/* lgam()
*
* Natural logarithm of gamma function
*
*
*
* SYNOPSIS:
*
* double x, y, __lgamma_r();
* int* sgngam;
* y = __lgamma_r( x, sgngam );
*
* double x, y, lgamma();
* y = lgamma( 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 sgngam.
*
* For arguments greater than 13, 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 MAXLGM return MAXNUM and an error
* message. MAXLGM = 2.035093e36 for DEC
* arithmetic or 2.556348e305 for IEEE arithmetic.
*
*
*
* ACCURACY:
*
*
* arithmetic domain # trials peak rms
* DEC 0, 3 7000 5.2e-17 1.3e-17
* DEC 2.718, 2.035e36 5000 3.9e-17 9.9e-18
* IEEE 0, 3 28000 5.4e-16 1.1e-16
* IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17
* The error criterion was relative when the function magnitude
* was greater than one but absolute when it was less than one.
*
* The following test used the relative error criterion, though
* at certain points the relative error could be much higher than
* indicated.
* IEEE -200, -4 10000 4.8e-16 1.3e-16
*
*/
/*
* Cephes Math Library Release 2.8: June, 2000
* Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
*/
/*
* 26-11-2002 Modified for mingw.
* Danny Smith <dannysmith@users.sourceforge.net>
*/
#ifndef __MINGW32__
#include "mconf.h"
#ifdef ANSIPROT
extern double pow ( double, double );
extern double log ( double );
extern double exp ( double );
extern double sin ( double );
extern double polevl ( double, void *, int );
extern double p1evl ( double, void *, int );
extern double floor ( double );
extern double fabs ( double );
extern int isnan ( double );
extern int isfinite ( double );
#else
double pow(), log(), exp(), sin(), polevl(), p1evl(), floor(), fabs();
int isnan(), isfinite();
#endif
#ifdef INFINITIES
extern double INFINITY;
#endif
#ifdef NANS
extern double NAN;
#endif
#else /* __MINGW32__ */
#include "cephes_mconf.h"
#endif /* __MINGW32__ */
/* A[]: Stirling's formula expansion of log gamma
* B[], C[]: log gamma function between 2 and 3
*/
#ifdef UNK
static double A[] = {
8.11614167470508450300E-4,
-5.95061904284301438324E-4,
7.93650340457716943945E-4,
-2.77777777730099687205E-3,
8.33333333333331927722E-2
};
static double B[] = {
-1.37825152569120859100E3,
-3.88016315134637840924E4,
-3.31612992738871184744E5,
-1.16237097492762307383E6,
-1.72173700820839662146E6,
-8.53555664245765465627E5
};
static double C[] = {
/* 1.00000000000000000000E0, */
-3.51815701436523470549E2,
-1.70642106651881159223E4,
-2.20528590553854454839E5,
-1.13933444367982507207E6,
-2.53252307177582951285E6,
-2.01889141433532773231E6
};
/* log( sqrt( 2*pi ) ) */
static double LS2PI = 0.91893853320467274178;
#define MAXLGM 2.556348e305
static double LOGPI = 1.14472988584940017414;
#endif
#ifdef DEC
static const unsigned short A[] = {
0035524,0141201,0034633,0031405,
0135433,0176755,0126007,0045030,
0035520,0006371,0003342,0172730,
0136066,0005540,0132605,0026407,
0037252,0125252,0125252,0125132
};
static const unsigned short B[] = {
0142654,0044014,0077633,0035410,
0144027,0110641,0125335,0144760,
0144641,0165637,0142204,0047447,
0145215,0162027,0146246,0155211,
0145322,0026110,0010317,0110130,
0145120,0061472,0120300,0025363
};
static const unsigned short C[] = {
/*0040200,0000000,0000000,0000000*/
0142257,0164150,0163630,0112622,
0143605,0050153,0156116,0135272,
0144527,0056045,0145642,0062332,
0145213,0012063,0106250,0001025,
0145432,0111254,0044577,0115142,
0145366,0071133,0050217,0005122
};
/* log( sqrt( 2*pi ) ) */
static const unsigned short LS2P[] = {040153,037616,041445,0172645,};
#define LS2PI *(double *)LS2P
#define MAXLGM 2.035093e36
static const unsigned short LPI[4] = {
0040222,0103202,0043475,0006750,
};
#define LOGPI *(double *)LPI
#endif
#ifdef IBMPC
static const unsigned short A[] = {
0x6661,0x2733,0x9850,0x3f4a,
0xe943,0xb580,0x7fbd,0xbf43,
0x5ebb,0x20dc,0x019f,0x3f4a,
0xa5a1,0x16b0,0xc16c,0xbf66,
0x554b,0x5555,0x5555,0x3fb5
};
static const unsigned short B[] = {
0x6761,0x8ff3,0x8901,0xc095,
0xb93e,0x355b,0xf234,0xc0e2,
0x89e5,0xf890,0x3d73,0xc114,
0xdb51,0xf994,0xbc82,0xc131,
0xf20b,0x0219,0x4589,0xc13a,
0x055e,0x5418,0x0c67,0xc12a
};
static const unsigned short C[] = {
/*0x0000,0x0000,0x0000,0x3ff0,*/
0x12b2,0x1cf3,0xfd0d,0xc075,
0xd757,0x7b89,0xaa0d,0xc0d0,
0x4c9b,0xb974,0xeb84,0xc10a,
0x0043,0x7195,0x6286,0xc131,
0xf34c,0x892f,0x5255,0xc143,
0xe14a,0x6a11,0xce4b,0xc13e
};
/* log( sqrt( 2*pi ) ) */
static const union
{
unsigned short s[4];
double d;
} ls2p = {{0xbeb5,0xc864,0x67f1,0x3fed}};
#define LS2PI (ls2p.d)
#define MAXLGM 2.556348e305
/* log (pi) */
static const union
{
unsigned short s[4];
double d;
} lpi = {{0xa1bd,0x48e7,0x50d0,0x3ff2}};
#define LOGPI (lpi.d)
#endif
#ifdef MIEEE
static const unsigned short A[] = {
0x3f4a,0x9850,0x2733,0x6661,
0xbf43,0x7fbd,0xb580,0xe943,
0x3f4a,0x019f,0x20dc,0x5ebb,
0xbf66,0xc16c,0x16b0,0xa5a1,
0x3fb5,0x5555,0x5555,0x554b
};
static const unsigned short B[] = {
0xc095,0x8901,0x8ff3,0x6761,
0xc0e2,0xf234,0x355b,0xb93e,
0xc114,0x3d73,0xf890,0x89e5,
0xc131,0xbc82,0xf994,0xdb51,
0xc13a,0x4589,0x0219,0xf20b,
0xc12a,0x0c67,0x5418,0x055e
};
static const unsigned short C[] = {
0xc075,0xfd0d,0x1cf3,0x12b2,
0xc0d0,0xaa0d,0x7b89,0xd757,
0xc10a,0xeb84,0xb974,0x4c9b,
0xc131,0x6286,0x7195,0x0043,
0xc143,0x5255,0x892f,0xf34c,
0xc13e,0xce4b,0x6a11,0xe14a
};
/* log( sqrt( 2*pi ) ) */
static const union
{
unsigned short s[4];
double d;
} ls2p = {{0x3fed,0x67f1,0xc864,0xbeb5}};
#define LS2PI ls2p.d
#define MAXLGM 2.556348e305
/* log (pi) */
static const union
{
unsigned short s[4];
double d;
} lpi = {{0x3ff2, 0x50d0, 0x48e7, 0xa1bd}};
#define LOGPI (lpi.d)
#endif
/* Logarithm of gamma function */
/* Reentrant version */
double __lgamma_r(double x, int* sgngam)
{
double p, q, u, w, z;
int i;
*sgngam = 1;
#ifdef NANS
if( isnan(x) )
return(x);
#endif
#ifdef INFINITIES
if( !isfinite(x) )
return(INFINITY);
#endif
if( x < -34.0 )
{
q = -x;
w = __lgamma_r(q, sgngam); /* note this modifies sgngam! */
p = floor(q);
if( p == q )
{
lgsing:
_SET_ERRNO(EDOM);
mtherr( "lgam", SING );
#ifdef INFINITIES
return (INFINITY);
#else
return (MAXNUM);
#endif
}
i = p;
if( (i & 1) == 0 )
*sgngam = -1;
else
*sgngam = 1;
z = q - p;
if( z > 0.5 )
{
p += 1.0;
z = p - q;
}
z = q * sin( PI * z );
if( z == 0.0 )
goto lgsing;
/* z = log(PI) - log( z ) - w;*/
z = LOGPI - log( z ) - w;
return( z );
}
if( x < 13.0 )
{
z = 1.0;
p = 0.0;
u = x;
while( u >= 3.0 )
{
p -= 1.0;
u = x + p;
z *= u;
}
while( u < 2.0 )
{
if( u == 0.0 )
goto lgsing;
z /= u;
p += 1.0;
u = x + p;
}
if( z < 0.0 )
{
*sgngam = -1;
z = -z;
}
else
*sgngam = 1;
if( u == 2.0 )
return( log(z) );
p -= 2.0;
x = x + p;
p = x * polevl( x, B, 5 ) / p1evl( x, C, 6);
return( log(z) + p );
}
if( x > MAXLGM )
{
_SET_ERRNO(ERANGE);
mtherr( "lgamma", OVERFLOW );
#ifdef INFINITIES
return( *sgngam * INFINITY );
#else
return( *sgngam * MAXNUM );
#endif
}
q = ( x - 0.5 ) * log(x) - x + LS2PI;
if( x > 1.0e8 )
return( q );
p = 1.0/(x*x);
if( x >= 1000.0 )
q += (( 7.9365079365079365079365e-4 * p
- 2.7777777777777777777778e-3) *p
+ 0.0833333333333333333333) / x;
else
q += polevl( p, A, 4 ) / x;
return( q );
}
/* This is the C99 version */
double lgamma(double x)
{
int local_sgngam=0;
return (__lgamma_r(x, &local_sgngam));
}