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path: root/src/math/tgamma.c
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 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 ``` ``````/* "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 > 0 && isnormal(x) assumption */ 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: /* sin(0-x) and -sin(x) have different sign at 0 */ return __sin(0-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) { double absx, y, dy, z, r; /* special cases */ if (!isfinite(x)) /* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */ return x + INFINITY; /* integer arguments */ /* raise inexact when non-integer */ if (x == floor(x)) { if (x == 0) /* tgamma(+-0)=+-inf with divide-by-zero */ return 1/x; if (x < 0) return 0/0.0; if (x <= sizeof fact/sizeof *fact) return fact[(int)x - 1]; } absx = fabs(x); /* x ~ 0: tgamma(x) ~ 1/x */ if (absx < 0x1p-54) return 1/x; /* x >= 172: tgamma(x)=inf with overflow */ /* x =< -184: tgamma(x)=+-0 with underflow */ if (absx >= 184) { if (x < 0) { if (floor(x) * 0.5 == floor(x * 0.5)) return 0; return -0.0; } x *= 0x1p1023; return 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 */ r = -pi / (sinpi(absx) * absx * r); dy = -dy; z = -z; } r += dy * (gmhalf+0.5) * r / y; z = pow(y, 0.5*z); r = r * z * z; return r; } #if 0 double __lgamma_r(double x, int *sign) { double r, absx, z, zz, w; *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 ``````