1 /* $OpenBSD: e_tgammal.c,v 1.2 2011/07/20 21:02:51 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 /* 64 tgamma(x+2) = tgamma(x+2) P(x)/Q(x) 65 0 <= x <= 1 66 Relative error 67 n=7, d=8 68 Peak error = 1.83e-20 69 Relative error spread = 8.4e-23 70 */ 71 72 static long double P[8] = { 73 4.212760487471622013093E-5L, 74 4.542931960608009155600E-4L, 75 4.092666828394035500949E-3L, 76 2.385363243461108252554E-2L, 77 1.113062816019361559013E-1L, 78 3.629515436640239168939E-1L, 79 8.378004301573126728826E-1L, 80 1.000000000000000000009E0L, 81 }; 82 static long double Q[9] = { 83 -1.397148517476170440917E-5L, 84 2.346584059160635244282E-4L, 85 -1.237799246653152231188E-3L, 86 -7.955933682494738320586E-4L, 87 2.773706565840072979165E-2L, 88 -4.633887671244534213831E-2L, 89 -2.243510905670329164562E-1L, 90 4.150160950588455434583E-1L, 91 9.999999999999999999908E-1L, 92 }; 93 94 /* 95 static long double P[] = { 96 -3.01525602666895735709e0L, 97 -3.25157411956062339893e1L, 98 -2.92929976820724030353e2L, 99 -1.70730828800510297666e3L, 100 -7.96667499622741999770e3L, 101 -2.59780216007146401957e4L, 102 -5.99650230220855581642e4L, 103 -7.15743521530849602425e4L 104 }; 105 static long double Q[] = { 106 1.00000000000000000000e0L, 107 -1.67955233807178858919e1L, 108 8.85946791747759881659e1L, 109 5.69440799097468430177e1L, 110 -1.98526250512761318471e3L, 111 3.31667508019495079814e3L, 112 1.60577839621734713377e4L, 113 -2.97045081369399940529e4L, 114 -7.15743521530849602412e4L 115 }; 116 */ 117 #define MAXGAML 1755.455L 118 /*static const long double LOGPI = 1.14472988584940017414L;*/ 119 120 /* Stirling's formula for the gamma function 121 tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x)) 122 z(x) = x 123 13 <= x <= 1024 124 Relative error 125 n=8, d=0 126 Peak error = 9.44e-21 127 Relative error spread = 8.8e-4 128 */ 129 130 static long double STIR[9] = { 131 7.147391378143610789273E-4L, 132 -2.363848809501759061727E-5L, 133 -5.950237554056330156018E-4L, 134 6.989332260623193171870E-5L, 135 7.840334842744753003862E-4L, 136 -2.294719747873185405699E-4L, 137 -2.681327161876304418288E-3L, 138 3.472222222230075327854E-3L, 139 8.333333333333331800504E-2L, 140 }; 141 142 #define MAXSTIR 1024.0L 143 static const long double SQTPI = 2.50662827463100050242E0L; 144 145 /* 1/tgamma(x) = z P(z) 146 * z(x) = 1/x 147 * 0 < x < 0.03125 148 * Peak relative error 4.2e-23 149 */ 150 151 static long double S[9] = { 152 -1.193945051381510095614E-3L, 153 7.220599478036909672331E-3L, 154 -9.622023360406271645744E-3L, 155 -4.219773360705915470089E-2L, 156 1.665386113720805206758E-1L, 157 -4.200263503403344054473E-2L, 158 -6.558780715202540684668E-1L, 159 5.772156649015328608253E-1L, 160 1.000000000000000000000E0L, 161 }; 162 163 /* 1/tgamma(-x) = z P(z) 164 * z(x) = 1/x 165 * 0 < x < 0.03125 166 * Peak relative error 5.16e-23 167 * Relative error spread = 2.5e-24 168 */ 169 170 static long double SN[9] = { 171 1.133374167243894382010E-3L, 172 7.220837261893170325704E-3L, 173 9.621911155035976733706E-3L, 174 -4.219773343731191721664E-2L, 175 -1.665386113944413519335E-1L, 176 -4.200263503402112910504E-2L, 177 6.558780715202536547116E-1L, 178 5.772156649015328608727E-1L, 179 -1.000000000000000000000E0L, 180 }; 181 182 static const long double PIL = 3.1415926535897932384626L; 183 184 extern long double __polevll(long double, void *, int); 185 static long double stirf ( long double ); 186 187 /* Gamma function computed by Stirling's formula. 188 */ 189 static long double stirf(long double x) 190 { 191 long double y, w, v; 192 193 w = 1.0L/x; 194 /* For large x, use rational coefficients from the analytical expansion. */ 195 if( x > 1024.0L ) 196 w = (((((6.97281375836585777429E-5L * w 197 + 7.84039221720066627474E-4L) * w 198 - 2.29472093621399176955E-4L) * w 199 - 2.68132716049382716049E-3L) * w 200 + 3.47222222222222222222E-3L) * w 201 + 8.33333333333333333333E-2L) * w 202 + 1.0L; 203 else 204 w = 1.0L + w * __polevll( w, STIR, 8 ); 205 y = expl(x); 206 if( x > MAXSTIR ) 207 { /* Avoid overflow in pow() */ 208 v = powl( x, 0.5L * x - 0.25L ); 209 y = v * (v / y); 210 } 211 else 212 { 213 y = powl( x, x - 0.5L ) / y; 214 } 215 y = SQTPI * y * w; 216 return( y ); 217 } 218 219 long double 220 tgammal(long double x) 221 { 222 long double p, q, z; 223 int i; 224 225 signgam = 1; 226 if( isnan(x) ) 227 return(NAN); 228 if(x == INFINITY) 229 return(INFINITY); 230 if(x == -INFINITY) 231 return(x - x); 232 q = fabsl(x); 233 234 if( q > 13.0L ) 235 { 236 if( q > MAXGAML ) 237 goto goverf; 238 if( x < 0.0L ) 239 { 240 p = floorl(q); 241 if( p == q ) 242 return (x - x) / (x - x); 243 i = p; 244 if( (i & 1) == 0 ) 245 signgam = -1; 246 z = q - p; 247 if( z > 0.5L ) 248 { 249 p += 1.0L; 250 z = q - p; 251 } 252 z = q * sinl( PIL * z ); 253 z = fabsl(z) * stirf(q); 254 if( z <= PIL/LDBL_MAX ) 255 { 256 goverf: 257 return( signgam * INFINITY); 258 } 259 z = PIL/z; 260 } 261 else 262 { 263 z = stirf(x); 264 } 265 return( signgam * z ); 266 } 267 268 z = 1.0L; 269 while( x >= 3.0L ) 270 { 271 x -= 1.0L; 272 z *= x; 273 } 274 275 while( x < -0.03125L ) 276 { 277 z /= x; 278 x += 1.0L; 279 } 280 281 if( x <= 0.03125L ) 282 goto small; 283 284 while( x < 2.0L ) 285 { 286 z /= x; 287 x += 1.0L; 288 } 289 290 if( x == 2.0L ) 291 return(z); 292 293 x -= 2.0L; 294 p = __polevll( x, P, 7 ); 295 q = __polevll( x, Q, 8 ); 296 z = z * p / q; 297 if( z < 0 ) 298 signgam = -1; 299 return z; 300 301 small: 302 if( x == 0.0L ) 303 return (x - x) / (x - x); 304 else 305 { 306 if( x < 0.0L ) 307 { 308 x = -x; 309 q = z / (x * __polevll( x, SN, 8 )); 310 signgam = -1; 311 } 312 else 313 q = z / (x * __polevll( x, S, 8 )); 314 } 315 return q; 316 } 317