From 7ce331c01ce6eb7b3f5c715a38a24359da9c6ee2 Mon Sep 17 00:00:00 2001 From: Eric Andersen Date: Thu, 22 Nov 2001 14:04:29 +0000 Subject: Totally rework the math library, this time based on the MacOs X math library (which is itself based on the math lib from FreeBSD). -Erik --- libm/float/Makefile | 59 - libm/float/README.txt | 4721 ------------------------------------------------- libm/float/acoshf.c | 97 - libm/float/airyf.c | 377 ---- libm/float/asinf.c | 186 -- libm/float/asinhf.c | 88 - libm/float/atanf.c | 190 -- libm/float/atanhf.c | 92 - libm/float/bdtrf.c | 247 --- libm/float/betaf.c | 122 -- libm/float/cbrtf.c | 119 -- libm/float/chbevlf.c | 86 - libm/float/chdtrf.c | 210 --- libm/float/clogf.c | 669 ------- libm/float/cmplxf.c | 407 ----- libm/float/constf.c | 20 - libm/float/coshf.c | 67 - libm/float/dawsnf.c | 168 -- libm/float/ellief.c | 115 -- libm/float/ellikf.c | 113 -- libm/float/ellpef.c | 105 -- libm/float/ellpjf.c | 161 -- libm/float/ellpkf.c | 128 -- libm/float/exp10f.c | 115 -- libm/float/exp2f.c | 116 -- libm/float/expf.c | 122 -- libm/float/expnf.c | 207 --- libm/float/facf.c | 106 -- libm/float/fdtrf.c | 214 --- libm/float/floorf.c | 526 ------ libm/float/fresnlf.c | 173 -- libm/float/gammaf.c | 423 ----- libm/float/gdtrf.c | 144 -- libm/float/hyp2f1f.c | 442 ----- libm/float/hypergf.c | 384 ---- libm/float/i0f.c | 160 -- libm/float/i1f.c | 177 -- libm/float/igamf.c | 223 --- libm/float/igamif.c | 112 -- libm/float/incbetf.c | 424 ----- libm/float/incbif.c | 197 --- libm/float/ivf.c | 114 -- libm/float/j0f.c | 228 --- libm/float/j0tst.c | 43 - libm/float/j1f.c | 211 --- libm/float/jnf.c | 124 -- libm/float/jvf.c | 848 --------- libm/float/k0f.c | 175 -- libm/float/k1f.c | 174 -- libm/float/knf.c | 252 --- libm/float/log10f.c | 129 -- libm/float/log2f.c | 129 -- libm/float/logf.c | 128 -- libm/float/mtherr.c | 99 -- libm/float/nantst.c | 54 - libm/float/nbdtrf.c | 141 -- libm/float/ndtrf.c | 281 --- libm/float/ndtrif.c | 186 -- libm/float/pdtrf.c | 188 -- libm/float/polevlf.c | 99 -- libm/float/polynf.c | 520 ------ libm/float/powf.c | 338 ---- libm/float/powif.c | 156 -- libm/float/powtst.c | 41 - libm/float/psif.c | 153 -- libm/float/rgammaf.c | 130 -- libm/float/setprec.c | 10 - libm/float/shichif.c | 212 --- libm/float/sicif.c | 279 --- libm/float/sindgf.c | 232 --- libm/float/sinf.c | 283 --- libm/float/sinhf.c | 87 - libm/float/spencef.c | 135 -- libm/float/sqrtf.c | 140 -- libm/float/stdtrf.c | 154 -- libm/float/struvef.c | 315 ---- libm/float/tandgf.c | 206 --- libm/float/tanf.c | 192 -- libm/float/tanhf.c | 88 - libm/float/ynf.c | 120 -- libm/float/zetacf.c | 266 --- libm/float/zetaf.c | 175 -- 82 files changed, 20647 deletions(-) delete mode 100644 libm/float/Makefile delete mode 100644 libm/float/README.txt delete mode 100644 libm/float/acoshf.c delete mode 100644 libm/float/airyf.c delete mode 100644 libm/float/asinf.c delete mode 100644 libm/float/asinhf.c delete mode 100644 libm/float/atanf.c delete mode 100644 libm/float/atanhf.c delete mode 100644 libm/float/bdtrf.c delete mode 100644 libm/float/betaf.c delete mode 100644 libm/float/cbrtf.c delete mode 100644 libm/float/chbevlf.c delete mode 100644 libm/float/chdtrf.c delete mode 100644 libm/float/clogf.c delete mode 100644 libm/float/cmplxf.c delete mode 100644 libm/float/constf.c delete mode 100644 libm/float/coshf.c delete mode 100644 libm/float/dawsnf.c delete mode 100644 libm/float/ellief.c delete mode 100644 libm/float/ellikf.c delete mode 100644 libm/float/ellpef.c delete mode 100644 libm/float/ellpjf.c delete mode 100644 libm/float/ellpkf.c delete mode 100644 libm/float/exp10f.c delete mode 100644 libm/float/exp2f.c delete mode 100644 libm/float/expf.c delete mode 100644 libm/float/expnf.c delete mode 100644 libm/float/facf.c delete mode 100644 libm/float/fdtrf.c delete mode 100644 libm/float/floorf.c delete mode 100644 libm/float/fresnlf.c delete mode 100644 libm/float/gammaf.c delete mode 100644 libm/float/gdtrf.c delete mode 100644 libm/float/hyp2f1f.c delete mode 100644 libm/float/hypergf.c delete mode 100644 libm/float/i0f.c delete mode 100644 libm/float/i1f.c delete mode 100644 libm/float/igamf.c delete mode 100644 libm/float/igamif.c delete mode 100644 libm/float/incbetf.c delete mode 100644 libm/float/incbif.c delete mode 100644 libm/float/ivf.c delete mode 100644 libm/float/j0f.c delete mode 100644 libm/float/j0tst.c delete mode 100644 libm/float/j1f.c delete mode 100644 libm/float/jnf.c delete mode 100644 libm/float/jvf.c delete mode 100644 libm/float/k0f.c delete mode 100644 libm/float/k1f.c delete mode 100644 libm/float/knf.c delete mode 100644 libm/float/log10f.c delete mode 100644 libm/float/log2f.c delete mode 100644 libm/float/logf.c delete mode 100644 libm/float/mtherr.c delete mode 100644 libm/float/nantst.c delete mode 100644 libm/float/nbdtrf.c delete mode 100644 libm/float/ndtrf.c delete mode 100644 libm/float/ndtrif.c delete mode 100644 libm/float/pdtrf.c delete mode 100644 libm/float/polevlf.c delete mode 100644 libm/float/polynf.c delete mode 100644 libm/float/powf.c delete mode 100644 libm/float/powif.c delete mode 100644 libm/float/powtst.c delete mode 100644 libm/float/psif.c delete mode 100644 libm/float/rgammaf.c delete mode 100644 libm/float/setprec.c delete mode 100644 libm/float/shichif.c delete mode 100644 libm/float/sicif.c delete mode 100644 libm/float/sindgf.c delete mode 100644 libm/float/sinf.c delete mode 100644 libm/float/sinhf.c delete mode 100644 libm/float/spencef.c delete mode 100644 libm/float/sqrtf.c delete mode 100644 libm/float/stdtrf.c delete mode 100644 libm/float/struvef.c delete mode 100644 libm/float/tandgf.c delete mode 100644 libm/float/tanf.c delete mode 100644 libm/float/tanhf.c delete mode 100644 libm/float/ynf.c delete mode 100644 libm/float/zetacf.c delete mode 100644 libm/float/zetaf.c (limited to 'libm/float') diff --git a/libm/float/Makefile b/libm/float/Makefile deleted file mode 100644 index 80f7aa1ff..000000000 --- a/libm/float/Makefile +++ /dev/null @@ -1,59 +0,0 @@ -# Makefile for uClibc's math library -# Copyright (C) 2001 by Lineo, inc. -# -# This math library is derived primarily from the Cephes Math Library, -# copyright by Stephen L. Moshier -# -# This program is free software; you can redistribute it and/or modify it under -# the terms of the GNU Library General Public License as published by the Free -# Software Foundation; either version 2 of the License, or (at your option) any -# later version. -# -# This program is distributed in the hope that it will be useful, but WITHOUT -# ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS -# FOR A PARTICULAR PURPOSE. See the GNU Library General Public License for more -# details. -# -# You should have received a copy of the GNU Library General Public License -# along with this program; if not, write to the Free Software Foundation, Inc., -# 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA -# - -TOPDIR=../../ -include $(TOPDIR)Rules.mak - -LIBM=../libm.a -TARGET_CC= $(TOPDIR)/extra/gcc-uClibc/$(TARGET_ARCH)-uclibc-gcc - -CSRC= acoshf.c airyf.c asinf.c asinhf.c atanf.c \ - atanhf.c bdtrf.c betaf.c cbrtf.c chbevlf.c chdtrf.c \ - clogf.c cmplxf.c constf.c coshf.c dawsnf.c ellief.c \ - ellikf.c ellpef.c ellpkf.c ellpjf.c expf.c exp2f.c \ - exp10f.c expnf.c facf.c fdtrf.c floorf.c fresnlf.c \ - gammaf.c gdtrf.c hypergf.c hyp2f1f.c igamf.c igamif.c \ - incbetf.c incbif.c i0f.c i1f.c ivf.c j0f.c j1f.c \ - jnf.c jvf.c k0f.c k1f.c knf.c logf.c log2f.c \ - log10f.c nbdtrf.c ndtrf.c ndtrif.c pdtrf.c polynf.c \ - powif.c powf.c psif.c rgammaf.c shichif.c sicif.c \ - sindgf.c sinf.c sinhf.c spencef.c sqrtf.c stdtrf.c \ - struvef.c tandgf.c tanf.c tanhf.c ynf.c zetaf.c \ - zetacf.c polevlf.c setprec.c mtherr.c -COBJS=$(patsubst %.c,%.o, $(CSRC)) -OBJS=$(COBJS) - -all: $(OBJS) $(LIBM) - -$(LIBM): ar-target - -ar-target: $(OBJS) - $(AR) $(ARFLAGS) $(LIBM) $(OBJS) - -$(COBJS): %.o : %.c - $(TARGET_CC) $(TARGET_CFLAGS) -c $< -o $@ - $(STRIPTOOL) -x -R .note -R .comment $*.o - -$(OBJ): Makefile - -clean: - rm -f *.[oa] *~ core - diff --git a/libm/float/README.txt b/libm/float/README.txt deleted file mode 100644 index 30a10b083..000000000 --- a/libm/float/README.txt +++ /dev/null @@ -1,4721 +0,0 @@ -/* acoshf.c - * - * Inverse hyperbolic cosine - * - * - * - * SYNOPSIS: - * - * float x, y, acoshf(); - * - * y = acoshf( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic cosine of argument. - * - * If 1 <= x < 1.5, a polynomial approximation - * - * sqrt(z) * P(z) - * - * where z = x-1, is used. Otherwise, - * - * acosh(x) = log( x + sqrt( (x-1)(x+1) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE 1,3 100000 1.8e-7 3.9e-8 - * IEEE 1,2000 100000 3.0e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * acoshf domain |x| < 1 0.0 - * - */ - -/* airy.c - * - * Airy function - * - * - * - * SYNOPSIS: - * - * float x, ai, aip, bi, bip; - * int airyf(); - * - * airyf( x, _&ai, _&aip, _&bi, _&bip ); - * - * - * - * DESCRIPTION: - * - * Solution of the differential equation - * - * y"(x) = xy. - * - * The function returns the two independent solutions Ai, Bi - * and their first derivatives Ai'(x), Bi'(x). - * - * Evaluation is by power series summation for small x, - * by rational minimax approximations for large x. - * - * - * - * ACCURACY: - * Error criterion is absolute when function <= 1, relative - * when function > 1, except * denotes relative error criterion. - * For large negative x, the absolute error increases as x^1.5. - * For large positive x, the relative error increases as x^1.5. - * - * Arithmetic domain function # trials peak rms - * IEEE -10, 0 Ai 50000 7.0e-7 1.2e-7 - * IEEE 0, 10 Ai 50000 9.9e-6* 6.8e-7* - * IEEE -10, 0 Ai' 50000 2.4e-6 3.5e-7 - * IEEE 0, 10 Ai' 50000 8.7e-6* 6.2e-7* - * IEEE -10, 10 Bi 100000 2.2e-6 2.6e-7 - * IEEE -10, 10 Bi' 50000 2.2e-6 3.5e-7 - * - */ - -/* asinf.c - * - * Inverse circular sine - * - * - * - * SYNOPSIS: - * - * float x, y, asinf(); - * - * y = asinf( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between -pi/2 and +pi/2 whose sine is x. - * - * A polynomial of the form x + x**3 P(x**2) - * is used for |x| in the interval [0, 0.5]. If |x| > 0.5 it is - * transformed by the identity - * - * asin(x) = pi/2 - 2 asin( sqrt( (1-x)/2 ) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -1, 1 100000 2.5e-7 5.0e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * asinf domain |x| > 1 0.0 - * - */ - /* acosf() - * - * Inverse circular cosine - * - * - * - * SYNOPSIS: - * - * float x, y, acosf(); - * - * y = acosf( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between -pi/2 and +pi/2 whose cosine - * is x. - * - * Analytically, acos(x) = pi/2 - asin(x). However if |x| is - * near 1, there is cancellation error in subtracting asin(x) - * from pi/2. Hence if x < -0.5, - * - * acos(x) = pi - 2.0 * asin( sqrt((1+x)/2) ); - * - * or if x > +0.5, - * - * acos(x) = 2.0 * asin( sqrt((1-x)/2) ). - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -1, 1 100000 1.4e-7 4.2e-8 - * - * - * ERROR MESSAGES: - * - * message condition value returned - * acosf domain |x| > 1 0.0 - */ - -/* asinhf.c - * - * Inverse hyperbolic sine - * - * - * - * SYNOPSIS: - * - * float x, y, asinhf(); - * - * y = asinhf( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic sine of argument. - * - * If |x| < 0.5, the function is approximated by a rational - * form x + x**3 P(x)/Q(x). Otherwise, - * - * asinh(x) = log( x + sqrt(1 + x*x) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -3,3 100000 2.4e-7 4.1e-8 - * - */ - -/* atanf.c - * - * Inverse circular tangent - * (arctangent) - * - * - * - * SYNOPSIS: - * - * float x, y, atanf(); - * - * y = atanf( x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle between -pi/2 and +pi/2 whose tangent - * is x. - * - * Range reduction is from four intervals into the interval - * from zero to tan( pi/8 ). A polynomial approximates - * the function in this basic interval. - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10, 10 100000 1.9e-7 4.1e-8 - * - */ - /* atan2f() - * - * Quadrant correct inverse circular tangent - * - * - * - * SYNOPSIS: - * - * float x, y, z, atan2f(); - * - * z = atan2f( y, x ); - * - * - * - * DESCRIPTION: - * - * Returns radian angle whose tangent is y/x. - * Define compile time symbol ANSIC = 1 for ANSI standard, - * range -PI < z <= +PI, args (y,x); else ANSIC = 0 for range - * 0 to 2PI, args (x,y). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -10, 10 100000 1.9e-7 4.1e-8 - * See atan.c. - * - */ - -/* atanhf.c - * - * Inverse hyperbolic tangent - * - * - * - * SYNOPSIS: - * - * float x, y, atanhf(); - * - * y = atanhf( x ); - * - * - * - * DESCRIPTION: - * - * Returns inverse hyperbolic tangent of argument in the range - * MINLOGF to MAXLOGF. - * - * If |x| < 0.5, a polynomial approximation is used. - * Otherwise, - * atanh(x) = 0.5 * log( (1+x)/(1-x) ). - * - * - * - * ACCURACY: - * - * Relative error: - * arithmetic domain # trials peak rms - * IEEE -1,1 100000 1.4e-7 3.1e-8 - * - */ - -/* bdtrf.c - * - * Binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, bdtrf(); - * - * y = bdtrf( k, n, p ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms 0 through k of the Binomial - * probability density: - * - * k - * -- ( n ) j n-j - * > ( ) p (1-p) - * -- ( j ) - * j=0 - * - * The terms are not summed directly; instead the incomplete - * beta integral is employed, according to the formula - * - * y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * - * - * ACCURACY: - * - * Relative error (p varies from 0 to 1): - * arithmetic domain # trials peak rms - * IEEE 0,100 2000 6.9e-5 1.1e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * bdtrf domain k < 0 0.0 - * n < k - * x < 0, x > 1 - * - */ - /* bdtrcf() - * - * Complemented binomial distribution - * - * - * - * SYNOPSIS: - * - * int k, n; - * float p, y, bdtrcf(); - * - * y = bdtrcf( k, n, p ); - * - * - * - * DESCRIPTION: - * - * Returns the sum of the terms k+1 through n of the Binomial - * probability density: - * - * n - * -- ( n ) j n-j - * > ( ) p (1-p) - * -- ( j ) - * j=k+1 - * - * The terms are not summed directly; instead the incomplete - * beta integral is employed, according to the formula - * - * y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ). - * - * The arguments must be positive, with p ranging from 0 to 1. - * - * - * - * ACCURACY: - * - * Relative error (p varies from 0 to 1): - * arithmetic domain # trials peak rms - * IEEE 0,100 2000 6.0e-5 1.2e-5 - * - * ERROR MESSAGES: - * - * message condition value returned - * bdtrcf domain x<0, x>1, n 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 - * -