1 /* $OpenBSD: e_tgammal.c,v 1.4 2013/11/12 20:35:19 martynas Exp $ */ 2 3 /* 4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 5 * 6 * Permission to use, copy, modify, and distribute this software for any 7 * purpose with or without fee is hereby granted, provided that the above 8 * copyright notice and this permission notice appear in all copies. 9 * 10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 17 */ 18 19 /* tgammal.c 20 * 21 * Gamma function 22 * 23 * 24 * 25 * SYNOPSIS: 26 * 27 * long double x, y, tgammal(); 28 * extern int signgam; 29 * 30 * y = tgammal( x ); 31 * 32 * 33 * 34 * DESCRIPTION: 35 * 36 * Returns gamma function of the argument. The result is 37 * correctly signed, and the sign (+1 or -1) is also 38 * returned in a global (extern) variable named signgam. 39 * This variable is also filled in by the logarithmic gamma 40 * function lgamma(). 41 * 42 * Arguments |x| <= 13 are reduced by recurrence and the function 43 * approximated by a rational function of degree 7/8 in the 44 * interval (2,3). Large arguments are handled by Stirling's 45 * formula. Large negative arguments are made positive using 46 * a reflection formula. 47 * 48 * 49 * ACCURACY: 50 * 51 * Relative error: 52 * arithmetic domain # trials peak rms 53 * IEEE -40,+40 10000 3.6e-19 7.9e-20 54 * IEEE -1755,+1755 10000 4.8e-18 6.5e-19 55 * 56 * Accuracy for large arguments is dominated by error in powl(). 57 * 58 */ 59 60 #include <float.h> 61 #include <math.h> 62 63 #include "math_private.h" 64 65 /* 66 tgamma(x+2) = tgamma(x+2) P(x)/Q(x) 67 0 <= x <= 1 68 Relative error 69 n=7, d=8 70 Peak error = 1.83e-20 71 Relative error spread = 8.4e-23 72 */ 73 74 static long double P[8] = { 75 4.212760487471622013093E-5L, 76 4.542931960608009155600E-4L, 77 4.092666828394035500949E-3L, 78 2.385363243461108252554E-2L, 79 1.113062816019361559013E-1L, 80 3.629515436640239168939E-1L, 81 8.378004301573126728826E-1L, 82 1.000000000000000000009E0L, 83 }; 84 static long double Q[9] = { 85 -1.397148517476170440917E-5L, 86 2.346584059160635244282E-4L, 87 -1.237799246653152231188E-3L, 88 -7.955933682494738320586E-4L, 89 2.773706565840072979165E-2L, 90 -4.633887671244534213831E-2L, 91 -2.243510905670329164562E-1L, 92 4.150160950588455434583E-1L, 93 9.999999999999999999908E-1L, 94 }; 95 96 /* 97 static long double P[] = { 98 -3.01525602666895735709e0L, 99 -3.25157411956062339893e1L, 100 -2.92929976820724030353e2L, 101 -1.70730828800510297666e3L, 102 -7.96667499622741999770e3L, 103 -2.59780216007146401957e4L, 104 -5.99650230220855581642e4L, 105 -7.15743521530849602425e4L 106 }; 107 static long double Q[] = { 108 1.00000000000000000000e0L, 109 -1.67955233807178858919e1L, 110 8.85946791747759881659e1L, 111 5.69440799097468430177e1L, 112 -1.98526250512761318471e3L, 113 3.31667508019495079814e3L, 114 1.60577839621734713377e4L, 115 -2.97045081369399940529e4L, 116 -7.15743521530849602412e4L 117 }; 118 */ 119 #define MAXGAML 1755.455L 120 /*static const long double LOGPI = 1.14472988584940017414L;*/ 121 122 /* Stirling's formula for the gamma function 123 tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x)) 124 z(x) = x 125 13 <= x <= 1024 126 Relative error 127 n=8, d=0 128 Peak error = 9.44e-21 129 Relative error spread = 8.8e-4 130 */ 131 132 static long double STIR[9] = { 133 7.147391378143610789273E-4L, 134 -2.363848809501759061727E-5L, 135 -5.950237554056330156018E-4L, 136 6.989332260623193171870E-5L, 137 7.840334842744753003862E-4L, 138 -2.294719747873185405699E-4L, 139 -2.681327161876304418288E-3L, 140 3.472222222230075327854E-3L, 141 8.333333333333331800504E-2L, 142 }; 143 144 #define MAXSTIR 1024.0L 145 static const long double SQTPI = 2.50662827463100050242E0L; 146 147 /* 1/tgamma(x) = z P(z) 148 * z(x) = 1/x 149 * 0 < x < 0.03125 150 * Peak relative error 4.2e-23 151 */ 152 153 static long double S[9] = { 154 -1.193945051381510095614E-3L, 155 7.220599478036909672331E-3L, 156 -9.622023360406271645744E-3L, 157 -4.219773360705915470089E-2L, 158 1.665386113720805206758E-1L, 159 -4.200263503403344054473E-2L, 160 -6.558780715202540684668E-1L, 161 5.772156649015328608253E-1L, 162 1.000000000000000000000E0L, 163 }; 164 165 /* 1/tgamma(-x) = z P(z) 166 * z(x) = 1/x 167 * 0 < x < 0.03125 168 * Peak relative error 5.16e-23 169 * Relative error spread = 2.5e-24 170 */ 171 172 static long double SN[9] = { 173 1.133374167243894382010E-3L, 174 7.220837261893170325704E-3L, 175 9.621911155035976733706E-3L, 176 -4.219773343731191721664E-2L, 177 -1.665386113944413519335E-1L, 178 -4.200263503402112910504E-2L, 179 6.558780715202536547116E-1L, 180 5.772156649015328608727E-1L, 181 -1.000000000000000000000E0L, 182 }; 183 184 static const long double PIL = 3.1415926535897932384626L; 185 186 static long double stirf ( long double ); 187 188 /* Gamma function computed by Stirling's formula. 189 */ 190 static long double stirf(long double x) 191 { 192 long double y, w, v; 193 194 w = 1.0L/x; 195 /* For large x, use rational coefficients from the analytical expansion. */ 196 if( x > 1024.0L ) 197 w = (((((6.97281375836585777429E-5L * w 198 + 7.84039221720066627474E-4L) * w 199 - 2.29472093621399176955E-4L) * w 200 - 2.68132716049382716049E-3L) * w 201 + 3.47222222222222222222E-3L) * w 202 + 8.33333333333333333333E-2L) * w 203 + 1.0L; 204 else 205 w = 1.0L + w * __polevll( w, STIR, 8 ); 206 y = expl(x); 207 if( x > MAXSTIR ) 208 { /* Avoid overflow in pow() */ 209 v = powl( x, 0.5L * x - 0.25L ); 210 y = v * (v / y); 211 } 212 else 213 { 214 y = powl( x, x - 0.5L ) / y; 215 } 216 y = SQTPI * y * w; 217 return( y ); 218 } 219 220 long double 221 tgammal(long double x) 222 { 223 long double p, q, z; 224 int i; 225 226 signgam = 1; 227 if( isnan(x) ) 228 return(NAN); 229 if(x == INFINITY) 230 return(INFINITY); 231 if(x == -INFINITY) 232 return(x - x); 233 if( x == 0.0L ) 234 return( 1.0L / x ); 235 q = fabsl(x); 236 237 if( q > 13.0L ) 238 { 239 if( q > MAXGAML ) 240 goto goverf; 241 if( x < 0.0L ) 242 { 243 p = floorl(q); 244 if( p == q ) 245 return (x - x) / (x - x); 246 i = p; 247 if( (i & 1) == 0 ) 248 signgam = -1; 249 z = q - p; 250 if( z > 0.5L ) 251 { 252 p += 1.0L; 253 z = q - p; 254 } 255 z = q * sinl( PIL * z ); 256 z = fabsl(z) * stirf(q); 257 if( z <= PIL/LDBL_MAX ) 258 { 259 goverf: 260 return( signgam * INFINITY); 261 } 262 z = PIL/z; 263 } 264 else 265 { 266 z = stirf(x); 267 } 268 return( signgam * z ); 269 } 270 271 z = 1.0L; 272 while( x >= 3.0L ) 273 { 274 x -= 1.0L; 275 z *= x; 276 } 277 278 while( x < -0.03125L ) 279 { 280 z /= x; 281 x += 1.0L; 282 } 283 284 if( x <= 0.03125L ) 285 goto small; 286 287 while( x < 2.0L ) 288 { 289 z /= x; 290 x += 1.0L; 291 } 292 293 if( x == 2.0L ) 294 return(z); 295 296 x -= 2.0L; 297 p = __polevll( x, P, 7 ); 298 q = __polevll( x, Q, 8 ); 299 z = z * p / q; 300 if( z < 0 ) 301 signgam = -1; 302 return z; 303 304 small: 305 if( x == 0.0L ) 306 return (x - x) / (x - x); 307 else 308 { 309 if( x < 0.0L ) 310 { 311 x = -x; 312 q = z / (x * __polevll( x, SN, 8 )); 313 signgam = -1; 314 } 315 else 316 q = z / (x * __polevll( x, S, 8 )); 317 } 318 return q; 319 } 320