OpenLibm/ld80/e_tgammal.c

/*	$OpenBSD: e_tgammal.c,v 1.4 2013/11/12 20:35:19 martynas Exp $	*/

/*
 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
 *
 * Permission to use, copy, modify, and distribute this software for any
 * purpose with or without fee is hereby granted, provided that the above
 * copyright notice and this permission notice appear in all copies.
 *
 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
 */

/*							tgammal.c
 *
 *	Gamma function
 *
 *
 *
 * SYNOPSIS:
 *
 * long double x, y, tgammal();
 *
 * y = tgammal( x );
 *
 *
 *
 * DESCRIPTION:
 *
 * Returns gamma function of the argument.  The result is correctly
 * signed.  This variable is also filled in by the logarithmic gamma
 * function lgamma().
 *
 * Arguments |x| <= 13 are reduced by recurrence and the function
 * approximated by a rational function of degree 7/8 in the
 * interval (2,3).  Large arguments are handled by Stirling's
 * formula. Large negative arguments are made positive using
 * a reflection formula.
 *
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE     -40,+40      10000       3.6e-19     7.9e-20
 *    IEEE    -1755,+1755   10000       4.8e-18     6.5e-19
 *
 * Accuracy for large arguments is dominated by error in powl().
 *
 */

#include <float.h>
#include <openlibm_math.h>

#include "math_private.h"

/*
tgamma(x+2)  = tgamma(x+2) P(x)/Q(x)
0 <= x <= 1
Relative error
n=7, d=8
Peak error =  1.83e-20
Relative error spread =  8.4e-23
*/

static long double P[8] = {
 4.212760487471622013093E-5L,
 4.542931960608009155600E-4L,
 4.092666828394035500949E-3L,
 2.385363243461108252554E-2L,
 1.113062816019361559013E-1L,
 3.629515436640239168939E-1L,
 8.378004301573126728826E-1L,
 1.000000000000000000009E0L,
};
static long double Q[9] = {
-1.397148517476170440917E-5L,
 2.346584059160635244282E-4L,
-1.237799246653152231188E-3L,
-7.955933682494738320586E-4L,
 2.773706565840072979165E-2L,
-4.633887671244534213831E-2L,
-2.243510905670329164562E-1L,
 4.150160950588455434583E-1L,
 9.999999999999999999908E-1L,
};

/*
static long double P[] = {
-3.01525602666895735709e0L,
-3.25157411956062339893e1L,
-2.92929976820724030353e2L,
-1.70730828800510297666e3L,
-7.96667499622741999770e3L,
-2.59780216007146401957e4L,
-5.99650230220855581642e4L,
-7.15743521530849602425e4L
};
static long double Q[] = {
 1.00000000000000000000e0L,
-1.67955233807178858919e1L,
 8.85946791747759881659e1L,
 5.69440799097468430177e1L,
-1.98526250512761318471e3L,
 3.31667508019495079814e3L,
 1.60577839621734713377e4L,
-2.97045081369399940529e4L,
-7.15743521530849602412e4L
};
*/
#define MAXGAML 1755.455L
/*static const long double LOGPI = 1.14472988584940017414L;*/

/* Stirling's formula for the gamma function
tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x))
z(x) = x
13 <= x <= 1024
Relative error
n=8, d=0
Peak error =  9.44e-21
Relative error spread =  8.8e-4
*/

static long double STIR[9] = {
 7.147391378143610789273E-4L,
-2.363848809501759061727E-5L,
-5.950237554056330156018E-4L,
 6.989332260623193171870E-5L,
 7.840334842744753003862E-4L,
-2.294719747873185405699E-4L,
-2.681327161876304418288E-3L,
 3.472222222230075327854E-3L,
 8.333333333333331800504E-2L,
};

#define MAXSTIR 1024.0L
static const long double SQTPI = 2.50662827463100050242E0L;

/* 1/tgamma(x) = z P(z)
 * z(x) = 1/x
 * 0 < x < 0.03125
 * Peak relative error 4.2e-23
 */

static long double S[9] = {
-1.193945051381510095614E-3L,
 7.220599478036909672331E-3L,
-9.622023360406271645744E-3L,
-4.219773360705915470089E-2L,
 1.665386113720805206758E-1L,
-4.200263503403344054473E-2L,
-6.558780715202540684668E-1L,
 5.772156649015328608253E-1L,
 1.000000000000000000000E0L,
};

/* 1/tgamma(-x) = z P(z)
 * z(x) = 1/x
 * 0 < x < 0.03125
 * Peak relative error 5.16e-23
 * Relative error spread =  2.5e-24
 */

static long double SN[9] = {
 1.133374167243894382010E-3L,
 7.220837261893170325704E-3L,
 9.621911155035976733706E-3L,
-4.219773343731191721664E-2L,
-1.665386113944413519335E-1L,
-4.200263503402112910504E-2L,
 6.558780715202536547116E-1L,
 5.772156649015328608727E-1L,
-1.000000000000000000000E0L,
};

static const long double PIL = 3.1415926535897932384626L;

static long double stirf ( long double );

/* Gamma function computed by Stirling's formula.
 */
static long double stirf(long double x)
{
long double y, w, v;

w = 1.0L/x;
/* For large x, use rational coefficients from the analytical expansion.  */
if( x > 1024.0L )
	w = (((((6.97281375836585777429E-5L * w
		+ 7.84039221720066627474E-4L) * w
		- 2.29472093621399176955E-4L) * w
		- 2.68132716049382716049E-3L) * w
		+ 3.47222222222222222222E-3L) * w
		+ 8.33333333333333333333E-2L) * w
		+ 1.0L;
else
	w = 1.0L + w * __polevll( w, STIR, 8 );
y = expl(x);
if( x > MAXSTIR )
	{ /* Avoid overflow in pow() */
	v = powl( x, 0.5L * x - 0.25L );
	y = v * (v / y);
	}
else
	{
	y = powl( x, x - 0.5L ) / y;
	}
y = SQTPI * y * w;
return( y );
}

long double
tgammal(long double x)
{
long double p, q, z;
int i;

if( isnan(x) )
	return(NAN);
if(x == INFINITY)
	return(INFINITY);
if(x == -INFINITY)
	return(x - x);
if( x == 0.0L )
	return( 1.0L / x );
q = fabsl(x);

if( q > 13.0L )
	{
	int sign = 1;
	if( q > MAXGAML )
		goto goverf;
	if( x < 0.0L )
		{
		p = floorl(q);
		if( p == q )
			return (x - x) / (x - x);
		i = p;
		if( (i & 1) == 0 )
			sign = -1;
		z = q - p;
		if( z > 0.5L )
			{
			p += 1.0L;
			z = q - p;
			}
		z = q * sinl( PIL * z );
		z = fabsl(z) * stirf(q);
		if( z <= PIL/LDBL_MAX )
			{
goverf:
			return( sign * INFINITY);
			}
		z = PIL/z;
		}
	else
		{
		z = stirf(x);
		}
	return( sign * z );
	}

z = 1.0L;
while( x >= 3.0L )
	{
	x -= 1.0L;
	z *= x;
	}

while( x < -0.03125L )
	{
	z /= x;
	x += 1.0L;
	}

if( x <= 0.03125L )
	goto small;

while( x < 2.0L )
	{
	z /= x;
	x += 1.0L;
	}

if( x == 2.0L )
	return(z);

x -= 2.0L;
p = __polevll( x, P, 7 );
q = __polevll( x, Q, 8 );
z = z * p / q;
return z;

small:
if( x == 0.0L )
	return (x - x) / (x - x);
else
	{
	if( x < 0.0L )
		{
		x = -x;
		q = z / (x * __polevll( x, SN, 8 ));
		}
	else
		q = z / (x * __polevll( x, S, 8 ));
	}
return q;
}