From 1077fa4d772832f77a677ce7fb7c2d513b959e3f Mon Sep 17 00:00:00 2001 From: Eric Andersen Date: Thu, 10 May 2001 00:40:28 +0000 Subject: uClibc now has a math library. muahahahaha! -Erik --- libm/float/Makefile | 62 + 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, 20650 insertions(+) create mode 100644 libm/float/Makefile create mode 100644 libm/float/README.txt create mode 100644 libm/float/acoshf.c create mode 100644 libm/float/airyf.c create mode 100644 libm/float/asinf.c create mode 100644 libm/float/asinhf.c create mode 100644 libm/float/atanf.c create mode 100644 libm/float/atanhf.c create mode 100644 libm/float/bdtrf.c create mode 100644 libm/float/betaf.c create mode 100644 libm/float/cbrtf.c create mode 100644 libm/float/chbevlf.c create mode 100644 libm/float/chdtrf.c create mode 100644 libm/float/clogf.c create mode 100644 libm/float/cmplxf.c create mode 100644 libm/float/constf.c create mode 100644 libm/float/coshf.c create mode 100644 libm/float/dawsnf.c create mode 100644 libm/float/ellief.c create mode 100644 libm/float/ellikf.c create mode 100644 libm/float/ellpef.c create mode 100644 libm/float/ellpjf.c create mode 100644 libm/float/ellpkf.c create mode 100644 libm/float/exp10f.c create mode 100644 libm/float/exp2f.c create mode 100644 libm/float/expf.c create mode 100644 libm/float/expnf.c create mode 100644 libm/float/facf.c create mode 100644 libm/float/fdtrf.c create mode 100644 libm/float/floorf.c create mode 100644 libm/float/fresnlf.c create mode 100644 libm/float/gammaf.c create mode 100644 libm/float/gdtrf.c create mode 100644 libm/float/hyp2f1f.c create mode 100644 libm/float/hypergf.c create mode 100644 libm/float/i0f.c create mode 100644 libm/float/i1f.c create mode 100644 libm/float/igamf.c create mode 100644 libm/float/igamif.c create mode 100644 libm/float/incbetf.c create mode 100644 libm/float/incbif.c create mode 100644 libm/float/ivf.c create mode 100644 libm/float/j0f.c create mode 100644 libm/float/j0tst.c create mode 100644 libm/float/j1f.c create mode 100644 libm/float/jnf.c create mode 100644 libm/float/jvf.c create mode 100644 libm/float/k0f.c create mode 100644 libm/float/k1f.c create mode 100644 libm/float/knf.c create mode 100644 libm/float/log10f.c create mode 100644 libm/float/log2f.c create mode 100644 libm/float/logf.c create mode 100644 libm/float/mtherr.c create mode 100644 libm/float/nantst.c create mode 100644 libm/float/nbdtrf.c create mode 100644 libm/float/ndtrf.c create mode 100644 libm/float/ndtrif.c create mode 100644 libm/float/pdtrf.c create mode 100644 libm/float/polevlf.c create mode 100644 libm/float/polynf.c create mode 100644 libm/float/powf.c create mode 100644 libm/float/powif.c create mode 100644 libm/float/powtst.c create mode 100644 libm/float/psif.c create mode 100644 libm/float/rgammaf.c create mode 100644 libm/float/setprec.c create mode 100644 libm/float/shichif.c create mode 100644 libm/float/sicif.c create mode 100644 libm/float/sindgf.c create mode 100644 libm/float/sinf.c create mode 100644 libm/float/sinhf.c create mode 100644 libm/float/spencef.c create mode 100644 libm/float/sqrtf.c create mode 100644 libm/float/stdtrf.c create mode 100644 libm/float/struvef.c create mode 100644 libm/float/tandgf.c create mode 100644 libm/float/tanf.c create mode 100644 libm/float/tanhf.c create mode 100644 libm/float/ynf.c create mode 100644 libm/float/zetacf.c create mode 100644 libm/float/zetaf.c (limited to 'libm/float') diff --git a/libm/float/Makefile b/libm/float/Makefile new file mode 100644 index 000000000..389ac1a5d --- /dev/null +++ b/libm/float/Makefile @@ -0,0 +1,62 @@ +# Makefile for uClibc's math library +# +# Copyright (C) 2001 by Lineo, inc. +# +# 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 +# +# Derived in part from the Linux-8086 C library, the GNU C Library, and several +# other sundry sources. Files within this library are copyright by their +# respective copyright holders. + +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) $(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 new file mode 100644 index 000000000..30a10b083 --- /dev/null +++ b/libm/float/README.txt @@ -0,0 +1,4721 @@ +/* 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 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 fun