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 54 long double tgammal(long double x) 55 { 56 return tgamma(x); 57 } 58 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 59 /* 60 tgamma(x+2) = tgamma(x+2) P(x)/Q(x) 61 0 <= x <= 1 62 Relative error 63 n=7, d=8 64 Peak error = 1.83e-20 65 Relative error spread = 8.4e-23 66 */ 67 static const long double P[8] = { 68 4.212760487471622013093E-5L, 69 4.542931960608009155600E-4L, 70 4.092666828394035500949E-3L, 71 2.385363243461108252554E-2L, 72 1.113062816019361559013E-1L, 73 3.629515436640239168939E-1L, 74 8.378004301573126728826E-1L, 75 1.000000000000000000009E0L, 76 }; 77 static const long double Q[9] = { 78 -1.397148517476170440917E-5L, 79 2.346584059160635244282E-4L, 80 -1.237799246653152231188E-3L, 81 -7.955933682494738320586E-4L, 82 2.773706565840072979165E-2L, 83 -4.633887671244534213831E-2L, 84 -2.243510905670329164562E-1L, 85 4.150160950588455434583E-1L, 86 9.999999999999999999908E-1L, 87 }; 88 89 /* 90 static const long double P[] = { 91 -3.01525602666895735709e0L, 92 -3.25157411956062339893e1L, 93 -2.92929976820724030353e2L, 94 -1.70730828800510297666e3L, 95 -7.96667499622741999770e3L, 96 -2.59780216007146401957e4L, 97 -5.99650230220855581642e4L, 98 -7.15743521530849602425e4L 99 }; 100 static const long double Q[] = { 101 1.00000000000000000000e0L, 102 -1.67955233807178858919e1L, 103 8.85946791747759881659e1L, 104 5.69440799097468430177e1L, 105 -1.98526250512761318471e3L, 106 3.31667508019495079814e3L, 107 1.60577839621734713377e4L, 108 -2.97045081369399940529e4L, 109 -7.15743521530849602412e4L 110 }; 111 */ 112 #define MAXGAML 1755.455L 113 /*static const long double LOGPI = 1.14472988584940017414L;*/ 114 115 /* Stirling's formula for the gamma function 116 tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x)) 117 z(x) = x 118 13 <= x <= 1024 119 Relative error 120 n=8, d=0 121 Peak error = 9.44e-21 122 Relative error spread = 8.8e-4 123 */ 124 static const long double STIR[9] = { 125 7.147391378143610789273E-4L, 126 -2.363848809501759061727E-5L, 127 -5.950237554056330156018E-4L, 128 6.989332260623193171870E-5L, 129 7.840334842744753003862E-4L, 130 -2.294719747873185405699E-4L, 131 -2.681327161876304418288E-3L, 132 3.472222222230075327854E-3L, 133 8.333333333333331800504E-2L, 134 }; 135 136 #define MAXSTIR 1024.0L 137 static const long double SQTPI = 2.50662827463100050242E0L; 138 139 /* 1/tgamma(x) = z P(z) 140 * z(x) = 1/x 141 * 0 < x < 0.03125 142 * Peak relative error 4.2e-23 143 */ 144 static const long double S[9] = { 145 -1.193945051381510095614E-3L, 146 7.220599478036909672331E-3L, 147 -9.622023360406271645744E-3L, 148 -4.219773360705915470089E-2L, 149 1.665386113720805206758E-1L, 150 -4.200263503403344054473E-2L, 151 -6.558780715202540684668E-1L, 152 5.772156649015328608253E-1L, 153 1.000000000000000000000E0L, 154 }; 155 156 /* 1/tgamma(-x) = z P(z) 157 * z(x) = 1/x 158 * 0 < x < 0.03125 159 * Peak relative error 5.16e-23 160 * Relative error spread = 2.5e-24 161 */ 162 static const long double SN[9] = { 163 1.133374167243894382010E-3L, 164 7.220837261893170325704E-3L, 165 9.621911155035976733706E-3L, 166 -4.219773343731191721664E-2L, 167 -1.665386113944413519335E-1L, 168 -4.200263503402112910504E-2L, 169 6.558780715202536547116E-1L, 170 5.772156649015328608727E-1L, 171 -1.000000000000000000000E0L, 172 }; 173 174 static const long double PIL = 3.1415926535897932384626L; 175 176 /* Gamma function computed by Stirling's formula. 177 */ 178 static long double stirf(long double x) 179 { 180 long double y, w, v; 181 182 w = 1.0/x; 183 /* For large x, use rational coefficients from the analytical expansion. */ 184 if (x > 1024.0) 185 w = (((((6.97281375836585777429E-5L * w 186 + 7.84039221720066627474E-4L) * w 187 - 2.29472093621399176955E-4L) * w 188 - 2.68132716049382716049E-3L) * w 189 + 3.47222222222222222222E-3L) * w 190 + 8.33333333333333333333E-2L) * w 191 + 1.0; 192 else 193 w = 1.0 + w * __polevll(w, STIR, 8); 194 y = expl(x); 195 if (x > MAXSTIR) { /* Avoid overflow in pow() */ 196 v = powl(x, 0.5L * x - 0.25L); 197 y = v * (v / y); 198 } else { 199 y = powl(x, x - 0.5L) / y; 200 } 201 y = SQTPI * y * w; 202 return y; 203 } 204 205 long double tgammal(long double x) 206 { 207 long double p, q, z; 208 209 if (!isfinite(x)) 210 return x + INFINITY; 211 212 q = fabsl(x); 213 if (q > 13.0) { 214 if (x < 0.0) { 215 p = floorl(q); 216 z = q - p; 217 if (z == 0) 218 return 0 / z; 219 if (q > MAXGAML) { 220 z = 0; 221 } else { 222 if (z > 0.5) { 223 p += 1.0; 224 z = q - p; 225 } 226 z = q * sinl(PIL * z); 227 z = fabsl(z) * stirf(q); 228 z = PIL/z; 229 } 230 if (0.5 * p == floorl(q * 0.5)) 231 z = -z; 232 } else if (x > MAXGAML) { 233 z = x * 0x1p16383L; 234 } else { 235 z = stirf(x); 236 } 237 return z; 238 } 239 240 z = 1.0; 241 while (x >= 3.0) { 242 x -= 1.0; 243 z *= x; 244 } 245 while (x < -0.03125L) { 246 z /= x; 247 x += 1.0; 248 } 249 if (x <= 0.03125L) 250 goto small; 251 while (x < 2.0) { 252 z /= x; 253 x += 1.0; 254 } 255 if (x == 2.0) 256 return z; 257 258 x -= 2.0; 259 p = __polevll(x, P, 7); 260 q = __polevll(x, Q, 8); 261 z = z * p / q; 262 return z; 263 264 small: 265 /* z==1 if x was originally +-0 */ 266 if (x == 0 && z != 1) 267 return x / x; 268 if (x < 0.0) { 269 x = -x; 270 q = z / (x * __polevll(x, SN, 8)); 271 } else 272 q = z / (x * __polevll(x, S, 8)); 273 return q; 274 } 275 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 276 // TODO: broken implementation to make things compile 277 long double tgammal(long double x) 278 { 279 return tgamma(x); 280 } 281 #endif 282