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gamma.c

/*                                        gamma.c
 *
 *    Gamma function
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, gamma();
 * extern int sgngam;
 *
 * y = gamma( 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 sgngam.
 * This variable is also filled in by the logarithmic gamma
 * function lgam().
 *
 * Arguments |x| <= 34 are reduced by recurrence and the function
 * approximated by a rational function of degree 6/7 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
 *    DEC      -34, 34      10000       1.3e-16     2.5e-17
 *    IEEE    -170,-33      20000       2.3e-15     3.3e-16
 *    IEEE     -33,  33     20000       9.4e-16     2.2e-16
 *    IEEE      33, 171.6   20000       2.3e-15     3.2e-16
 *
 * Error for arguments outside the test range will be larger
 * owing to error amplification by the exponential function.
 *
 */
/*                                        lgam()
 *
 *    Natural logarithm of gamma function
 *
 *
 *
 * SYNOPSIS:
 *
 * double x, y, lgam();
 * extern int sgngam;
 *
 * y = lgam( 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 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
 *
 */

/*                                        gamma.c     */
/*    gamma function    */

/*
Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
*/


#include "mconf.h"

#ifdef UNK
static double P[] = {
  1.60119522476751861407E-4,
  1.19135147006586384913E-3,
  1.04213797561761569935E-2,
  4.76367800457137231464E-2,
  2.07448227648435975150E-1,
  4.94214826801497100753E-1,
  9.99999999999999996796E-1
};
static double Q[] = {
-2.31581873324120129819E-5,
 5.39605580493303397842E-4,
-4.45641913851797240494E-3,
 1.18139785222060435552E-2,
 3.58236398605498653373E-2,
-2.34591795718243348568E-1,
 7.14304917030273074085E-2,
 1.00000000000000000320E0
};
#define MAXGAM 171.624376956302725
static double LOGPI = 1.14472988584940017414;
#endif

#ifdef DEC
static unsigned short P[] = {
0035047,0162701,0146301,0005234,
0035634,0023437,0032065,0176530,
0036452,0137157,0047330,0122574,
0037103,0017310,0143041,0017232,
0037524,0066516,0162563,0164605,
0037775,0004671,0146237,0014222,
0040200,0000000,0000000,0000000
};
static unsigned short Q[] = {
0134302,0041724,0020006,0116565,
0035415,0072121,0044251,0025634,
0136222,0003447,0035205,0121114,
0036501,0107552,0154335,0104271,
0037022,0135717,0014776,0171471,
0137560,0034324,0165024,0037021,
0037222,0045046,0047151,0161213,
0040200,0000000,0000000,0000000
};
#define MAXGAM 34.84425627277176174
static unsigned short LPI[4] = {
0040222,0103202,0043475,0006750,
};
#define LOGPI *(double *)LPI
#endif

#ifdef IBMPC
static unsigned short P[] = {
0x2153,0x3998,0xfcb8,0x3f24,
0xbfab,0xe686,0x84e3,0x3f53,
0x14b0,0xe9db,0x57cd,0x3f85,
0x23d3,0x18c4,0x63d9,0x3fa8,
0x7d31,0xdcae,0x8da9,0x3fca,
0xe312,0x3993,0xa137,0x3fdf,
0x0000,0x0000,0x0000,0x3ff0
};
static unsigned short Q[] = {
0xd3af,0x8400,0x487a,0xbef8,
0x2573,0x2915,0xae8a,0x3f41,
0xb44a,0xe750,0x40e4,0xbf72,
0xb117,0x5b1b,0x31ed,0x3f88,
0xde67,0xe33f,0x5779,0x3fa2,
0x87c2,0x9d42,0x071a,0xbfce,
0x3c51,0xc9cd,0x4944,0x3fb2,
0x0000,0x0000,0x0000,0x3ff0
};
#define MAXGAM 171.624376956302725
static unsigned short LPI[4] = {
0xa1bd,0x48e7,0x50d0,0x3ff2,
};
#define LOGPI *(double *)LPI
#endif 

#ifdef MIEEE
static unsigned short P[] = {
0x3f24,0xfcb8,0x3998,0x2153,
0x3f53,0x84e3,0xe686,0xbfab,
0x3f85,0x57cd,0xe9db,0x14b0,
0x3fa8,0x63d9,0x18c4,0x23d3,
0x3fca,0x8da9,0xdcae,0x7d31,
0x3fdf,0xa137,0x3993,0xe312,
0x3ff0,0x0000,0x0000,0x0000
};
static unsigned short Q[] = {
0xbef8,0x487a,0x8400,0xd3af,
0x3f41,0xae8a,0x2915,0x2573,
0xbf72,0x40e4,0xe750,0xb44a,
0x3f88,0x31ed,0x5b1b,0xb117,
0x3fa2,0x5779,0xe33f,0xde67,
0xbfce,0x071a,0x9d42,0x87c2,
0x3fb2,0x4944,0xc9cd,0x3c51,
0x3ff0,0x0000,0x0000,0x0000
};
#define MAXGAM 171.624376956302725
static unsigned short LPI[4] = {
0x3ff2,0x50d0,0x48e7,0xa1bd,
};
#define LOGPI *(double *)LPI
#endif 

/* Stirling's formula for the gamma function */
#if UNK
static double STIR[5] = {
 7.87311395793093628397E-4,
-2.29549961613378126380E-4,
-2.68132617805781232825E-3,
 3.47222221605458667310E-3,
 8.33333333333482257126E-2,
};
#define MAXSTIR 143.01608
static double SQTPI = 2.50662827463100050242E0;
#endif
#if DEC
static unsigned short STIR[20] = {
0035516,0061622,0144553,0112224,
0135160,0131531,0037460,0165740,
0136057,0134460,0037242,0077270,
0036143,0107070,0156306,0027751,
0037252,0125252,0125252,0146064,
};
#define MAXSTIR 26.77
static unsigned short SQT[4] = {
0040440,0066230,0177661,0034055,
};
#define SQTPI *(double *)SQT
#endif
#if IBMPC
static unsigned short STIR[20] = {
0x7293,0x592d,0xcc72,0x3f49,
0x1d7c,0x27e6,0x166b,0xbf2e,
0x4fd7,0x07d4,0xf726,0xbf65,
0xc5fd,0x1b98,0x71c7,0x3f6c,
0x5986,0x5555,0x5555,0x3fb5,
};
#define MAXSTIR 143.01608
static unsigned short SQT[4] = {
0x2706,0x1ff6,0x0d93,0x4004,
};
#define SQTPI *(double *)SQT
#endif
#if MIEEE
static unsigned short STIR[20] = {
0x3f49,0xcc72,0x592d,0x7293,
0xbf2e,0x166b,0x27e6,0x1d7c,
0xbf65,0xf726,0x07d4,0x4fd7,
0x3f6c,0x71c7,0x1b98,0xc5fd,
0x3fb5,0x5555,0x5555,0x5986,
};
#define MAXSTIR 143.01608
static unsigned short SQT[4] = {
0x4004,0x0d93,0x1ff6,0x2706,
};
#define SQTPI *(double *)SQT
#endif

int sgngam = 0;
extern int sgngam;
extern double MAXLOG, MAXNUM, PI;
#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 );
static double stirf ( double );
double lgam ( double );
#else
double pow(), log(), exp(), sin(), polevl(), p1evl(), floor(), fabs();
int isnan(), isfinite();
static double stirf();
double lgam();
#endif
#ifdef INFINITIES
extern double INFINITY;
#endif
#ifdef NANS
extern double NAN;
#endif

/* Gamma function computed by Stirling's formula.
 * The polynomial STIR is valid for 33 <= x <= 172.
 */
static double stirf(x)
double x;
{
double y, w, v;

w = 1.0/x;
w = 1.0 + w * polevl( w, STIR, 4 );
y = exp(x);
if( x > MAXSTIR )
      { /* Avoid overflow in pow() */
      v = pow( x, 0.5 * x - 0.25 );
      y = v * (v / y);
      }
else
      {
      y = pow( x, x - 0.5 ) / y;
      }
y = SQTPI * y * w;
return( y );
}



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

sgngam = 1;
#ifdef NANS
if( isnan(x) )
      return(x);
#endif
#ifdef INFINITIES
#ifdef NANS
if( x == INFINITY )
      return(x);
if( x == -INFINITY )
      return(NAN);
#else
if( !isfinite(x) )
      return(x);
#endif
#endif
q = fabs(x);

if( q > 33.0 )
      {
      if( x < 0.0 )
            {
            p = floor(q);
            if( p == q )
                  {
#ifdef NANS
gamnan:
                  mtherr( "gamma", DOMAIN );
                  return (NAN);
#else
                  goto goverf;
#endif
                  }
            i = p;
            if( (i & 1) == 0 )
                  sgngam = -1;
            z = q - p;
            if( z > 0.5 )
                  {
                  p += 1.0;
                  z = q - p;
                  }
            z = q * sin( PI * z );
            if( z == 0.0 )
                  {
#ifdef INFINITIES
                  return( sgngam * INFINITY);
#else
goverf:
                  mtherr( "gamma", OVERFLOW );
                  return( sgngam * MAXNUM);
#endif
                  }
            z = fabs(z);
            z = PI/(z * stirf(q) );
            }
      else
            {
            z = stirf(x);
            }
      return( sgngam * z );
      }

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

while( x < 0.0 )
      {
      if( x > -1.E-9 )
            goto small;
      z /= x;
      x += 1.0;
      }

while( x < 2.0 )
      {
      if( x < 1.e-9 )
            goto small;
      z /= x;
      x += 1.0;
      }

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

x -= 2.0;
p = polevl( x, P, 6 );
q = polevl( x, Q, 7 );
return( z * p / q );

small:
if( x == 0.0 )
      {
#ifdef INFINITIES
#ifdef NANS
        goto gamnan;
#else
        return( INFINITY );
#endif
#else
      mtherr( "gamma", SING );
      return( MAXNUM );
#endif
      }
else
      return( z/((1.0 + 0.5772156649015329 * x) * x) );
}



/* 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
#endif

#ifdef DEC
static unsigned short A[] = {
0035524,0141201,0034633,0031405,
0135433,0176755,0126007,0045030,
0035520,0006371,0003342,0172730,
0136066,0005540,0132605,0026407,
0037252,0125252,0125252,0125132
};
static 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 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 unsigned short LS2P[] = {040153,037616,041445,0172645,};
#define LS2PI *(double *)LS2P
#define MAXLGM 2.035093e36
#endif

#ifdef IBMPC
static unsigned short A[] = {
0x6661,0x2733,0x9850,0x3f4a,
0xe943,0xb580,0x7fbd,0xbf43,
0x5ebb,0x20dc,0x019f,0x3f4a,
0xa5a1,0x16b0,0xc16c,0xbf66,
0x554b,0x5555,0x5555,0x3fb5
};
static 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 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 unsigned short LS2P[] = {
0xbeb5,0xc864,0x67f1,0x3fed
};
#define LS2PI *(double *)LS2P
#define MAXLGM 2.556348e305
#endif

#ifdef MIEEE
static unsigned short A[] = {
0x3f4a,0x9850,0x2733,0x6661,
0xbf43,0x7fbd,0xb580,0xe943,
0x3f4a,0x019f,0x20dc,0x5ebb,
0xbf66,0xc16c,0x16b0,0xa5a1,
0x3fb5,0x5555,0x5555,0x554b
};
static 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 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 unsigned short LS2P[] = {
0x3fed,0x67f1,0xc864,0xbeb5
};
#define LS2PI *(double *)LS2P
#define MAXLGM 2.556348e305
#endif


/* Logarithm of gamma function */


double lgam(x)
double x;
{
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 = lgam(q); /* note this modifies sgngam! */
      p = floor(q);
      if( p == q )
            {
lgsing:
#ifdef INFINITIES
            mtherr( "lgam", SING );
            return (INFINITY);
#else
            goto loverf;
#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 )
      {
#ifdef INFINITIES
      return( sgngam * INFINITY );
#else
loverf:
      mtherr( "lgam", OVERFLOW );
      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 );
}

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