1/* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_tgammal.c */ 2/* 3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 4 * 5 * Permission to use, copy, modify, and distribute this software for any 6 * purpose with or without fee is hereby granted, provided that the above 7 * copyright notice and this permission notice appear in all copies. 8 * 9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 11 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 12 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 13 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 14 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 15 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 16 */ 17/* 18 * Gamma function 19 * 20 * 21 * SYNOPSIS: 22 * 23 * long double x, y, tgammal(); 24 * 25 * y = tgammal( x ); 26 * 27 * 28 * DESCRIPTION: 29 * 30 * Returns gamma function of the argument. The result is 31 * correctly signed. 32 * 33 * Arguments |x| <= 13 are reduced by recurrence and the function 34 * approximated by a rational function of degree 7/8 in the 35 * interval (2,3). Large arguments are handled by Stirling's 36 * formula. Large negative arguments are made positive using 37 * a reflection formula. 38 * 39 * 40 * ACCURACY: 41 * 42 * Relative error: 43 * arithmetic domain # trials peak rms 44 * IEEE -40,+40 10000 3.6e-19 7.9e-20 45 * IEEE -1755,+1755 10000 4.8e-18 6.5e-19 46 * 47 * Accuracy for large arguments is dominated by error in powl(). 48 * 49 */ 50 51#include "libm.h" 52 53#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 54long double tgammal(long double x) { 55 return tgamma(x); 56} 57#elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 58/* 59tgamma(x+2) = tgamma(x+2) P(x)/Q(x) 600 <= x <= 1 61Relative error 62n=7, d=8 63Peak error = 1.83e-20 64Relative error spread = 8.4e-23 65*/ 66static const long double P[8] = { 67 4.212760487471622013093E-5L, 4.542931960608009155600E-4L, 4.092666828394035500949E-3L, 68 2.385363243461108252554E-2L, 1.113062816019361559013E-1L, 3.629515436640239168939E-1L, 69 8.378004301573126728826E-1L, 1.000000000000000000009E0L, 70}; 71static const long double Q[9] = { 72 -1.397148517476170440917E-5L, 2.346584059160635244282E-4L, -1.237799246653152231188E-3L, 73 -7.955933682494738320586E-4L, 2.773706565840072979165E-2L, -4.633887671244534213831E-2L, 74 -2.243510905670329164562E-1L, 4.150160950588455434583E-1L, 9.999999999999999999908E-1L, 75}; 76 77/* 78static const long double P[] = { 79-3.01525602666895735709e0L, 80-3.25157411956062339893e1L, 81-2.92929976820724030353e2L, 82-1.70730828800510297666e3L, 83-7.96667499622741999770e3L, 84-2.59780216007146401957e4L, 85-5.99650230220855581642e4L, 86-7.15743521530849602425e4L 87}; 88static const long double Q[] = { 89 1.00000000000000000000e0L, 90-1.67955233807178858919e1L, 91 8.85946791747759881659e1L, 92 5.69440799097468430177e1L, 93-1.98526250512761318471e3L, 94 3.31667508019495079814e3L, 95 1.60577839621734713377e4L, 96-2.97045081369399940529e4L, 97-7.15743521530849602412e4L 98}; 99*/ 100#define MAXGAML 1755.455L 101/*static const long double LOGPI = 1.14472988584940017414L;*/ 102 103/* Stirling's formula for the gamma function 104tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x)) 105z(x) = x 10613 <= x <= 1024 107Relative error 108n=8, d=0 109Peak error = 9.44e-21 110Relative error spread = 8.8e-4 111*/ 112static const long double STIR[9] = { 113 7.147391378143610789273E-4L, -2.363848809501759061727E-5L, -5.950237554056330156018E-4L, 114 6.989332260623193171870E-5L, 7.840334842744753003862E-4L, -2.294719747873185405699E-4L, 115 -2.681327161876304418288E-3L, 3.472222222230075327854E-3L, 8.333333333333331800504E-2L, 116}; 117 118#define MAXSTIR 1024.0L 119static const long double SQTPI = 2.50662827463100050242E0L; 120 121/* 1/tgamma(x) = z P(z) 122 * z(x) = 1/x 123 * 0 < x < 0.03125 124 * Peak relative error 4.2e-23 125 */ 126static const long double S[9] = { 127 -1.193945051381510095614E-3L, 7.220599478036909672331E-3L, -9.622023360406271645744E-3L, 128 -4.219773360705915470089E-2L, 1.665386113720805206758E-1L, -4.200263503403344054473E-2L, 129 -6.558780715202540684668E-1L, 5.772156649015328608253E-1L, 1.000000000000000000000E0L, 130}; 131 132/* 1/tgamma(-x) = z P(z) 133 * z(x) = 1/x 134 * 0 < x < 0.03125 135 * Peak relative error 5.16e-23 136 * Relative error spread = 2.5e-24 137 */ 138static const long double SN[9] = { 139 1.133374167243894382010E-3L, 7.220837261893170325704E-3L, 9.621911155035976733706E-3L, 140 -4.219773343731191721664E-2L, -1.665386113944413519335E-1L, -4.200263503402112910504E-2L, 141 6.558780715202536547116E-1L, 5.772156649015328608727E-1L, -1.000000000000000000000E0L, 142}; 143 144static const long double PIL = 3.1415926535897932384626L; 145 146/* Gamma function computed by Stirling's formula. 147 */ 148static long double stirf(long double x) { 149 long double y, w, v; 150 151 w = 1.0 / x; 152 /* For large x, use rational coefficients from the analytical expansion. */ 153 if (x > 1024.0) 154 w = (((((6.97281375836585777429E-5L * w + 7.84039221720066627474E-4L) * w - 155 2.29472093621399176955E-4L) * 156 w - 157 2.68132716049382716049E-3L) * 158 w + 159 3.47222222222222222222E-3L) * 160 w + 161 8.33333333333333333333E-2L) * 162 w + 163 1.0; 164 else 165 w = 1.0 + w * __polevll(w, STIR, 8); 166 y = expl(x); 167 if (x > MAXSTIR) { /* Avoid overflow in pow() */ 168 v = powl(x, 0.5L * x - 0.25L); 169 y = v * (v / y); 170 } else { 171 y = powl(x, x - 0.5L) / y; 172 } 173 y = SQTPI * y * w; 174 return y; 175} 176 177long double tgammal(long double x) { 178 long double p, q, z; 179 180 if (!isfinite(x)) 181 return x + INFINITY; 182 183 q = fabsl(x); 184 if (q > 13.0) { 185 if (x < 0.0) { 186 p = floorl(q); 187 z = q - p; 188 if (z == 0) 189 return 0 / z; 190 if (q > MAXGAML) { 191 z = 0; 192 } else { 193 if (z > 0.5) { 194 p += 1.0; 195 z = q - p; 196 } 197 z = q * sinl(PIL * z); 198 z = fabsl(z) * stirf(q); 199 z = PIL / z; 200 } 201 if (0.5 * p == floorl(q * 0.5)) 202 z = -z; 203 } else if (x > MAXGAML) { 204 z = x * 0x1p16383L; 205 } else { 206 z = stirf(x); 207 } 208 return z; 209 } 210 211 z = 1.0; 212 while (x >= 3.0) { 213 x -= 1.0; 214 z *= x; 215 } 216 while (x < -0.03125L) { 217 z /= x; 218 x += 1.0; 219 } 220 if (x <= 0.03125L) 221 goto small; 222 while (x < 2.0) { 223 z /= x; 224 x += 1.0; 225 } 226 if (x == 2.0) 227 return z; 228 229 x -= 2.0; 230 p = __polevll(x, P, 7); 231 q = __polevll(x, Q, 8); 232 z = z * p / q; 233 return z; 234 235small: 236 /* z==1 if x was originally +-0 */ 237 if (x == 0 && z != 1) 238 return x / x; 239 if (x < 0.0) { 240 x = -x; 241 q = z / (x * __polevll(x, SN, 8)); 242 } else 243 q = z / (x * __polevll(x, S, 8)); 244 return q; 245} 246#elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 247// TODO: broken implementation to make things compile 248long double tgammal(long double x) { 249 return tgamma(x); 250} 251#endif 252