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370 lines
8.2 KiB
C
370 lines
8.2 KiB
C
/* lgam()
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*
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* Natural logarithm of gamma function
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*
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*
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*
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* SYNOPSIS:
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*
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* double x, y, __lgamma_r();
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* int* sgngam;
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* y = __lgamma_r( x, sgngam );
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*
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* double x, y, lgamma();
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* y = lgamma( x);
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*
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*
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*
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* DESCRIPTION:
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*
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* Returns the base e (2.718...) logarithm of the absolute
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* value of the gamma function of the argument. In the reentrant
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* version, the sign (+1 or -1) of the gamma function is returned
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* in the variable referenced by sgngam.
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*
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* For arguments greater than 13, the logarithm of the gamma
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* function is approximated by the logarithmic version of
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* Stirling's formula using a polynomial approximation of
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* degree 4. Arguments between -33 and +33 are reduced by
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* recurrence to the interval [2,3] of a rational approximation.
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* The cosecant reflection formula is employed for arguments
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* less than -33.
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*
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* Arguments greater than MAXLGM return MAXNUM and an error
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* message. MAXLGM = 2.035093e36 for DEC
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* arithmetic or 2.556348e305 for IEEE arithmetic.
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*
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*
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*
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* ACCURACY:
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*
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*
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* arithmetic domain # trials peak rms
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* DEC 0, 3 7000 5.2e-17 1.3e-17
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* DEC 2.718, 2.035e36 5000 3.9e-17 9.9e-18
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* IEEE 0, 3 28000 5.4e-16 1.1e-16
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* IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17
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* The error criterion was relative when the function magnitude
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* was greater than one but absolute when it was less than one.
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*
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* The following test used the relative error criterion, though
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* at certain points the relative error could be much higher than
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* indicated.
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* IEEE -200, -4 10000 4.8e-16 1.3e-16
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*
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*/
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/*
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* Cephes Math Library Release 2.8: June, 2000
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* Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
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*/
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/*
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* 26-11-2002 Modified for mingw.
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* Danny Smith <dannysmith@users.sourceforge.net>
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*/
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#ifndef __MINGW32__
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#include "mconf.h"
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#ifdef ANSIPROT
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extern double pow ( double, double );
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extern double log ( double );
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extern double exp ( double );
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extern double sin ( double );
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extern double polevl ( double, void *, int );
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extern double p1evl ( double, void *, int );
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extern double floor ( double );
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extern double fabs ( double );
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extern int isnan ( double );
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extern int isfinite ( double );
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#else
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double pow(), log(), exp(), sin(), polevl(), p1evl(), floor(), fabs();
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int isnan(), isfinite();
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#endif
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#ifdef INFINITIES
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extern double INFINITY;
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#endif
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#ifdef NANS
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extern double NAN;
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#endif
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#else /* __MINGW32__ */
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#include "cephes_mconf.h"
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#endif /* __MINGW32__ */
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/* A[]: Stirling's formula expansion of log gamma
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* B[], C[]: log gamma function between 2 and 3
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*/
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#ifdef UNK
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static double A[] = {
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8.11614167470508450300E-4,
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-5.95061904284301438324E-4,
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7.93650340457716943945E-4,
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-2.77777777730099687205E-3,
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8.33333333333331927722E-2
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};
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static double B[] = {
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-1.37825152569120859100E3,
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-3.88016315134637840924E4,
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-3.31612992738871184744E5,
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-1.16237097492762307383E6,
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-1.72173700820839662146E6,
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-8.53555664245765465627E5
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};
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static double C[] = {
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/* 1.00000000000000000000E0, */
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-3.51815701436523470549E2,
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-1.70642106651881159223E4,
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-2.20528590553854454839E5,
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-1.13933444367982507207E6,
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-2.53252307177582951285E6,
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-2.01889141433532773231E6
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};
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/* log( sqrt( 2*pi ) ) */
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static double LS2PI = 0.91893853320467274178;
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#define MAXLGM 2.556348e305
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static double LOGPI = 1.14472988584940017414;
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#endif
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#ifdef DEC
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static const unsigned short A[] = {
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0035524,0141201,0034633,0031405,
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0135433,0176755,0126007,0045030,
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0035520,0006371,0003342,0172730,
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0136066,0005540,0132605,0026407,
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0037252,0125252,0125252,0125132
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};
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static const unsigned short B[] = {
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0142654,0044014,0077633,0035410,
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0144027,0110641,0125335,0144760,
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0144641,0165637,0142204,0047447,
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0145215,0162027,0146246,0155211,
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0145322,0026110,0010317,0110130,
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0145120,0061472,0120300,0025363
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};
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static const unsigned short C[] = {
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/*0040200,0000000,0000000,0000000*/
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0142257,0164150,0163630,0112622,
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0143605,0050153,0156116,0135272,
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0144527,0056045,0145642,0062332,
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0145213,0012063,0106250,0001025,
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0145432,0111254,0044577,0115142,
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0145366,0071133,0050217,0005122
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};
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/* log( sqrt( 2*pi ) ) */
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static const unsigned short LS2P[] = {040153,037616,041445,0172645,};
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#define LS2PI *(double *)LS2P
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#define MAXLGM 2.035093e36
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static const unsigned short LPI[4] = {
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0040222,0103202,0043475,0006750,
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};
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#define LOGPI *(double *)LPI
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#endif
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#ifdef IBMPC
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static const unsigned short A[] = {
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0x6661,0x2733,0x9850,0x3f4a,
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0xe943,0xb580,0x7fbd,0xbf43,
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0x5ebb,0x20dc,0x019f,0x3f4a,
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0xa5a1,0x16b0,0xc16c,0xbf66,
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0x554b,0x5555,0x5555,0x3fb5
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};
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static const unsigned short B[] = {
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0x6761,0x8ff3,0x8901,0xc095,
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0xb93e,0x355b,0xf234,0xc0e2,
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0x89e5,0xf890,0x3d73,0xc114,
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0xdb51,0xf994,0xbc82,0xc131,
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0xf20b,0x0219,0x4589,0xc13a,
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0x055e,0x5418,0x0c67,0xc12a
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};
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static const unsigned short C[] = {
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/*0x0000,0x0000,0x0000,0x3ff0,*/
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0x12b2,0x1cf3,0xfd0d,0xc075,
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0xd757,0x7b89,0xaa0d,0xc0d0,
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0x4c9b,0xb974,0xeb84,0xc10a,
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0x0043,0x7195,0x6286,0xc131,
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0xf34c,0x892f,0x5255,0xc143,
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0xe14a,0x6a11,0xce4b,0xc13e
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};
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/* log( sqrt( 2*pi ) ) */
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static const union
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{
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unsigned short s[4];
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double d;
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} ls2p = {{0xbeb5,0xc864,0x67f1,0x3fed}};
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#define LS2PI (ls2p.d)
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#define MAXLGM 2.556348e305
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/* log (pi) */
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static const union
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{
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unsigned short s[4];
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double d;
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} lpi = {{0xa1bd,0x48e7,0x50d0,0x3ff2}};
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#define LOGPI (lpi.d)
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#endif
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#ifdef MIEEE
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static const unsigned short A[] = {
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0x3f4a,0x9850,0x2733,0x6661,
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0xbf43,0x7fbd,0xb580,0xe943,
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0x3f4a,0x019f,0x20dc,0x5ebb,
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0xbf66,0xc16c,0x16b0,0xa5a1,
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0x3fb5,0x5555,0x5555,0x554b
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};
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static const unsigned short B[] = {
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0xc095,0x8901,0x8ff3,0x6761,
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0xc0e2,0xf234,0x355b,0xb93e,
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0xc114,0x3d73,0xf890,0x89e5,
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0xc131,0xbc82,0xf994,0xdb51,
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0xc13a,0x4589,0x0219,0xf20b,
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0xc12a,0x0c67,0x5418,0x055e
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};
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static const unsigned short C[] = {
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0xc075,0xfd0d,0x1cf3,0x12b2,
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0xc0d0,0xaa0d,0x7b89,0xd757,
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0xc10a,0xeb84,0xb974,0x4c9b,
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0xc131,0x6286,0x7195,0x0043,
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0xc143,0x5255,0x892f,0xf34c,
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0xc13e,0xce4b,0x6a11,0xe14a
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};
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/* log( sqrt( 2*pi ) ) */
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static const union
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{
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unsigned short s[4];
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double d;
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} ls2p = {{0x3fed,0x67f1,0xc864,0xbeb5}};
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#define LS2PI ls2p.d
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#define MAXLGM 2.556348e305
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/* log (pi) */
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static const union
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{
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unsigned short s[4];
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double d;
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} lpi = {{0x3ff2, 0x50d0, 0x48e7, 0xa1bd}};
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#define LOGPI (lpi.d)
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#endif
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/* Logarithm of gamma function */
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/* Reentrant version */
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double __lgamma_r(double x, int* sgngam)
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{
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double p, q, u, w, z;
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int i;
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*sgngam = 1;
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#ifdef NANS
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if( isnan(x) )
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return(x);
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#endif
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#ifdef INFINITIES
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if( !isfinite(x) )
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return(INFINITY);
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#endif
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if( x < -34.0 )
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{
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q = -x;
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w = __lgamma_r(q, sgngam); /* note this modifies sgngam! */
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p = floor(q);
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if( p == q )
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{
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lgsing:
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_SET_ERRNO(EDOM);
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mtherr( "lgam", SING );
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#ifdef INFINITIES
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return (INFINITY);
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#else
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return (MAXNUM);
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#endif
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}
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i = p;
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if( (i & 1) == 0 )
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*sgngam = -1;
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else
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*sgngam = 1;
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z = q - p;
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if( z > 0.5 )
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{
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p += 1.0;
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z = p - q;
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}
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z = q * sin( PI * z );
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if( z == 0.0 )
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goto lgsing;
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/* z = log(PI) - log( z ) - w;*/
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z = LOGPI - log( z ) - w;
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return( z );
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}
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if( x < 13.0 )
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{
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z = 1.0;
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p = 0.0;
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u = x;
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while( u >= 3.0 )
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{
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p -= 1.0;
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u = x + p;
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z *= u;
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}
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while( u < 2.0 )
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{
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if( u == 0.0 )
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goto lgsing;
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z /= u;
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p += 1.0;
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u = x + p;
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}
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if( z < 0.0 )
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{
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*sgngam = -1;
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z = -z;
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}
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else
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*sgngam = 1;
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if( u == 2.0 )
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return( log(z) );
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p -= 2.0;
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x = x + p;
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p = x * polevl( x, B, 5 ) / p1evl( x, C, 6);
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return( log(z) + p );
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}
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if( x > MAXLGM )
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{
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_SET_ERRNO(ERANGE);
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mtherr( "lgamma", OVERFLOW );
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#ifdef INFINITIES
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return( *sgngam * INFINITY );
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#else
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return( *sgngam * MAXNUM );
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#endif
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}
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q = ( x - 0.5 ) * log(x) - x + LS2PI;
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if( x > 1.0e8 )
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return( q );
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p = 1.0/(x*x);
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if( x >= 1000.0 )
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q += (( 7.9365079365079365079365e-4 * p
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- 2.7777777777777777777778e-3) *p
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+ 0.0833333333333333333333) / x;
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else
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q += polevl( p, A, 4 ) / x;
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return( q );
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}
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/* This is the C99 version */
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double lgamma(double x)
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{
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int local_sgngam=0;
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return (__lgamma_r(x, &local_sgngam));
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}
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