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cephes.c

#include <stdio.h>
#include "mconf.h"

double p1evl(double, double [], int);
double polevl(double, double [], int);

#ifndef ANSIPROT
double lgam(), exp(), log(), fabs(), igam(), igamc();
#endif

int merror = 0;

/* Notice: the order of appearance of the following
 * messages is bound to the error codes defined
 * in mconf.h.
 */
static char *ermsg[7] = {
"unknown",      /* error code 0 */
"domain",       /* error code 1 */
"singularity",  /* et seq.      */
"overflow",
"underflow",
"total loss of precision",
"partial loss of precision"
};



/*                                                      const.c
 *
 *     Globally declared constants
 *
 *
 *
 * SYNOPSIS:
 *
 * extern double nameofconstant;
 *
 *
 *
 *
 * DESCRIPTION:
 *
 * This file contains a number of mathematical constants and
 * also some needed size parameters of the computer arithmetic.
 * The values are supplied as arrays of hexadecimal integers
 * for IEEE arithmetic; arrays of octal constants for DEC
 * arithmetic; and in a normal decimal scientific notation for
 * other machines.  The particular notation used is determined
 * by a symbol (DEC, IBMPC, or UNK) defined in the include file
 * mconf.h.
 *
 * The default size parameters are as follows.
 *
 * For DEC and UNK modes:
 * MACHEP =  1.38777878078144567553E-17       2**-56
 * MAXLOG =  8.8029691931113054295988E1       log(2**127)
 * MINLOG = -8.872283911167299960540E1        log(2**-128)
 * MAXNUM =  1.701411834604692317316873e38    2**127
 *
 * For IEEE arithmetic (IBMPC):
 * MACHEP =  1.11022302462515654042E-16       2**-53
 * MAXLOG =  7.09782712893383996843E2         log(2**1024)
 * MINLOG = -7.08396418532264106224E2         log(2**-1022)
 * MAXNUM =  1.7976931348623158E308           2**1024
 *
 * The global symbols for mathematical constants are
 * PI     =  3.14159265358979323846           pi
 * PIO2   =  1.57079632679489661923           pi/2
 * PIO4   =  7.85398163397448309616E-1        pi/4
 * SQRT2  =  1.41421356237309504880           sqrt(2)
 * SQRTH  =  7.07106781186547524401E-1        sqrt(2)/2
 * LOG2E  =  1.4426950408889634073599         1/log(2)
 * SQ2OPI =  7.9788456080286535587989E-1      sqrt( 2/pi )
 * LOGE2  =  6.93147180559945309417E-1        log(2)
 * LOGSQ2 =  3.46573590279972654709E-1        log(2)/2
 * THPIO4 =  2.35619449019234492885           3*pi/4
 * TWOOPI =  6.36619772367581343075535E-1     2/pi
 *
 * These lists are subject to change.
 */

/*                                                      const.c */

/*
Cephes Math Library Release 2.3:  March, 1995
Copyright 1984, 1995 by Stephen L. Moshier
*/

#ifdef UNK
#if 1
double MACHEP =  1.11022302462515654042E-16;   /* 2**-53 */
#else
double MACHEP =  1.38777878078144567553E-17;   /* 2**-56 */
#endif
double UFLOWTHRESH =  2.22507385850720138309E-308; /* 2**-1022 */
#ifdef DENORMAL
double MAXLOG =  7.09782712893383996732E2;     /* log(MAXNUM) */
/* double MINLOG = -7.44440071921381262314E2; */     /* log(2**-1074) */
double MINLOG = -7.451332191019412076235E2;     /* log(2**-1075) */
#else
double MAXLOG =  7.08396418532264106224E2;     /* log 2**1022 */
double MINLOG = -7.08396418532264106224E2;     /* log 2**-1022 */
#endif
double MAXNUM =  1.79769313486231570815E308;    /* 2**1024*(1-MACHEP) */
double PI     =  3.14159265358979323846;       /* pi */
double PIO2   =  1.57079632679489661923;       /* pi/2 */
double PIO4   =  7.85398163397448309616E-1;    /* pi/4 */
double SQRT2  =  1.41421356237309504880;       /* sqrt(2) */
double SQRTH  =  7.07106781186547524401E-1;    /* sqrt(2)/2 */
double LOG2E  =  1.4426950408889634073599;     /* 1/log(2) */
double SQ2OPI =  7.9788456080286535587989E-1;  /* sqrt( 2/pi ) */
double LOGE2  =  6.93147180559945309417E-1;    /* log(2) */
double LOGSQ2 =  3.46573590279972654709E-1;    /* log(2)/2 */
double THPIO4 =  2.35619449019234492885;       /* 3*pi/4 */
double TWOOPI =  6.36619772367581343075535E-1; /* 2/pi */
#ifdef INFINITIES
double INFINITY = 1.0/0.0;  /* 99e999; */
#else
double INFINITY =  1.79769313486231570815E308;    /* 2**1024*(1-MACHEP) */
#endif
#ifdef NANS
double NAN = 1.0/0.0 - 1.0/0.0;
#else
double NAN = 0.0;
#endif
#ifdef MINUSZERO
double NEGZERO = -0.0;
#else
double NEGZERO = 0.0;
#endif
#endif

#ifdef IBMPC
                       /* 2**-53 =  1.11022302462515654042E-16 */
unsigned short MACHEP[4] = {0x0000,0x0000,0x0000,0x3ca0};
unsigned short UFLOWTHRESH[4] = {0x0000,0x0000,0x0000,0x0010};
#ifdef DENORMAL
                       /* log(MAXNUM) =  7.09782712893383996732224E2 */
unsigned short MAXLOG[4] = {0x39ef,0xfefa,0x2e42,0x4086};
                       /* log(2**-1074) = - -7.44440071921381262314E2 */
/*unsigned short MINLOG[4] = {0x71c3,0x446d,0x4385,0xc087};*/
unsigned short MINLOG[4] = {0x3052,0xd52d,0x4910,0xc087};
#else
                       /* log(2**1022) =   7.08396418532264106224E2 */
unsigned short MAXLOG[4] = {0xbcd2,0xdd7a,0x232b,0x4086};
                       /* log(2**-1022) = - 7.08396418532264106224E2 */
unsigned short MINLOG[4] = {0xbcd2,0xdd7a,0x232b,0xc086};
#endif
                       /* 2**1024*(1-MACHEP) =  1.7976931348623158E308 */
unsigned short MAXNUM[4] = {0xffff,0xffff,0xffff,0x7fef};
unsigned short PI[4]     = {0x2d18,0x5444,0x21fb,0x4009};
unsigned short PIO2[4]   = {0x2d18,0x5444,0x21fb,0x3ff9};
unsigned short PIO4[4]   = {0x2d18,0x5444,0x21fb,0x3fe9};
unsigned short SQRT2[4]  = {0x3bcd,0x667f,0xa09e,0x3ff6};
unsigned short SQRTH[4]  = {0x3bcd,0x667f,0xa09e,0x3fe6};
unsigned short LOG2E[4]  = {0x82fe,0x652b,0x1547,0x3ff7};
unsigned short SQ2OPI[4] = {0x3651,0x33d4,0x8845,0x3fe9};
unsigned short LOGE2[4]  = {0x39ef,0xfefa,0x2e42,0x3fe6};
unsigned short LOGSQ2[4] = {0x39ef,0xfefa,0x2e42,0x3fd6};
unsigned short THPIO4[4] = {0x21d2,0x7f33,0xd97c,0x4002};
unsigned short TWOOPI[4] = {0xc883,0x6dc9,0x5f30,0x3fe4};
#ifdef INFINITIES
unsigned short INFINITY[4] = {0x0000,0x0000,0x0000,0x7ff0};
#else
unsigned short INFINITY[4] = {0xffff,0xffff,0xffff,0x7fef};
#endif
#ifdef NANS
unsigned short NAN[4] = {0x0000,0x0000,0x0000,0x7ffc};
#else
unsigned short NAN[4] = {0x0000,0x0000,0x0000,0x0000};
#endif
#ifdef MINUSZERO
unsigned short NEGZERO[4] = {0x0000,0x0000,0x0000,0x8000};
#else
unsigned short NEGZERO[4] = {0x0000,0x0000,0x0000,0x0000};
#endif
#endif

#ifdef MIEEE
                       /* 2**-53 =  1.11022302462515654042E-16 */
unsigned short MACHEP[4] = {0x3ca0,0x0000,0x0000,0x0000};
unsigned short UFLOWTHRESH[4] = {0x0010,0x0000,0x0000,0x0000};
#ifdef DENORMAL
                       /* log(2**1024) =   7.09782712893383996843E2 */
unsigned short MAXLOG[4] = {0x4086,0x2e42,0xfefa,0x39ef};
                       /* log(2**-1074) = - -7.44440071921381262314E2 */
/* unsigned short MINLOG[4] = {0xc087,0x4385,0x446d,0x71c3}; */
unsigned short MINLOG[4] = {0xc087,0x4910,0xd52d,0x3052};
#else
                       /* log(2**1022) =  7.08396418532264106224E2 */
unsigned short MAXLOG[4] = {0x4086,0x232b,0xdd7a,0xbcd2};
                       /* log(2**-1022) = - 7.08396418532264106224E2 */
unsigned short MINLOG[4] = {0xc086,0x232b,0xdd7a,0xbcd2};
#endif
                       /* 2**1024*(1-MACHEP) =  1.7976931348623158E308 */
unsigned short MAXNUM[4] = {0x7fef,0xffff,0xffff,0xffff};
unsigned short PI[4]     = {0x4009,0x21fb,0x5444,0x2d18};
unsigned short PIO2[4]   = {0x3ff9,0x21fb,0x5444,0x2d18};
unsigned short PIO4[4]   = {0x3fe9,0x21fb,0x5444,0x2d18};
unsigned short SQRT2[4]  = {0x3ff6,0xa09e,0x667f,0x3bcd};
unsigned short SQRTH[4]  = {0x3fe6,0xa09e,0x667f,0x3bcd};
unsigned short LOG2E[4]  = {0x3ff7,0x1547,0x652b,0x82fe};
unsigned short SQ2OPI[4] = {0x3fe9,0x8845,0x33d4,0x3651};
unsigned short LOGE2[4]  = {0x3fe6,0x2e42,0xfefa,0x39ef};
unsigned short LOGSQ2[4] = {0x3fd6,0x2e42,0xfefa,0x39ef};
unsigned short THPIO4[4] = {0x4002,0xd97c,0x7f33,0x21d2};
unsigned short TWOOPI[4] = {0x3fe4,0x5f30,0x6dc9,0xc883};
#ifdef INFINITIES
unsigned short INFINITY[4] = {0x7ff0,0x0000,0x0000,0x0000};
#else
unsigned short INFINITY[4] = {0x7fef,0xffff,0xffff,0xffff};
#endif
#ifdef NANS
unsigned short NAN[4] = {0x7ff8,0x0000,0x0000,0x0000};
#else
unsigned short NAN[4] = {0x0000,0x0000,0x0000,0x0000};
#endif
#ifdef MINUSZERO
unsigned short NEGZERO[4] = {0x8000,0x0000,0x0000,0x0000};
#else
unsigned short NEGZERO[4] = {0x0000,0x0000,0x0000,0x0000};
#endif
#endif

#ifdef DEC
                       /* 2**-56 =  1.38777878078144567553E-17 */
unsigned short MACHEP[4] = {0022200,0000000,0000000,0000000};
unsigned short UFLOWTHRESH[4] = {0x0080,0x0000,0x0000,0x0000};
                       /* log 2**127 = 88.029691931113054295988 */
unsigned short MAXLOG[4] = {041660,007463,0143742,025733,};
                       /* log 2**-128 = -88.72283911167299960540 */
unsigned short MINLOG[4] = {0141661,071027,0173721,0147572,};
                       /* 2**127 = 1.701411834604692317316873e38 */
unsigned short MAXNUM[4] = {077777,0177777,0177777,0177777,};
unsigned short PI[4]     = {040511,007732,0121041,064302,};
unsigned short PIO2[4]   = {040311,007732,0121041,064302,};
unsigned short PIO4[4]   = {040111,007732,0121041,064302,};
unsigned short SQRT2[4]  = {040265,002363,031771,0157145,};
unsigned short SQRTH[4]  = {040065,002363,031771,0157144,};
unsigned short LOG2E[4]  = {040270,0125073,024534,013761,};
unsigned short SQ2OPI[4] = {040114,041051,0117241,0131204,};
unsigned short LOGE2[4]  = {040061,071027,0173721,0147572,};
unsigned short LOGSQ2[4] = {037661,071027,0173721,0147572,};
unsigned short THPIO4[4] = {040426,0145743,0174631,007222,};
unsigned short TWOOPI[4] = {040042,0174603,067116,042025,};
/* Approximate infinity by MAXNUM.  */
unsigned short INFINITY[4] = {077777,0177777,0177777,0177777,};
unsigned short NAN[4] = {0000000,0000000,0000000,0000000};
#ifdef MINUSZERO
unsigned short NEGZERO[4] = {0000000,0000000,0000000,0100000};
#else
unsigned short NEGZERO[4] = {0000000,0000000,0000000,0000000};
#endif
#endif

#ifndef UNK
extern unsigned short MACHEP[];
extern unsigned short UFLOWTHRESH[];
extern unsigned short MAXLOG[];
extern unsigned short UNDLOG[];
extern unsigned short MINLOG[];
extern unsigned short MAXNUM[];
extern unsigned short PI[];
extern unsigned short PIO2[];
extern unsigned short PIO4[];
extern unsigned short SQRT2[];
extern unsigned short SQRTH[];
extern unsigned short LOG2E[];
extern unsigned short SQ2OPI[];
extern unsigned short LOGE2[];
extern unsigned short LOGSQ2[];
extern unsigned short THPIO4[];
extern unsigned short TWOOPI[];
extern unsigned short INFINITY[];
extern unsigned short NAN[];
extern unsigned short NEGZERO[];
#endif

extern double MACHEP, MAXLOG; 
static double big = 4.503599627370496e15;
static double biginv =  2.22044604925031308085e-16;

#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;
#ifndef ANSIPROT
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

/*                                                      igam.c
 *
 *     Incomplete gamma integral
 *
 *
 *
 * SYNOPSIS:
 *
 * double a, x, y, igam();
 *
 * y = igam( a, x );
 *
 * DESCRIPTION:
 *
 * The function is defined by
 *
 *                           x
 *                            -
 *                   1       | |  -t  a-1
 *  igam(a,x)  =   -----     |   e   t   dt.
 *                  -      | |
 *                 | (a)    -
 *                           0
 *
 *
 * In this implementation both arguments must be positive.
 * The integral is evaluated by either a power series or
 * continued fraction expansion, depending on the relative
 * values of a and x.
 *
 * ACCURACY:
 *
 *                      Relative error:
 * arithmetic   domain     # trials      peak         rms
 *    IEEE      0,30       200000       3.6e-14     2.9e-15
 *    IEEE      0,100      300000       9.9e-14     1.5e-14
 */
/*                                                     igamc()
 *
 *     Complemented incomplete gamma integral
 *
 *
 *
 * SYNOPSIS:
 *
 * double a, x, y, igamc();
 *
 * y = igamc( a, x );
 *
 * DESCRIPTION:
 *
 * The function is defined by
 *
 *
 *  igamc(a,x)   =   1 - igam(a,x)
 *
 *                            inf.
 *                              -
 *                     1       | |  -t  a-1
 *               =   -----     |   e   t   dt.
 *                    -      | |
 *                   | (a)    -
 *                             x
 *
 *
 * In this implementation both arguments must be positive.
 * The integral is evaluated by either a power series or
 * continued fraction expansion, depending on the relative
 * values of a and x.
 *
 * ACCURACY:
 *
 * Tested at random a, x.
 *                a         x                      Relative error:
 * arithmetic   domain   domain     # trials      peak         rms
 *    IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
 *    IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15
 */

/*
Cephes Math Library Release 2.0:  April, 1987
Copyright 1985, 1987 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/

double igamc( a, x )
double a, x;
{
double ans, ax, c, yc, r, t, y, z;
double pk, pkm1, pkm2, qk, qkm1, qkm2;

if( (x <= 0) || ( a <= 0) )
       return( 1.0 );

if( (x < 1.0) || (x < a) )
       return( 1.e0 - igam(a,x) );

ax = a * log(x) - x - lgam(a);
if( ax < -MAXLOG )
       {
       mtherr( "igamc", UNDERFLOW );
       return( 0.0 );
       }
ax = exp(ax);

/* continued fraction */
y = 1.0 - a;
z = x + y + 1.0;
c = 0.0;
pkm2 = 1.0;
qkm2 = x;
pkm1 = x + 1.0;
qkm1 = z * x;
ans = pkm1/qkm1;

do
       {
       c += 1.0;
       y += 1.0;
       z += 2.0;
       yc = y * c;
       pk = pkm1 * z  -  pkm2 * yc;
       qk = qkm1 * z  -  qkm2 * yc;
       if( qk != 0 )
               {
               r = pk/qk;
               t = fabs( (ans - r)/r );
               ans = r;
               }
       else
               t = 1.0;
       pkm2 = pkm1;
       pkm1 = pk;
       qkm2 = qkm1;
       qkm1 = qk;
       if( fabs(pk) > big )
               {
               pkm2 *= biginv;
               pkm1 *= biginv;
               qkm2 *= biginv;
               qkm1 *= biginv;
               }
       }
while( t > MACHEP );

return( ans * ax );
}



/* left tail of incomplete gamma function:
 *
 *          inf.      k
 *   a  -x   -       x
 *  x  e     >   ----------
 *           -     -
 *          k=0   | (a+k+1)
 *
 */

double igam( a, x )
double a, x;
{
double ans, ax, c, r;

if( (x <= 0) || ( a <= 0) )
       return( 0.0 );

if( (x > 1.0) && (x > a ) )
       return( 1.e0 - igamc(a,x) );

/* Compute  x**a * exp(-x) / gamma(a)  */
ax = a * log(x) - x - lgam(a);
if( ax < -MAXLOG )
       {
       mtherr( "igam", UNDERFLOW );
       return( 0.0 );
       }
ax = exp(ax);

/* power series */
r = a;
c = 1.0;
ans = 1.0;

do
       {
       r += 1.0;
       c *= x/r;
       ans += c;
       }
while( c/ans > MACHEP );

return( ans * ax/a );
}

/*                                                      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 2.2:  July, 1992
Copyright 1984, 1987, 1989, 1992 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/

/* 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 );
}

/*                                                      mtherr.c
 *
 *     Library common error handling routine
 *
 *
 *
 * SYNOPSIS:
 *
 * char *fctnam;
 * int code;
 * int mtherr();
 *
 * mtherr( fctnam, code );
 *
 *
 *
 * DESCRIPTION:
 *
 * This routine may be called to report one of the following
 * error conditions (in the include file mconf.h).
 *
 *   Mnemonic        Value          Significance
 *
 *    DOMAIN            1       argument domain error
 *    SING              2       function singularity
 *    OVERFLOW          3       overflow range error
 *    UNDERFLOW         4       underflow range error
 *    TLOSS             5       total loss of precision
 *    PLOSS             6       partial loss of precision
 *    EDOM             33       Unix domain error code
 *    ERANGE           34       Unix range error code
 *
 * The default version of the file prints the function name,
 * passed to it by the pointer fctnam, followed by the
 * error condition.  The display is directed to the standard
 * output device.  The routine then returns to the calling
 * program.  Users may wish to modify the program to abort by
 * calling exit() under severe error conditions such as domain
 * errors.
 *
 * Since all error conditions pass control to this function,
 * the display may be easily changed, eliminated, or directed
 * to an error logging device.
 *
 * SEE ALSO:
 *
 * mconf.h
 *
 */

/*
Cephes Math Library Release 2.0:  April, 1987
Copyright 1984, 1987 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/

int mtherr( name, code )
char *name;
int code;
{

/* Display string passed by calling program,
 * which is supposed to be the name of the
 * function in which the error occurred:
 */
printf("\n%s ", name);

/* Set global error message word */
merror = code;

/* Display error message defined
 * by the code argument.
 */
if( (code <= 0) || (code >= 7) )
       code = 0;
printf( "%s error\n", ermsg[code] );

/* Return to calling
 * program
 */
return( 0 );
}

/*                                                      polevl.c
 *                                                     p1evl.c
 *
 *     Evaluate polynomial
 *
 *
 *
 * SYNOPSIS:
 *
 * int N;
 * double x, y, coef[N+1], polevl[];
 *
 * y = polevl( x, coef, N );
 *
 *
 *
 * DESCRIPTION:
 *
 * Evaluates polynomial of degree N:
 *
 *                     2          N
 * y  =  C  + C x + C x  +...+ C x
 *        0    1     2          N
 *
 * Coefficients are stored in reverse order:
 *
 * coef[0] = C  , ..., coef[N] = C  .
 *            N                   0
 *
 *  The function p1evl() assumes that coef[N] = 1.0 and is
 * omitted from the array.  Its calling arguments are
 * otherwise the same as polevl().
 *
 *
 * SPEED:
 *
 * In the interest of speed, there are no checks for out
 * of bounds arithmetic.  This routine is used by most of
 * the functions in the library.  Depending on available
 * equipment features, the user may wish to rewrite the
 * program in microcode or assembly language.
 *
 */


/*
Cephes Math Library Release 2.1:  December, 1988
Copyright 1984, 1987, 1988 by Stephen L. Moshier
Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/

double polevl( x, coef, N )
double x;
double coef[];
int N;
{
double ans;
int i;
double *p;

p = coef;
ans = *p++;
i = N;

do
       ans = ans * x  +  *p++;
while( --i );

return( ans );
}

/*                                                      p1evl() */
/*                                          N
 * Evaluate polynomial when coefficient of x  is 1.0.
 * Otherwise same as polevl.
 */

double p1evl( x, coef, N )
double x;
double coef[];
int N;
{
double ans;
double *p;
int i;

p = coef;
ans = x + *p++;
i = N-1;

do
       ans = ans * x  + *p++;
while( --i );

return( ans );
}

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