From 7ce331c01ce6eb7b3f5c715a38a24359da9c6ee2 Mon Sep 17 00:00:00 2001
From: Eric Andersen <andersen@codepoet.org>
Date: Thu, 22 Nov 2001 14:04:29 +0000
Subject: 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
---
libm/float/Makefile | 59 -
libm/float/README.txt | 4721 -------------------------------------------------
libm/float/acoshf.c | 97 -
libm/float/airyf.c | 377 ----
libm/float/asinf.c | 186 --
libm/float/asinhf.c | 88 -
libm/float/atanf.c | 190 --
libm/float/atanhf.c | 92 -
libm/float/bdtrf.c | 247 ---
libm/float/betaf.c | 122 --
libm/float/cbrtf.c | 119 --
libm/float/chbevlf.c | 86 -
libm/float/chdtrf.c | 210 ---
libm/float/clogf.c | 669 -------
libm/float/cmplxf.c | 407 -----
libm/float/constf.c | 20 -
libm/float/coshf.c | 67 -
libm/float/dawsnf.c | 168 --
libm/float/ellief.c | 115 --
libm/float/ellikf.c | 113 --
libm/float/ellpef.c | 105 --
libm/float/ellpjf.c | 161 --
libm/float/ellpkf.c | 128 --
libm/float/exp10f.c | 115 --
libm/float/exp2f.c | 116 --
libm/float/expf.c | 122 --
libm/float/expnf.c | 207 ---
libm/float/facf.c | 106 --
libm/float/fdtrf.c | 214 ---
libm/float/floorf.c | 526 ------
libm/float/fresnlf.c | 173 --
libm/float/gammaf.c | 423 -----
libm/float/gdtrf.c | 144 --
libm/float/hyp2f1f.c | 442 -----
libm/float/hypergf.c | 384 ----
libm/float/i0f.c | 160 --
libm/float/i1f.c | 177 --
libm/float/igamf.c | 223 ---
libm/float/igamif.c | 112 --
libm/float/incbetf.c | 424 -----
libm/float/incbif.c | 197 ---
libm/float/ivf.c | 114 --
libm/float/j0f.c | 228 ---
libm/float/j0tst.c | 43 -
libm/float/j1f.c | 211 ---
libm/float/jnf.c | 124 --
libm/float/jvf.c | 848 ---------
libm/float/k0f.c | 175 --
libm/float/k1f.c | 174 --
libm/float/knf.c | 252 ---
libm/float/log10f.c | 129 --
libm/float/log2f.c | 129 --
libm/float/logf.c | 128 --
libm/float/mtherr.c | 99 --
libm/float/nantst.c | 54 -
libm/float/nbdtrf.c | 141 --
libm/float/ndtrf.c | 281 ---
libm/float/ndtrif.c | 186 --
libm/float/pdtrf.c | 188 --
libm/float/polevlf.c | 99 --
libm/float/polynf.c | 520 ------
libm/float/powf.c | 338 ----
libm/float/powif.c | 156 --
libm/float/powtst.c | 41 -
libm/float/psif.c | 153 --
libm/float/rgammaf.c | 130 --
libm/float/setprec.c | 10 -
libm/float/shichif.c | 212 ---
libm/float/sicif.c | 279 ---
libm/float/sindgf.c | 232 ---
libm/float/sinf.c | 283 ---
libm/float/sinhf.c | 87 -
libm/float/spencef.c | 135 --
libm/float/sqrtf.c | 140 --
libm/float/stdtrf.c | 154 --
libm/float/struvef.c | 315 ----
libm/float/tandgf.c | 206 ---
libm/float/tanf.c | 192 --
libm/float/tanhf.c | 88 -
libm/float/ynf.c | 120 --
libm/float/zetacf.c | 266 ---
libm/float/zetaf.c | 175 --
82 files changed, 20647 deletions(-)
delete mode 100644 libm/float/Makefile
delete mode 100644 libm/float/README.txt
delete mode 100644 libm/float/acoshf.c
delete mode 100644 libm/float/airyf.c
delete mode 100644 libm/float/asinf.c
delete mode 100644 libm/float/asinhf.c
delete mode 100644 libm/float/atanf.c
delete mode 100644 libm/float/atanhf.c
delete mode 100644 libm/float/bdtrf.c
delete mode 100644 libm/float/betaf.c
delete mode 100644 libm/float/cbrtf.c
delete mode 100644 libm/float/chbevlf.c
delete mode 100644 libm/float/chdtrf.c
delete mode 100644 libm/float/clogf.c
delete mode 100644 libm/float/cmplxf.c
delete mode 100644 libm/float/constf.c
delete mode 100644 libm/float/coshf.c
delete mode 100644 libm/float/dawsnf.c
delete mode 100644 libm/float/ellief.c
delete mode 100644 libm/float/ellikf.c
delete mode 100644 libm/float/ellpef.c
delete mode 100644 libm/float/ellpjf.c
delete mode 100644 libm/float/ellpkf.c
delete mode 100644 libm/float/exp10f.c
delete mode 100644 libm/float/exp2f.c
delete mode 100644 libm/float/expf.c
delete mode 100644 libm/float/expnf.c
delete mode 100644 libm/float/facf.c
delete mode 100644 libm/float/fdtrf.c
delete mode 100644 libm/float/floorf.c
delete mode 100644 libm/float/fresnlf.c
delete mode 100644 libm/float/gammaf.c
delete mode 100644 libm/float/gdtrf.c
delete mode 100644 libm/float/hyp2f1f.c
delete mode 100644 libm/float/hypergf.c
delete mode 100644 libm/float/i0f.c
delete mode 100644 libm/float/i1f.c
delete mode 100644 libm/float/igamf.c
delete mode 100644 libm/float/igamif.c
delete mode 100644 libm/float/incbetf.c
delete mode 100644 libm/float/incbif.c
delete mode 100644 libm/float/ivf.c
delete mode 100644 libm/float/j0f.c
delete mode 100644 libm/float/j0tst.c
delete mode 100644 libm/float/j1f.c
delete mode 100644 libm/float/jnf.c
delete mode 100644 libm/float/jvf.c
delete mode 100644 libm/float/k0f.c
delete mode 100644 libm/float/k1f.c
delete mode 100644 libm/float/knf.c
delete mode 100644 libm/float/log10f.c
delete mode 100644 libm/float/log2f.c
delete mode 100644 libm/float/logf.c
delete mode 100644 libm/float/mtherr.c
delete mode 100644 libm/float/nantst.c
delete mode 100644 libm/float/nbdtrf.c
delete mode 100644 libm/float/ndtrf.c
delete mode 100644 libm/float/ndtrif.c
delete mode 100644 libm/float/pdtrf.c
delete mode 100644 libm/float/polevlf.c
delete mode 100644 libm/float/polynf.c
delete mode 100644 libm/float/powf.c
delete mode 100644 libm/float/powif.c
delete mode 100644 libm/float/powtst.c
delete mode 100644 libm/float/psif.c
delete mode 100644 libm/float/rgammaf.c
delete mode 100644 libm/float/setprec.c
delete mode 100644 libm/float/shichif.c
delete mode 100644 libm/float/sicif.c
delete mode 100644 libm/float/sindgf.c
delete mode 100644 libm/float/sinf.c
delete mode 100644 libm/float/sinhf.c
delete mode 100644 libm/float/spencef.c
delete mode 100644 libm/float/sqrtf.c
delete mode 100644 libm/float/stdtrf.c
delete mode 100644 libm/float/struvef.c
delete mode 100644 libm/float/tandgf.c
delete mode 100644 libm/float/tanf.c
delete mode 100644 libm/float/tanhf.c
delete mode 100644 libm/float/ynf.c
delete mode 100644 libm/float/zetacf.c
delete mode 100644 libm/float/zetaf.c
(limited to 'libm/float')
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
- *
-