/* 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 <dannysmith@users.sourceforge.net> */ #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)); }