/* gammal.c * * Gamma function * * * * SYNOPSIS: * * long double x, y, gammal(); * extern int sgngam; * * y = gammal( 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| <= 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(). * */ /* lgaml() * * Natural logarithm of gamma function * * * * SYNOPSIS: * * long double x, y, lgaml(); * extern int sgngam; * * y = lgaml( 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 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. * */ /* gamma.c */ /* gamma function */ /* Copyright 1994 by Stephen L. Moshier */ #include /* 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 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 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 short 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 short 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 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 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 long double P[] = { -3.01525602666895735709e0L, -3.25157411956062339893e1L, -2.92929976820724030353e2L, -1.70730828800510297666e3L, -7.96667499622741999770e3L, -2.59780216007146401957e4L, -5.99650230220855581642e4L, -7.15743521530849602425e4L }; static 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 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 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 short 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 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 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 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 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 /* 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 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 short 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 int sgngaml = 0; extern int sgngaml; 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 /* 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 gammal(x) long double x; { long double p, q, z; int i; sgngaml = 1; #ifdef NANS if( isnanl(x) ) return(NANL); #endif #ifdef INFINITIES if(x == INFINITYL) return(INFINITYL); #ifdef NANS if(x == -INFINITYL) goto gamnan; #endif #endif q = fabsl(x); if( q > 13.0L ) { if( q > MAXGAML ) goto goverf; if( x < 0.0L ) { p = floorl(q); if( p == q ) { gamnan: #ifdef NANS mtherr( "gammal", DOMAIN ); return (NANL); #else goto goverf; #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: #ifdef INFINITIES return( sgngaml * INFINITYL); #else mtherr( "gammal", OVERFLOW ); 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 gamnan; } else { if( x < 0.0L ) { x = -x; q = z / (x * polevll( x, SN, 8 )); } else q = z / (x * polevll( x, S, 8 )); } return q; } /* 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 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 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 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 long double LS2PI = 0.91893853320467274178L; #define MAXLGM 1.04848146839019521116e+4928L /* Logarithm of gamma function */ long double lgaml(x) long double x; { 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 = lgaml(q); /* note this modifies sgngam! */ p = floorl(q); if( p == q ) { #ifdef INFINITIES mtherr( "lgaml", SING ); return (INFINITYL); #else goto loverf; #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 loverf; /* 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 ) { loverf: #ifdef INFINITIES return( sgngaml * INFINITYL ); #else mtherr( "lgaml", OVERFLOW ); 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 loverf; 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); }