - ans = incbdf( a, b, x ); - t = (b-1.0) * logf(t); - } - -t += a*logf(x) + lgamf(a+b) - lgamf(a) - lgamf(b); -t += logf( ans/a ); - -if( t < MINLOGF ) - { - t = 0.0; - if( flag == 0 ) - { - mtherr( "incbetf", UNDERFLOW ); - } - } -else - { - t = expf(t); - } -bdone: - -if( flag ) - t = 1.0 - t; - -return( t ); -} - -/* Continued fraction expansion #1 - * for incomplete beta integral - */ - -static float incbcff( float aa, float bb, float xx ) -{ -float a, b, x, xk, pk, pkm1, pkm2, qk, qkm1, qkm2; -float k1, k2, k3, k4, k5, k6, k7, k8; -float r, t, ans; -static float big = BIG; -int n; - -a = aa; -b = bb; -x = xx; -k1 = a; -k2 = a + b; -k3 = a; -k4 = a + 1.0; -k5 = 1.0; -k6 = b - 1.0; -k7 = k4; -k8 = a + 2.0; - -pkm2 = 0.0; -qkm2 = 1.0; -pkm1 = 1.0; -qkm1 = 1.0; -ans = 1.0; -r = 0.0; -n = 0; -do - { - - xk = -( x * k1 * k2 )/( k3 * k4 ); - pk = pkm1 + pkm2 * xk; - qk = qkm1 + qkm2 * xk; - pkm2 = pkm1; - pkm1 = pk; - qkm2 = qkm1; - qkm1 = qk; - - xk = ( x * k5 * k6 )/( k7 * k8 ); - pk = pkm1 + pkm2 * xk; - qk = qkm1 + qkm2 * xk; - pkm2 = pkm1; - pkm1 = pk; - qkm2 = qkm1; - qkm1 = qk; - - if( qk != 0 ) - r = pk/qk; - if( r != 0 ) - { - t = fabsf( (ans - r)/r ); - ans = r; - } - else - t = 1.0; - - if( t < MACHEPF ) - goto cdone; - - k1 += 1.0; - k2 += 1.0; - k3 += 2.0; - k4 += 2.0; - k5 += 1.0; - k6 -= 1.0; - k7 += 2.0; - k8 += 2.0; - - if( (fabsf(qk) + fabsf(pk)) > big ) - { - pkm2 *= MACHEPF; - pkm1 *= MACHEPF; - qkm2 *= MACHEPF; - qkm1 *= MACHEPF; - } - if( (fabsf(qk) < MACHEPF) || (fabsf(pk) < MACHEPF) ) - { - pkm2 *= big; - pkm1 *= big; - qkm2 *= big; - qkm1 *= big; - } - } -while( ++n < 100 ); - -cdone: -return(ans); -} - - -/* Continued fraction expansion #2 - * for incomplete beta integral - */ - -static float incbdf( float aa, float bb, float xx ) -{ -float a, b, x, xk, pk, pkm1, pkm2, qk, qkm1, qkm2; -float k1, k2, k3, k4, k5, k6, k7, k8; -float r, t, ans, z; -static float big = BIG; -int n; - -a = aa; -b = bb; -x = xx; -k1 = a; -k2 = b - 1.0; -k3 = a; -k4 = a + 1.0; -k5 = 1.0; -k6 = a + b; -k7 = a + 1.0;; -k8 = a + 2.0; - -pkm2 = 0.0; -qkm2 = 1.0; -pkm1 = 1.0; -qkm1 = 1.0; -z = x / (1.0-x); -ans = 1.0; -r = 0.0; -n = 0; -do - { - - xk = -( z * k1 * k2 )/( k3 * k4 ); - pk = pkm1 + pkm2 * xk; - qk = qkm1 + qkm2 * xk; - pkm2 = pkm1; - pkm1 = pk; - qkm2 = qkm1; - qkm1 = qk; - - xk = ( z * k5 * k6 )/( k7 * k8 ); - pk = pkm1 + pkm2 * xk; - qk = qkm1 + qkm2 * xk; - pkm2 = pkm1; - pkm1 = pk; - qkm2 = qkm1; - qkm1 = qk; - - if( qk != 0 ) - r = pk/qk; - if( r != 0 ) - { - t = fabsf( (ans - r)/r ); - ans = r; - } - else - t = 1.0; - - if( t < MACHEPF ) - goto cdone; - - k1 += 1.0; - k2 -= 1.0; - k3 += 2.0; - k4 += 2.0; - k5 += 1.0; - k6 += 1.0; - k7 += 2.0; - k8 += 2.0; - - if( (fabsf(qk) + fabsf(pk)) > big ) - { - pkm2 *= MACHEPF; - pkm1 *= MACHEPF; - qkm2 *= MACHEPF; - qkm1 *= MACHEPF; - } - if( (fabsf(qk) < MACHEPF) || (fabsf(pk) < MACHEPF) ) - { - pkm2 *= big; - pkm1 *= big; - qkm2 *= big; - qkm1 *= big; - } - } -while( ++n < 100 ); - -cdone: -return(ans); -} - - -/* power series */ -float incbpsf( float aa, float bb, float xx ) -{ -float a, b, x, t, u, y, s; - -a = aa; -b = bb; -x = xx; - -y = a * logf(x) + (b-1.0)*logf(1.0-x) - logf(a); -y -= lgamf(a) + lgamf(b); -y += lgamf(a+b); - - -t = x / (1.0 - x); -s = 0.0; -u = 1.0; -do - { - b -= 1.0; - if( b == 0.0 ) - break; - a += 1.0; - u *= t*b/a; - s += u; - } -while( fabsf(u) > MACHEPF ); - -if( y < MINLOGF ) - { - mtherr( "incbetf", UNDERFLOW ); - s = 0.0; - } -else - s = expf(y) * (1.0 + s); -/*printf( "incbpsf: %.4e\n", s );*/ -return(s); -} diff --git a/libm/float/incbif.c b/libm/float/incbif.c deleted file mode 100644 index 4d8c0652e..000000000 --- a/libm/float/incbif.c +++ /dev/null @@ -1,197 +0,0 @@ -/* incbif() - * - * Inverse of imcomplete beta integral - * - * - * - * SYNOPSIS: - * - * float a, b, x, y, incbif(); - * - * x = incbif( a, b, y ); - * - * - * - * DESCRIPTION: - * - * Given y, the function finds x such that - * - * incbet( a, b, x ) = y. - * - * the routine performs up to 10 Newton iterations to find the - * root of incbet(a,b,x) - y = 0. - * - * - * ACCURACY: - * - * Relative error: - * x a,b - * arithmetic domain domain # trials peak rms - * IEEE 0,1 0,100 5000 2.8e-4 8.3e-6 - * - * Overflow and larger errors may occur for one of a or b near zero - * and the other large. - */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -extern float MACHEPF, MINLOGF; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float incbetf(float, float, float); -float ndtrif(float), expf(float), logf(float), sqrtf(float), lgamf(float); -#else -float incbetf(); -float ndtrif(), expf(), logf(), sqrtf(), lgamf(); -#endif - -float incbif( float aaa, float bbb, float yyy0 ) -{ -float aa, bb, yy0, a, b, y0; -float d, y, x, x0, x1, lgm, yp, di; -int i, rflg; - - -aa = aaa; -bb = bbb; -yy0 = yyy0; -if( yy0 <= 0 ) - return(0.0); -if( yy0 >= 1.0 ) - return(1.0); - -/* approximation to inverse function */ - -yp = -ndtrif(yy0); - -if( yy0 > 0.5 ) - { - rflg = 1; - a = bb; - b = aa; - y0 = 1.0 - yy0; - yp = -yp; - } -else - { - rflg = 0; - a = aa; - b = bb; - y0 = yy0; - } - - -if( (aa <= 1.0) || (bb <= 1.0) ) - { - y = 0.5 * yp * yp; - } -else - { - lgm = (yp * yp - 3.0)* 0.16666666666666667; - x0 = 2.0/( 1.0/(2.0*a-1.0) + 1.0/(2.0*b-1.0) ); - y = yp * sqrtf( x0 + lgm ) / x0 - - ( 1.0/(2.0*b-1.0) - 1.0/(2.0*a-1.0) ) - * (lgm + 0.833333333333333333 - 2.0/(3.0*x0)); - y = 2.0 * y; - if( y < MINLOGF ) - { - x0 = 1.0; - goto under; - } - } - -x = a/( a + b * expf(y) ); -y = incbetf( a, b, x ); -yp = (y - y0)/y0; -if( fabsf(yp) < 0.1 ) - goto newt; - -/* Resort to interval halving if not close enough */ -x0 = 0.0; -x1 = 1.0; -di = 0.5; - -for( i=0; i<20; i++ ) - { - if( i != 0 ) - { - x = di * x1 + (1.0-di) * x0; - y = incbetf( a, b, x ); - yp = (y - y0)/y0; - if( fabsf(yp) < 1.0e-3 ) - goto newt; - } - - if( y < y0 ) - { - x0 = x; - di = 0.5; - } - else - { - x1 = x; - di *= di; - if( di == 0.0 ) - di = 0.5; - } - } - -if( x0 == 0.0 ) - { -under: - mtherr( "incbif", UNDERFLOW ); - goto done; - } - -newt: - -x0 = x; -lgm = lgamf(a+b) - lgamf(a) - lgamf(b); - -for( i=0; i<10; i++ ) - { -/* compute the function at this point */ - if( i != 0 ) - y = incbetf(a,b,x0); -/* compute the derivative of the function at this point */ - d = (a - 1.0) * logf(x0) + (b - 1.0) * logf(1.0-x0) + lgm; - if( d < MINLOGF ) - { - x0 = 0.0; - goto under; - } - d = expf(d); -/* compute the step to the next approximation of x */ - d = (y - y0)/d; - x = x0; - x0 = x0 - d; - if( x0 <= 0.0 ) - { - x0 = 0.0; - goto under; - } - if( x0 >= 1.0 ) - { - x0 = 1.0; - goto under; - } - if( i < 2 ) - continue; - if( fabsf(d/x0) < 256.0 * MACHEPF ) - goto done; - } - -done: -if( rflg ) - x0 = 1.0 - x0; -return( x0 ); -} diff --git a/libm/float/ivf.c b/libm/float/ivf.c deleted file mode 100644 index b7ab2b619..000000000 --- a/libm/float/ivf.c +++ /dev/null @@ -1,114 +0,0 @@ -/* ivf.c - * - * Modified Bessel function of noninteger order - * - * - * - * SYNOPSIS: - * - * float v, x, y, ivf(); - * - * y = ivf( v, x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of order v of the - * argument. If x is negative, v must be integer valued. - * - * The function is defined as Iv(x) = Jv( ix ). It is - * here computed in terms of the confluent hypergeometric - * function, according to the formula - * - * v -x - * Iv(x) = (x/2) e hyperg( v+0.5, 2v+1, 2x ) / gamma(v+1) - * - * If v is a negative integer, then v is replaced by -v. - * - * - * ACCURACY: - * - * Tested at random points (v, x), with v between 0 and - * 30, x between 0 and 28. - * arithmetic domain # trials peak rms - * Relative error: - * IEEE 0,15 3000 4.7e-6 5.4e-7 - * Absolute error (relative when function > 1) - * IEEE 0,30 5000 8.5e-6 1.3e-6 - * - * Accuracy is diminished if v is near a negative integer. - * The useful domain for relative error is limited by overflow - * of the single precision exponential function. - * - * See also hyperg.c. - * - */ - /* iv.c */ -/* Modified Bessel function of noninteger order */ -/* If x < 0, then v must be an integer. */ - - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> - -extern float MAXNUMF; -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -float hypergf(float, float, float); -float expf(float), gammaf(float), logf(float), floorf(float); - -float ivf( float v, float x ) -{ -int sign; -float t, ax; - -/* If v is a negative integer, invoke symmetry */ -t = floorf(v); -if( v < 0.0 ) - { - if( t == v ) - { - v = -v; /* symmetry */ - t = -t; - } - } -/* If x is negative, require v to be an integer */ -sign = 1; -if( x < 0.0 ) - { - if( t != v ) - { - mtherr( "ivf", DOMAIN ); - return( 0.0 ); - } - if( v != 2.0 * floorf(v/2.0) ) - sign = -1; - } - -/* Avoid logarithm singularity */ -if( x == 0.0 ) - { - if( v == 0.0 ) - return( 1.0 ); - if( v < 0.0 ) - { - mtherr( "ivf", OVERFLOW ); - return( MAXNUMF ); - } - else - return( 0.0 ); - } - -ax = fabsf(x); -t = v * logf( 0.5 * ax ) - x; -t = sign * expf(t) / gammaf( v + 1.0 ); -ax = v + 0.5; -return( t * hypergf( ax, 2.0 * ax, 2.0 * x ) ); -} diff --git a/libm/float/j0f.c b/libm/float/j0f.c deleted file mode 100644 index 2b0d4a5a4..000000000 --- a/libm/float/j0f.c +++ /dev/null @@ -1,228 +0,0 @@ -/* j0f.c - * - * Bessel function of order zero - * - * - * - * SYNOPSIS: - * - * float x, y, j0f(); - * - * y = j0f( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order zero of the argument. - * - * The domain is divided into the intervals [0, 2] and - * (2, infinity). In the first interval the following polynomial - * approximation is used: - * - * - * 2 2 2 - * (w - r ) (w - r ) (w - r ) P(w) - * 1 2 3 - * - * 2 - * where w = x and the three r's are zeros of the function. - * - * In the second interval, the modulus and phase are approximated - * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) - * and Phase(x) = x + 1/x R(1/x^2) - pi/4. The function is - * - * j0(x) = Modulus(x) cos( Phase(x) ). - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic domain # trials peak rms - * IEEE 0, 2 100000 1.3e-7 3.6e-8 - * IEEE 2, 32 100000 1.9e-7 5.4e-8 - * - */ - /* y0f.c - * - * Bessel function of the second kind, order zero - * - * - * - * SYNOPSIS: - * - * float x, y, y0f(); - * - * y = y0f( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of the second kind, of order - * zero, of the argument. - * - * The domain is divided into the intervals [0, 2] and - * (2, infinity). In the first interval a rational approximation - * R(x) is employed to compute - * - * 2 2 2 - * y0(x) = (w - r ) (w - r ) (w - r ) R(x) + 2/pi ln(x) j0(x). - * 1 2 3 - * - * Thus a call to j0() is required. The three zeros are removed - * from R(x) to improve its numerical stability. - * - * In the second interval, the modulus and phase are approximated - * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) - * and Phase(x) = x + 1/x S(1/x^2) - pi/4. Then the function is - * - * y0(x) = Modulus(x) sin( Phase(x) ). - * - * - * - * - * ACCURACY: - * - * Absolute error, when y0(x) < 1; else relative error: - * - * arithmetic domain # trials peak rms - * IEEE 0, 2 100000 2.4e-7 3.4e-8 - * IEEE 2, 32 100000 1.8e-7 5.3e-8 - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> - -static float MO[8] = { --6.838999669318810E-002f, - 1.864949361379502E-001f, --2.145007480346739E-001f, - 1.197549369473540E-001f, --3.560281861530129E-003f, --4.969382655296620E-002f, --3.355424622293709E-006f, - 7.978845717621440E-001f -}; - -static float PH[8] = { - 3.242077816988247E+001f, --3.630592630518434E+001f, - 1.756221482109099E+001f, --4.974978466280903E+000f, - 1.001973420681837E+000f, --1.939906941791308E-001f, - 6.490598792654666E-002f, --1.249992184872738E-001f -}; - -static float YP[5] = { - 9.454583683980369E-008f, --9.413212653797057E-006f, - 5.344486707214273E-004f, --1.584289289821316E-002f, - 1.707584643733568E-001f -}; - -float YZ1 = 0.43221455686510834878f; -float YZ2 = 22.401876406482861405f; -float YZ3 = 64.130620282338755553f; - -static float DR1 = 5.78318596294678452118f; -/* -static float DR2 = 30.4712623436620863991; -static float DR3 = 74.887006790695183444889; -*/ - -static float JP[5] = { --6.068350350393235E-008f, - 6.388945720783375E-006f, --3.969646342510940E-004f, - 1.332913422519003E-002f, --1.729150680240724E-001f -}; -extern float PIO4F; - - -float polevlf(float, float *, int); -float logf(float), sinf(float), cosf(float), sqrtf(float); - -float j0f( float xx ) -{ -float x, w, z, p, q, xn; - - -if( xx < 0 ) - x = -xx; -else - x = xx; - -if( x <= 2.0f ) - { - z = x * x; - if( x < 1.0e-3f ) - return( 1.0f - 0.25f*z ); - - p = (z-DR1) * polevlf( z, JP, 4); - return( p ); - } - -q = 1.0f/x; -w = sqrtf(q); - -p = w * polevlf( q, MO, 7); -w = q*q; -xn = q * polevlf( w, PH, 7) - PIO4F; -p = p * cosf(xn + x); -return(p); -} - -/* y0() 2 */ -/* Bessel function of second kind, order zero */ - -/* Rational approximation coefficients YP[] are used for x < 6.5. - * The function computed is y0(x) - 2 ln(x) j0(x) / pi, - * whose value at x = 0 is 2 * ( log(0.5) + EUL ) / pi - * = 0.073804295108687225 , EUL is Euler's constant. - */ - -static float TWOOPI = 0.636619772367581343075535f; /* 2/pi */ -extern float MAXNUMF; - -float y0f( float xx ) -{ -float x, w, z, p, q, xn; - - -x = xx; -if( x <= 2.0f ) - { - if( x <= 0.0f ) - { - mtherr( "y0f", DOMAIN ); - return( -MAXNUMF ); - } - z = x * x; -/* w = (z-YZ1)*(z-YZ2)*(z-YZ3) * polevlf( z, YP, 4);*/ - w = (z-YZ1) * polevlf( z, YP, 4); - w += TWOOPI * logf(x) * j0f(x); - return( w ); - } - -q = 1.0f/x; -w = sqrtf(q); - -p = w * polevlf( q, MO, 7); -w = q*q; -xn = q * polevlf( w, PH, 7) - PIO4F; -p = p * sinf(xn + x); -return( p ); -} diff --git a/libm/float/j0tst.c b/libm/float/j0tst.c deleted file mode 100644 index e5a5607d7..000000000 --- a/libm/float/j0tst.c +++ /dev/null @@ -1,43 +0,0 @@ -float z[20] = { -2.4048254489898681641, -5.5200781822204589844, -8.6537275314331054687, -11.791533470153808594, -14.930917739868164062, -18.071063995361328125, -21.211637496948242188, -24.352472305297851563, -27.493478775024414062, -30.634607315063476562, -33.775821685791015625, -36.9170989990234375, -40.0584259033203125, -43.19979095458984375, -46.3411865234375, -49.482608795166015625, -52.624050140380859375, -55.76551055908203125, -58.906982421875, -62.04846954345703125, -}; - -/* #if ANSIC */ -#if __STDC__ -float j0f(float); -#else -float j0f(); -#endif - -int main() -{ -float y; -int i; - -for (i = 0; i< 20; i++) - { - y = j0f(z[i]); - printf("%.9e\n", y); - } -exit(0); -} - diff --git a/libm/float/j1f.c b/libm/float/j1f.c deleted file mode 100644 index 4306e9747..000000000 --- a/libm/float/j1f.c +++ /dev/null @@ -1,211 +0,0 @@ -/* j1f.c - * - * Bessel function of order one - * - * - * - * SYNOPSIS: - * - * float x, y, j1f(); - * - * y = j1f( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order one of the argument. - * - * The domain is divided into the intervals [0, 2] and - * (2, infinity). In the first interval a polynomial approximation - * 2 - * (w - r ) x P(w) - * 1 - * 2 - * is used, where w = x and r is the first zero of the function. - * - * In the second interval, the modulus and phase are approximated - * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) - * and Phase(x) = x + 1/x R(1/x^2) - 3pi/4. The function is - * - * j0(x) = Modulus(x) cos( Phase(x) ). - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic domain # trials peak rms - * IEEE 0, 2 100000 1.2e-7 2.5e-8 - * IEEE 2, 32 100000 2.0e-7 5.3e-8 - * - * - */ - /* y1.c - * - * Bessel function of second kind of order one - * - * - * - * SYNOPSIS: - * - * double x, y, y1(); - * - * y = y1( x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of the second kind of order one - * of the argument. - * - * The domain is divided into the intervals [0, 2] and - * (2, infinity). In the first interval a rational approximation - * R(x) is employed to compute - * - * 2 - * y0(x) = (w - r ) x R(x^2) + 2/pi (ln(x) j1(x) - 1/x) . - * 1 - * - * Thus a call to j1() is required. - * - * In the second interval, the modulus and phase are approximated - * by polynomials of the form Modulus(x) = sqrt(1/x) Q(1/x) - * and Phase(x) = x + 1/x S(1/x^2) - 3pi/4. Then the function is - * - * y0(x) = Modulus(x) sin( Phase(x) ). - * - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic domain # trials peak rms - * IEEE 0, 2 100000 2.2e-7 4.6e-8 - * IEEE 2, 32 100000 1.9e-7 5.3e-8 - * - * (error criterion relative when |y1| > 1). - * - */ - - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> - - -static float JP[5] = { --4.878788132172128E-009f, - 6.009061827883699E-007f, --4.541343896997497E-005f, - 1.937383947804541E-003f, --3.405537384615824E-002f -}; - -static float YP[5] = { - 8.061978323326852E-009f, --9.496460629917016E-007f, - 6.719543806674249E-005f, --2.641785726447862E-003f, - 4.202369946500099E-002f -}; - -static float MO1[8] = { - 6.913942741265801E-002f, --2.284801500053359E-001f, - 3.138238455499697E-001f, --2.102302420403875E-001f, - 5.435364690523026E-003f, - 1.493389585089498E-001f, - 4.976029650847191E-006f, - 7.978845453073848E-001f -}; - -static float PH1[8] = { --4.497014141919556E+001f, - 5.073465654089319E+001f, --2.485774108720340E+001f, - 7.222973196770240E+000f, --1.544842782180211E+000f, - 3.503787691653334E-001f, --1.637986776941202E-001f, - 3.749989509080821E-001f -}; - -static float YO1 = 4.66539330185668857532f; -static float Z1 = 1.46819706421238932572E1f; - -static float THPIO4F = 2.35619449019234492885f; /* 3*pi/4 */ -static float TWOOPI = 0.636619772367581343075535f; /* 2/pi */ -extern float PIO4; - - -float polevlf(float, float *, int); -float logf(float), sinf(float), cosf(float), sqrtf(float); - -float j1f( float xx ) -{ -float x, w, z, p, q, xn; - - -x = xx; -if( x < 0 ) - x = -xx; - -if( x <= 2.0f ) - { - z = x * x; - p = (z-Z1) * x * polevlf( z, JP, 4 ); - return( p ); - } - -q = 1.0f/x; -w = sqrtf(q); - -p = w * polevlf( q, MO1, 7); -w = q*q; -xn = q * polevlf( w, PH1, 7) - THPIO4F; -p = p * cosf(xn + x); -return(p); -} - - - - -extern float MAXNUMF; - -float y1f( float xx ) -{ -float x, w, z, p, q, xn; - - -x = xx; -if( x <= 2.0f ) - { - if( x <= 0.0f ) - { - mtherr( "y1f", DOMAIN ); - return( -MAXNUMF ); - } - z = x * x; - w = (z - YO1) * x * polevlf( z, YP, 4 ); - w += TWOOPI * ( j1f(x) * logf(x) - 1.0f/x ); - return( w ); - } - -q = 1.0f/x; -w = sqrtf(q); - -p = w * polevlf( q, MO1, 7); -w = q*q; -xn = q * polevlf( w, PH1, 7) - THPIO4F; -p = p * sinf(xn + x); -return(p); -} diff --git a/libm/float/jnf.c b/libm/float/jnf.c deleted file mode 100644 index de358e0ef..000000000 --- a/libm/float/jnf.c +++ /dev/null @@ -1,124 +0,0 @@ -/* jnf.c - * - * Bessel function of integer order - * - * - * - * SYNOPSIS: - * - * int n; - * float x, y, jnf(); - * - * y = jnf( n, x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order n, where n is a - * (possibly negative) integer. - * - * The ratio of jn(x) to j0(x) is computed by backward - * recurrence. First the ratio jn/jn-1 is found by a - * continued fraction expansion. Then the recurrence - * relating successive orders is applied until j0 or j1 is - * reached. - * - * If n = 0 or 1 the routine for j0 or j1 is called - * directly. - * - * - * - * ACCURACY: - * - * Absolute error: - * arithmetic range # trials peak rms - * IEEE 0, 15 30000 3.6e-7 3.6e-8 - * - * - * Not suitable for large n or x. Use jvf() instead. - * - */ - -/* jn.c -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ -#include <math.h> - -extern float MACHEPF; - -float j0f(float), j1f(float); - -float jnf( int n, float xx ) -{ -float x, pkm2, pkm1, pk, xk, r, ans, xinv, sign; -int k; - -x = xx; -sign = 1.0; -if( n < 0 ) - { - n = -n; - if( (n & 1) != 0 ) /* -1**n */ - sign = -1.0; - } - -if( n == 0 ) - return( sign * j0f(x) ); -if( n == 1 ) - return( sign * j1f(x) ); -if( n == 2 ) - return( sign * (2.0 * j1f(x) / x - j0f(x)) ); - -/* -if( x < MACHEPF ) - return( 0.0 ); -*/ - -/* continued fraction */ -k = 24; -pk = 2 * (n + k); -ans = pk; -xk = x * x; - -do - { - pk -= 2.0; - ans = pk - (xk/ans); - } -while( --k > 0 ); -/*ans = x/ans;*/ - -/* backward recurrence */ - -pk = 1.0; -/*pkm1 = 1.0/ans;*/ -xinv = 1.0/x; -pkm1 = ans * xinv; -k = n-1; -r = (float )(2 * k); - -do - { - pkm2 = (pkm1 * r - pk * x) * xinv; - pk = pkm1; - pkm1 = pkm2; - r -= 2.0; - } -while( --k > 0 ); - -r = pk; -if( r < 0 ) - r = -r; -ans = pkm1; -if( ans < 0 ) - ans = -ans; - -if( r > ans ) /* if( fabs(pk) > fabs(pkm1) ) */ - ans = sign * j1f(x)/pk; -else - ans = sign * j0f(x)/pkm1; -return( ans ); -} diff --git a/libm/float/jvf.c b/libm/float/jvf.c deleted file mode 100644 index 268a8e4eb..000000000 --- a/libm/float/jvf.c +++ /dev/null @@ -1,848 +0,0 @@ -/* jvf.c - * - * Bessel function of noninteger order - * - * - * - * SYNOPSIS: - * - * float v, x, y, jvf(); - * - * y = jvf( v, x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order v of the argument, - * where v is real. Negative x is allowed if v is an integer. - * - * Several expansions are included: the ascending power - * series, the Hankel expansion, and two transitional - * expansions for large v. If v is not too large, it - * is reduced by recurrence to a region of best accuracy. - * - * The single precision routine accepts negative v, but with - * reduced accuracy. - * - * - * - * ACCURACY: - * Results for integer v are indicated by *. - * Error criterion is absolute, except relative when |jv()| > 1. - * - * arithmetic domain # trials peak rms - * v x - * IEEE 0,125 0,125 30000 2.0e-6 2.0e-7 - * IEEE -17,0 0,125 30000 1.1e-5 4.0e-7 - * IEEE -100,0 0,125 3000 1.5e-4 7.8e-6 - */ - - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> -#define DEBUG 0 - -extern float MAXNUMF, MACHEPF, MINLOGF, MAXLOGF, PIF; -extern int sgngamf; - -/* BIG = 1/MACHEPF */ -#define BIG 16777216. - -#ifdef ANSIC -float floorf(float), j0f(float), j1f(float); -static float jnxf(float, float); -static float jvsf(float, float); -static float hankelf(float, float); -static float jntf(float, float); -static float recurf( float *, float, float * ); -float sqrtf(float), sinf(float), cosf(float); -float lgamf(float), expf(float), logf(float), powf(float, float); -float gammaf(float), cbrtf(float), acosf(float); -int airyf(float, float *, float *, float *, float *); -float polevlf(float, float *, int); -#else -float floorf(), j0f(), j1f(); -float sqrtf(), sinf(), cosf(); -float lgamf(), expf(), logf(), powf(), gammaf(); -float cbrtf(), polevlf(), acosf(); -void airyf(); -static float recurf(), jvsf(), hankelf(), jnxf(), jntf(), jvsf(); -#endif - - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -float jvf( float nn, float xx ) -{ -float n, x, k, q, t, y, an, sign; -int i, nint; - -n = nn; -x = xx; -nint = 0; /* Flag for integer n */ -sign = 1.0; /* Flag for sign inversion */ -an = fabsf( n ); -y = floorf( an ); -if( y == an ) - { - nint = 1; - i = an - 16384.0 * floorf( an/16384.0 ); - if( n < 0.0 ) - { - if( i & 1 ) - sign = -sign; - n = an; - } - if( x < 0.0 ) - { - if( i & 1 ) - sign = -sign; - x = -x; - } - if( n == 0.0 ) - return( j0f(x) ); - if( n == 1.0 ) - return( sign * j1f(x) ); - } - -if( (x < 0.0) && (y != an) ) - { - mtherr( "jvf", DOMAIN ); - y = 0.0; - goto done; - } - -y = fabsf(x); - -if( y < MACHEPF ) - goto underf; - -/* Easy cases - x small compared to n */ -t = 3.6 * sqrtf(an); -if( y < t ) - return( sign * jvsf(n,x) ); - -/* x large compared to n */ -k = 3.6 * sqrtf(y); -if( (an < k) && (y > 6.0) ) - return( sign * hankelf(n,x) ); - -if( (n > -100) && (n < 14.0) ) - { -/* Note: if x is too large, the continued - * fraction will fail; but then the - * Hankel expansion can be used. - */ - if( nint != 0 ) - { - k = 0.0; - q = recurf( &n, x, &k ); - if( k == 0.0 ) - { - y = j0f(x)/q; - goto done; - } - if( k == 1.0 ) - { - y = j1f(x)/q; - goto done; - } - } - - if( n >= 0.0 ) - { -/* Recur backwards from a larger value of n - */ - if( y > 1.3 * an ) - goto recurdwn; - if( an > 1.3 * y ) - goto recurdwn; - k = n; - y = 2.0*(y+an+1.0); - if( (y - n) > 33.0 ) - y = n + 33.0; - y = n + floorf(y-n); - q = recurf( &y, x, &k ); - y = jvsf(y,x) * q; - goto done; - } -recurdwn: - if( an > (k + 3.0) ) - { -/* Recur backwards from n to k - */ - if( n < 0.0 ) - k = -k; - q = n - floorf(n); - k = floorf(k) + q; - if( n > 0.0 ) - q = recurf( &n, x, &k ); - else - { - t = k; - k = n; - q = recurf( &t, x, &k ); - k = t; - } - if( q == 0.0 ) - { -underf: - y = 0.0; - goto done; - } - } - else - { - k = n; - q = 1.0; - } - -/* boundary between convergence of - * power series and Hankel expansion - */ - t = fabsf(k); - if( t < 26.0 ) - t = (0.0083*t + 0.09)*t + 12.9; - else - t = 0.9 * t; - - if( y > t ) /* y = |x| */ - y = hankelf(k,x); - else - y = jvsf(k,x); -#if DEBUG -printf( "y = %.16e, q = %.16e\n", y, q ); -#endif - if( n > 0.0 ) - y /= q; - else - y *= q; - } - -else - { -/* For large positive n, use the uniform expansion - * or the transitional expansion. - * But if x is of the order of n**2, - * these may blow up, whereas the - * Hankel expansion will then work. - */ - if( n < 0.0 ) - { - mtherr( "jvf", TLOSS ); - y = 0.0; - goto done; - } - t = y/an; - t /= an; - if( t > 0.3 ) - y = hankelf(n,x); - else - y = jnxf(n,x); - } - -done: return( sign * y); -} - -/* Reduce the order by backward recurrence. - * AMS55 #9.1.27 and 9.1.73. - */ - -static float recurf( float *n, float xx, float *newn ) -{ -float x, pkm2, pkm1, pk, pkp1, qkm2, qkm1; -float k, ans, qk, xk, yk, r, t, kf, xinv; -static float big = BIG; -int nflag, ctr; - -x = xx; -/* continued fraction for Jn(x)/Jn-1(x) */ -if( *n < 0.0 ) - nflag = 1; -else - nflag = 0; - -fstart: - -#if DEBUG -printf( "n = %.6e, newn = %.6e, cfrac = ", *n, *newn ); -#endif - -pkm2 = 0.0; -qkm2 = 1.0; -pkm1 = x; -qkm1 = *n + *n; -xk = -x * x; -yk = qkm1; -ans = 1.0; -ctr = 0; -do - { - yk += 2.0; - pk = pkm1 * yk + pkm2 * xk; - qk = qkm1 * yk + qkm2 * xk; - pkm2 = pkm1; - pkm1 = pk; - qkm2 = qkm1; - qkm1 = qk; - if( qk != 0 ) - r = pk/qk; - else - r = 0.0; - if( r != 0 ) - { - t = fabsf( (ans - r)/r ); - ans = r; - } - else - t = 1.0; - - if( t < MACHEPF ) - goto done; - - if( fabsf(pk) > big ) - { - pkm2 *= MACHEPF; - pkm1 *= MACHEPF; - qkm2 *= MACHEPF; - qkm1 *= MACHEPF; - } - } -while( t > MACHEPF ); - -done: - -#if DEBUG -printf( "%.6e\n", ans ); -#endif - -/* Change n to n-1 if n < 0 and the continued fraction is small - */ -if( nflag > 0 ) - { - if( fabsf(ans) < 0.125 ) - { - nflag = -1; - *n = *n - 1.0; - goto fstart; - } - } - - -kf = *newn; - -/* backward recurrence - * 2k - * J (x) = --- J (x) - J (x) - * k-1 x k k+1 - */ - -pk = 1.0; -pkm1 = 1.0/ans; -k = *n - 1.0; -r = 2 * k; -xinv = 1.0/x; -do - { - pkm2 = (pkm1 * r - pk * x) * xinv; - pkp1 = pk; - pk = pkm1; - pkm1 = pkm2; - r -= 2.0; -#if 0 - t = fabsf(pkp1) + fabsf(pk); - if( (k > (kf + 2.5)) && (fabsf(pkm1) < 0.25*t) ) - { - k -= 1.0; - t = x*x; - pkm2 = ( (r*(r+2.0)-t)*pk - r*x*pkp1 )/t; - pkp1 = pk; - pk = pkm1; - pkm1 = pkm2; - r -= 2.0; - } -#endif - k -= 1.0; - } -while( k > (kf + 0.5) ); - -#if 0 -/* Take the larger of the last two iterates - * on the theory that it may have less cancellation error. - */ -if( (kf >= 0.0) && (fabsf(pk) > fabsf(pkm1)) ) - { - k += 1.0; - pkm2 = pk; - } -#endif - -*newn = k; -#if DEBUG -printf( "newn %.6e\n", k ); -#endif -return( pkm2 ); -} - - - -/* Ascending power series for Jv(x). - * AMS55 #9.1.10. - */ - -static float jvsf( float nn, float xx ) -{ -float n, x, t, u, y, z, k, ay; - -#if DEBUG -printf( "jvsf: " ); -#endif -n = nn; -x = xx; -z = -0.25 * x * x; -u = 1.0; -y = u; -k = 1.0; -t = 1.0; - -while( t > MACHEPF ) - { - u *= z / (k * (n+k)); - y += u; - k += 1.0; - t = fabsf(u); - if( (ay = fabsf(y)) > 1.0 ) - t /= ay; - } - -if( x < 0.0 ) - { - y = y * powf( 0.5 * x, n ) / gammaf( n + 1.0 ); - } -else - { - t = n * logf(0.5*x) - lgamf(n + 1.0); - if( t < -MAXLOGF ) - { - return( 0.0 ); - } - if( t > MAXLOGF ) - { - t = logf(y) + t; - if( t > MAXLOGF ) - { - mtherr( "jvf", OVERFLOW ); - return( MAXNUMF ); - } - else - { - y = sgngamf * expf(t); - return(y); - } - } - y = sgngamf * y * expf( t ); - } -#if DEBUG -printf( "y = %.8e\n", y ); -#endif -return(y); -} - -/* Hankel's asymptotic expansion - * for large x. - * AMS55 #9.2.5. - */ -static float hankelf( float nn, float xx ) -{ -float n, x, t, u, z, k, sign, conv; -float p, q, j, m, pp, qq; -int flag; - -#if DEBUG -printf( "hankelf: " ); -#endif -n = nn; -x = xx; -m = 4.0*n*n; -j = 1.0; -z = 8.0 * x; -k = 1.0; -p = 1.0; -u = (m - 1.0)/z; -q = u; -sign = 1.0; -conv = 1.0; -flag = 0; -t = 1.0; -pp = 1.0e38; -qq = 1.0e38; - -while( t > MACHEPF ) - { - k += 2.0; - j += 1.0; - sign = -sign; - u *= (m - k * k)/(j * z); - p += sign * u; - k += 2.0; - j += 1.0; - u *= (m - k * k)/(j * z); - q += sign * u; - t = fabsf(u/p); - if( t < conv ) - { - conv = t; - qq = q; - pp = p; - flag = 1; - } -/* stop if the terms start getting larger */ - if( (flag != 0) && (t > conv) ) - { -#if DEBUG - printf( "Hankel: convergence to %.4E\n", conv ); -#endif - goto hank1; - } - } - -hank1: -u = x - (0.5*n + 0.25) * PIF; -t = sqrtf( 2.0/(PIF*x) ) * ( pp * cosf(u) - qq * sinf(u) ); -return( t ); -} - - -/* Asymptotic expansion for large n. - * AMS55 #9.3.35. - */ - -static float lambda[] = { - 1.0, - 1.041666666666666666666667E-1, - 8.355034722222222222222222E-2, - 1.282265745563271604938272E-1, - 2.918490264641404642489712E-1, - 8.816272674437576524187671E-1, - 3.321408281862767544702647E+0, - 1.499576298686255465867237E+1, - 7.892301301158651813848139E+1, - 4.744515388682643231611949E+2, - 3.207490090890661934704328E+3 -}; -static float mu[] = { - 1.0, - -1.458333333333333333333333E-1, - -9.874131944444444444444444E-2, - -1.433120539158950617283951E-1, - -3.172272026784135480967078E-1, - -9.424291479571202491373028E-1, - -3.511203040826354261542798E+0, - -1.572726362036804512982712E+1, - -8.228143909718594444224656E+1, - -4.923553705236705240352022E+2, - -3.316218568547972508762102E+3 -}; -static float P1[] = { - -2.083333333333333333333333E-1, - 1.250000000000000000000000E-1 -}; -static float P2[] = { - 3.342013888888888888888889E-1, - -4.010416666666666666666667E-1, - 7.031250000000000000000000E-2 -}; -static float P3[] = { - -1.025812596450617283950617E+0, - 1.846462673611111111111111E+0, - -8.912109375000000000000000E-1, - 7.324218750000000000000000E-2 -}; -static float P4[] = { - 4.669584423426247427983539E+0, - -1.120700261622299382716049E+1, - 8.789123535156250000000000E+0, - -2.364086914062500000000000E+0, - 1.121520996093750000000000E-1 -}; -static float P5[] = { - -2.8212072558200244877E1, - 8.4636217674600734632E1, - -9.1818241543240017361E1, - 4.2534998745388454861E1, - -7.3687943594796316964E0, - 2.27108001708984375E-1 -}; -static float P6[] = { - 2.1257013003921712286E2, - -7.6525246814118164230E2, - 1.0599904525279998779E3, - -6.9957962737613254123E2, - 2.1819051174421159048E2, - -2.6491430486951555525E1, - 5.7250142097473144531E-1 -}; -static float P7[] = { - -1.9194576623184069963E3, - 8.0617221817373093845E3, - -1.3586550006434137439E4, - 1.1655393336864533248E4, - -5.3056469786134031084E3, - 1.2009029132163524628E3, - -1.0809091978839465550E2, - 1.7277275025844573975E0 -}; - - -static float jnxf( float nn, float xx ) -{ -float n, x, zeta, sqz, zz, zp, np; -float cbn, n23, t, z, sz; -float pp, qq, z32i, zzi; -float ak, bk, akl, bkl; -int sign, doa, dob, nflg, k, s, tk, tkp1, m; -static float u[8]; -static float ai, aip, bi, bip; - -n = nn; -x = xx; -/* Test for x very close to n. - * Use expansion for transition region if so. - */ -cbn = cbrtf(n); -z = (x - n)/cbn; -if( (fabsf(z) <= 0.7) || (n < 0.0) ) - return( jntf(n,x) ); -z = x/n; -zz = 1.0 - z*z; -if( zz == 0.0 ) - return(0.0); - -if( zz > 0.0 ) - { - sz = sqrtf( zz ); - t = 1.5 * (logf( (1.0+sz)/z ) - sz ); /* zeta ** 3/2 */ - zeta = cbrtf( t * t ); - nflg = 1; - } -else - { - sz = sqrtf(-zz); - t = 1.5 * (sz - acosf(1.0/z)); - zeta = -cbrtf( t * t ); - nflg = -1; - } -z32i = fabsf(1.0/t); -sqz = cbrtf(t); - -/* Airy function */ -n23 = cbrtf( n * n ); -t = n23 * zeta; - -#if DEBUG -printf("zeta %.5E, Airyf(%.5E)\n", zeta, t ); -#endif -airyf( t, &ai, &aip, &bi, &bip ); - -/* polynomials in expansion */ -u[0] = 1.0; -zzi = 1.0/zz; -u[1] = polevlf( zzi, P1, 1 )/sz; -u[2] = polevlf( zzi, P2, 2 )/zz; -u[3] = polevlf( zzi, P3, 3 )/(sz*zz); -pp = zz*zz; -u[4] = polevlf( zzi, P4, 4 )/pp; -u[5] = polevlf( zzi, P5, 5 )/(pp*sz); -pp *= zz; -u[6] = polevlf( zzi, P6, 6 )/pp; -u[7] = polevlf( zzi, P7, 7 )/(pp*sz); - -#if DEBUG -for( k=0; k<=7; k++ ) - printf( "u[%d] = %.5E\n", k, u[k] ); -#endif - -pp = 0.0; -qq = 0.0; -np = 1.0; -/* flags to stop when terms get larger */ -doa = 1; -dob = 1; -akl = MAXNUMF; -bkl = MAXNUMF; - -for( k=0; k<=3; k++ ) - { - tk = 2 * k; - tkp1 = tk + 1; - zp = 1.0; - ak = 0.0; - bk = 0.0; - for( s=0; s<=tk; s++ ) - { - if( doa ) - { - if( (s & 3) > 1 ) - sign = nflg; - else - sign = 1; - ak += sign * mu[s] * zp * u[tk-s]; - } - - if( dob ) - { - m = tkp1 - s; - if( ((m+1) & 3) > 1 ) - sign = nflg; - else - sign = 1; - bk += sign * lambda[s] * zp * u[m]; - } - zp *= z32i; - } - - if( doa ) - { - ak *= np; - t = fabsf(ak); - if( t < akl ) - { - akl = t; - pp += ak; - } - else - doa = 0; - } - - if( dob ) - { - bk += lambda[tkp1] * zp * u[0]; - bk *= -np/sqz; - t = fabsf(bk); - if( t < bkl ) - { - bkl = t; - qq += bk; - } - else - dob = 0; - } -#if DEBUG - printf("a[%d] %.5E, b[%d] %.5E\n", k, ak, k, bk ); -#endif - if( np < MACHEPF ) - break; - np /= n*n; - } - -/* normalizing factor ( 4*zeta/(1 - z**2) )**1/4 */ -t = 4.0 * zeta/zz; -t = sqrtf( sqrtf(t) ); - -t *= ai*pp/cbrtf(n) + aip*qq/(n23*n); -return(t); -} - -/* Asymptotic expansion for transition region, - * n large and x close to n. - * AMS55 #9.3.23. - */ - -static float PF2[] = { - -9.0000000000000000000e-2, - 8.5714285714285714286e-2 -}; -static float PF3[] = { - 1.3671428571428571429e-1, - -5.4920634920634920635e-2, - -4.4444444444444444444e-3 -}; -static float PF4[] = { - 1.3500000000000000000e-3, - -1.6036054421768707483e-1, - 4.2590187590187590188e-2, - 2.7330447330447330447e-3 -}; -static float PG1[] = { - -2.4285714285714285714e-1, - 1.4285714285714285714e-2 -}; -static float PG2[] = { - -9.0000000000000000000e-3, - 1.9396825396825396825e-1, - -1.1746031746031746032e-2 -}; -static float PG3[] = { - 1.9607142857142857143e-2, - -1.5983694083694083694e-1, - 6.3838383838383838384e-3 -}; - - -static float jntf( float nn, float xx ) -{ -float n, x, z, zz, z3; -float cbn, n23, cbtwo; -float ai, aip, bi, bip; /* Airy functions */ -float nk, fk, gk, pp, qq; -float F[5], G[4]; -int k; - -n = nn; -x = xx; -cbn = cbrtf(n); -z = (x - n)/cbn; -cbtwo = cbrtf( 2.0 ); - -/* Airy function */ -zz = -cbtwo * z; -airyf( zz, &ai, &aip, &bi, &bip ); - -/* polynomials in expansion */ -zz = z * z; -z3 = zz * z; -F[0] = 1.0; -F[1] = -z/5.0; -F[2] = polevlf( z3, PF2, 1 ) * zz; -F[3] = polevlf( z3, PF3, 2 ); -F[4] = polevlf( z3, PF4, 3 ) * z; -G[0] = 0.3 * zz; -G[1] = polevlf( z3, PG1, 1 ); -G[2] = polevlf( z3, PG2, 2 ) * z; -G[3] = polevlf( z3, PG3, 2 ) * zz; -#if DEBUG -for( k=0; k<=4; k++ ) - printf( "F[%d] = %.5E\n", k, F[k] ); -for( k=0; k<=3; k++ ) - printf( "G[%d] = %.5E\n", k, G[k] ); -#endif -pp = 0.0; -qq = 0.0; -nk = 1.0; -n23 = cbrtf( n * n ); - -for( k=0; k<=4; k++ ) - { - fk = F[k]*nk; - pp += fk; - if( k != 4 ) - { - gk = G[k]*nk; - qq += gk; - } -#if DEBUG - printf("fk[%d] %.5E, gk[%d] %.5E\n", k, fk, k, gk ); -#endif - nk /= n23; - } - -fk = cbtwo * ai * pp/cbn + cbrtf(4.0) * aip * qq/n; -return(fk); -} diff --git a/libm/float/k0f.c b/libm/float/k0f.c deleted file mode 100644 index e0e0698ac..000000000 --- a/libm/float/k0f.c +++ /dev/null @@ -1,175 +0,0 @@ -/* k0f.c - * - * Modified Bessel function, third kind, order zero - * - * - * - * SYNOPSIS: - * - * float x, y, k0f(); - * - * y = k0f( x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of the third kind - * of order zero of the argument. - * - * The range is partitioned into the two intervals [0,8] and - * (8, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Tested at 2000 random points between 0 and 8. Peak absolute - * error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15. - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 7.8e-7 8.5e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * K0 domain x <= 0 MAXNUM - * - */ - /* k0ef() - * - * Modified Bessel function, third kind, order zero, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * float x, y, k0ef(); - * - * y = k0ef( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of the third kind of order zero of the argument. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 8.1e-7 7.8e-8 - * See k0(). - * - */ - -/* -Cephes Math Library Release 2.0: April, 1987 -Copyright 1984, 1987 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -/* Chebyshev coefficients for K0(x) + log(x/2) I0(x) - * in the interval [0,2]. The odd order coefficients are all - * zero; only the even order coefficients are listed. - * - * lim(x->0){ K0(x) + log(x/2) I0(x) } = -EUL. - */ - -static float A[] = -{ - 1.90451637722020886025E-9f, - 2.53479107902614945675E-7f, - 2.28621210311945178607E-5f, - 1.26461541144692592338E-3f, - 3.59799365153615016266E-2f, - 3.44289899924628486886E-1f, --5.35327393233902768720E-1f -}; - - - -/* Chebyshev coefficients for exp(x) sqrt(x) K0(x) - * in the inverted interval [2,infinity]. - * - * lim(x->inf){ exp(x) sqrt(x) K0(x) } = sqrt(pi/2). - */ - -static float B[] = { --1.69753450938905987466E-9f, - 8.57403401741422608519E-9f, --4.66048989768794782956E-8f, - 2.76681363944501510342E-7f, --1.83175552271911948767E-6f, - 1.39498137188764993662E-5f, --1.28495495816278026384E-4f, - 1.56988388573005337491E-3f, --3.14481013119645005427E-2f, - 2.44030308206595545468E0f -}; - -/* k0.c */ - -extern float MAXNUMF; - -#ifdef ANSIC -float chbevlf(float, float *, int); -float expf(float), i0f(float), logf(float), sqrtf(float); -#else -float chbevlf(), expf(), i0f(), logf(), sqrtf(); -#endif - - -float k0f( float xx ) -{ -float x, y, z; - -x = xx; -if( x <= 0.0f ) - { - mtherr( "k0f", DOMAIN ); - return( MAXNUMF ); - } - -if( x <= 2.0f ) - { - y = x * x - 2.0f; - y = chbevlf( y, A, 7 ) - logf( 0.5f * x ) * i0f(x); - return( y ); - } -z = 8.0f/x - 2.0f; -y = expf(-x) * chbevlf( z, B, 10 ) / sqrtf(x); -return(y); -} - - - -float k0ef( float xx ) -{ -float x, y; - - -x = xx; -if( x <= 0.0f ) - { - mtherr( "k0ef", DOMAIN ); - return( MAXNUMF ); - } - -if( x <= 2.0f ) - { - y = x * x - 2.0f; - y = chbevlf( y, A, 7 ) - logf( 0.5f * x ) * i0f(x); - return( y * expf(x) ); - } - -y = chbevlf( 8.0f/x - 2.0f, B, 10 ) / sqrtf(x); -return(y); -} diff --git a/libm/float/k1f.c b/libm/float/k1f.c deleted file mode 100644 index d5b9bdfce..000000000 --- a/libm/float/k1f.c +++ /dev/null @@ -1,174 +0,0 @@ -/* k1f.c - * - * Modified Bessel function, third kind, order one - * - * - * - * SYNOPSIS: - * - * float x, y, k1f(); - * - * y = k1f( x ); - * - * - * - * DESCRIPTION: - * - * Computes the modified Bessel function of the third kind - * of order one of the argument. - * - * The range is partitioned into the two intervals [0,2] and - * (2, infinity). Chebyshev polynomial expansions are employed - * in each interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 4.6e-7 7.6e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * k1 domain x <= 0 MAXNUM - * - */ - /* k1ef.c - * - * Modified Bessel function, third kind, order one, - * exponentially scaled - * - * - * - * SYNOPSIS: - * - * float x, y, k1ef(); - * - * y = k1ef( x ); - * - * - * - * DESCRIPTION: - * - * Returns exponentially scaled modified Bessel function - * of the third kind of order one of the argument: - * - * k1e(x) = exp(x) * k1(x). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0, 30 30000 4.9e-7 6.7e-8 - * See k1(). - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -/* Chebyshev coefficients for x(K1(x) - log(x/2) I1(x)) - * in the interval [0,2]. - * - * lim(x->0){ x(K1(x) - log(x/2) I1(x)) } = 1. - */ - -#define MINNUMF 6.0e-39 -static float A[] = -{ --2.21338763073472585583E-8f, --2.43340614156596823496E-6f, --1.73028895751305206302E-4f, --6.97572385963986435018E-3f, --1.22611180822657148235E-1f, --3.53155960776544875667E-1f, - 1.52530022733894777053E0f -}; - - - - -/* Chebyshev coefficients for exp(x) sqrt(x) K1(x) - * in the interval [2,infinity]. - * - * lim(x->inf){ exp(x) sqrt(x) K1(x) } = sqrt(pi/2). - */ - -static float B[] = -{ - 2.01504975519703286596E-9f, --1.03457624656780970260E-8f, - 5.74108412545004946722E-8f, --3.50196060308781257119E-7f, - 2.40648494783721712015E-6f, --1.93619797416608296024E-5f, - 1.95215518471351631108E-4f, --2.85781685962277938680E-3f, - 1.03923736576817238437E-1f, - 2.72062619048444266945E0f -}; - - - -extern float MAXNUMF; -#ifdef ANSIC -float chbevlf(float, float *, int); -float expf(float), i1f(float), logf(float), sqrtf(float); -#else -float chbevlf(), expf(), i1f(), logf(), sqrtf(); -#endif - -float k1f(float xx) -{ -float x, y; - -x = xx; -if( x <= MINNUMF ) - { - mtherr( "k1f", DOMAIN ); - return( MAXNUMF ); - } - -if( x <= 2.0f ) - { - y = x * x - 2.0f; - y = logf( 0.5f * x ) * i1f(x) + chbevlf( y, A, 7 ) / x; - return( y ); - } - -return( expf(-x) * chbevlf( 8.0f/x - 2.0f, B, 10 ) / sqrtf(x) ); - -} - - - -float k1ef( float xx ) -{ -float x, y; - -x = xx; -if( x <= 0.0f ) - { - mtherr( "k1ef", DOMAIN ); - return( MAXNUMF ); - } - -if( x <= 2.0f ) - { - y = x * x - 2.0f; - y = logf( 0.5f * x ) * i1f(x) + chbevlf( y, A, 7 ) / x; - return( y * expf(x) ); - } - -return( chbevlf( 8.0f/x - 2.0f, B, 10 ) / sqrtf(x) ); - -} diff --git a/libm/float/knf.c b/libm/float/knf.c deleted file mode 100644 index 85e297390..000000000 --- a/libm/float/knf.c +++ /dev/null @@ -1,252 +0,0 @@ -/* knf.c - * - * Modified Bessel function, third kind, integer order - * - * - * - * SYNOPSIS: - * - * float x, y, knf(); - * int n; - * - * y = knf( n, x ); - * - * - * - * DESCRIPTION: - * - * Returns modified Bessel function of the third kind - * of order n of the argument. - * - * The range is partitioned into the two intervals [0,9.55] and - * (9.55, infinity). An ascending power series is used in the - * low range, and an asymptotic expansion in the high range. - * - * - * - * ACCURACY: - * - * Absolute error, relative when function > 1: - * arithmetic domain # trials peak rms - * IEEE 0,30 10000 2.0e-4 3.8e-6 - * - * Error is high only near the crossover point x = 9.55 - * between the two expansions used. - */ - - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 - -*/ - - -/* -Algorithm for Kn. - n-1 - -n - (n-k-1)! 2 k -K (x) = 0.5 (x/2) > -------- (-x /4) - n - k! - k=0 - - inf. 2 k - n n - (x /4) - + (-1) 0.5(x/2) > {p(k+1) + p(n+k+1) - 2log(x/2)} --------- - - k! (n+k)! - k=0 - -where p(m) is the psi function: p(1) = -EUL and - - m-1 - - - p(m) = -EUL + > 1/k - - - k=1 - -For large x, - 2 2 2 - u-1 (u-1 )(u-3 ) -K (z) = sqrt(pi/2z) exp(-z) { 1 + ------- + ------------ + ...} - v 1 2 - 1! (8z) 2! (8z) -asymptotically, where - - 2 - u = 4 v . - -*/ - -#include <math.h> - -#define EUL 5.772156649015328606065e-1 -#define MAXFAC 31 -extern float MACHEPF, MAXNUMF, MAXLOGF, PIF; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -float expf(float), logf(float), sqrtf(float); - -float knf( int nnn, float xx ) -{ -float x, k, kf, nk1f, nkf, zn, t, s, z0, z; -float ans, fn, pn, pk, zmn, tlg, tox; -int i, n, nn; - -nn = nnn; -x = xx; -if( nn < 0 ) - n = -nn; -else - n = nn; - -if( n > MAXFAC ) - { -overf: - mtherr( "knf", OVERFLOW ); - return( MAXNUMF ); - } - -if( x <= 0.0 ) - { - if( x < 0.0 ) - mtherr( "knf", DOMAIN ); - else - mtherr( "knf", SING ); - return( MAXNUMF ); - } - - -if( x > 9.55 ) - goto asymp; - -ans = 0.0; -z0 = 0.25 * x * x; -fn = 1.0; -pn = 0.0; -zmn = 1.0; -tox = 2.0/x; - -if( n > 0 ) - { - /* compute factorial of n and psi(n) */ - pn = -EUL; - k = 1.0; - for( i=1; i<n; i++ ) - { - pn += 1.0/k; - k += 1.0; - fn *= k; - } - - zmn = tox; - - if( n == 1 ) - { - ans = 1.0/x; - } - else - { - nk1f = fn/n; - kf = 1.0; - s = nk1f; - z = -z0; - zn = 1.0; - for( i=1; i<n; i++ ) - { - nk1f = nk1f/(n-i); - kf = kf * i; - zn *= z; - t = nk1f * zn / kf; - s += t; - if( (MAXNUMF - fabsf(t)) < fabsf(s) ) - goto overf; - if( (tox > 1.0) && ((MAXNUMF/tox) < zmn) ) - goto overf; - zmn *= tox; - } - s *= 0.5; - t = fabsf(s); - if( (zmn > 1.0) && ((MAXNUMF/zmn) < t) ) - goto overf; - if( (t > 1.0) && ((MAXNUMF/t) < zmn) ) - goto overf; - ans = s * zmn; - } - } - - -tlg = 2.0 * logf( 0.5 * x ); -pk = -EUL; -if( n == 0 ) - { - pn = pk; - t = 1.0; - } -else - { - pn = pn + 1.0/n; - t = 1.0/fn; - } -s = (pk+pn-tlg)*t; -k = 1.0; -do - { - t *= z0 / (k * (k+n)); - pk += 1.0/k; - pn += 1.0/(k+n); - s += (pk+pn-tlg)*t; - k += 1.0; - } -while( fabsf(t/s) > MACHEPF ); - -s = 0.5 * s / zmn; -if( n & 1 ) - s = -s; -ans += s; - -return(ans); - - - -/* Asymptotic expansion for Kn(x) */ -/* Converges to 1.4e-17 for x > 18.4 */ - -asymp: - -if( x > MAXLOGF ) - { - mtherr( "knf", UNDERFLOW ); - return(0.0); - } -k = n; -pn = 4.0 * k * k; -pk = 1.0; -z0 = 8.0 * x; -fn = 1.0; -t = 1.0; -s = t; -nkf = MAXNUMF; -i = 0; -do - { - z = pn - pk * pk; - t = t * z /(fn * z0); - nk1f = fabsf(t); - if( (i >= n) && (nk1f > nkf) ) - { - goto adone; - } - nkf = nk1f; - s += t; - fn += 1.0; - pk += 2.0; - i += 1; - } -while( fabsf(t/s) > MACHEPF ); - -adone: -ans = expf(-x) * sqrtf( PIF/(2.0*x) ) * s; -return(ans); -} diff --git a/libm/float/log10f.c b/libm/float/log10f.c deleted file mode 100644 index 6cb2e4d87..000000000 --- a/libm/float/log10f.c +++ /dev/null @@ -1,129 +0,0 @@ -/* log10f.c - * - * Common logarithm - * - * - * - * SYNOPSIS: - * - * float x, y, log10f(); - * - * y = log10f( x ); - * - * - * - * DESCRIPTION: - * - * Returns logarithm to the base 10 of x. - * - * The argument is separated into its exponent and fractional - * parts. The logarithm of the fraction is approximated by - * - * log(1+x) = x - 0.5 x**2 + x**3 P(x). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0.5, 2.0 100000 1.3e-7 3.4e-8 - * IEEE 0, MAXNUMF 100000 1.3e-7 2.6e-8 - * - * In the tests over the interval [0, MAXNUM], the logarithms - * of the random arguments were uniformly distributed over - * [-MAXL10, MAXL10]. - * - * ERROR MESSAGES: - * - * log10f singularity: x = 0; returns -MAXL10 - * log10f domain: x < 0; returns -MAXL10 - * MAXL10 = 38.230809449325611792 - */ - -/* -Cephes Math Library Release 2.1: December, 1988 -Copyright 1984, 1987, 1988 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> -static char fname[] = {"log10"}; - -/* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x)/Q(x) - * 1/sqrt(2) <= x < sqrt(2) - */ -static float P[] = { - 7.0376836292E-2, --1.1514610310E-1, - 1.1676998740E-1, --1.2420140846E-1, - 1.4249322787E-1, --1.6668057665E-1, - 2.0000714765E-1, --2.4999993993E-1, - 3.3333331174E-1 -}; - - -#define SQRTH 0.70710678118654752440 -#define L102A 3.0078125E-1 -#define L102B 2.48745663981195213739E-4 -#define L10EA 4.3359375E-1 -#define L10EB 7.00731903251827651129E-4 - -static float MAXL10 = 38.230809449325611792; - -float frexpf(float, int *), polevlf(float, float *, int); - -float log10f(float xx) -{ -float x, y, z; -int e; - -x = xx; -/* Test for domain */ -if( x <= 0.0 ) - { - if( x == 0.0 ) - mtherr( fname, SING ); - else - mtherr( fname, DOMAIN ); - return( -MAXL10 ); - } - -/* separate mantissa from exponent */ - -x = frexpf( x, &e ); - -/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x) */ - -if( x < SQRTH ) - { - e -= 1; - x = 2.0*x - 1.0; - } -else - { - x = x - 1.0; - } - - -/* rational form */ -z = x*x; -y = x * ( z * polevlf( x, P, 8 ) ); -y = y - 0.5 * z; /* y - 0.5 * x**2 */ - -/* multiply log of fraction by log10(e) - * and base 2 exponent by log10(2) - */ -z = (x + y) * L10EB; /* accumulate terms in order of size */ -z += y * L10EA; -z += x * L10EA; -x = e; -z += x * L102B; -z += x * L102A; - - -return( z ); -} diff --git a/libm/float/log2f.c b/libm/float/log2f.c deleted file mode 100644 index 5cd5f4838..000000000 --- a/libm/float/log2f.c +++ /dev/null @@ -1,129 +0,0 @@ -/* log2f.c - * - * Base 2 logarithm - * - * - * - * SYNOPSIS: - * - * float x, y, log2f(); - * - * y = log2f( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base 2 logarithm of x. - * - * The argument is separated into its exponent and fractional - * parts. If the exponent is between -1 and +1, the base e - * logarithm of the fraction is approximated by - * - * log(1+x) = x - 0.5 x**2 + x**3 P(x)/Q(x). - * - * Otherwise, setting z = 2(x-1)/x+1), - * - * log(x) = z + z**3 P(z)/Q(z). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE exp(+-88) 100000 1.1e-7 2.4e-8 - * IEEE 0.5, 2.0 100000 1.1e-7 3.0e-8 - * - * In the tests over the interval [exp(+-88)], the logarithms - * of the random arguments were uniformly distributed. - * - * ERROR MESSAGES: - * - * log singularity: x = 0; returns MINLOGF/log(2) - * log domain: x < 0; returns MINLOGF/log(2) - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> -static char fname[] = {"log2"}; - -/* Coefficients for log(1+x) = x - x**2/2 + x**3 P(x) - * 1/sqrt(2) <= x < sqrt(2) - */ - -static float P[] = { - 7.0376836292E-2, --1.1514610310E-1, - 1.1676998740E-1, --1.2420140846E-1, - 1.4249322787E-1, --1.6668057665E-1, - 2.0000714765E-1, --2.4999993993E-1, - 3.3333331174E-1 -}; - -#define LOG2EA 0.44269504088896340735992 -#define SQRTH 0.70710678118654752440 -extern float MINLOGF, LOGE2F; - -float frexpf(float, int *), polevlf(float, float *, int); - -float log2f(float xx) -{ -float x, y, z; -int e; - -x = xx; -/* Test for domain */ -if( x <= 0.0 ) - { - if( x == 0.0 ) - mtherr( fname, SING ); - else - mtherr( fname, DOMAIN ); - return( MINLOGF/LOGE2F ); - } - -/* separate mantissa from exponent */ -x = frexpf( x, &e ); - - -/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */ - -if( x < SQRTH ) - { - e -= 1; - x = 2.0*x - 1.0; - } -else - { - x = x - 1.0; - } - -z = x*x; -y = x * ( z * polevlf( x, P, 8 ) ); -y = y - 0.5 * z; /* y - 0.5 * x**2 */ - - -/* Multiply log of fraction by log2(e) - * and base 2 exponent by 1 - * - * ***CAUTION*** - * - * This sequence of operations is critical and it may - * be horribly defeated by some compiler optimizers. - */ -z = y * LOG2EA; -z += x * LOG2EA; -z += y; -z += x; -z += (float )e; -return( z ); -} diff --git a/libm/float/logf.c b/libm/float/logf.c deleted file mode 100644 index 750138564..000000000 --- a/libm/float/logf.c +++ /dev/null @@ -1,128 +0,0 @@ -/* logf.c - * - * Natural logarithm - * - * - * - * SYNOPSIS: - * - * float x, y, logf(); - * - * y = logf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the base e (2.718...) logarithm of x. - * - * The argument is separated into its exponent and fractional - * parts. If the exponent is between -1 and +1, the logarithm - * of the fraction is approximated by - * - * log(1+x) = x - 0.5 x**2 + x**3 P(x) - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0.5, 2.0 100000 7.6e-8 2.7e-8 - * IEEE 1, MAXNUMF 100000 2.6e-8 - * - * In the tests over the interval [1, MAXNUM], the logarithms - * of the random arguments were uniformly distributed over - * [0, MAXLOGF]. - * - * ERROR MESSAGES: - * - * logf singularity: x = 0; returns MINLOG - * logf domain: x < 0; returns MINLOG - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision natural logarithm - * test interval: [sqrt(2)/2, sqrt(2)] - * trials: 10000 - * peak relative error: 7.1e-8 - * rms relative error: 2.7e-8 - */ - -#include <math.h> -extern float MINLOGF, SQRTHF; - - -float frexpf( float, int * ); - -float logf( float xx ) -{ -register float y; -float x, z, fe; -int e; - -x = xx; -fe = 0.0; -/* Test for domain */ -if( x <= 0.0 ) - { - if( x == 0.0 ) - mtherr( "logf", SING ); - else - mtherr( "logf", DOMAIN ); - return( MINLOGF ); - } - -x = frexpf( x, &e ); -if( x < SQRTHF ) - { - e -= 1; - x = x + x - 1.0; /* 2x - 1 */ - } -else - { - x = x - 1.0; - } -z = x * x; -/* 3.4e-9 */ -/* -p = logfcof; -y = *p++ * x; -for( i=0; i<8; i++ ) - { - y += *p++; - y *= x; - } -y *= z; -*/ - -y = -(((((((( 7.0376836292E-2 * x -- 1.1514610310E-1) * x -+ 1.1676998740E-1) * x -- 1.2420140846E-1) * x -+ 1.4249322787E-1) * x -- 1.6668057665E-1) * x -+ 2.0000714765E-1) * x -- 2.4999993993E-1) * x -+ 3.3333331174E-1) * x * z; - -if( e ) - { - fe = e; - y += -2.12194440e-4 * fe; - } - -y += -0.5 * z; /* y - 0.5 x^2 */ -z = x + y; /* ... + x */ - -if( e ) - z += 0.693359375 * fe; - -return( z ); -} diff --git a/libm/float/mtherr.c b/libm/float/mtherr.c deleted file mode 100644 index d67dc042e..000000000 --- a/libm/float/mtherr.c +++ /dev/null @@ -1,99 +0,0 @@ -/* mtherr.c - * - * Library common error handling routine - * - * - * - * SYNOPSIS: - * - * char *fctnam; - * int code; - * void mtherr(); - * - * mtherr( fctnam, code ); - * - * - * - * DESCRIPTION: - * - * This routine may be called to report one of the following - * error conditions (in the include file math.h). - * - * Mnemonic Value Significance - * - * DOMAIN 1 argument domain error - * SING 2 function singularity - * OVERFLOW 3 overflow range error - * UNDERFLOW 4 underflow range error - * TLOSS 5 total loss of precision - * PLOSS 6 partial loss of precision - * EDOM 33 Unix domain error code - * ERANGE 34 Unix range error code - * - * The default version of the file prints the function name, - * passed to it by the pointer fctnam, followed by the - * error condition. The display is directed to the standard - * output device. The routine then returns to the calling - * program. Users may wish to modify the program to abort by - * calling exit() under severe error conditions such as domain - * errors. - * - * Since all error conditions pass control to this function, - * the display may be easily changed, eliminated, or directed - * to an error logging device. - * - * SEE ALSO: - * - * math.h - * - */ - -/* -Cephes Math Library Release 2.0: April, 1987 -Copyright 1984, 1987 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -/* Notice: the order of appearance of the following - * messages is bound to the error codes defined - * in math.h. - */ -static char *ermsg[7] = { -"unknown", /* error code 0 */ -"domain", /* error code 1 */ -"singularity", /* et seq. */ -"overflow", -"underflow", -"total loss of precision", -"partial loss of precision" -}; - - -void printf(); - -int mtherr( name, code ) -char *name; -int code; -{ - -/* Display string passed by calling program, - * which is supposed to be the name of the - * function in which the error occurred: - */ -printf( "\n%s ", name ); - /* exit(2); */ - -/* Display error message defined - * by the code argument. - */ -if( (code <= 0) || (code >= 6) ) - code = 0; -printf( "%s error\n", ermsg[code] ); - -/* Return to calling - * program - */ -return 0; -} diff --git a/libm/float/nantst.c b/libm/float/nantst.c deleted file mode 100644 index 7edd992ae..000000000 --- a/libm/float/nantst.c +++ /dev/null @@ -1,54 +0,0 @@ -float inf = 1.0f/0.0f; -float nnn = 1.0f/0.0f - 1.0f/0.0f; -float fin = 1.0f; -float neg = -1.0f; -float nn2; - -int isnanf(), isfinitef(), signbitf(); - -void pvalue (char *str, float x) -{ -union - { - float f; - unsigned int i; - }u; - -printf("%s ", str); -u.f = x; -printf("%08x\n", u.i); -} - - -int -main() -{ - -if (!isnanf(nnn)) - abort(); -pvalue("nnn", nnn); -pvalue("inf", inf); -nn2 = inf - inf; -pvalue("inf - inf", nn2); -if (isnanf(fin)) - abort(); -if (isnanf(inf)) - abort(); -if (!isfinitef(fin)) - abort(); -if (isfinitef(nnn)) - abort(); -if (isfinitef(inf)) - abort(); -if (!signbitf(neg)) - abort(); -if (signbitf(fin)) - abort(); -if (signbitf(inf)) - abort(); -/* -if (signbitf(nnn)) - abort(); - */ -exit (0); -} diff --git a/libm/float/nbdtrf.c b/libm/float/nbdtrf.c deleted file mode 100644 index e9b02753b..000000000 --- a/libm/float/nbdtrf.c +++ /dev/null @@ -1,141 +0,0 @@ -/* nbdtrf.c - * - * Negative binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, nbdtrf(); - * - * y = nbdtrf( k, n, p ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms 0 through k of the negative - * binomial distribution: - * - * k - * -- ( n+j-1 ) n j - * > ( ) p (1-p) - * -- ( j ) - * j=0 - * - * In a sequence of Bernoulli trials, this is the probability - * that k or fewer failures precede the nth success. - * - * The terms are not computed individually; instead the incomplete - * beta integral is employed, according to the formula - * - * y = nbdtr( k, n, p ) = incbet( n, k+1, p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 1.5e-4 1.9e-5 - * - */ - /* nbdtrcf.c - * - * Complemented negative binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, nbdtrcf(); - * - * y = nbdtrcf( k, n, p ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms k+1 to infinity of the negative - * binomial distribution: - * - * inf - * -- ( n+j-1 ) n j - * > ( ) p (1-p) - * -- ( j ) - * j=k+1 - * - * The terms are not computed individually; instead the incomplete - * beta integral is employed, according to the formula - * - * y = nbdtrc( k, n, p ) = incbet( k+1, n, 1-p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 1.4e-4 2.0e-5 - * - */ - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -#ifdef ANSIC -float incbetf(float, float, float); -#else -float incbetf(); -#endif - - -float nbdtrcf( int k, int n, float pp ) -{ -float dk, dn, p; - -p = pp; -if( (p < 0.0) || (p > 1.0) ) - goto domerr; -if( k < 0 ) - { -domerr: - mtherr( "nbdtrf", DOMAIN ); - return( 0.0 ); - } - -dk = k+1; -dn = n; -return( incbetf( dk, dn, 1.0 - p ) ); -} - - - -float nbdtrf( int k, int n, float pp ) -{ -float dk, dn, p; - -p = pp; -if( (p < 0.0) || (p > 1.0) ) - goto domerr; -if( k < 0 ) - { -domerr: - mtherr( "nbdtrf", DOMAIN ); - return( 0.0 ); - } -dk = k+1; -dn = n; -return( incbetf( dn, dk, p ) ); -} diff --git a/libm/float/ndtrf.c b/libm/float/ndtrf.c deleted file mode 100644 index c08d69eca..000000000 --- a/libm/float/ndtrf.c +++ /dev/null @@ -1,281 +0,0 @@ -/* ndtrf.c - * - * Normal distribution function - * - * - * - * SYNOPSIS: - * - * float x, y, ndtrf(); - * - * y = ndtrf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the area under the Gaussian probability density - * function, integrated from minus infinity to x: - * - * x - * - - * 1 | | 2 - * ndtr(x) = --------- | exp( - t /2 ) dt - * sqrt(2pi) | | - * - - * -inf. - * - * = ( 1 + erf(z) ) / 2 - * = erfc(z) / 2 - * - * where z = x/sqrt(2). Computation is via the functions - * erf and erfc. - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -13,0 50000 1.5e-5 2.6e-6 - * - * - * ERROR MESSAGES: - * - * See erfcf(). - * - */ - /* erff.c - * - * Error function - * - * - * - * SYNOPSIS: - * - * float x, y, erff(); - * - * y = erff( x ); - * - * - * - * DESCRIPTION: - * - * The integral is - * - * x - * - - * 2 | | 2 - * erf(x) = -------- | exp( - t ) dt. - * sqrt(pi) | | - * - - * 0 - * - * The magnitude of x is limited to 9.231948545 for DEC - * arithmetic; 1 or -1 is returned outside this range. - * - * For 0 <= |x| < 1, erf(x) = x * P(x**2); otherwise - * erf(x) = 1 - erfc(x). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -9.3,9.3 50000 1.7e-7 2.8e-8 - * - */ - /* erfcf.c - * - * Complementary error function - * - * - * - * SYNOPSIS: - * - * float x, y, erfcf(); - * - * y = erfcf( x ); - * - * - * - * DESCRIPTION: - * - * - * 1 - erf(x) = - * - * inf. - * - - * 2 | | 2 - * erfc(x) = -------- | exp( - t ) dt - * sqrt(pi) | | - * - - * x - * - * - * For small x, erfc(x) = 1 - erf(x); otherwise polynomial - * approximations 1/x P(1/x**2) are computed. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -9.3,9.3 50000 3.9e-6 7.2e-7 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * erfcf underflow x**2 > MAXLOGF 0.0 - * - * - */ - - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1988 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> - -extern float MAXLOGF, SQRTHF; - - -/* erfc(x) = exp(-x^2) P(1/x), 1 < x < 2 */ -static float P[] = { - 2.326819970068386E-002, --1.387039388740657E-001, - 3.687424674597105E-001, --5.824733027278666E-001, - 6.210004621745983E-001, --4.944515323274145E-001, - 3.404879937665872E-001, --2.741127028184656E-001, - 5.638259427386472E-001 -}; - -/* erfc(x) = exp(-x^2) 1/x P(1/x^2), 2 < x < 14 */ -static float R[] = { --1.047766399936249E+001, - 1.297719955372516E+001, --7.495518717768503E+000, - 2.921019019210786E+000, --1.015265279202700E+000, - 4.218463358204948E-001, --2.820767439740514E-001, - 5.641895067754075E-001 -}; - -/* erf(x) = x P(x^2), 0 < x < 1 */ -static float T[] = { - 7.853861353153693E-005, --8.010193625184903E-004, - 5.188327685732524E-003, --2.685381193529856E-002, - 1.128358514861418E-001, --3.761262582423300E-001, - 1.128379165726710E+000 -}; - -/*#define UTHRESH 37.519379347*/ - -#define UTHRESH 14.0 - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float polevlf(float, float *, int); -float expf(float), logf(float), erff(float), erfcf(float); -#else -float polevlf(), expf(), logf(), erff(), erfcf(); -#endif - - - -float ndtrf(float aa) -{ -float x, y, z; - -x = aa; -x *= SQRTHF; -z = fabsf(x); - -if( z < SQRTHF ) - y = 0.5 + 0.5 * erff(x); -else - { - y = 0.5 * erfcf(z); - - if( x > 0 ) - y = 1.0 - y; - } - -return(y); -} - - -float erfcf(float aa) -{ -float a, p,q,x,y,z; - - -a = aa; -x = fabsf(a); - -if( x < 1.0 ) - return( 1.0 - erff(a) ); - -z = -a * a; - -if( z < -MAXLOGF ) - { -under: - mtherr( "erfcf", UNDERFLOW ); - if( a < 0 ) - return( 2.0 ); - else - return( 0.0 ); - } - -z = expf(z); -q = 1.0/x; -y = q * q; -if( x < 2.0 ) - { - p = polevlf( y, P, 8 ); - } -else - { - p = polevlf( y, R, 7 ); - } - -y = z * q * p; - -if( a < 0 ) - y = 2.0 - y; - -if( y == 0.0 ) - goto under; - -return(y); -} - - -float erff(float xx) -{ -float x, y, z; - -x = xx; -if( fabsf(x) > 1.0 ) - return( 1.0 - erfcf(x) ); - -z = x * x; -y = x * polevlf( z, T, 6 ); -return( y ); - -} diff --git a/libm/float/ndtrif.c b/libm/float/ndtrif.c deleted file mode 100644 index 3e33bc2c5..000000000 --- a/libm/float/ndtrif.c +++ /dev/null @@ -1,186 +0,0 @@ -/* ndtrif.c - * - * Inverse of Normal distribution function - * - * - * - * SYNOPSIS: - * - * float x, y, ndtrif(); - * - * x = ndtrif( y ); - * - * - * - * DESCRIPTION: - * - * Returns the argument, x, for which the area under the - * Gaussian probability density function (integrated from - * minus infinity to x) is equal to y. - * - * - * For small arguments 0 < y < exp(-2), the program computes - * z = sqrt( -2.0 * log(y) ); then the approximation is - * x = z - log(z)/z - (1/z) P(1/z) / Q(1/z). - * There are two rational functions P/Q, one for 0 < y < exp(-32) - * and the other for y up to exp(-2). For larger arguments, - * w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 1e-38, 1 30000 3.6e-7 5.0e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * ndtrif domain x <= 0 -MAXNUM - * ndtrif domain x >= 1 MAXNUM - * - */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> -extern float MAXNUMF; - -/* sqrt(2pi) */ -static float s2pi = 2.50662827463100050242; - -/* approximation for 0 <= |y - 0.5| <= 3/8 */ -static float P0[5] = { --5.99633501014107895267E1, - 9.80010754185999661536E1, --5.66762857469070293439E1, - 1.39312609387279679503E1, --1.23916583867381258016E0, -}; -static float Q0[8] = { -/* 1.00000000000000000000E0,*/ - 1.95448858338141759834E0, - 4.67627912898881538453E0, - 8.63602421390890590575E1, --2.25462687854119370527E2, - 2.00260212380060660359E2, --8.20372256168333339912E1, - 1.59056225126211695515E1, --1.18331621121330003142E0, -}; - -/* Approximation for interval z = sqrt(-2 log y ) between 2 and 8 - * i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14. - */ -static float P1[9] = { - 4.05544892305962419923E0, - 3.15251094599893866154E1, - 5.71628192246421288162E1, - 4.40805073893200834700E1, - 1.46849561928858024014E1, - 2.18663306850790267539E0, --1.40256079171354495875E-1, --3.50424626827848203418E-2, --8.57456785154685413611E-4, -}; -static float Q1[8] = { -/* 1.00000000000000000000E0,*/ - 1.57799883256466749731E1, - 4.53907635128879210584E1, - 4.13172038254672030440E1, - 1.50425385692907503408E1, - 2.50464946208309415979E0, --1.42182922854787788574E-1, --3.80806407691578277194E-2, --9.33259480895457427372E-4, -}; - - -/* Approximation for interval z = sqrt(-2 log y ) between 8 and 64 - * i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890. - */ - -static float P2[9] = { - 3.23774891776946035970E0, - 6.91522889068984211695E0, - 3.93881025292474443415E0, - 1.33303460815807542389E0, - 2.01485389549179081538E-1, - 1.23716634817820021358E-2, - 3.01581553508235416007E-4, - 2.65806974686737550832E-6, - 6.23974539184983293730E-9, -}; -static float Q2[8] = { -/* 1.00000000000000000000E0,*/ - 6.02427039364742014255E0, - 3.67983563856160859403E0, - 1.37702099489081330271E0, - 2.16236993594496635890E-1, - 1.34204006088543189037E-2, - 3.28014464682127739104E-4, - 2.89247864745380683936E-6, - 6.79019408009981274425E-9, -}; - -#ifdef ANSIC -float polevlf(float, float *, int); -float p1evlf(float, float *, int); -float logf(float), sqrtf(float); -#else -float polevlf(), p1evlf(), logf(), sqrtf(); -#endif - - -float ndtrif(float yy0) -{ -float y0, x, y, z, y2, x0, x1; -int code; - -y0 = yy0; -if( y0 <= 0.0 ) - { - mtherr( "ndtrif", DOMAIN ); - return( -MAXNUMF ); - } -if( y0 >= 1.0 ) - { - mtherr( "ndtrif", DOMAIN ); - return( MAXNUMF ); - } -code = 1; -y = y0; -if( y > (1.0 - 0.13533528323661269189) ) /* 0.135... = exp(-2) */ - { - y = 1.0 - y; - code = 0; - } - -if( y > 0.13533528323661269189 ) - { - y = y - 0.5; - y2 = y * y; - x = y + y * (y2 * polevlf( y2, P0, 4)/p1evlf( y2, Q0, 8 )); - x = x * s2pi; - return(x); - } - -x = sqrtf( -2.0 * logf(y) ); -x0 = x - logf(x)/x; - -z = 1.0/x; -if( x < 8.0 ) /* y > exp(-32) = 1.2664165549e-14 */ - x1 = z * polevlf( z, P1, 8 )/p1evlf( z, Q1, 8 ); -else - x1 = z * polevlf( z, P2, 8 )/p1evlf( z, Q2, 8 ); -x = x0 - x1; -if( code != 0 ) - x = -x; -return( x ); -} diff --git a/libm/float/pdtrf.c b/libm/float/pdtrf.c deleted file mode 100644 index 17a05ee13..000000000 --- a/libm/float/pdtrf.c +++ /dev/null @@ -1,188 +0,0 @@ -/* pdtrf.c - * - * Poisson distribution - * - * - * - * SYNOPSIS: - * - * int k; - * float m, y, pdtrf(); - * - * y = pdtrf( k, m ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the first k terms of the Poisson - * distribution: - * - * k j - * -- -m m - * > e -- - * -- j! - * j=0 - * - * The terms are not summed directly; instead the incomplete - * gamma integral is employed, according to the relation - * - * y = pdtr( k, m ) = igamc( k+1, m ). - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 6.9e-5 8.0e-6 - * - */ - /* pdtrcf() - * - * Complemented poisson distribution - * - * - * - * SYNOPSIS: - * - * int k; - * float m, y, pdtrcf(); - * - * y = pdtrcf( k, m ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms k+1 to infinity of the Poisson - * distribution: - * - * inf. j - * -- -m m - * > e -- - * -- j! - * j=k+1 - * - * The terms are not summed directly; instead the incomplete - * gamma integral is employed, according to the formula - * - * y = pdtrc( k, m ) = igam( k+1, m ). - * - * The arguments must both be positive. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 8.4e-5 1.2e-5 - * - */ - /* pdtrif() - * - * Inverse Poisson distribution - * - * - * - * SYNOPSIS: - * - * int k; - * float m, y, pdtrf(); - * - * m = pdtrif( k, y ); - * - * - * - * - * DESCRIPTION: - * - * Finds the Poisson variable x such that the integral - * from 0 to x of the Poisson density is equal to the - * given probability y. - * - * This is accomplished using the inverse gamma integral - * function and the relation - * - * m = igami( k+1, y ). - * - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,100 5000 8.7e-6 1.4e-6 - * - * ERROR MESSAGES: - * - * message condition value returned - * pdtri domain y < 0 or y >= 1 0.0 - * k < 0 - * - */ - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -#ifdef ANSIC -float igamf(float, float), igamcf(float, float), igamif(float, float); -#else -float igamf(), igamcf(), igamif(); -#endif - - -float pdtrcf( int k, float mm ) -{ -float v, m; - -m = mm; -if( (k < 0) || (m <= 0.0) ) - { - mtherr( "pdtrcf", DOMAIN ); - return( 0.0 ); - } -v = k+1; -return( igamf( v, m ) ); -} - - - -float pdtrf( int k, float mm ) -{ -float v, m; - -m = mm; -if( (k < 0) || (m <= 0.0) ) - { - mtherr( "pdtr", DOMAIN ); - return( 0.0 ); - } -v = k+1; -return( igamcf( v, m ) ); -} - - -float pdtrif( int k, float yy ) -{ -float v, y; - -y = yy; -if( (k < 0) || (y < 0.0) || (y >= 1.0) ) - { - mtherr( "pdtrif", DOMAIN ); - return( 0.0 ); - } -v = k+1; -v = igamif( v, y ); -return( v ); -} diff --git a/libm/float/polevlf.c b/libm/float/polevlf.c deleted file mode 100644 index 7d7b4d0b7..000000000 --- a/libm/float/polevlf.c +++ /dev/null @@ -1,99 +0,0 @@ -/* polevlf.c - * p1evlf.c - * - * Evaluate polynomial - * - * - * - * SYNOPSIS: - * - * int N; - * float x, y, coef[N+1], polevlf[]; - * - * y = polevlf( x, coef, N ); - * - * - * - * DESCRIPTION: - * - * Evaluates polynomial of degree N: - * - * 2 N - * y = C + C x + C x +...+ C x - * 0 1 2 N - * - * Coefficients are stored in reverse order: - * - * coef[0] = C , ..., coef[N] = C . - * N 0 - * - * The function p1evl() assumes that coef[N] = 1.0 and is - * omitted from the array. Its calling arguments are - * otherwise the same as polevl(). - * - * - * SPEED: - * - * In the interest of speed, there are no checks for out - * of bounds arithmetic. This routine is used by most of - * the functions in the library. Depending on available - * equipment features, the user may wish to rewrite the - * program in microcode or assembly language. - * - */ - - -/* -Cephes Math Library Release 2.1: December, 1988 -Copyright 1984, 1987, 1988 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -float polevlf( float xx, float *coef, int N ) -{ -float ans, x; -float *p; -int i; - -x = xx; -p = coef; -ans = *p++; - -/* -for( i=0; i<N; i++ ) - ans = ans * x + *p++; -*/ - -i = N; -do - ans = ans * x + *p++; -while( --i ); - -return( ans ); -} - -/* p1evl() */ -/* N - * Evaluate polynomial when coefficient of x is 1.0. - * Otherwise same as polevl. - */ - -float p1evlf( float xx, float *coef, int N ) -{ -float ans, x; -float *p; -int i; - -x = xx; -p = coef; -ans = x + *p++; -i = N-1; - -do - ans = ans * x + *p++; -while( --i ); - -return( ans ); -} diff --git a/libm/float/polynf.c b/libm/float/polynf.c deleted file mode 100644 index 48c6675d4..000000000 --- a/libm/float/polynf.c +++ /dev/null @@ -1,520 +0,0 @@ -/* polynf.c - * polyrf.c - * Arithmetic operations on polynomials - * - * In the following descriptions a, b, c are polynomials of degree - * na, nb, nc respectively. The degree of a polynomial cannot - * exceed a run-time value MAXPOLF. An operation that attempts - * to use or generate a polynomial of higher degree may produce a - * result that suffers truncation at degree MAXPOL. The value of - * MAXPOL is set by calling the function - * - * polinif( maxpol ); - * - * where maxpol is the desired maximum degree. This must be - * done prior to calling any of the other functions in this module. - * Memory for internal temporary polynomial storage is allocated - * by polinif(). - * - * Each polynomial is represented by an array containing its - * coefficients, together with a separately declared integer equal - * to the degree of the polynomial. The coefficients appear in - * ascending order; that is, - * - * 2 na - * a(x) = a[0] + a[1] * x + a[2] * x + ... + a[na] * x . - * - * - * - * sum = poleva( a, na, x ); Evaluate polynomial a(t) at t = x. - * polprtf( a, na, D ); Print the coefficients of a to D digits. - * polclrf( a, na ); Set a identically equal to zero, up to a[na]. - * polmovf( a, na, b ); Set b = a. - * poladdf( a, na, b, nb, c ); c = b + a, nc = max(na,nb) - * polsubf( a, na, b, nb, c ); c = b - a, nc = max(na,nb) - * polmulf( a, na, b, nb, c ); c = b * a, nc = na+nb - * - * - * Division: - * - * i = poldivf( a, na, b, nb, c ); c = b / a, nc = MAXPOL - * - * returns i = the degree of the first nonzero coefficient of a. - * The computed quotient c must be divided by x^i. An error message - * is printed if a is identically zero. - * - * - * Change of variables: - * If a and b are polynomials, and t = a(x), then - * c(t) = b(a(x)) - * is a polynomial found by substituting a(x) for t. The - * subroutine call for this is - * - * polsbtf( a, na, b, nb, c ); - * - * - * Notes: - * poldivf() is an integer routine; polevaf() is float. - * Any of the arguments a, b, c may refer to the same array. - * - */ - -#ifndef NULL -#define NULL 0 -#endif -#include <math.h> - -#ifdef ANSIC -void printf(), sprintf(), exit(); -void free(void *); -void *malloc(int); -#else -void printf(), sprintf(), free(), exit(); -void *malloc(); -#endif -/* near pointer version of malloc() */ -/*#define malloc _nmalloc*/ -/*#define free _nfree*/ - -/* Pointers to internal arrays. Note poldiv() allocates - * and deallocates some temporary arrays every time it is called. - */ -static float *pt1 = 0; -static float *pt2 = 0; -static float *pt3 = 0; - -/* Maximum degree of polynomial. */ -int MAXPOLF = 0; -extern int MAXPOLF; - -/* Number of bytes (chars) in maximum size polynomial. */ -static int psize = 0; - - -/* Initialize max degree of polynomials - * and allocate temporary storage. - */ -#ifdef ANSIC -void polinif( int maxdeg ) -#else -int polinif( maxdeg ) -int maxdeg; -#endif -{ - -MAXPOLF = maxdeg; -psize = (maxdeg + 1) * sizeof(float); - -/* Release previously allocated memory, if any. */ -if( pt3 ) - free(pt3); -if( pt2 ) - free(pt2); -if( pt1 ) - free(pt1); - -/* Allocate new arrays */ -pt1 = (float * )malloc(psize); /* used by polsbtf */ -pt2 = (float * )malloc(psize); /* used by polsbtf */ -pt3 = (float * )malloc(psize); /* used by polmul */ - -/* Report if failure */ -if( (pt1 == NULL) || (pt2 == NULL) || (pt3 == NULL) ) - { - mtherr( "polinif", ERANGE ); - exit(1); - } -#if !ANSIC -return 0; -#endif -} - - - -/* Print the coefficients of a, with d decimal precision. - */ -static char *form = "abcdefghijk"; - -#ifdef ANSIC -void polprtf( float *a, int na, int d ) -#else -int polprtf( a, na, d ) -float a[]; -int na, d; -#endif -{ -int i, j, d1; -char *p; - -/* Create format descriptor string for the printout. - * Do this partly by hand, since sprintf() may be too - * bug-ridden to accomplish this feat by itself. - */ -p = form; -*p++ = '%'; -d1 = d + 8; -(void )sprintf( p, "%d ", d1 ); -p += 1; -if( d1 >= 10 ) - p += 1; -*p++ = '.'; -(void )sprintf( p, "%d ", d ); -p += 1; -if( d >= 10 ) - p += 1; -*p++ = 'e'; -*p++ = ' '; -*p++ = '\0'; - - -/* Now do the printing. - */ -d1 += 1; -j = 0; -for( i=0; i<=na; i++ ) - { -/* Detect end of available line */ - j += d1; - if( j >= 78 ) - { - printf( "\n" ); - j = d1; - } - printf( form, a[i] ); - } -printf( "\n" ); -#if !ANSIC -return 0; -#endif -} - - - -/* Set a = 0. - */ -#ifdef ANSIC -void polclrf( register float *a, int n ) -#else -int polclrf( a, n ) -register float *a; -int n; -#endif -{ -int i; - -if( n > MAXPOLF ) - n = MAXPOLF; -for( i=0; i<=n; i++ ) - *a++ = 0.0; -#if !ANSIC -return 0; -#endif -} - - - -/* Set b = a. - */ -#ifdef ANSIC -void polmovf( register float *a, int na, register float *b ) -#else -int polmovf( a, na, b ) -register float *a, *b; -int na; -#endif -{ -int i; - -if( na > MAXPOLF ) - na = MAXPOLF; - -for( i=0; i<= na; i++ ) - { - *b++ = *a++; - } -#if !ANSIC -return 0; -#endif -} - - -/* c = b * a. - */ -#ifdef ANSIC -void polmulf( float a[], int na, float b[], int nb, float c[] ) -#else -int polmulf( a, na, b, nb, c ) -float a[], b[], c[]; -int na, nb; -#endif -{ -int i, j, k, nc; -float x; - -nc = na + nb; -polclrf( pt3, MAXPOLF ); - -for( i=0; i<=na; i++ ) - { - x = a[i]; - for( j=0; j<=nb; j++ ) - { - k = i + j; - if( k > MAXPOLF ) - break; - pt3[k] += x * b[j]; - } - } - -if( nc > MAXPOLF ) - nc = MAXPOLF; -for( i=0; i<=nc; i++ ) - c[i] = pt3[i]; -#if !ANSIC -return 0; -#endif -} - - - - -/* c = b + a. - */ -#ifdef ANSIC -void poladdf( float a[], int na, float b[], int nb, float c[] ) -#else -int poladdf( a, na, b, nb, c ) -float a[], b[], c[]; -int na, nb; -#endif -{ -int i, n; - - -if( na > nb ) - n = na; -else - n = nb; - -if( n > MAXPOLF ) - n = MAXPOLF; - -for( i=0; i<=n; i++ ) - { - if( i > na ) - c[i] = b[i]; - else if( i > nb ) - c[i] = a[i]; - else - c[i] = b[i] + a[i]; - } -#if !ANSIC -return 0; -#endif -} - -/* c = b - a. - */ -#ifdef ANSIC -void polsubf( float a[], int na, float b[], int nb, float c[] ) -#else -int polsubf( a, na, b, nb, c ) -float a[], b[], c[]; -int na, nb; -#endif -{ -int i, n; - - -if( na > nb ) - n = na; -else - n = nb; - -if( n > MAXPOLF ) - n = MAXPOLF; - -for( i=0; i<=n; i++ ) - { - if( i > na ) - c[i] = b[i]; - else if( i > nb ) - c[i] = -a[i]; - else - c[i] = b[i] - a[i]; - } -#if !ANSIC -return 0; -#endif -} - - - -/* c = b/a - */ -#ifdef ANSIC -int poldivf( float a[], int na, float b[], int nb, float c[] ) -#else -int poldivf( a, na, b, nb, c ) -float a[], b[], c[]; -int na, nb; -#endif -{ -float quot; -float *ta, *tb, *tq; -int i, j, k, sing; - -sing = 0; - -/* Allocate temporary arrays. This would be quicker - * if done automatically on the stack, but stack space - * may be hard to obtain on a small computer. - */ -ta = (float * )malloc( psize ); -polclrf( ta, MAXPOLF ); -polmovf( a, na, ta ); - -tb = (float * )malloc( psize ); -polclrf( tb, MAXPOLF ); -polmovf( b, nb, tb ); - -tq = (float * )malloc( psize ); -polclrf( tq, MAXPOLF ); - -/* What to do if leading (constant) coefficient - * of denominator is zero. - */ -if( a[0] == 0.0 ) - { - for( i=0; i<=na; i++ ) - { - if( ta[i] != 0.0 ) - goto nzero; - } - mtherr( "poldivf", SING ); - goto done; - -nzero: -/* Reduce the degree of the denominator. */ - for( i=0; i<na; i++ ) - ta[i] = ta[i+1]; - ta[na] = 0.0; - - if( b[0] != 0.0 ) - { -/* Optional message: - printf( "poldivf singularity, divide quotient by x\n" ); -*/ - sing += 1; - } - else - { -/* Reduce degree of numerator. */ - for( i=0; i<nb; i++ ) - tb[i] = tb[i+1]; - tb[nb] = 0.0; - } -/* Call self, using reduced polynomials. */ - sing += poldivf( ta, na, tb, nb, c ); - goto done; - } - -/* Long division algorithm. ta[0] is nonzero. - */ -for( i=0; i<=MAXPOLF; i++ ) - { - quot = tb[i]/ta[0]; - for( j=0; j<=MAXPOLF; j++ ) - { - k = j + i; - if( k > MAXPOLF ) - break; - tb[k] -= quot * ta[j]; - } - tq[i] = quot; - } -/* Send quotient to output array. */ -polmovf( tq, MAXPOLF, c ); - -done: - -/* Restore allocated memory. */ -free(tq); -free(tb); -free(ta); -return( sing ); -} - - - - -/* Change of variables - * Substitute a(y) for the variable x in b(x). - * x = a(y) - * c(x) = b(x) = b(a(y)). - */ - -#ifdef ANSIC -void polsbtf( float a[], int na, float b[], int nb, float c[] ) -#else -int polsbtf( a, na, b, nb, c ) -float a[], b[], c[]; -int na, nb; -#endif -{ -int i, j, k, n2; -float x; - -/* 0th degree term: - */ -polclrf( pt1, MAXPOLF ); -pt1[0] = b[0]; - -polclrf( pt2, MAXPOLF ); -pt2[0] = 1.0; -n2 = 0; - -for( i=1; i<=nb; i++ ) - { -/* Form ith power of a. */ - polmulf( a, na, pt2, n2, pt2 ); - n2 += na; - x = b[i]; -/* Add the ith coefficient of b times the ith power of a. */ - for( j=0; j<=n2; j++ ) - { - if( j > MAXPOLF ) - break; - pt1[j] += x * pt2[j]; - } - } - -k = n2 + nb; -if( k > MAXPOLF ) - k = MAXPOLF; -for( i=0; i<=k; i++ ) - c[i] = pt1[i]; -#if !ANSIC -return 0; -#endif -} - - - - -/* Evaluate polynomial a(t) at t = x. - */ -float polevaf( float *a, int na, float xx ) -{ -float x, s; -int i; - -x = xx; -s = a[na]; -for( i=na-1; i>=0; i-- ) - { - s = s * x + a[i]; - } -return(s); -} - diff --git a/libm/float/powf.c b/libm/float/powf.c deleted file mode 100644 index 367a39ad4..000000000 --- a/libm/float/powf.c +++ /dev/null @@ -1,338 +0,0 @@ -/* powf.c - * - * Power function - * - * - * - * SYNOPSIS: - * - * float x, y, z, powf(); - * - * z = powf( x, y ); - * - * - * - * DESCRIPTION: - * - * Computes x raised to the yth power. Analytically, - * - * x**y = exp( y log(x) ). - * - * Following Cody and Waite, this program uses a lookup table - * of 2**-i/16 and pseudo extended precision arithmetic to - * obtain an extra three bits of accuracy in both the logarithm - * and the exponential. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10,10 100,000 1.4e-7 3.6e-8 - * 1/10 < x < 10, x uniformly distributed. - * -10 < y < 10, y uniformly distributed. - * - * - * ERROR MESSAGES: - * - * message condition value returned - * powf overflow x**y > MAXNUMF MAXNUMF - * powf underflow x**y < 1/MAXNUMF 0.0 - * powf domain x<0 and y noninteger 0.0 - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1988 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> -static char fname[] = {"powf"}; - - -/* 2^(-i/16) - * The decimal values are rounded to 24-bit precision - */ -static float A[] = { - 1.00000000000000000000E0, - 9.57603275775909423828125E-1, - 9.17004048824310302734375E-1, - 8.78126084804534912109375E-1, - 8.40896427631378173828125E-1, - 8.05245161056518554687500E-1, - 7.71105408668518066406250E-1, - 7.38413095474243164062500E-1, - 7.07106769084930419921875E-1, - 6.77127778530120849609375E-1, - 6.48419797420501708984375E-1, - 6.20928883552551269531250E-1, - 5.94603538513183593750000E-1, - 5.69394290447235107421875E-1, - 5.45253872871398925781250E-1, - 5.22136867046356201171875E-1, - 5.00000000000000000000E-1 -}; -/* continuation, for even i only - * 2^(i/16) = A[i] + B[i/2] - */ -static float B[] = { - 0.00000000000000000000E0, --5.61963907099083340520586E-9, --1.23776636307969995237668E-8, - 4.03545234539989593104537E-9, - 1.21016171044789693621048E-8, --2.00949968760174979411038E-8, - 1.89881769396087499852802E-8, --6.53877009617774467211965E-9, - 0.00000000000000000000E0 -}; - -/* 1 / A[i] - * The decimal values are full precision - */ -static float Ainv[] = { - 1.00000000000000000000000E0, - 1.04427378242741384032197E0, - 1.09050773266525765920701E0, - 1.13878863475669165370383E0, - 1.18920711500272106671750E0, - 1.24185781207348404859368E0, - 1.29683955465100966593375E0, - 1.35425554693689272829801E0, - 1.41421356237309504880169E0, - 1.47682614593949931138691E0, - 1.54221082540794082361229E0, - 1.61049033194925430817952E0, - 1.68179283050742908606225E0, - 1.75625216037329948311216E0, - 1.83400808640934246348708E0, - 1.91520656139714729387261E0, - 2.00000000000000000000000E0 -}; - -#ifdef DEC -#define MEXP 2032.0 -#define MNEXP -2032.0 -#else -#define MEXP 2048.0 -#define MNEXP -2400.0 -#endif - -/* log2(e) - 1 */ -#define LOG2EA 0.44269504088896340736F -extern float MAXNUMF; - -#define F W -#define Fa Wa -#define Fb Wb -#define G W -#define Ga Wa -#define Gb u -#define H W -#define Ha Wb -#define Hb Wb - - -#ifdef ANSIC -float floorf( float ); -float frexpf( float, int *); -float ldexpf( float, int ); -float powif( float, int ); -#else -float floorf(), frexpf(), ldexpf(), powif(); -#endif - -/* Find a multiple of 1/16 that is within 1/16 of x. */ -#define reduc(x) 0.0625 * floorf( 16 * (x) ) - -#ifdef ANSIC -float powf( float x, float y ) -#else -float powf( x, y ) -float x, y; -#endif -{ -float u, w, z, W, Wa, Wb, ya, yb; -/* float F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */ -int e, i, nflg; - - -nflg = 0; /* flag = 1 if x<0 raised to integer power */ -w = floorf(y); -if( w < 0 ) - z = -w; -else - z = w; -if( (w == y) && (z < 32768.0) ) - { - i = w; - w = powif( x, i ); - return( w ); - } - - -if( x <= 0.0F ) - { - if( x == 0.0 ) - { - if( y == 0.0 ) - return( 1.0 ); /* 0**0 */ - else - return( 0.0 ); /* 0**y */ - } - else - { - if( w != y ) - { /* noninteger power of negative number */ - mtherr( fname, DOMAIN ); - return(0.0); - } - nflg = 1; - if( x < 0 ) - x = -x; - } - } - -/* separate significand from exponent */ -x = frexpf( x, &e ); - -/* find significand in antilog table A[] */ -i = 1; -if( x <= A[9] ) - i = 9; -if( x <= A[i+4] ) - i += 4; -if( x <= A[i+2] ) - i += 2; -if( x >= A[1] ) - i = -1; -i += 1; - - -/* Find (x - A[i])/A[i] - * in order to compute log(x/A[i]): - * - * log(x) = log( a x/a ) = log(a) + log(x/a) - * - * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a - */ -x -= A[i]; -x -= B[ i >> 1 ]; -x *= Ainv[i]; - - -/* rational approximation for log(1+v): - * - * log(1+v) = v - 0.5 v^2 + v^3 P(v) - * Theoretical relative error of the approximation is 3.5e-11 - * on the interval 2^(1/16) - 1 > v > 2^(-1/16) - 1 - */ -z = x*x; -w = (((-0.1663883081054895 * x - + 0.2003770364206271) * x - - 0.2500006373383951) * x - + 0.3333331095506474) * x * z; -w -= 0.5 * z; - -/* Convert to base 2 logarithm: - * multiply by log2(e) - */ -w = w + LOG2EA * w; -/* Note x was not yet added in - * to above rational approximation, - * so do it now, while multiplying - * by log2(e). - */ -z = w + LOG2EA * x; -z = z + x; - -/* Compute exponent term of the base 2 logarithm. */ -w = -i; -w *= 0.0625; /* divide by 16 */ -w += e; -/* Now base 2 log of x is w + z. */ - -/* Multiply base 2 log by y, in extended precision. */ - -/* separate y into large part ya - * and small part yb less than 1/16 - */ -ya = reduc(y); -yb = y - ya; - - -F = z * y + w * yb; -Fa = reduc(F); -Fb = F - Fa; - -G = Fa + w * ya; -Ga = reduc(G); -Gb = G - Ga; - -H = Fb + Gb; -Ha = reduc(H); -w = 16 * (Ga + Ha); - -/* Test the power of 2 for overflow */ -if( w > MEXP ) - { - mtherr( fname, OVERFLOW ); - return( MAXNUMF ); - } - -if( w < MNEXP ) - { - mtherr( fname, UNDERFLOW ); - return( 0.0 ); - } - -e = w; -Hb = H - Ha; - -if( Hb > 0.0 ) - { - e += 1; - Hb -= 0.0625; - } - -/* Now the product y * log2(x) = Hb + e/16.0. - * - * Compute base 2 exponential of Hb, - * where -0.0625 <= Hb <= 0. - * Theoretical relative error of the approximation is 2.8e-12. - */ -/* z = 2**Hb - 1 */ -z = ((( 9.416993633606397E-003 * Hb - + 5.549356188719141E-002) * Hb - + 2.402262883964191E-001) * Hb - + 6.931471791490764E-001) * Hb; - -/* Express e/16 as an integer plus a negative number of 16ths. - * Find lookup table entry for the fractional power of 2. - */ -if( e < 0 ) - i = -( -e >> 4 ); -else - i = (e >> 4) + 1; -e = (i << 4) - e; -w = A[e]; -z = w + w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */ -z = ldexpf( z, i ); /* multiply by integer power of 2 */ - -if( nflg ) - { -/* For negative x, - * find out if the integer exponent - * is odd or even. - */ - w = 2 * floorf( (float) 0.5 * w ); - if( w != y ) - z = -z; /* odd exponent */ - } - -return( z ); -} diff --git a/libm/float/powif.c b/libm/float/powif.c deleted file mode 100644 index d226896ba..000000000 --- a/libm/float/powif.c +++ /dev/null @@ -1,156 +0,0 @@ -/* powif.c - * - * Real raised to integer power - * - * - * - * SYNOPSIS: - * - * float x, y, powif(); - * int n; - * - * y = powif( x, n ); - * - * - * - * DESCRIPTION: - * - * Returns argument x raised to the nth power. - * The routine efficiently decomposes n as a sum of powers of - * two. The desired power is a product of two-to-the-kth - * powers of x. Thus to compute the 32767 power of x requires - * 28 multiplications instead of 32767 multiplications. - * - * - * - * ACCURACY: - * - * - * Relative error: - * arithmetic x domain n domain # trials peak rms - * IEEE .04,26 -26,26 100000 1.1e-6 2.0e-7 - * IEEE 1,2 -128,128 100000 1.1e-5 1.0e-6 - * - * Returns MAXNUMF on overflow, zero on underflow. - * - */ - -/* powi.c */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1989 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> -extern float MAXNUMF, MAXLOGF, MINLOGF, LOGE2F; - -float frexpf( float, int * ); - -float powif( float x, int nn ) -{ -int n, e, sign, asign, lx; -float w, y, s; - -if( x == 0.0 ) - { - if( nn == 0 ) - return( 1.0 ); - else if( nn < 0 ) - return( MAXNUMF ); - else - return( 0.0 ); - } - -if( nn == 0 ) - return( 1.0 ); - - -if( x < 0.0 ) - { - asign = -1; - x = -x; - } -else - asign = 0; - - -if( nn < 0 ) - { - sign = -1; - n = -nn; -/* - x = 1.0/x; -*/ - } -else - { - sign = 0; - n = nn; - } - -/* Overflow detection */ - -/* Calculate approximate logarithm of answer */ -s = frexpf( x, &lx ); -e = (lx - 1)*n; -if( (e == 0) || (e > 64) || (e < -64) ) - { - s = (s - 7.0710678118654752e-1) / (s + 7.0710678118654752e-1); - s = (2.9142135623730950 * s - 0.5 + lx) * nn * LOGE2F; - } -else - { - s = LOGE2F * e; - } - -if( s > MAXLOGF ) - { - mtherr( "powi", OVERFLOW ); - y = MAXNUMF; - goto done; - } - -if( s < MINLOGF ) - return(0.0); - -/* Handle tiny denormal answer, but with less accuracy - * since roundoff error in 1.0/x will be amplified. - * The precise demarcation should be the gradual underflow threshold. - */ -if( s < (-MAXLOGF+2.0) ) - { - x = 1.0/x; - sign = 0; - } - -/* First bit of the power */ -if( n & 1 ) - y = x; - -else - { - y = 1.0; - asign = 0; - } - -w = x; -n >>= 1; -while( n ) - { - w = w * w; /* arg to the 2-to-the-kth power */ - if( n & 1 ) /* if that bit is set, then include in product */ - y *= w; - n >>= 1; - } - - -done: - -if( asign ) - y = -y; /* odd power of negative number */ -if( sign ) - y = 1.0/y; -return(y); -} diff --git a/libm/float/powtst.c b/libm/float/powtst.c deleted file mode 100644 index ff4845de2..000000000 --- a/libm/float/powtst.c +++ /dev/null @@ -1,41 +0,0 @@ -#include <stdio.h> -#include <math.h> -extern float MAXNUMF, MAXLOGF, MINLOGF; - -int -main() -{ -float exp1, minnum, x, y, z, e; -exp1 = expf(1.0F); - -minnum = powif(2.0F,-149); - -x = exp1; -y = MINLOGF + logf(0.501); -/*y = MINLOGF - 0.405;*/ -z = powf(x,y); -e = (z - minnum) / minnum; -printf("%.16e %.16e\n", z, e); - -x = exp1; -y = MAXLOGF; -z = powf(x,y); -e = (z - MAXNUMF) / MAXNUMF; -printf("%.16e %.16e\n", z, e); - -x = MAXNUMF; -y = 1.0F/MAXLOGF; -z = powf(x,y); -e = (z - exp1) / exp1; -printf("%.16e %.16e\n", z, e); - - -x = exp1; -y = MINLOGF; -z = powf(x,y); -e = (z - minnum) / minnum; -printf("%.16e %.16e\n", z, e); - - -exit(0); -} diff --git a/libm/float/psif.c b/libm/float/psif.c deleted file mode 100644 index 2d9187c67..000000000 --- a/libm/float/psif.c +++ /dev/null @@ -1,153 +0,0 @@ -/* psif.c - * - * Psi (digamma) function - * - * - * SYNOPSIS: - * - * float x, y, psif(); - * - * y = psif( x ); - * - * - * DESCRIPTION: - * - * d - - * psi(x) = -- ln | (x) - * dx - * - * is the logarithmic derivative of the gamma function. - * For integer x, - * n-1 - * - - * psi(n) = -EUL + > 1/k. - * - - * k=1 - * - * This formula is used for 0 < n <= 10. If x is negative, it - * is transformed to a positive argument by the reflection - * formula psi(1-x) = psi(x) + pi cot(pi x). - * For general positive x, the argument is made greater than 10 - * using the recurrence psi(x+1) = psi(x) + 1/x. - * Then the following asymptotic expansion is applied: - * - * inf. B - * - 2k - * psi(x) = log(x) - 1/2x - > ------- - * - 2k - * k=1 2k x - * - * where the B2k are Bernoulli numbers. - * - * ACCURACY: - * Absolute error, relative when |psi| > 1 : - * arithmetic domain # trials peak rms - * IEEE -33,0 30000 8.2e-7 1.2e-7 - * IEEE 0,33 100000 7.3e-7 7.7e-8 - * - * ERROR MESSAGES: - * message condition value returned - * psi singularity x integer <=0 MAXNUMF - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - - -static float A[] = { --4.16666666666666666667E-3, - 3.96825396825396825397E-3, --8.33333333333333333333E-3, - 8.33333333333333333333E-2 -}; - - -#define EUL 0.57721566490153286061 - -extern float PIF, MAXNUMF; - - - -float floorf(float), logf(float), tanf(float); -float polevlf(float, float *, int); - -float psif(float xx) -{ -float p, q, nz, x, s, w, y, z; -int i, n, negative; - - -x = xx; -nz = 0.0; -negative = 0; -if( x <= 0.0 ) - { - negative = 1; - q = x; - p = floorf(q); - if( p == q ) - { - mtherr( "psif", SING ); - return( MAXNUMF ); - } - nz = q - p; - if( nz != 0.5 ) - { - if( nz > 0.5 ) - { - p += 1.0; - nz = q - p; - } - nz = PIF/tanf(PIF*nz); - } - else - { - nz = 0.0; - } - x = 1.0 - x; - } - -/* check for positive integer up to 10 */ -if( (x <= 10.0) && (x == floorf(x)) ) - { - y = 0.0; - n = x; - for( i=1; i<n; i++ ) - { - w = i; - y += 1.0/w; - } - y -= EUL; - goto done; - } - -s = x; -w = 0.0; -while( s < 10.0 ) - { - w += 1.0/s; - s += 1.0; - } - -if( s < 1.0e8 ) - { - z = 1.0/(s * s); - y = z * polevlf( z, A, 3 ); - } -else - y = 0.0; - -y = logf(s) - (0.5/s) - y - w; - -done: -if( negative ) - { - y -= nz; - } -return(y); -} diff --git a/libm/float/rgammaf.c b/libm/float/rgammaf.c deleted file mode 100644 index 5afa25e91..000000000 --- a/libm/float/rgammaf.c +++ /dev/null @@ -1,130 +0,0 @@ -/* rgammaf.c - * - * Reciprocal gamma function - * - * - * - * SYNOPSIS: - * - * float x, y, rgammaf(); - * - * y = rgammaf( x ); - * - * - * - * DESCRIPTION: - * - * Returns one divided by the gamma function of the argument. - * - * The function is approximated by a Chebyshev expansion in - * the interval [0,1]. Range reduction is by recurrence - * for arguments between -34.034 and +34.84425627277176174. - * 1/MAXNUMF is returned for positive arguments outside this - * range. - * - * The reciprocal gamma function has no singularities, - * but overflow and underflow may occur for large arguments. - * These conditions return either MAXNUMF or 1/MAXNUMF with - * appropriate sign. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -34,+34 100000 8.9e-7 1.1e-7 - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1985, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -/* Chebyshev coefficients for reciprocal gamma function - * in interval 0 to 1. Function is 1/(x gamma(x)) - 1 - */ - -static float R[] = { - 1.08965386454418662084E-9, --3.33964630686836942556E-8, - 2.68975996440595483619E-7, - 2.96001177518801696639E-6, --8.04814124978471142852E-5, - 4.16609138709688864714E-4, - 5.06579864028608725080E-3, --6.41925436109158228810E-2, --4.98558728684003594785E-3, - 1.27546015610523951063E-1 -}; - - -static char name[] = "rgammaf"; - -extern float PIF, MAXLOGF, MAXNUMF; - - - -float chbevlf(float, float *, int); -float expf(float), logf(float), sinf(float), lgamf(float); - -float rgammaf(float xx) -{ -float x, w, y, z; -int sign; - -x = xx; -if( x > 34.84425627277176174) - { - mtherr( name, UNDERFLOW ); - return(1.0/MAXNUMF); - } -if( x < -34.034 ) - { - w = -x; - z = sinf( PIF*w ); - if( z == 0.0 ) - return(0.0); - if( z < 0.0 ) - { - sign = 1; - z = -z; - } - else - sign = -1; - - y = logf( w * z / PIF ) + lgamf(w); - if( y < -MAXLOGF ) - { - mtherr( name, UNDERFLOW ); - return( sign * 1.0 / MAXNUMF ); - } - if( y > MAXLOGF ) - { - mtherr( name, OVERFLOW ); - return( sign * MAXNUMF ); - } - return( sign * expf(y)); - } -z = 1.0; -w = x; - -while( w > 1.0 ) /* Downward recurrence */ - { - w -= 1.0; - z *= w; - } -while( w < 0.0 ) /* Upward recurrence */ - { - z /= w; - w += 1.0; - } -if( w == 0.0 ) /* Nonpositive integer */ - return(0.0); -if( w == 1.0 ) /* Other integer */ - return( 1.0/z ); - -y = w * ( 1.0 + chbevlf( 4.0*w-2.0, R, 10 ) ) / z; -return(y); -} diff --git a/libm/float/setprec.c b/libm/float/setprec.c deleted file mode 100644 index a5222ae73..000000000 --- a/libm/float/setprec.c +++ /dev/null @@ -1,10 +0,0 @@ -/* Null stubs for coprocessor precision settings */ - -int -sprec() {return 0; } - -int -dprec() {return 0; } - -int -ldprec() {return 0; } diff --git a/libm/float/shichif.c b/libm/float/shichif.c deleted file mode 100644 index ae98021a9..000000000 --- a/libm/float/shichif.c +++ /dev/null @@ -1,212 +0,0 @@ -/* shichif.c - * - * Hyperbolic sine and cosine integrals - * - * - * - * SYNOPSIS: - * - * float x, Chi, Shi; - * - * shichi( x, &Chi, &Shi ); - * - * - * DESCRIPTION: - * - * Approximates the integrals - * - * x - * - - * | | cosh t - 1 - * Chi(x) = eul + ln x + | ----------- dt, - * | | t - * - - * 0 - * - * x - * - - * | | sinh t - * Shi(x) = | ------ dt - * | | t - * - - * 0 - * - * where eul = 0.57721566490153286061 is Euler's constant. - * The integrals are evaluated by power series for x < 8 - * and by Chebyshev expansions for x between 8 and 88. - * For large x, both functions approach exp(x)/2x. - * Arguments greater than 88 in magnitude return MAXNUM. - * - * - * ACCURACY: - * - * Test interval 0 to 88. - * Relative error: - * arithmetic function # trials peak rms - * IEEE Shi 20000 3.5e-7 7.0e-8 - * Absolute error, except relative when |Chi| > 1: - * IEEE Chi 20000 3.8e-7 7.6e-8 - */ - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -#include <math.h> - -/* x exp(-x) shi(x), inverted interval 8 to 18 */ -static float S1[] = { --3.56699611114982536845E-8, - 1.44818877384267342057E-7, - 7.82018215184051295296E-7, --5.39919118403805073710E-6, --3.12458202168959833422E-5, - 8.90136741950727517826E-5, - 2.02558474743846862168E-3, - 2.96064440855633256972E-2, - 1.11847751047257036625E0 -}; - -/* x exp(-x) shi(x), inverted interval 18 to 88 */ -static float S2[] = { - 1.69050228879421288846E-8, - 1.25391771228487041649E-7, - 1.16229947068677338732E-6, - 1.61038260117376323993E-5, - 3.49810375601053973070E-4, - 1.28478065259647610779E-2, - 1.03665722588798326712E0 -}; - - -/* x exp(-x) chin(x), inverted interval 8 to 18 */ -static float C1[] = { - 1.31458150989474594064E-8, --4.75513930924765465590E-8, --2.21775018801848880741E-7, - 1.94635531373272490962E-6, - 4.33505889257316408893E-6, --6.13387001076494349496E-5, --3.13085477492997465138E-4, - 4.97164789823116062801E-4, - 2.64347496031374526641E-2, - 1.11446150876699213025E0 -}; - -/* x exp(-x) chin(x), inverted interval 18 to 88 */ -static float C2[] = { --3.00095178028681682282E-9, - 7.79387474390914922337E-8, - 1.06942765566401507066E-6, - 1.59503164802313196374E-5, - 3.49592575153777996871E-4, - 1.28475387530065247392E-2, - 1.03665693917934275131E0 -}; - - - -/* Sine and cosine integrals */ - -#define EUL 0.57721566490153286061 -extern float MACHEPF, MAXNUMF; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float logf(float ), expf(float), chbevlf(float, float *, int); -#else -float logf(), expf(), chbevlf(); -#endif - - - -int shichif( float xx, float *si, float *ci ) -{ -float x, k, z, c, s, a; -short sign; - -x = xx; -if( x < 0.0 ) - { - sign = -1; - x = -x; - } -else - sign = 0; - - -if( x == 0.0 ) - { - *si = 0.0; - *ci = -MAXNUMF; - return( 0 ); - } - -if( x >= 8.0 ) - goto chb; - -z = x * x; - -/* Direct power series expansion */ - -a = 1.0; -s = 1.0; -c = 0.0; -k = 2.0; - -do - { - a *= z/k; - c += a/k; - k += 1.0; - a /= k; - s += a/k; - k += 1.0; - } -while( fabsf(a/s) > MACHEPF ); - -s *= x; -goto done; - - -chb: - -if( x < 18.0 ) - { - a = (576.0/x - 52.0)/10.0; - k = expf(x) / x; - s = k * chbevlf( a, S1, 9 ); - c = k * chbevlf( a, C1, 10 ); - goto done; - } - -if( x <= 88.0 ) - { - a = (6336.0/x - 212.0)/70.0; - k = expf(x) / x; - s = k * chbevlf( a, S2, 7 ); - c = k * chbevlf( a, C2, 7 ); - goto done; - } -else - { - if( sign ) - *si = -MAXNUMF; - else - *si = MAXNUMF; - *ci = MAXNUMF; - return(0); - } -done: -if( sign ) - s = -s; - -*si = s; - -*ci = EUL + logf(x) + c; -return(0); -} diff --git a/libm/float/sicif.c b/libm/float/sicif.c deleted file mode 100644 index 04633ee68..000000000 --- a/libm/float/sicif.c +++ /dev/null @@ -1,279 +0,0 @@ -/* sicif.c - * - * Sine and cosine integrals - * - * - * - * SYNOPSIS: - * - * float x, Ci, Si; - * - * sicif( x, &Si, &Ci ); - * - * - * DESCRIPTION: - * - * Evaluates the integrals - * - * x - * - - * | cos t - 1 - * Ci(x) = eul + ln x + | --------- dt, - * | t - * - - * 0 - * x - * - - * | sin t - * Si(x) = | ----- dt - * | t - * - - * 0 - * - * where eul = 0.57721566490153286061 is Euler's constant. - * The integrals are approximated by rational functions. - * For x > 8 auxiliary functions f(x) and g(x) are employed - * such that - * - * Ci(x) = f(x) sin(x) - g(x) cos(x) - * Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x) - * - * - * ACCURACY: - * Test interval = [0,50]. - * Absolute error, except relative when > 1: - * arithmetic function # trials peak rms - * IEEE Si 30000 2.1e-7 4.3e-8 - * IEEE Ci 30000 3.9e-7 2.2e-8 - */ - -/* -Cephes Math Library Release 2.1: January, 1989 -Copyright 1984, 1987, 1989 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -static float SN[] = { --8.39167827910303881427E-11, - 4.62591714427012837309E-8, --9.75759303843632795789E-6, - 9.76945438170435310816E-4, --4.13470316229406538752E-2, - 1.00000000000000000302E0, -}; -static float SD[] = { - 2.03269266195951942049E-12, - 1.27997891179943299903E-9, - 4.41827842801218905784E-7, - 9.96412122043875552487E-5, - 1.42085239326149893930E-2, - 9.99999999999999996984E-1, -}; - -static float CN[] = { - 2.02524002389102268789E-11, --1.35249504915790756375E-8, - 3.59325051419993077021E-6, --4.74007206873407909465E-4, - 2.89159652607555242092E-2, --1.00000000000000000080E0, -}; -static float CD[] = { - 4.07746040061880559506E-12, - 3.06780997581887812692E-9, - 1.23210355685883423679E-6, - 3.17442024775032769882E-4, - 5.10028056236446052392E-2, - 4.00000000000000000080E0, -}; - - -static float FN4[] = { - 4.23612862892216586994E0, - 5.45937717161812843388E0, - 1.62083287701538329132E0, - 1.67006611831323023771E-1, - 6.81020132472518137426E-3, - 1.08936580650328664411E-4, - 5.48900223421373614008E-7, -}; -static float FD4[] = { -/* 1.00000000000000000000E0,*/ - 8.16496634205391016773E0, - 7.30828822505564552187E0, - 1.86792257950184183883E0, - 1.78792052963149907262E-1, - 7.01710668322789753610E-3, - 1.10034357153915731354E-4, - 5.48900252756255700982E-7, -}; - - -static float FN8[] = { - 4.55880873470465315206E-1, - 7.13715274100146711374E-1, - 1.60300158222319456320E-1, - 1.16064229408124407915E-2, - 3.49556442447859055605E-4, - 4.86215430826454749482E-6, - 3.20092790091004902806E-8, - 9.41779576128512936592E-11, - 9.70507110881952024631E-14, -}; -static float FD8[] = { -/* 1.00000000000000000000E0,*/ - 9.17463611873684053703E-1, - 1.78685545332074536321E-1, - 1.22253594771971293032E-2, - 3.58696481881851580297E-4, - 4.92435064317881464393E-6, - 3.21956939101046018377E-8, - 9.43720590350276732376E-11, - 9.70507110881952025725E-14, -}; - -static float GN4[] = { - 8.71001698973114191777E-2, - 6.11379109952219284151E-1, - 3.97180296392337498885E-1, - 7.48527737628469092119E-2, - 5.38868681462177273157E-3, - 1.61999794598934024525E-4, - 1.97963874140963632189E-6, - 7.82579040744090311069E-9, -}; -static float GD4[] = { -/* 1.00000000000000000000E0,*/ - 1.64402202413355338886E0, - 6.66296701268987968381E-1, - 9.88771761277688796203E-2, - 6.22396345441768420760E-3, - 1.73221081474177119497E-4, - 2.02659182086343991969E-6, - 7.82579218933534490868E-9, -}; - -static float GN8[] = { - 6.97359953443276214934E-1, - 3.30410979305632063225E-1, - 3.84878767649974295920E-2, - 1.71718239052347903558E-3, - 3.48941165502279436777E-5, - 3.47131167084116673800E-7, - 1.70404452782044526189E-9, - 3.85945925430276600453E-12, - 3.14040098946363334640E-15, -}; -static float GD8[] = { -/* 1.00000000000000000000E0,*/ - 1.68548898811011640017E0, - 4.87852258695304967486E-1, - 4.67913194259625806320E-2, - 1.90284426674399523638E-3, - 3.68475504442561108162E-5, - 3.57043223443740838771E-7, - 1.72693748966316146736E-9, - 3.87830166023954706752E-12, - 3.14040098946363335242E-15, -}; - -#define EUL 0.57721566490153286061 -extern float MAXNUMF, PIO2F, MACHEPF; - - - -#ifdef ANSIC -float logf(float), sinf(float), cosf(float); -float polevlf(float, float *, int); -float p1evlf(float, float *, int); -#else -float logf(), sinf(), cosf(), polevlf(), p1evlf(); -#endif - - -int sicif( float xx, float *si, float *ci ) -{ -float x, z, c, s, f, g; -int sign; - -x = xx; -if( x < 0.0 ) - { - sign = -1; - x = -x; - } -else - sign = 0; - - -if( x == 0.0 ) - { - *si = 0.0; - *ci = -MAXNUMF; - return( 0 ); - } - - -if( x > 1.0e9 ) - { - *si = PIO2F - cosf(x)/x; - *ci = sinf(x)/x; - return( 0 ); - } - - - -if( x > 4.0 ) - goto asympt; - -z = x * x; -s = x * polevlf( z, SN, 5 ) / polevlf( z, SD, 5 ); -c = z * polevlf( z, CN, 5 ) / polevlf( z, CD, 5 ); - -if( sign ) - s = -s; -*si = s; -*ci = EUL + logf(x) + c; /* real part if x < 0 */ -return(0); - - - -/* The auxiliary functions are: - * - * - * *si = *si - PIO2; - * c = cos(x); - * s = sin(x); - * - * t = *ci * s - *si * c; - * a = *ci * c + *si * s; - * - * *si = t; - * *ci = -a; - */ - - -asympt: - -s = sinf(x); -c = cosf(x); -z = 1.0/(x*x); -if( x < 8.0 ) - { - f = polevlf( z, FN4, 6 ) / (x * p1evlf( z, FD4, 7 )); - g = z * polevlf( z, GN4, 7 ) / p1evlf( z, GD4, 7 ); - } -else - { - f = polevlf( z, FN8, 8 ) / (x * p1evlf( z, FD8, 8 )); - g = z * polevlf( z, GN8, 8 ) / p1evlf( z, GD8, 9 ); - } -*si = PIO2F - f * c - g * s; -if( sign ) - *si = -( *si ); -*ci = f * s - g * c; - -return(0); -} diff --git a/libm/float/sindgf.c b/libm/float/sindgf.c deleted file mode 100644 index a3f5851c8..000000000 --- a/libm/float/sindgf.c +++ /dev/null @@ -1,232 +0,0 @@ -/* sindgf.c - * - * Circular sine of angle in degrees - * - * - * - * SYNOPSIS: - * - * float x, y, sindgf(); - * - * y = sindgf( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of 45 degrees. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the sine is approximated by - * x + x**3 P(x**2). - * Between pi/4 and pi/2 the cosine is represented as - * 1 - x**2 Q(x**2). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-3600 100,000 1.2e-7 3.0e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * sin total loss x > 2^24 0.0 - * - */ - -/* cosdgf.c - * - * Circular cosine of angle in degrees - * - * - * - * SYNOPSIS: - * - * float x, y, cosdgf(); - * - * y = cosdgf( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of 45 degrees. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the cosine is approximated by - * 1 - x**2 Q(x**2). - * Between pi/4 and pi/2 the sine is represented as - * x + x**3 P(x**2). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8 - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1985, 1987, 1988, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -/* Single precision circular sine - * test interval: [-pi/4, +pi/4] - * trials: 10000 - * peak relative error: 6.8e-8 - * rms relative error: 2.6e-8 - */ -#include <math.h> - - -/*static float FOPI = 1.27323954473516;*/ - -extern float PIO4F; - -/* These are for a 24-bit significand: */ -static float T24M1 = 16777215.; - -static float PI180 = 0.0174532925199432957692; /* pi/180 */ - -float sindgf( float xx ) -{ -float x, y, z; -long j; -int sign; - -sign = 1; -x = xx; -if( xx < 0 ) - { - sign = -1; - x = -xx; - } -if( x > T24M1 ) - { - mtherr( "sindgf", TLOSS ); - return(0.0); - } -j = 0.022222222222222222222 * x; /* integer part of x/45 */ -y = j; -/* map zeros to origin */ -if( j & 1 ) - { - j += 1; - y += 1.0; - } -j &= 7; /* octant modulo 360 degrees */ -/* reflect in x axis */ -if( j > 3) - { - sign = -sign; - j -= 4; - } - -x = x - y * 45.0; -x *= PI180; /* multiply by pi/180 to convert to radians */ - -z = x * x; -if( (j==1) || (j==2) ) - { -/* - y = ((( 2.4462803166E-5 * z - - 1.3887580023E-3) * z - + 4.1666650433E-2) * z - - 4.9999999968E-1) * z - + 1.0; -*/ - -/* measured relative error in +/- pi/4 is 7.8e-8 */ - y = (( 2.443315711809948E-005 * z - - 1.388731625493765E-003) * z - + 4.166664568298827E-002) * z * z; - y -= 0.5 * z; - y += 1.0; - } -else - { -/* Theoretical relative error = 3.8e-9 in [-pi/4, +pi/4] */ - y = ((-1.9515295891E-4 * z - + 8.3321608736E-3) * z - - 1.6666654611E-1) * z * x; - y += x; - } - -if(sign < 0) - y = -y; -return( y); -} - - -/* Single precision circular cosine - * test interval: [-pi/4, +pi/4] - * trials: 10000 - * peak relative error: 8.3e-8 - * rms relative error: 2.2e-8 - */ - -float cosdgf( float xx ) -{ -register float x, y, z; -int j, sign; - -/* make argument positive */ -sign = 1; -x = xx; -if( x < 0 ) - x = -x; - -if( x > T24M1 ) - { - mtherr( "cosdgf", TLOSS ); - return(0.0); - } - -j = 0.02222222222222222222222 * x; /* integer part of x/PIO4 */ -y = j; -/* integer and fractional part modulo one octant */ -if( j & 1 ) /* map zeros to origin */ - { - j += 1; - y += 1.0; - } -j &= 7; -if( j > 3) - { - j -=4; - sign = -sign; - } - -if( j > 1 ) - sign = -sign; - -x = x - y * 45.0; /* x mod 45 degrees */ -x *= PI180; /* multiply by pi/180 to convert to radians */ - -z = x * x; - -if( (j==1) || (j==2) ) - { - y = (((-1.9515295891E-4 * z - + 8.3321608736E-3) * z - - 1.6666654611E-1) * z * x) - + x; - } -else - { - y = (( 2.443315711809948E-005 * z - - 1.388731625493765E-003) * z - + 4.166664568298827E-002) * z * z; - y -= 0.5 * z; - y += 1.0; - } -if(sign < 0) - y = -y; -return( y ); -} - diff --git a/libm/float/sinf.c b/libm/float/sinf.c deleted file mode 100644 index 2f1bb45b8..000000000 --- a/libm/float/sinf.c +++ /dev/null @@ -1,283 +0,0 @@ -/* sinf.c - * - * Circular sine - * - * - * - * SYNOPSIS: - * - * float x, y, sinf(); - * - * y = sinf( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of pi/4. The reduction - * error is nearly eliminated by contriving an extended precision - * modular arithmetic. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the sine is approximated by - * x + x**3 P(x**2). - * Between pi/4 and pi/2 the cosine is represented as - * 1 - x**2 Q(x**2). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -4096,+4096 100,000 1.2e-7 3.0e-8 - * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * sin total loss x > 2^24 0.0 - * - * Partial loss of accuracy begins to occur at x = 2^13 - * = 8192. Results may be meaningless for x >= 2^24 - * The routine as implemented flags a TLOSS error - * for x >= 2^24 and returns 0.0. - */ - -/* cosf.c - * - * Circular cosine - * - * - * - * SYNOPSIS: - * - * float x, y, cosf(); - * - * y = cosf( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of pi/4. The reduction - * error is nearly eliminated by contriving an extended precision - * modular arithmetic. - * - * Two polynomial approximating functions are employed. - * Between 0 and pi/4 the cosine is approximated by - * 1 - x**2 Q(x**2). - * Between pi/4 and pi/2 the sine is represented as - * x + x**3 P(x**2). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -8192,+8192 100,000 3.0e-7 3.0e-8 - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1985, 1987, 1988, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - - -/* Single precision circular sine - * test interval: [-pi/4, +pi/4] - * trials: 10000 - * peak relative error: 6.8e-8 - * rms relative error: 2.6e-8 - */ -#include <math.h> - - -static float FOPI = 1.27323954473516; - -extern float PIO4F; -/* Note, these constants are for a 32-bit significand: */ -/* -static float DP1 = 0.7853851318359375; -static float DP2 = 1.30315311253070831298828125e-5; -static float DP3 = 3.03855025325309630e-11; -static float lossth = 65536.; -*/ - -/* These are for a 24-bit significand: */ -static float DP1 = 0.78515625; -static float DP2 = 2.4187564849853515625e-4; -static float DP3 = 3.77489497744594108e-8; -static float lossth = 8192.; -static float T24M1 = 16777215.; - -static float sincof[] = { --1.9515295891E-4, - 8.3321608736E-3, --1.6666654611E-1 -}; -static float coscof[] = { - 2.443315711809948E-005, --1.388731625493765E-003, - 4.166664568298827E-002 -}; - -float sinf( float xx ) -{ -float *p; -float x, y, z; -register unsigned long j; -register int sign; - -sign = 1; -x = xx; -if( xx < 0 ) - { - sign = -1; - x = -xx; - } -if( x > T24M1 ) - { - mtherr( "sinf", TLOSS ); - return(0.0); - } -j = FOPI * x; /* integer part of x/(PI/4) */ -y = j; -/* map zeros to origin */ -if( j & 1 ) - { - j += 1; - y += 1.0; - } -j &= 7; /* octant modulo 360 degrees */ -/* reflect in x axis */ -if( j > 3) - { - sign = -sign; - j -= 4; - } - -if( x > lossth ) - { - mtherr( "sinf", PLOSS ); - x = x - y * PIO4F; - } -else - { -/* Extended precision modular arithmetic */ - x = ((x - y * DP1) - y * DP2) - y * DP3; - } -/*einits();*/ -z = x * x; -if( (j==1) || (j==2) ) - { -/* measured relative error in +/- pi/4 is 7.8e-8 */ -/* - y = (( 2.443315711809948E-005 * z - - 1.388731625493765E-003) * z - + 4.166664568298827E-002) * z * z; -*/ - p = coscof; - y = *p++; - y = y * z + *p++; - y = y * z + *p++; - y *= z * z; - y -= 0.5 * z; - y += 1.0; - } -else - { -/* Theoretical relative error = 3.8e-9 in [-pi/4, +pi/4] */ -/* - y = ((-1.9515295891E-4 * z - + 8.3321608736E-3) * z - - 1.6666654611E-1) * z * x; - y += x; -*/ - p = sincof; - y = *p++; - y = y * z + *p++; - y = y * z + *p++; - y *= z * x; - y += x; - } -/*einitd();*/ -if(sign < 0) - y = -y; -return( y); -} - - -/* Single precision circular cosine - * test interval: [-pi/4, +pi/4] - * trials: 10000 - * peak relative error: 8.3e-8 - * rms relative error: 2.2e-8 - */ - -float cosf( float xx ) -{ -float x, y, z; -int j, sign; - -/* make argument positive */ -sign = 1; -x = xx; -if( x < 0 ) - x = -x; - -if( x > T24M1 ) - { - mtherr( "cosf", TLOSS ); - return(0.0); - } - -j = FOPI * x; /* integer part of x/PIO4 */ -y = j; -/* integer and fractional part modulo one octant */ -if( j & 1 ) /* map zeros to origin */ - { - j += 1; - y += 1.0; - } -j &= 7; -if( j > 3) - { - j -=4; - sign = -sign; - } - -if( j > 1 ) - sign = -sign; - -if( x > lossth ) - { - mtherr( "cosf", PLOSS ); - x = x - y * PIO4F; - } -else -/* Extended precision modular arithmetic */ - x = ((x - y * DP1) - y * DP2) - y * DP3; - -z = x * x; - -if( (j==1) || (j==2) ) - { - y = (((-1.9515295891E-4 * z - + 8.3321608736E-3) * z - - 1.6666654611E-1) * z * x) - + x; - } -else - { - y = (( 2.443315711809948E-005 * z - - 1.388731625493765E-003) * z - + 4.166664568298827E-002) * z * z; - y -= 0.5 * z; - y += 1.0; - } -if(sign < 0) - y = -y; -return( y ); -} - diff --git a/libm/float/sinhf.c b/libm/float/sinhf.c deleted file mode 100644 index e8baaf4fa..000000000 --- a/libm/float/sinhf.c +++ /dev/null @@ -1,87 +0,0 @@ -/* sinhf.c - * - * Hyperbolic sine - * - * - * - * SYNOPSIS: - * - * float x, y, sinhf(); - * - * y = sinhf( x ); - * - * - * - * DESCRIPTION: - * - * Returns hyperbolic sine of argument in the range MINLOGF to - * MAXLOGF. - * - * The range is partitioned into two segments. If |x| <= 1, a - * polynomial approximation is used. - * Otherwise the calculation is sinh(x) = ( exp(x) - exp(-x) )/2. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-MAXLOG 100000 1.1e-7 2.9e-8 - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision hyperbolic sine - * test interval: [-1, +1] - * trials: 10000 - * peak relative error: 9.0e-8 - * rms relative error: 3.0e-8 - */ -#include <math.h> -extern float MAXLOGF, MAXNUMF; - -float expf( float ); - -float sinhf( float xx ) -{ -register float z; -float x; - -x = xx; -if( xx < 0 ) - z = -x; -else - z = x; - -if( z > MAXLOGF ) - { - mtherr( "sinhf", DOMAIN ); - if( x > 0 ) - return( MAXNUMF ); - else - return( -MAXNUMF ); - } -if( z > 1.0 ) - { - z = expf(z); - z = 0.5*z - (0.5/z); - if( x < 0 ) - z = -z; - } -else - { - z = x * x; - z = - (( 2.03721912945E-4 * z - + 8.33028376239E-3) * z - + 1.66667160211E-1) * z * x - + x; - } -return( z ); -} diff --git a/libm/float/spencef.c b/libm/float/spencef.c deleted file mode 100644 index 52799babe..000000000 --- a/libm/float/spencef.c +++ /dev/null @@ -1,135 +0,0 @@ -/* spencef.c - * - * Dilogarithm - * - * - * - * SYNOPSIS: - * - * float x, y, spencef(); - * - * y = spencef( x ); - * - * - * - * DESCRIPTION: - * - * Computes the integral - * - * x - * - - * | | log t - * spence(x) = - | ----- dt - * | | t - 1 - * - - * 1 - * - * for x >= 0. A rational approximation gives the integral in - * the interval (0.5, 1.5). Transformation formulas for 1/x - * and 1-x are employed outside the basic expansion range. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,4 30000 4.4e-7 6.3e-8 - * - * - */ - -/* spence.c */ - - -/* -Cephes Math Library Release 2.1: January, 1989 -Copyright 1985, 1987, 1989 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -static float A[8] = { - 4.65128586073990045278E-5, - 7.31589045238094711071E-3, - 1.33847639578309018650E-1, - 8.79691311754530315341E-1, - 2.71149851196553469920E0, - 4.25697156008121755724E0, - 3.29771340985225106936E0, - 1.00000000000000000126E0, -}; -static float B[8] = { - 6.90990488912553276999E-4, - 2.54043763932544379113E-2, - 2.82974860602568089943E-1, - 1.41172597751831069617E0, - 3.63800533345137075418E0, - 5.03278880143316990390E0, - 3.54771340985225096217E0, - 9.99999999999999998740E-1, -}; - -extern float PIF, MACHEPF; - -/* pi * pi / 6 */ -#define PIFS 1.64493406684822643647 - - -float logf(float), polevlf(float, float *, int); -float spencef(float xx) -{ -float x, w, y, z; -int flag; - -x = xx; -if( x < 0.0 ) - { - mtherr( "spencef", DOMAIN ); - return(0.0); - } - -if( x == 1.0 ) - return( 0.0 ); - -if( x == 0.0 ) - return( PIFS ); - -flag = 0; - -if( x > 2.0 ) - { - x = 1.0/x; - flag |= 2; - } - -if( x > 1.5 ) - { - w = (1.0/x) - 1.0; - flag |= 2; - } - -else if( x < 0.5 ) - { - w = -x; - flag |= 1; - } - -else - w = x - 1.0; - - -y = -w * polevlf( w, A, 7) / polevlf( w, B, 7 ); - -if( flag & 1 ) - y = PIFS - logf(x) * logf(1.0-x) - y; - -if( flag & 2 ) - { - z = logf(x); - y = -0.5 * z * z - y; - } - -return( y ); -} diff --git a/libm/float/sqrtf.c b/libm/float/sqrtf.c deleted file mode 100644 index bc75a907b..000000000 --- a/libm/float/sqrtf.c +++ /dev/null @@ -1,140 +0,0 @@ -/* sqrtf.c - * - * Square root - * - * - * - * SYNOPSIS: - * - * float x, y, sqrtf(); - * - * y = sqrtf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the square root of x. - * - * Range reduction involves isolating the power of two of the - * argument and using a polynomial approximation to obtain - * a rough value for the square root. Then Heron's iteration - * is used three times to converge to an accurate value. - * - * - * - * ACCURACY: - * - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,1.e38 100000 8.7e-8 2.9e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * sqrtf domain x < 0 0.0 - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1988, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision square root - * test interval: [sqrt(2)/2, sqrt(2)] - * trials: 30000 - * peak relative error: 8.8e-8 - * rms relative error: 3.3e-8 - * - * test interval: [0.01, 100.0] - * trials: 50000 - * peak relative error: 8.7e-8 - * rms relative error: 3.3e-8 - * - * Copyright (C) 1989 by Stephen L. Moshier. All rights reserved. - */ -#include <math.h> - -#ifdef ANSIC -float frexpf( float, int * ); -float ldexpf( float, int ); - -float sqrtf( float xx ) -#else -float frexpf(), ldexpf(); - -float sqrtf(xx) -float xx; -#endif -{ -float f, x, y; -int e; - -f = xx; -if( f <= 0.0 ) - { - if( f < 0.0 ) - mtherr( "sqrtf", DOMAIN ); - return( 0.0 ); - } - -x = frexpf( f, &e ); /* f = x * 2**e, 0.5 <= x < 1.0 */ -/* If power of 2 is odd, double x and decrement the power of 2. */ -if( e & 1 ) - { - x = x + x; - e -= 1; - } - -e >>= 1; /* The power of 2 of the square root. */ - -if( x > 1.41421356237 ) - { -/* x is between sqrt(2) and 2. */ - x = x - 2.0; - y = - ((((( -9.8843065718E-4 * x - + 7.9479950957E-4) * x - - 3.5890535377E-3) * x - + 1.1028809744E-2) * x - - 4.4195203560E-2) * x - + 3.5355338194E-1) * x - + 1.41421356237E0; - goto sqdon; - } - -if( x > 0.707106781187 ) - { -/* x is between sqrt(2)/2 and sqrt(2). */ - x = x - 1.0; - y = - ((((( 1.35199291026E-2 * x - - 2.26657767832E-2) * x - + 2.78720776889E-2) * x - - 3.89582788321E-2) * x - + 6.24811144548E-2) * x - - 1.25001503933E-1) * x * x - + 0.5 * x - + 1.0; - goto sqdon; - } - -/* x is between 0.5 and sqrt(2)/2. */ -x = x - 0.5; -y = -((((( -3.9495006054E-1 * x - + 5.1743034569E-1) * x - - 4.3214437330E-1) * x - + 3.5310730460E-1) * x - - 3.5354581892E-1) * x - + 7.0710676017E-1) * x - + 7.07106781187E-1; - -sqdon: -y = ldexpf( y, e ); /* y = y * 2**e */ -return( y); -} diff --git a/libm/float/stdtrf.c b/libm/float/stdtrf.c deleted file mode 100644 index 76b14c1f6..000000000 --- a/libm/float/stdtrf.c +++ /dev/null @@ -1,154 +0,0 @@ -/* stdtrf.c - * - * Student's t distribution - * - * - * - * SYNOPSIS: - * - * float t, stdtrf(); - * short k; - * - * y = stdtrf( k, t ); - * - * - * DESCRIPTION: - * - * Computes the integral from minus infinity to t of the Student - * t distribution with integer k > 0 degrees of freedom: - * - * t - * - - * | | - * - | 2 -(k+1)/2 - * | ( (k+1)/2 ) | ( x ) - * ---------------------- | ( 1 + --- ) dx - * - | ( k ) - * sqrt( k pi ) | ( k/2 ) | - * | | - * - - * -inf. - * - * Relation to incomplete beta integral: - * - * 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z ) - * where - * z = k/(k + t**2). - * - * For t < -1, this is the method of computation. For higher t, - * a direct method is derived from integration by parts. - * Since the function is symmetric about t=0, the area under the - * right tail of the density is found by calling the function - * with -t instead of t. - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +/- 100 5000 2.3e-5 2.9e-6 - */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - -extern float PIF, MACHEPF; - -#ifdef ANSIC -float sqrtf(float), atanf(float), incbetf(float, float, float); -#else -float sqrtf(), atanf(), incbetf(); -#endif - - - -float stdtrf( int k, float tt ) -{ -float t, x, rk, z, f, tz, p, xsqk; -int j; - -t = tt; -if( k <= 0 ) - { - mtherr( "stdtrf", DOMAIN ); - return(0.0); - } - -if( t == 0 ) - return( 0.5 ); - -if( t < -1.0 ) - { - rk = k; - z = rk / (rk + t * t); - p = 0.5 * incbetf( 0.5*rk, 0.5, z ); - return( p ); - } - -/* compute integral from -t to + t */ - -if( t < 0 ) - x = -t; -else - x = t; - -rk = k; /* degrees of freedom */ -z = 1.0 + ( x * x )/rk; - -/* test if k is odd or even */ -if( (k & 1) != 0) - { - - /* computation for odd k */ - - xsqk = x/sqrtf(rk); - p = atanf( xsqk ); - if( k > 1 ) - { - f = 1.0; - tz = 1.0; - j = 3; - while( (j<=(k-2)) && ( (tz/f) > MACHEPF ) ) - { - tz *= (j-1)/( z * j ); - f += tz; - j += 2; - } - p += f * xsqk/z; - } - p *= 2.0/PIF; - } - - -else - { - - /* computation for even k */ - - f = 1.0; - tz = 1.0; - j = 2; - - while( ( j <= (k-2) ) && ( (tz/f) > MACHEPF ) ) - { - tz *= (j - 1)/( z * j ); - f += tz; - j += 2; - } - p = f * x/sqrtf(z*rk); - } - -/* common exit */ - - -if( t < 0 ) - p = -p; /* note destruction of relative accuracy */ - - p = 0.5 + 0.5 * p; -return(p); -} diff --git a/libm/float/struvef.c b/libm/float/struvef.c deleted file mode 100644 index 4cf8854ed..000000000 --- a/libm/float/struvef.c +++ /dev/null @@ -1,315 +0,0 @@ -/* struvef.c - * - * Struve function - * - * - * - * SYNOPSIS: - * - * float v, x, y, struvef(); - * - * y = struvef( v, x ); - * - * - * - * DESCRIPTION: - * - * Computes the Struve function Hv(x) of order v, argument x. - * Negative x is rejected unless v is an integer. - * - * This module also contains the hypergeometric functions 1F2 - * and 3F0 and a routine for the Bessel function Yv(x) with - * noninteger v. - * - * - * - * ACCURACY: - * - * v varies from 0 to 10. - * Absolute error (relative error when |Hv(x)| > 1): - * arithmetic domain # trials peak rms - * IEEE -10,10 100000 9.0e-5 4.0e-6 - * - */ - - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1989 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> -#define DEBUG 0 - -extern float MACHEPF, MAXNUMF, PIF; - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - -#ifdef ANSIC -float gammaf(float), powf(float, float), sqrtf(float); -float yvf(float, float); -float floorf(float), ynf(int, float); -float jvf(float, float); -float sinf(float), cosf(float); -#else -float gammaf(), powf(), sqrtf(), yvf(); -float floorf(), ynf(), jvf(), sinf(), cosf(); -#endif - -float onef2f( float aa, float bb, float cc, float xx, float *err ) -{ -float a, b, c, x, n, a0, sum, t; -float an, bn, cn, max, z; - -a = aa; -b = bb; -c = cc; -x = xx; -an = a; -bn = b; -cn = c; -a0 = 1.0; -sum = 1.0; -n = 1.0; -t = 1.0; -max = 0.0; - -do - { - if( an == 0 ) - goto done; - if( bn == 0 ) - goto error; - if( cn == 0 ) - goto error; - if( (a0 > 1.0e34) || (n > 200) ) - goto error; - a0 *= (an * x) / (bn * cn * n); - sum += a0; - an += 1.0; - bn += 1.0; - cn += 1.0; - n += 1.0; - z = fabsf( a0 ); - if( z > max ) - max = z; - if( sum != 0 ) - t = fabsf( a0 / sum ); - else - t = z; - } -while( t > MACHEPF ); - -done: - -*err = fabsf( MACHEPF*max /sum ); - -#if DEBUG - printf(" onef2f cancellation error %.5E\n", *err ); -#endif - -goto xit; - -error: -#if DEBUG -printf("onef2f does not converge\n"); -#endif -*err = MAXNUMF; - -xit: - -#if DEBUG -printf("onef2( %.2E %.2E %.2E %.5E ) = %.3E %.6E\n", a, b, c, x, n, sum); -#endif -return(sum); -} - - - -float threef0f( float aa, float bb, float cc, float xx, float *err ) -{ -float a, b, c, x, n, a0, sum, t, conv, conv1; -float an, bn, cn, max, z; - -a = aa; -b = bb; -c = cc; -x = xx; -an = a; -bn = b; -cn = c; -a0 = 1.0; -sum = 1.0; -n = 1.0; -t = 1.0; -max = 0.0; -conv = 1.0e38; -conv1 = conv; - -do - { - if( an == 0.0 ) - goto done; - if( bn == 0.0 ) - goto done; - if( cn == 0.0 ) - goto done; - if( (a0 > 1.0e34) || (n > 200) ) - goto error; - a0 *= (an * bn * cn * x) / n; - an += 1.0; - bn += 1.0; - cn += 1.0; - n += 1.0; - z = fabsf( a0 ); - if( z > max ) - max = z; - if( z >= conv ) - { - if( (z < max) && (z > conv1) ) - goto done; - } - conv1 = conv; - conv = z; - sum += a0; - if( sum != 0 ) - t = fabsf( a0 / sum ); - else - t = z; - } -while( t > MACHEPF ); - -done: - -t = fabsf( MACHEPF*max/sum ); -#if DEBUG - printf(" threef0f cancellation error %.5E\n", t ); -#endif - -max = fabsf( conv/sum ); -if( max > t ) - t = max; -#if DEBUG - printf(" threef0f convergence %.5E\n", max ); -#endif - -goto xit; - -error: -#if DEBUG -printf("threef0f does not converge\n"); -#endif -t = MAXNUMF; - -xit: - -#if DEBUG -printf("threef0f( %.2E %.2E %.2E %.5E ) = %.3E %.6E\n", a, b, c, x, n, sum); -#endif - -*err = t; -return(sum); -} - - - - -float struvef( float vv, float xx ) -{ -float v, x, y, ya, f, g, h, t; -float onef2err, threef0err; - -v = vv; -x = xx; -f = floorf(v); -if( (v < 0) && ( v-f == 0.5 ) ) - { - y = jvf( -v, x ); - f = 1.0 - f; - g = 2.0 * floorf(0.5*f); - if( g != f ) - y = -y; - return(y); - } -t = 0.25*x*x; -f = fabsf(x); -g = 1.5 * fabsf(v); -if( (f > 30.0) && (f > g) ) - { - onef2err = MAXNUMF; - y = 0.0; - } -else - { - y = onef2f( 1.0, 1.5, 1.5+v, -t, &onef2err ); - } - -if( (f < 18.0) || (x < 0.0) ) - { - threef0err = MAXNUMF; - ya = 0.0; - } -else - { - ya = threef0f( 1.0, 0.5, 0.5-v, -1.0/t, &threef0err ); - } - -f = sqrtf( PIF ); -h = powf( 0.5*x, v-1.0 ); - -if( onef2err <= threef0err ) - { - g = gammaf( v + 1.5 ); - y = y * h * t / ( 0.5 * f * g ); - return(y); - } -else - { - g = gammaf( v + 0.5 ); - ya = ya * h / ( f * g ); - ya = ya + yvf( v, x ); - return(ya); - } -} - - - - -/* Bessel function of noninteger order - */ - -float yvf( float vv, float xx ) -{ -float v, x, y, t; -int n; - -v = vv; -x = xx; -y = floorf( v ); -if( y == v ) - { - n = v; - y = ynf( n, x ); - return( y ); - } -t = PIF * v; -y = (cosf(t) * jvf( v, x ) - jvf( -v, x ))/sinf(t); -return( y ); -} - -/* Crossover points between ascending series and asymptotic series - * for Struve function - * - * v x - * - * 0 19.2 - * 1 18.95 - * 2 19.15 - * 3 19.3 - * 5 19.7 - * 10 21.35 - * 20 26.35 - * 30 32.31 - * 40 40.0 - */ diff --git a/libm/float/tandgf.c b/libm/float/tandgf.c deleted file mode 100644 index dc55ad5e4..000000000 --- a/libm/float/tandgf.c +++ /dev/null @@ -1,206 +0,0 @@ -/* tandgf.c - * - * Circular tangent of angle in degrees - * - * - * - * SYNOPSIS: - * - * float x, y, tandgf(); - * - * y = tandgf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the circular tangent of the radian argument x. - * - * Range reduction is into intervals of 45 degrees. - * - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-2^24 50000 2.4e-7 4.8e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * tanf total loss x > 2^24 0.0 - * - */ - /* cotdgf.c - * - * Circular cotangent of angle in degrees - * - * - * - * SYNOPSIS: - * - * float x, y, cotdgf(); - * - * y = cotdgf( x ); - * - * - * - * DESCRIPTION: - * - * Range reduction is into intervals of 45 degrees. - * A common routine computes either the tangent or cotangent. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-2^24 50000 2.4e-7 4.8e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * cot total loss x > 2^24 0.0 - * cot singularity x = 0 MAXNUMF - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision circular tangent - * test interval: [-pi/4, +pi/4] - * trials: 10000 - * peak relative error: 8.7e-8 - * rms relative error: 2.8e-8 - */ -#include <math.h> - -extern float MAXNUMF; - -static float T24M1 = 16777215.; -static float PI180 = 0.0174532925199432957692; /* pi/180 */ - -static float tancotf( float xx, int cotflg ) -{ -float x, y, z, zz; -long j; -int sign; - - -/* make argument positive but save the sign */ -if( xx < 0.0 ) - { - x = -xx; - sign = -1; - } -else - { - x = xx; - sign = 1; - } - -if( x > T24M1 ) - { - if( cotflg ) - mtherr( "cotdgf", TLOSS ); - else - mtherr( "tandgf", TLOSS ); - return(0.0); - } - -/* compute x mod PIO4 */ -j = 0.022222222222222222222 * x; /* integer part of x/45 */ -y = j; - -/* map zeros and singularities to origin */ -if( j & 1 ) - { - j += 1; - y += 1.0; - } - -z = x - y * 45.0; -z *= PI180; /* multiply by pi/180 to convert to radians */ - -zz = z * z; - -if( x > 1.0e-4 ) - { -/* 1.7e-8 relative error in [-pi/4, +pi/4] */ - y = - ((((( 9.38540185543E-3 * zz - + 3.11992232697E-3) * zz - + 2.44301354525E-2) * zz - + 5.34112807005E-2) * zz - + 1.33387994085E-1) * zz - + 3.33331568548E-1) * zz * z - + z; - } -else - { - y = z; - } - -if( j & 2 ) - { - if( cotflg ) - y = -y; - else - { - if( y != 0.0 ) - { - y = -1.0/y; - } - else - { - mtherr( "tandgf", SING ); - y = MAXNUMF; - } - } - } -else - { - if( cotflg ) - { - if( y != 0.0 ) - y = 1.0/y; - else - { - mtherr( "cotdgf", SING ); - y = MAXNUMF; - } - } - } - -if( sign < 0 ) - y = -y; - -return( y ); -} - - -float tandgf( float x ) -{ - -return( tancotf(x,0) ); -} - -float cotdgf( float x ) -{ - -if( x == 0.0 ) - { - mtherr( "cotdgf", SING ); - return( MAXNUMF ); - } -return( tancotf(x,1) ); -} - diff --git a/libm/float/tanf.c b/libm/float/tanf.c deleted file mode 100644 index 5bbf43075..000000000 --- a/libm/float/tanf.c +++ /dev/null @@ -1,192 +0,0 @@ -/* tanf.c - * - * Circular tangent - * - * - * - * SYNOPSIS: - * - * float x, y, tanf(); - * - * y = tanf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the circular tangent of the radian argument x. - * - * Range reduction is modulo pi/4. A polynomial approximation - * is employed in the basic interval [0, pi/4]. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-4096 100000 3.3e-7 4.5e-8 - * - * ERROR MESSAGES: - * - * message condition value returned - * tanf total loss x > 2^24 0.0 - * - */ - /* cotf.c - * - * Circular cotangent - * - * - * - * SYNOPSIS: - * - * float x, y, cotf(); - * - * y = cotf( x ); - * - * - * - * DESCRIPTION: - * - * Returns the circular cotangent of the radian argument x. - * A common routine computes either the tangent or cotangent. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE +-4096 100000 3.0e-7 4.5e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * cot total loss x > 2^24 0.0 - * cot singularity x = 0 MAXNUMF - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1989 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision circular tangent - * test interval: [-pi/4, +pi/4] - * trials: 10000 - * peak relative error: 8.7e-8 - * rms relative error: 2.8e-8 - */ -#include <math.h> - -extern float MAXNUMF; - -static float DP1 = 0.78515625; -static float DP2 = 2.4187564849853515625e-4; -static float DP3 = 3.77489497744594108e-8; -float FOPI = 1.27323954473516; /* 4/pi */ -static float lossth = 8192.; -/*static float T24M1 = 16777215.;*/ - - -static float tancotf( float xx, int cotflg ) -{ -float x, y, z, zz; -long j; -int sign; - - -/* make argument positive but save the sign */ -if( xx < 0.0 ) - { - x = -xx; - sign = -1; - } -else - { - x = xx; - sign = 1; - } - -if( x > lossth ) - { - if( cotflg ) - mtherr( "cotf", TLOSS ); - else - mtherr( "tanf", TLOSS ); - return(0.0); - } - -/* compute x mod PIO4 */ -j = FOPI * x; /* integer part of x/(PI/4) */ -y = j; - -/* map zeros and singularities to origin */ -if( j & 1 ) - { - j += 1; - y += 1.0; - } - -z = ((x - y * DP1) - y * DP2) - y * DP3; - -zz = z * z; - -if( x > 1.0e-4 ) - { -/* 1.7e-8 relative error in [-pi/4, +pi/4] */ - y = - ((((( 9.38540185543E-3 * zz - + 3.11992232697E-3) * zz - + 2.44301354525E-2) * zz - + 5.34112807005E-2) * zz - + 1.33387994085E-1) * zz - + 3.33331568548E-1) * zz * z - + z; - } -else - { - y = z; - } - -if( j & 2 ) - { - if( cotflg ) - y = -y; - else - y = -1.0/y; - } -else - { - if( cotflg ) - y = 1.0/y; - } - -if( sign < 0 ) - y = -y; - -return( y ); -} - - -float tanf( float x ) -{ - -return( tancotf(x,0) ); -} - -float cotf( float x ) -{ - -if( x == 0.0 ) - { - mtherr( "cotf", SING ); - return( MAXNUMF ); - } -return( tancotf(x,1) ); -} - diff --git a/libm/float/tanhf.c b/libm/float/tanhf.c deleted file mode 100644 index 4636192c2..000000000 --- a/libm/float/tanhf.c +++ /dev/null @@ -1,88 +0,0 @@ -/* tanhf.c - * - * Hyperbolic tangent - * - * - * - * SYNOPSIS: - * - * float x, y, tanhf(); - * - * y = tanhf( x ); - * - * - * - * DESCRIPTION: - * - * Returns hyperbolic tangent of argument in the range MINLOG to - * MAXLOG. - * - * A polynomial approximation is used for |x| < 0.625. - * Otherwise, - * - * tanh(x) = sinh(x)/cosh(x) = 1 - 2/(exp(2x) + 1). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -2,2 100000 1.3e-7 2.6e-8 - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -/* Single precision hyperbolic tangent - * test interval: [-0.625, +0.625] - * trials: 10000 - * peak relative error: 7.2e-8 - * rms relative error: 2.6e-8 - */ -#include <math.h> - -extern float MAXLOGF; - -float expf( float ); - -float tanhf( float xx ) -{ -float x, z; - -if( xx < 0 ) - x = -xx; -else - x = xx; - -if( x > 0.5 * MAXLOGF ) - { - if( xx > 0 ) - return( 1.0 ); - else - return( -1.0 ); - } -if( x >= 0.625 ) - { - x = expf(x+x); - z = 1.0 - 2.0/(x + 1.0); - if( xx < 0 ) - z = -z; - } -else - { - z = x * x; - z = - (((( -5.70498872745E-3 * z - + 2.06390887954E-2) * z - - 5.37397155531E-2) * z - + 1.33314422036E-1) * z - - 3.33332819422E-1) * z * xx - + xx; - } -return( z ); -} diff --git a/libm/float/ynf.c b/libm/float/ynf.c deleted file mode 100644 index 55d984b26..000000000 --- a/libm/float/ynf.c +++ /dev/null @@ -1,120 +0,0 @@ -/* ynf.c - * - * Bessel function of second kind of integer order - * - * - * - * SYNOPSIS: - * - * float x, y, ynf(); - * int n; - * - * y = ynf( n, x ); - * - * - * - * DESCRIPTION: - * - * Returns Bessel function of order n, where n is a - * (possibly negative) integer. - * - * The function is evaluated by forward recurrence on - * n, starting with values computed by the routines - * y0() and y1(). - * - * If n = 0 or 1 the routine for y0 or y1 is called - * directly. - * - * - * - * ACCURACY: - * - * - * Absolute error, except relative when y > 1: - * - * arithmetic domain # trials peak rms - * IEEE 0, 30 10000 2.3e-6 3.4e-7 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * yn singularity x = 0 MAXNUMF - * yn overflow MAXNUMF - * - * Spot checked against tables for x, n between 0 and 100. - * - */ - -/* -Cephes Math Library Release 2.2: June, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> -extern float MAXNUMF, MAXLOGF; - -float y0f(float), y1f(float), logf(float); - -float ynf( int nn, float xx ) -{ -float x, an, anm1, anm2, r, xinv; -int k, n, sign; - -x = xx; -n = nn; -if( n < 0 ) - { - n = -n; - if( (n & 1) == 0 ) /* -1**n */ - sign = 1; - else - sign = -1; - } -else - sign = 1; - - -if( n == 0 ) - return( sign * y0f(x) ); -if( n == 1 ) - return( sign * y1f(x) ); - -/* test for overflow */ -if( x <= 0.0 ) - { - mtherr( "ynf", SING ); - return( -MAXNUMF ); - } -if( (x < 1.0) || (n > 29) ) - { - an = (float )n; - r = an * logf( an/x ); - if( r > MAXLOGF ) - { - mtherr( "ynf", OVERFLOW ); - return( -MAXNUMF ); - } - } - -/* forward recurrence on n */ - -anm2 = y0f(x); -anm1 = y1f(x); -k = 1; -r = 2 * k; -xinv = 1.0/x; -do - { - an = r * anm1 * xinv - anm2; - anm2 = anm1; - anm1 = an; - r += 2.0; - ++k; - } -while( k < n ); - - -return( sign * an ); -} diff --git a/libm/float/zetacf.c b/libm/float/zetacf.c deleted file mode 100644 index da2ace6a4..000000000 --- a/libm/float/zetacf.c +++ /dev/null @@ -1,266 +0,0 @@ - /* zetacf.c - * - * Riemann zeta function - * - * - * - * SYNOPSIS: - * - * float x, y, zetacf(); - * - * y = zetacf( x ); - * - * - * - * DESCRIPTION: - * - * - * - * inf. - * - -x - * zetac(x) = > k , x > 1, - * - - * k=2 - * - * is related to the Riemann zeta function by - * - * Riemann zeta(x) = zetac(x) + 1. - * - * Extension of the function definition for x < 1 is implemented. - * Zero is returned for x > log2(MAXNUM). - * - * An overflow error may occur for large negative x, due to the - * gamma function in the reflection formula. - * - * ACCURACY: - * - * Tabulated values have full machine accuracy. - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 1,50 30000 5.5e-7 7.5e-8 - * - * - */ - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> - - -/* Riemann zeta(x) - 1 - * for integer arguments between 0 and 30. - */ -static float azetacf[] = { --1.50000000000000000000E0, - 1.70141183460469231730E38, /* infinity. */ - 6.44934066848226436472E-1, - 2.02056903159594285400E-1, - 8.23232337111381915160E-2, - 3.69277551433699263314E-2, - 1.73430619844491397145E-2, - 8.34927738192282683980E-3, - 4.07735619794433937869E-3, - 2.00839282608221441785E-3, - 9.94575127818085337146E-4, - 4.94188604119464558702E-4, - 2.46086553308048298638E-4, - 1.22713347578489146752E-4, - 6.12481350587048292585E-5, - 3.05882363070204935517E-5, - 1.52822594086518717326E-5, - 7.63719763789976227360E-6, - 3.81729326499983985646E-6, - 1.90821271655393892566E-6, - 9.53962033872796113152E-7, - 4.76932986787806463117E-7, - 2.38450502727732990004E-7, - 1.19219925965311073068E-7, - 5.96081890512594796124E-8, - 2.98035035146522801861E-8, - 1.49015548283650412347E-8, - 7.45071178983542949198E-9, - 3.72533402478845705482E-9, - 1.86265972351304900640E-9, - 9.31327432419668182872E-10 -}; - - -/* 2**x (1 - 1/x) (zeta(x) - 1) = P(1/x)/Q(1/x), 1 <= x <= 10 */ -static float P[9] = { - 5.85746514569725319540E11, - 2.57534127756102572888E11, - 4.87781159567948256438E10, - 5.15399538023885770696E9, - 3.41646073514754094281E8, - 1.60837006880656492731E7, - 5.92785467342109522998E5, - 1.51129169964938823117E4, - 2.01822444485997955865E2, -}; -static float Q[8] = { -/* 1.00000000000000000000E0,*/ - 3.90497676373371157516E11, - 5.22858235368272161797E10, - 5.64451517271280543351E9, - 3.39006746015350418834E8, - 1.79410371500126453702E7, - 5.66666825131384797029E5, - 1.60382976810944131506E4, - 1.96436237223387314144E2, -}; - -/* log(zeta(x) - 1 - 2**-x), 10 <= x <= 50 */ -static float A[11] = { - 8.70728567484590192539E6, - 1.76506865670346462757E8, - 2.60889506707483264896E10, - 5.29806374009894791647E11, - 2.26888156119238241487E13, - 3.31884402932705083599E14, - 5.13778997975868230192E15, --1.98123688133907171455E15, --9.92763810039983572356E16, - 7.82905376180870586444E16, - 9.26786275768927717187E16, -}; -static float B[10] = { -/* 1.00000000000000000000E0,*/ --7.92625410563741062861E6, --1.60529969932920229676E8, --2.37669260975543221788E10, --4.80319584350455169857E11, --2.07820961754173320170E13, --2.96075404507272223680E14, --4.86299103694609136686E15, - 5.34589509675789930199E15, - 5.71464111092297631292E16, --1.79915597658676556828E16, -}; - -/* (1-x) (zeta(x) - 1), 0 <= x <= 1 */ - -static float R[6] = { --3.28717474506562731748E-1, - 1.55162528742623950834E1, --2.48762831680821954401E2, - 1.01050368053237678329E3, - 1.26726061410235149405E4, --1.11578094770515181334E5, -}; -static float S[5] = { -/* 1.00000000000000000000E0,*/ - 1.95107674914060531512E1, - 3.17710311750646984099E2, - 3.03835500874445748734E3, - 2.03665876435770579345E4, - 7.43853965136767874343E4, -}; - - -#define MAXL2 127 - -/* - * Riemann zeta function, minus one - */ - -extern float MACHEPF, PIO2F, MAXNUMF, PIF; - -#ifdef ANSIC -extern float sinf ( float xx ); -extern float floorf ( float x ); -extern float gammaf ( float xx ); -extern float powf ( float x, float y ); -extern float expf ( float xx ); -extern float polevlf ( float xx, float *coef, int N ); -extern float p1evlf ( float xx, float *coef, int N ); -#else -float sinf(), floorf(), gammaf(), powf(), expf(); -float polevlf(), p1evlf(); -#endif - -float zetacf(float xx) -{ -int i; -float x, a, b, s, w; - -x = xx; -if( x < 0.0 ) - { - if( x < -30.8148 ) - { - mtherr( "zetacf", OVERFLOW ); - return(0.0); - } - s = 1.0 - x; - w = zetacf( s ); - b = sinf(PIO2F*x) * powf(2.0*PIF, x) * gammaf(s) * (1.0 + w) / PIF; - return(b - 1.0); - } - -if( x >= MAXL2 ) - return(0.0); /* because first term is 2**-x */ - -/* Tabulated values for integer argument */ -w = floorf(x); -if( w == x ) - { - i = x; - if( i < 31 ) - { - return( azetacf[i] ); - } - } - - -if( x < 1.0 ) - { - w = 1.0 - x; - a = polevlf( x, R, 5 ) / ( w * p1evlf( x, S, 5 )); - return( a ); - } - -if( x == 1.0 ) - { - mtherr( "zetacf", SING ); - return( MAXNUMF ); - } - -if( x <= 10.0 ) - { - b = powf( 2.0, x ) * (x - 1.0); - w = 1.0/x; - s = (x * polevlf( w, P, 8 )) / (b * p1evlf( w, Q, 8 )); - return( s ); - } - -if( x <= 50.0 ) - { - b = powf( 2.0, -x ); - w = polevlf( x, A, 10 ) / p1evlf( x, B, 10 ); - w = expf(w) + b; - return(w); - } - - -/* Basic sum of inverse powers */ - - -s = 0.0; -a = 1.0; -do - { - a += 2.0; - b = powf( a, -x ); - s += b; - } -while( b/s > MACHEPF ); - -b = powf( 2.0, -x ); -s = (s + b)/(1.0-b); -return(s); -} diff --git a/libm/float/zetaf.c b/libm/float/zetaf.c deleted file mode 100644 index d01f1d2b2..000000000 --- a/libm/float/zetaf.c +++ /dev/null @@ -1,175 +0,0 @@ -/* zetaf.c - * - * Riemann zeta function of two arguments - * - * - * - * SYNOPSIS: - * - * float x, q, y, zetaf(); - * - * y = zetaf( x, q ); - * - * - * - * DESCRIPTION: - * - * - * - * inf. - * - -x - * zeta(x,q) = > (k+q) - * - - * k=0 - * - * where x > 1 and q is not a negative integer or zero. - * The Euler-Maclaurin summation formula is used to obtain - * the expansion - * - * n - * - -x - * zeta(x,q) = > (k+q) - * - - * k=1 - * - * 1-x inf. B x(x+1)...(x+2j) - * (n+q) 1 - 2j - * + --------- - ------- + > -------------------- - * x-1 x - x+2j+1 - * 2(n+q) j=1 (2j)! (n+q) - * - * where the B2j are Bernoulli numbers. Note that (see zetac.c) - * zeta(x,1) = zetac(x) + 1. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 0,25 10000 6.9e-7 1.0e-7 - * - * Large arguments may produce underflow in powf(), in which - * case the results are inaccurate. - * - * REFERENCE: - * - * Gradshteyn, I. S., and I. M. Ryzhik, Tables of Integrals, - * Series, and Products, p. 1073; Academic Press, 1980. - * - */ - -/* -Cephes Math Library Release 2.2: July, 1992 -Copyright 1984, 1987, 1992 by Stephen L. Moshier -Direct inquiries to 30 Frost Street, Cambridge, MA 02140 -*/ - -#include <math.h> -extern float MAXNUMF, MACHEPF; - -/* Expansion coefficients - * for Euler-Maclaurin summation formula - * (2k)! / B2k - * where B2k are Bernoulli numbers - */ -static float A[] = { -12.0, --720.0, -30240.0, --1209600.0, -47900160.0, --1.8924375803183791606e9, /*1.307674368e12/691*/ -7.47242496e10, --2.950130727918164224e12, /*1.067062284288e16/3617*/ -1.1646782814350067249e14, /*5.109094217170944e18/43867*/ --4.5979787224074726105e15, /*8.028576626982912e20/174611*/ -1.8152105401943546773e17, /*1.5511210043330985984e23/854513*/ --7.1661652561756670113e18 /*1.6938241367317436694528e27/236364091*/ -}; -/* 30 Nov 86 -- error in third coefficient fixed */ - - -#define fabsf(x) ( (x) < 0 ? -(x) : (x) ) - - -float powf( float, float ); -float zetaf(float xx, float qq) -{ -int i; -float x, q, a, b, k, s, w, t; - -x = xx; -q = qq; -if( x == 1.0 ) - return( MAXNUMF ); - -if( x < 1.0 ) - { - mtherr( "zetaf", DOMAIN ); - return(0.0); - } - - -/* Euler-Maclaurin summation formula */ -/* -if( x < 25.0 ) -{ -*/ -w = 9.0; -s = powf( q, -x ); -a = q; -for( i=0; i<9; i++ ) - { - a += 1.0; - b = powf( a, -x ); - s += b; - if( b/s < MACHEPF ) - goto done; - } - -w = a; -s += b*w/(x-1.0); -s -= 0.5 * b; -a = 1.0; -k = 0.0; -for( i=0; i<12; i++ ) - { - a *= x + k; - b /= w; - t = a*b/A[i]; - s = s + t; - t = fabsf(t/s); - if( t < MACHEPF ) - goto done; - k += 1.0; - a *= x + k; - b /= w; - k += 1.0; - } -done: -return(s); -/* -} -*/ - - -/* Basic sum of inverse powers */ -/* -pseres: - -s = powf( q, -x ); -a = q; -do - { - a += 2.0; - b = powf( a, -x ); - s += b; - } -while( b/s > MACHEPF ); - -b = powf( 2.0, -x ); -s = (s + b)/(1.0-b); -return(s); -*/ -} |