/* "A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964) "Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001) "An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004) approximation method: (x - 0.5) S(x) Gamma(x) = (x + g - 0.5) * ---------------- exp(x + g - 0.5) with a1 a2 a3 aN S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ] x + 1 x + 2 x + 3 x + N with a0, a1, a2, a3,.. aN constants which depend on g. for x < 0 the following reflection formula is used: Gamma(x)*Gamma(-x) = -pi/(x sin(pi x)) most ideas and constants are from boost and python */ #include "libm.h" static const double pi = 3.141592653589793238462643383279502884; /* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */ static double sinpi(double x) { int n; /* argument reduction: x = |x| mod 2 */ /* spurious inexact when x is odd int */ x = x * 0.5; x = 2 * (x - floor(x)); /* reduce x into [-.25,.25] */ n = 4 * x; n = (n+1)/2; x -= n * 0.5; x *= pi; switch (n) { default: /* case 4 */ case 0: return __sin(x, 0, 0); case 1: return __cos(x, 0); case 2: return __sin(-x, 0, 0); case 3: return -__cos(x, 0); } } #define N 12 //static const double g = 6.024680040776729583740234375; static const double gmhalf = 5.524680040776729583740234375; static const double Snum[N+1] = { 23531376880.410759688572007674451636754734846804940, 42919803642.649098768957899047001988850926355848959, 35711959237.355668049440185451547166705960488635843, 17921034426.037209699919755754458931112671403265390, 6039542586.3520280050642916443072979210699388420708, 1439720407.3117216736632230727949123939715485786772, 248874557.86205415651146038641322942321632125127801, 31426415.585400194380614231628318205362874684987640, 2876370.6289353724412254090516208496135991145378768, 186056.26539522349504029498971604569928220784236328, 8071.6720023658162106380029022722506138218516325024, 210.82427775157934587250973392071336271166969580291, 2.5066282746310002701649081771338373386264310793408, }; static const double Sden[N+1] = { 0, 39916800, 120543840, 150917976, 105258076, 45995730, 13339535, 2637558, 357423, 32670, 1925, 66, 1, }; /* n! for small integer n */ static const double fact[] = { 1, 1, 2, 6, 24, 120, 720, 5040.0, 40320.0, 362880.0, 3628800.0, 39916800.0, 479001600.0, 6227020800.0, 87178291200.0, 1307674368000.0, 20922789888000.0, 355687428096000.0, 6402373705728000.0, 121645100408832000.0, 2432902008176640000.0, 51090942171709440000.0, 1124000727777607680000.0, }; /* S(x) rational function for positive x */ static double S(double x) { double_t num = 0, den = 0; int i; /* to avoid overflow handle large x differently */ if (x < 8) for (i = N; i >= 0; i--) { num = num * x + Snum[i]; den = den * x + Sden[i]; } else for (i = 0; i <= N; i++) { num = num / x + Snum[i]; den = den / x + Sden[i]; } return num/den; } double tgamma(double x) { union {double f; uint64_t i;} u = {x}; double absx, y; double_t dy, z, r; uint32_t ix = u.i>>32 & 0x7fffffff; int sign = u.i>>63; /* special cases */ if (ix >= 0x7ff00000) /* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */ return x + INFINITY; if (ix < (0x3ff-54)<<20) /* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */ return 1/x; /* integer arguments */ /* raise inexact when non-integer */ if (x == floor(x)) { if (sign) return 0/0.0; if (x <= sizeof fact/sizeof *fact) return fact[(int)x - 1]; } /* x >= 172: tgamma(x)=inf with overflow */ /* x =< -184: tgamma(x)=+-0 with underflow */ if (ix >= 0x40670000) { /* |x| >= 184 */ if (sign) { FORCE_EVAL((float)(0x1p-126/x)); if (floor(x) * 0.5 == floor(x * 0.5)) return 0; return -0.0; } x *= 0x1p1023; return x; } absx = sign ? -x : x; /* handle the error of x + g - 0.5 */ y = absx + gmhalf; if (absx > gmhalf) { dy = y - absx; dy -= gmhalf; } else { dy = y - gmhalf; dy -= absx; } z = absx - 0.5; r = S(absx) * exp(-y); if (x < 0) { /* reflection formula for negative x */ /* sinpi(absx) is not 0, integers are already handled */ r = -pi / (sinpi(absx) * absx * r); dy = -dy; z = -z; } r += dy * (gmhalf+0.5) * r / y; z = pow(y, 0.5*z); y = r * z * z; return y; } #if 0 double __lgamma_r(double x, int *sign) { double r, absx; *sign = 1; /* special cases */ if (!isfinite(x)) /* lgamma(nan)=nan, lgamma(+-inf)=inf */ return x*x; /* integer arguments */ if (x == floor(x) && x <= 2) { /* n <= 0: lgamma(n)=inf with divbyzero */ /* n == 1,2: lgamma(n)=0 */ if (x <= 0) return 1/0.0; return 0; } absx = fabs(x); /* lgamma(x) ~ -log(|x|) for tiny |x| */ if (absx < 0x1p-54) { *sign = 1 - 2*!!signbit(x); return -log(absx); } /* use tgamma for smaller |x| */ if (absx < 128) { x = tgamma(x); *sign = 1 - 2*!!signbit(x); return log(fabs(x)); } /* second term (log(S)-g) could be more precise here.. */ /* or with stirling: (|x|-0.5)*(log(|x|)-1) + poly(1/|x|) */ r = (absx-0.5)*(log(absx+gmhalf)-1) + (log(S(absx)) - (gmhalf+0.5)); if (x < 0) { /* reflection formula for negative x */ x = sinpi(absx); *sign = 2*!!signbit(x) - 1; r = log(pi/(fabs(x)*absx)) - r; } return r; } weak_alias(__lgamma_r, lgamma_r); #endif