/* gammal.c * * Gamma function * * * * SYNOPSIS: * * long double x, y, __tgammal_r(); * int* sgngaml; * y = __tgammal_r( x, sgngaml ); * * long double x, y, tgammal(); * y = tgammal( x); * * * * DESCRIPTION: * * Returns gamma function of the argument. The result is * correctly signed. In the reentrant version the sign (+1 or -1) * is returned in the variable referenced by sgngamf. * * Arguments |x| <= 13 are reduced by recurrence and the function * approximated by a rational function of degree 7/8 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 * IEEE -40,+40 10000 3.6e-19 7.9e-20 * IEEE -1755,+1755 10000 4.8e-18 6.5e-19 * * Accuracy for large arguments is dominated by error in powl(). * */ /* Copyright 1994 by Stephen L. Moshier */ /* * 26-11-2002 Modified for mingw. * Danny Smith */ #ifndef __MINGW32__ #include "mconf.h" #else #include "cephes_mconf.h" #endif /* gamma(x+2) = gamma(x+2) P(x)/Q(x) 0 <= x <= 1 Relative error n=7, d=8 Peak error = 1.83e-20 Relative error spread = 8.4e-23 */ #if UNK static const long double P[8] = { 4.212760487471622013093E-5L, 4.542931960608009155600E-4L, 4.092666828394035500949E-3L, 2.385363243461108252554E-2L, 1.113062816019361559013E-1L, 3.629515436640239168939E-1L, 8.378004301573126728826E-1L, 1.000000000000000000009E0L, }; static const long double Q[9] = { -1.397148517476170440917E-5L, 2.346584059160635244282E-4L, -1.237799246653152231188E-3L, -7.955933682494738320586E-4L, 2.773706565840072979165E-2L, -4.633887671244534213831E-2L, -2.243510905670329164562E-1L, 4.150160950588455434583E-1L, 9.999999999999999999908E-1L, }; #endif #if IBMPC static const uLD P[] = { { { 0x434a,0x3f22,0x2bda,0xb0b2,0x3ff0, XPD } }, { { 0xf5aa,0xe82f,0x335b,0xee2e,0x3ff3, XPD } }, { { 0xbe6c,0x3757,0xc717,0x861b,0x3ff7, XPD } }, { { 0x7f43,0x5196,0xb166,0xc368,0x3ff9, XPD } }, { { 0x9549,0x8eb5,0x8c3a,0xe3f4,0x3ffb, XPD } }, { { 0x8d75,0x23af,0xc8e4,0xb9d4,0x3ffd, XPD } }, { { 0x29cf,0x19b3,0x16c8,0xd67a,0x3ffe, XPD } }, { { 0x0000,0x0000,0x0000,0x8000,0x3fff, XPD } } }; static const uLD Q[] = { { { 0x5473,0x2de8,0x1268,0xea67,0xbfee, XPD } }, { { 0x334b,0xc2f0,0xa2dd,0xf60e,0x3ff2, XPD } }, { { 0xbeed,0x1853,0xa691,0xa23d,0xbff5, XPD } }, { { 0x296e,0x7cb1,0x5dfd,0xd08f,0xbff4, XPD } }, { { 0x0417,0x7989,0xd7bc,0xe338,0x3ff9, XPD } }, { { 0x3295,0x3698,0xd580,0xbdcd,0xbffa, XPD } }, { { 0x75ef,0x3ab7,0x4ad3,0xe5bc,0xbffc, XPD } }, { { 0xe458,0x2ec7,0xfd57,0xd47c,0x3ffd, XPD } }, { { 0x0000,0x0000,0x0000,0x8000,0x3fff, XPD } } }; #endif #if MIEEE static const long P[24] = { 0x3ff00000,0xb0b22bda,0x3f22434a, 0x3ff30000,0xee2e335b,0xe82ff5aa, 0x3ff70000,0x861bc717,0x3757be6c, 0x3ff90000,0xc368b166,0x51967f43, 0x3ffb0000,0xe3f48c3a,0x8eb59549, 0x3ffd0000,0xb9d4c8e4,0x23af8d75, 0x3ffe0000,0xd67a16c8,0x19b329cf, 0x3fff0000,0x80000000,0x00000000, }; static const long Q[27] = { 0xbfee0000,0xea671268,0x2de85473, 0x3ff20000,0xf60ea2dd,0xc2f0334b, 0xbff50000,0xa23da691,0x1853beed, 0xbff40000,0xd08f5dfd,0x7cb1296e, 0x3ff90000,0xe338d7bc,0x79890417, 0xbffa0000,0xbdcdd580,0x36983295, 0xbffc0000,0xe5bc4ad3,0x3ab775ef, 0x3ffd0000,0xd47cfd57,0x2ec7e458, 0x3fff0000,0x80000000,0x00000000, }; #endif /* static const long double P[] = { -3.01525602666895735709e0L, -3.25157411956062339893e1L, -2.92929976820724030353e2L, -1.70730828800510297666e3L, -7.96667499622741999770e3L, -2.59780216007146401957e4L, -5.99650230220855581642e4L, -7.15743521530849602425e4L }; static const long double Q[] = { 1.00000000000000000000e0L, -1.67955233807178858919e1L, 8.85946791747759881659e1L, 5.69440799097468430177e1L, -1.98526250512761318471e3L, 3.31667508019495079814e3L, 1.60577839621734713377e4L, -2.97045081369399940529e4L, -7.15743521530849602412e4L }; */ #define MAXGAML 1755.455L /*static const long double LOGPI = 1.14472988584940017414L;*/ /* Stirling's formula for the gamma function gamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x)) z(x) = x 13 <= x <= 1024 Relative error n=8, d=0 Peak error = 9.44e-21 Relative error spread = 8.8e-4 */ #if UNK static const long double STIR[9] = { 7.147391378143610789273E-4L, -2.363848809501759061727E-5L, -5.950237554056330156018E-4L, 6.989332260623193171870E-5L, 7.840334842744753003862E-4L, -2.294719747873185405699E-4L, -2.681327161876304418288E-3L, 3.472222222230075327854E-3L, 8.333333333333331800504E-2L, }; #endif #if IBMPC static const uLD STIR[] = { { { 0x6ede,0x69f7,0x54e3,0xbb5d,0x3ff4, XPD } }, { { 0xc395,0x0295,0x4443,0xc64b,0xbfef, XPD } }, { { 0xba6f,0x7c59,0x5e47,0x9bfb,0xbff4, XPD } }, { { 0x5704,0x1a39,0xb11d,0x9293,0x3ff1, XPD } }, { { 0x30b7,0x1a21,0x98b2,0xcd87,0x3ff4, XPD } }, { { 0xbef3,0x7023,0x6a08,0xf09e,0xbff2, XPD } }, { { 0x3a1c,0x5ac8,0x3478,0xafb9,0xbff6, XPD } }, { { 0xc3c9,0x906e,0x38e3,0xe38e,0x3ff6, XPD } }, { { 0xa1d5,0xaaaa,0xaaaa,0xaaaa,0x3ffb, XPD } } }; #endif #if MIEEE static const long STIR[27] = { 0x3ff40000,0xbb5d54e3,0x69f76ede, 0xbfef0000,0xc64b4443,0x0295c395, 0xbff40000,0x9bfb5e47,0x7c59ba6f, 0x3ff10000,0x9293b11d,0x1a395704, 0x3ff40000,0xcd8798b2,0x1a2130b7, 0xbff20000,0xf09e6a08,0x7023bef3, 0xbff60000,0xafb93478,0x5ac83a1c, 0x3ff60000,0xe38e38e3,0x906ec3c9, 0x3ffb0000,0xaaaaaaaa,0xaaaaa1d5, }; #endif #define MAXSTIR 1024.0L static const long double SQTPI = 2.50662827463100050242E0L; /* 1/gamma(x) = z P(z) * z(x) = 1/x * 0 < x < 0.03125 * Peak relative error 4.2e-23 */ #if UNK static const 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 uLD 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 const 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 /* 1/gamma(-x) = z P(z) * z(x) = 1/x * 0 < x < 0.03125 * Peak relative error 5.16e-23 * Relative error spread = 2.5e-24 */ #if UNK static const 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 uLD 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 const 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 #ifndef __MINGW32__ extern long double MAXLOGL, MAXNUML, PIL; /* #define PIL 3.14159265358979323846L */ /* #define MAXNUML 1.189731495357231765021263853E4932L */ #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 ); static long double stirf ( long double ); #else long double fabsl(), lgaml(), logl(), expl(), gammal(), sinl(); long double floorl(), powl(), polevll(), p1evll(), isnanl(), isfinitel(); static long double stirf(); #endif #ifdef INFINITIES extern long double INFINITYL; #endif #ifdef NANS extern long double NANL; #endif #else /* __MINGW32__ */ static long double stirf ( long double ); #endif /* Gamma function computed by Stirling's formula. */ static long double stirf(x) long double x; { long double y, w, v; w = 1.0L/x; /* For large x, use rational coefficients from the analytical expansion. */ if( x > 1024.0L ) w = (((((6.97281375836585777429E-5L * w + 7.84039221720066627474E-4L) * w - 2.29472093621399176955E-4L) * w - 2.68132716049382716049E-3L) * w + 3.47222222222222222222E-3L) * w + 8.33333333333333333333E-2L) * w + 1.0L; else w = 1.0L + w * polevll( w, STIR, 8 ); y = expl(x); if( x > MAXSTIR ) { /* Avoid overflow in pow() */ v = powl( x, 0.5L * x - 0.25L ); y = v * (v / y); } else { y = powl( x, x - 0.5L ) / y; } y = SQTPI * y * w; return( y ); } long double __tgammal_r(long double x, int* sgngaml) { long double p, q, z; int i; *sgngaml = 1; #ifdef NANS if( isnanl(x) ) return(NANL); #endif #ifdef INFINITIES #ifdef NANS if( x == INFINITYL ) return(x); if( x == -INFINITYL ) return(NANL); #else if( !isfinite(x) ) return(x); #endif #endif q = fabsl(x); if( q > 13.0L ) { if( q > MAXGAML ) goto goverf; if( x < 0.0L ) { p = floorl(q); if( p == q ) { gsing: _SET_ERRNO(EDOM); mtherr( "tgammal", SING ); #ifdef INFINITIES return (INFINITYL); #else return( *sgngaml * MAXNUML); #endif } i = p; if( (i & 1) == 0 ) *sgngaml = -1; z = q - p; if( z > 0.5L ) { p += 1.0L; z = q - p; } z = q * sinl( PIL * z ); z = fabsl(z) * stirf(q); if( z <= PIL/MAXNUML ) { goverf: _SET_ERRNO(ERANGE); mtherr( "tgammal", OVERFLOW ); #ifdef INFINITIES return( *sgngaml * INFINITYL); #else return( *sgngaml * MAXNUML); #endif } z = PIL/z; } else { z = stirf(x); } return( *sgngaml * z ); } z = 1.0L; while( x >= 3.0L ) { x -= 1.0L; z *= x; } while( x < -0.03125L ) { z /= x; x += 1.0L; } if( x <= 0.03125L ) goto Small; while( x < 2.0L ) { z /= x; x += 1.0L; } if( x == 2.0L ) return(z); x -= 2.0L; p = polevll( x, P, 7 ); q = polevll( x, Q, 8 ); return( z * p / q ); Small: if( x == 0.0L ) { goto gsing; } else { if( x < 0.0L ) { x = -x; q = z / (x * polevll( x, SN, 8 )); } else q = z / (x * polevll( x, S, 8 )); } return q; } /* This is the C99 version. */ long double tgammal(long double x) { int local_sgngaml=0; return (__tgammal_r(x, &local_sgngaml)); }