314 lines
6.4 KiB
C
314 lines
6.4 KiB
C
/* $OpenBSD: e_tgammal.c,v 1.4 2013/11/12 20:35:19 martynas Exp $ */
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/*
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* Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
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*
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* Permission to use, copy, modify, and distribute this software for any
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* purpose with or without fee is hereby granted, provided that the above
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* copyright notice and this permission notice appear in all copies.
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*
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* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
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* ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
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* ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
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* OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
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*/
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/* tgammal.c
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*
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* Gamma function
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*
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*
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*
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* SYNOPSIS:
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*
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* long double x, y, tgammal();
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*
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* y = tgammal( x );
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*
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*
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*
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* DESCRIPTION:
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*
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* Returns gamma function of the argument. The result is correctly
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* signed. This variable is also filled in by the logarithmic gamma
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* function lgamma().
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*
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* Arguments |x| <= 13 are reduced by recurrence and the function
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* approximated by a rational function of degree 7/8 in the
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* interval (2,3). Large arguments are handled by Stirling's
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* formula. Large negative arguments are made positive using
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* a reflection formula.
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*
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*
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* ACCURACY:
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*
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* Relative error:
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* arithmetic domain # trials peak rms
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* IEEE -40,+40 10000 3.6e-19 7.9e-20
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* IEEE -1755,+1755 10000 4.8e-18 6.5e-19
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*
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* Accuracy for large arguments is dominated by error in powl().
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*
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*/
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#include <float.h>
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#include <openlibm_math.h>
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#include "math_private.h"
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/*
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tgamma(x+2) = tgamma(x+2) P(x)/Q(x)
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0 <= x <= 1
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Relative error
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n=7, d=8
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Peak error = 1.83e-20
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Relative error spread = 8.4e-23
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*/
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static long double P[8] = {
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4.212760487471622013093E-5L,
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4.542931960608009155600E-4L,
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4.092666828394035500949E-3L,
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2.385363243461108252554E-2L,
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1.113062816019361559013E-1L,
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3.629515436640239168939E-1L,
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8.378004301573126728826E-1L,
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1.000000000000000000009E0L,
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};
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static long double Q[9] = {
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-1.397148517476170440917E-5L,
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2.346584059160635244282E-4L,
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-1.237799246653152231188E-3L,
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-7.955933682494738320586E-4L,
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2.773706565840072979165E-2L,
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-4.633887671244534213831E-2L,
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-2.243510905670329164562E-1L,
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4.150160950588455434583E-1L,
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9.999999999999999999908E-1L,
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};
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/*
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static long double P[] = {
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-3.01525602666895735709e0L,
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-3.25157411956062339893e1L,
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-2.92929976820724030353e2L,
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-1.70730828800510297666e3L,
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-7.96667499622741999770e3L,
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-2.59780216007146401957e4L,
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-5.99650230220855581642e4L,
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-7.15743521530849602425e4L
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};
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static long double Q[] = {
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1.00000000000000000000e0L,
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-1.67955233807178858919e1L,
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8.85946791747759881659e1L,
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5.69440799097468430177e1L,
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-1.98526250512761318471e3L,
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3.31667508019495079814e3L,
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1.60577839621734713377e4L,
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-2.97045081369399940529e4L,
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-7.15743521530849602412e4L
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};
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*/
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#define MAXGAML 1755.455L
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/*static const long double LOGPI = 1.14472988584940017414L;*/
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/* Stirling's formula for the gamma function
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tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x))
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z(x) = x
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13 <= x <= 1024
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Relative error
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n=8, d=0
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Peak error = 9.44e-21
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Relative error spread = 8.8e-4
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*/
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static long double STIR[9] = {
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7.147391378143610789273E-4L,
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-2.363848809501759061727E-5L,
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-5.950237554056330156018E-4L,
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6.989332260623193171870E-5L,
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7.840334842744753003862E-4L,
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-2.294719747873185405699E-4L,
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-2.681327161876304418288E-3L,
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3.472222222230075327854E-3L,
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8.333333333333331800504E-2L,
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};
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#define MAXSTIR 1024.0L
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static const long double SQTPI = 2.50662827463100050242E0L;
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/* 1/tgamma(x) = z P(z)
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* z(x) = 1/x
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* 0 < x < 0.03125
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* Peak relative error 4.2e-23
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*/
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static long double S[9] = {
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-1.193945051381510095614E-3L,
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7.220599478036909672331E-3L,
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-9.622023360406271645744E-3L,
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-4.219773360705915470089E-2L,
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1.665386113720805206758E-1L,
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-4.200263503403344054473E-2L,
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-6.558780715202540684668E-1L,
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5.772156649015328608253E-1L,
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1.000000000000000000000E0L,
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};
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/* 1/tgamma(-x) = z P(z)
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* z(x) = 1/x
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* 0 < x < 0.03125
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* Peak relative error 5.16e-23
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* Relative error spread = 2.5e-24
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*/
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static long double SN[9] = {
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1.133374167243894382010E-3L,
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7.220837261893170325704E-3L,
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9.621911155035976733706E-3L,
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-4.219773343731191721664E-2L,
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-1.665386113944413519335E-1L,
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-4.200263503402112910504E-2L,
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6.558780715202536547116E-1L,
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5.772156649015328608727E-1L,
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-1.000000000000000000000E0L,
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};
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static const long double PIL = 3.1415926535897932384626L;
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static long double stirf ( long double );
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/* Gamma function computed by Stirling's formula.
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*/
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static long double stirf(long double x)
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{
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long double y, w, v;
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w = 1.0L/x;
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/* For large x, use rational coefficients from the analytical expansion. */
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if( x > 1024.0L )
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w = (((((6.97281375836585777429E-5L * w
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+ 7.84039221720066627474E-4L) * w
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- 2.29472093621399176955E-4L) * w
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- 2.68132716049382716049E-3L) * w
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+ 3.47222222222222222222E-3L) * w
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+ 8.33333333333333333333E-2L) * w
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+ 1.0L;
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else
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w = 1.0L + w * __polevll( w, STIR, 8 );
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y = expl(x);
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if( x > MAXSTIR )
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{ /* Avoid overflow in pow() */
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v = powl( x, 0.5L * x - 0.25L );
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y = v * (v / y);
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}
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else
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{
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y = powl( x, x - 0.5L ) / y;
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}
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y = SQTPI * y * w;
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return( y );
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}
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long double
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tgammal(long double x)
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{
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long double p, q, z;
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int i;
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if( isnan(x) )
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return(NAN);
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if(x == INFINITY)
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return(INFINITY);
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if(x == -INFINITY)
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return(x - x);
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if( x == 0.0L )
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return( 1.0L / x );
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q = fabsl(x);
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if( q > 13.0L )
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{
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int sign = 1;
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if( q > MAXGAML )
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goto goverf;
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if( x < 0.0L )
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{
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p = floorl(q);
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if( p == q )
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return (x - x) / (x - x);
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i = p;
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if( (i & 1) == 0 )
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sign = -1;
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z = q - p;
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if( z > 0.5L )
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{
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p += 1.0L;
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z = q - p;
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}
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z = q * sinl( PIL * z );
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z = fabsl(z) * stirf(q);
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if( z <= PIL/LDBL_MAX )
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{
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goverf:
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return( sign * INFINITY);
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}
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z = PIL/z;
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}
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else
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{
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z = stirf(x);
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}
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return( sign * z );
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}
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z = 1.0L;
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while( x >= 3.0L )
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{
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x -= 1.0L;
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z *= x;
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}
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while( x < -0.03125L )
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{
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z /= x;
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x += 1.0L;
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}
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if( x <= 0.03125L )
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goto small;
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while( x < 2.0L )
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{
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z /= x;
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x += 1.0L;
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}
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if( x == 2.0L )
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return(z);
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x -= 2.0L;
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p = __polevll( x, P, 7 );
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q = __polevll( x, Q, 8 );
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z = z * p / q;
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return z;
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small:
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if( x == 0.0L )
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return (x - x) / (x - x);
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else
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{
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if( x < 0.0L )
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{
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x = -x;
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q = z / (x * __polevll( x, SN, 8 ));
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}
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else
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q = z / (x * __polevll( x, S, 8 ));
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}
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return q;
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}
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