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/*							lgaml()

 *

 *	Natural logarithm of gamma function

 *

 *

 *

 * SYNOPSIS:

 *

 * long double x, y, __lgammal_r();

 * int* sgngaml;

 * y = __lgammal_r( x, sgngaml );

 *

 * long double x, y, lgammal();

 * y = lgammal( x);

 *

 *

 *

 * DESCRIPTION:

 *

 * Returns the base e (2.718...) logarithm of the absolute

 * value of the gamma function of the argument. In the reentrant

 * version, the sign (+1 or -1) of the gamma function is returned

 * in the variable referenced by sgngaml.

 *

 * For arguments greater than 33, the logarithm of the gamma

 * function is approximated by the logarithmic version of

 * Stirling's formula using a polynomial approximation of

 * degree 4. Arguments between -33 and +33 are reduced by

 * recurrence to the interval [2,3] of a rational approximation.

 * The cosecant reflection formula is employed for arguments

 * less than -33.

 *

 * Arguments greater than MAXLGML (10^4928) return MAXNUML.

 *

 *

 *

 * ACCURACY:

 *

 *

 * arithmetic      domain        # trials     peak         rms

 *    IEEE         -40, 40        100000     2.2e-19     4.6e-20

 *    IEEE    10^-2000,10^+2000    20000     1.6e-19     3.3e-20

 * The error criterion was relative when the function magnitude

 * was greater than one but absolute when it was less than one.

 *

 */

/*

 * Copyright 1994 by Stephen L. Moshier

 */

/*

 * 26-11-2002 Modified for mingw.

 * Danny Smith 

 */

#ifndef __MINGW32__

#include "mconf.h"

#ifdef ANSIPROT

extern long double fabsl ( long double );

extern long double lgaml ( long double );

extern long double logl ( long double );

extern long double expl ( long double );

extern long double gammal ( long double );

extern long double sinl ( long double );

extern long double floorl ( long double );

extern long double powl ( long double, long double );

extern long double polevll ( long double, void *, int );

extern long double p1evll ( long double, void *, int );

extern int isnanl ( long double );

extern int isfinitel ( long double );

#else

long double fabsl(), lgaml(), logl(), expl(), gammal(), sinl();

long double floorl(), powl(), polevll(), p1evll(), isnanl(), isfinitel();

#endif

#ifdef INFINITIES

extern long double INFINITYL;

#endif

#ifdef NANS

extern long double NANL;

#endif

#else /* __MINGW32__ */

#include "cephes_mconf.h"

#endif /* __MINGW32__ */

#if UNK

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,

};

#endif

#if IBMPC

static const unsigned short S[] = {

0xbaeb,0xd6d3,0x25e5,0x9c7e,0xbff5, XPD

0xfe9a,0xceb4,0xc74e,0xec9a,0x3ff7, XPD

0x9225,0xdfef,0xb0e9,0x9da5,0xbff8, XPD

0x10b0,0xec17,0x87dc,0xacd7,0xbffa, XPD

0x6b8d,0x7515,0x1905,0xaa89,0x3ffc, XPD

0xf183,0x126b,0xf47d,0xac0a,0xbffa, XPD

0x7bf6,0x57d1,0xa013,0xa7e7,0xbffe, XPD

0xc7a9,0x7db0,0x67e3,0x93c4,0x3ffe, XPD

0x0000,0x0000,0x0000,0x8000,0x3fff, XPD

};

#endif

#if MIEEE

static long S[27] = {

0xbff50000,0x9c7e25e5,0xd6d3baeb,

0x3ff70000,0xec9ac74e,0xceb4fe9a,

0xbff80000,0x9da5b0e9,0xdfef9225,

0xbffa0000,0xacd787dc,0xec1710b0,

0x3ffc0000,0xaa891905,0x75156b8d,

0xbffa0000,0xac0af47d,0x126bf183,

0xbffe0000,0xa7e7a013,0x57d17bf6,

0x3ffe0000,0x93c467e3,0x7db0c7a9,

0x3fff0000,0x80000000,0x00000000,

};

#endif

#if UNK

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,

};

#endif

#if IBMPC

static const unsigned SN[] = {

0x5dd1,0x02de,0xb9f7,0x948d,0x3ff5, XPD

0x989b,0xdd68,0xc5f1,0xec9c,0x3ff7, XPD

0x2ca1,0x18f0,0x386f,0x9da5,0x3ff8, XPD

0x783f,0x41dd,0x87d1,0xacd7,0xbffa, XPD

0x7a5b,0xd76d,0x1905,0xaa89,0xbffc, XPD

0x7f64,0x1234,0xf47d,0xac0a,0xbffa, XPD

0x5e26,0x57d1,0xa013,0xa7e7,0x3ffe, XPD

0xc7aa,0x7db0,0x67e3,0x93c4,0x3ffe, XPD

0x0000,0x0000,0x0000,0x8000,0xbfff, XPD

};

#endif

#if MIEEE

static long SN[27] = {

0x3ff50000,0x948db9f7,0x02de5dd1,

0x3ff70000,0xec9cc5f1,0xdd68989b,

0x3ff80000,0x9da5386f,0x18f02ca1,

0xbffa0000,0xacd787d1,0x41dd783f,

0xbffc0000,0xaa891905,0xd76d7a5b,

0xbffa0000,0xac0af47d,0x12347f64,

0x3ffe0000,0xa7e7a013,0x57d15e26,

0x3ffe0000,0x93c467e3,0x7db0c7aa,

0xbfff0000,0x80000000,0x00000000,

};

#endif

/* A[]: Stirling's formula expansion of log gamma

 * B[], C[]: log gamma function between 2 and 3

 */

/* log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x A(1/x^2)

 * x >= 8

 * Peak relative error 1.51e-21

 * Relative spread of error peaks 5.67e-21

 */

#if UNK

static long double A[7] = {

 4.885026142432270781165E-3L,

-1.880801938119376907179E-3L,

 8.412723297322498080632E-4L,

-5.952345851765688514613E-4L,

 7.936507795855070755671E-4L,

-2.777777777750349603440E-3L,

 8.333333333333331447505E-2L,

};

#endif

#if IBMPC

static const unsigned short A[] = {

0xd984,0xcc08,0x91c2,0xa012,0x3ff7, XPD

0x3d91,0x0304,0x3da1,0xf685,0xbff5, XPD

0x3bdc,0xaad1,0xd492,0xdc88,0x3ff4, XPD

0x8b20,0x9fce,0x844e,0x9c09,0xbff4, XPD

0xf8f2,0x30e5,0x0092,0xd00d,0x3ff4, XPD

0x4d88,0x03a8,0x60b6,0xb60b,0xbff6, XPD

0x9fcc,0xaaaa,0xaaaa,0xaaaa,0x3ffb, XPD

};

#endif

#if MIEEE

static long A[21] = {

0x3ff70000,0xa01291c2,0xcc08d984,

0xbff50000,0xf6853da1,0x03043d91,

0x3ff40000,0xdc88d492,0xaad13bdc,

0xbff40000,0x9c09844e,0x9fce8b20,

0x3ff40000,0xd00d0092,0x30e5f8f2,

0xbff60000,0xb60b60b6,0x03a84d88,

0x3ffb0000,0xaaaaaaaa,0xaaaa9fcc,

};

#endif

/* log gamma(x+2) = x B(x)/C(x)

 * 0 <= x <= 1

 * Peak relative error 7.16e-22

 * Relative spread of error peaks 4.78e-20

 */

#if UNK

static long double B[7] = {

-2.163690827643812857640E3L,

-8.723871522843511459790E4L,

-1.104326814691464261197E6L,

-6.111225012005214299996E6L,

-1.625568062543700591014E7L,

-2.003937418103815175475E7L,

-8.875666783650703802159E6L,

};

static long double C[7] = {

/* 1.000000000000000000000E0L,*/

-5.139481484435370143617E2L,

-3.403570840534304670537E4L,

-6.227441164066219501697E5L,

-4.814940379411882186630E6L,

-1.785433287045078156959E7L,

-3.138646407656182662088E7L,

-2.099336717757895876142E7L,

};

#endif

#if IBMPC

static const unsigned short B[] = {

0x9557,0x4995,0x0da1,0x873b,0xc00a, XPD

0xfe44,0x9af8,0x5b8c,0xaa63,0xc00f, XPD

0x5aa8,0x7cf5,0x3684,0x86ce,0xc013, XPD

0x259a,0x258c,0xf206,0xba7f,0xc015, XPD

0xbe18,0x1ca3,0xc0a0,0xf80a,0xc016, XPD

0x168f,0x2c42,0x6717,0x98e3,0xc017, XPD

0x2051,0x9d55,0x92c8,0x876e,0xc016, XPD

};

static const unsigned short C[] = {

/*0x0000,0x0000,0x0000,0x8000,0x3fff, XPD*/

0xaa77,0xcf2f,0xae76,0x807c,0xc008, XPD

0xb280,0x0d74,0xb55a,0x84f3,0xc00e, XPD

0xa505,0xcd30,0x81dc,0x9809,0xc012, XPD

0x3369,0x4246,0xb8c2,0x92f0,0xc015, XPD

0x63cf,0x6aee,0xbe6f,0x8837,0xc017, XPD

0x26bb,0xccc7,0xb009,0xef75,0xc017, XPD

0x462b,0xbae8,0xab96,0xa02a,0xc017, XPD

};

#endif

#if MIEEE

static long B[21] = {

0xc00a0000,0x873b0da1,0x49959557,

0xc00f0000,0xaa635b8c,0x9af8fe44,

0xc0130000,0x86ce3684,0x7cf55aa8,

0xc0150000,0xba7ff206,0x258c259a,

0xc0160000,0xf80ac0a0,0x1ca3be18,

0xc0170000,0x98e36717,0x2c42168f,

0xc0160000,0x876e92c8,0x9d552051,

};

static long C[21] = {

/*0x3fff0000,0x80000000,0x00000000,*/

0xc0080000,0x807cae76,0xcf2faa77,

0xc00e0000,0x84f3b55a,0x0d74b280,

0xc0120000,0x980981dc,0xcd30a505,

0xc0150000,0x92f0b8c2,0x42463369,

0xc0170000,0x8837be6f,0x6aee63cf,

0xc0170000,0xef75b009,0xccc726bb,

0xc0170000,0xa02aab96,0xbae8462b,

};

#endif

/* log( sqrt( 2*pi ) ) */

static const long double LS2PI  =  0.91893853320467274178L;

#define MAXLGM 1.04848146839019521116e+4928L

/* Logarithm of gamma function */

/* Reentrant version */

long double __lgammal_r(long double x, int* sgngaml)

{

long double p, q, w, z, f, nx;

int i;

*sgngaml = 1;

#ifdef NANS

if( isnanl(x) )

	return(NANL);

#endif

#ifdef INFINITIES

if( !isfinitel(x) )

	return(INFINITYL);

#endif

if( x < -34.0L )

	{

	q = -x;

	w = __lgammal_r(q, sgngaml); /* note this modifies sgngam! */

	p = floorl(q);

	if( p == q )

		{

lgsing:

		_SET_ERRNO(EDOM);

		mtherr( "lgammal", SING );

#ifdef INFINITIES

		return (INFINITYL);

#else

		return (MAXNUML);

#endif

		}

	i = p;

	if( (i & 1) == 0 )

		*sgngaml = -1;

	else

		*sgngaml = 1;

	z = q - p;

	if( z > 0.5L )

		{

		p += 1.0L;

		z = p - q;

		}

	z = q * sinl( PIL * z );

	if( z == 0.0L )

		goto lgsing;

/*	z = LOGPI - logl( z ) - w; */

	z = logl( PIL/z ) - w;

	return( z );

	}

if( x < 13.0L )

	{

	z = 1.0L;

	nx = floorl( x +  0.5L );

	f = x - nx;

	while( x >= 3.0L )

		{

		nx -= 1.0L;

		x = nx + f;

		z *= x;

		}

	while( x < 2.0L )

		{

		if( fabsl(x) <= 0.03125 )

			goto lsmall;

		z /= nx +  f;

		nx += 1.0L;

		x = nx + f;

		}

	if( z < 0.0L )

		{

		*sgngaml = -1;

		z = -z;

		}

	else

		*sgngaml = 1;

	if( x == 2.0L )

		return( logl(z) );

	x = (nx - 2.0L) + f;

	p = x * polevll( x, B, 6 ) / p1evll( x, C, 7);

	return( logl(z) + p );

	}

if( x > MAXLGM )

	{

	_SET_ERRNO(ERANGE);

	mtherr( "lgammal", OVERFLOW );

#ifdef INFINITIES

	return( *sgngaml * INFINITYL );

#else

	return( *sgngaml * MAXNUML );

#endif

	}

q = ( x - 0.5L ) * logl(x) - x + LS2PI;

if( x > 1.0e10L )

	return(q);

p = 1.0L/(x*x);

q += polevll( p, A, 6 ) / x;

return( q );

lsmall:

if( x == 0.0L )

	goto lgsing;

if( x < 0.0L )

	{

	x = -x;

	q = z / (x * polevll( x, SN, 8 ));

	}

else

	q = z / (x * polevll( x, S, 8 ));

if( q < 0.0L )

	{

	*sgngaml = -1;

	q = -q;

	}

else

	*sgngaml = 1;

q = logl( q );

return(q);

}

/* This is the C99 version */

long double lgammal(long double x)

{

  int local_sgngaml=0;

  return (__lgammal_r(x, &local_sgngaml));

}