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authorEric Andersen <andersen@codepoet.org>2001-05-10 00:40:28 +0000
committerEric Andersen <andersen@codepoet.org>2001-05-10 00:40:28 +0000
commit1077fa4d772832f77a677ce7fb7c2d513b959e3f (patch)
tree579bee13fb0b58d2800206366ec2caecbb15f3fc /libm/float/README.txt
parent22358dd7ce7bb49792204b698f01a6f69b9c8e08 (diff)
uClibc now has a math library. muahahahaha!
-Erik
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+/* 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.
+ *
+ */