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 * 29 * y = tgammal( x ); 30 * 31 * 32 * 33 * DESCRIPTION: 34 * 35 * Returns gamma function of the argument. The result is correctly 36 * signed. This variable is also filled in by the logarithmic gamma 37 * function lgamma(). 38 * 39 * Arguments |x| <= 13 are reduced by recurrence and the function 40 * approximated by a rational function of degree 7/8 in the 41 * interval (2,3). Large arguments are handled by Stirling's 42 * formula. Large negative arguments are made positive using 43 * a reflection formula. 44 * 45 * 46 * ACCURACY: 47 * 48 * Relative error: 49 * arithmetic domain # trials peak rms 50 * IEEE -40,+40 10000 3.6e-19 7.9e-20 51 * IEEE -1755,+1755 10000 4.8e-18 6.5e-19 52 * 53 * Accuracy for large arguments is dominated by error in powl(). 54 * 55 */ 56 57 #include <float.h> 58 #include <openlibm.h> 59 60 #include "math_private.h" 61 62 /* 63 tgamma(x+2) = tgamma(x+2) P(x)/Q(x) 64 0 <= x <= 1 65 Relative error 66 n=7, d=8 67 Peak error = 1.83e-20 68 Relative error spread = 8.4e-23 69 */ 70 71 static long double P[8] = { 72 4.212760487471622013093E-5L, 73 4.542931960608009155600E-4L, 74 4.092666828394035500949E-3L, 75 2.385363243461108252554E-2L, 76 1.113062816019361559013E-1L, 77 3.629515436640239168939E-1L, 78 8.378004301573126728826E-1L, 79 1.000000000000000000009E0L, 80 }; 81 static long double Q[9] = { 82 -1.397148517476170440917E-5L, 83 2.346584059160635244282E-4L, 84 -1.237799246653152231188E-3L, 85 -7.955933682494738320586E-4L, 86 2.773706565840072979165E-2L, 87 -4.633887671244534213831E-2L, 88 -2.243510905670329164562E-1L, 89 4.150160950588455434583E-1L, 90 9.999999999999999999908E-1L, 91 }; 92 93 /* 94 static long double P[] = { 95 -3.01525602666895735709e0L, 96 -3.25157411956062339893e1L, 97 -2.92929976820724030353e2L, 98 -1.70730828800510297666e3L, 99 -7.96667499622741999770e3L, 100 -2.59780216007146401957e4L, 101 -5.99650230220855581642e4L, 102 -7.15743521530849602425e4L 103 }; 104 static long double Q[] = { 105 1.00000000000000000000e0L, 106 -1.67955233807178858919e1L, 107 8.85946791747759881659e1L, 108 5.69440799097468430177e1L, 109 -1.98526250512761318471e3L, 110 3.31667508019495079814e3L, 111 1.60577839621734713377e4L, 112 -2.97045081369399940529e4L, 113 -7.15743521530849602412e4L 114 }; 115 */ 116 #define MAXGAML 1755.455L 117 /*static const long double LOGPI = 1.14472988584940017414L;*/ 118 119 /* Stirling's formula for the gamma function 120 tgamma(x) = sqrt(2 pi) x^(x-.5) exp(-x) (1 + 1/x P(1/x)) 121 z(x) = x 122 13 <= x <= 1024 123 Relative error 124 n=8, d=0 125 Peak error = 9.44e-21 126 Relative error spread = 8.8e-4 127 */ 128 129 static long double STIR[9] = { 130 7.147391378143610789273E-4L, 131 -2.363848809501759061727E-5L, 132 -5.950237554056330156018E-4L, 133 6.989332260623193171870E-5L, 134 7.840334842744753003862E-4L, 135 -2.294719747873185405699E-4L, 136 -2.681327161876304418288E-3L, 137 3.472222222230075327854E-3L, 138 8.333333333333331800504E-2L, 139 }; 140 141 #define MAXSTIR 1024.0L 142 static const long double SQTPI = 2.50662827463100050242E0L; 143 144 /* 1/tgamma(x) = z P(z) 145 * z(x) = 1/x 146 * 0 < x < 0.03125 147 * Peak relative error 4.2e-23 148 */ 149 150 static long double S[9] = { 151 -1.193945051381510095614E-3L, 152 7.220599478036909672331E-3L, 153 -9.622023360406271645744E-3L, 154 -4.219773360705915470089E-2L, 155 1.665386113720805206758E-1L, 156 -4.200263503403344054473E-2L, 157 -6.558780715202540684668E-1L, 158 5.772156649015328608253E-1L, 159 1.000000000000000000000E0L, 160 }; 161 162 /* 1/tgamma(-x) = z P(z) 163 * z(x) = 1/x 164 * 0 < x < 0.03125 165 * Peak relative error 5.16e-23 166 * Relative error spread = 2.5e-24 167 */ 168 169 static long double SN[9] = { 170 1.133374167243894382010E-3L, 171 7.220837261893170325704E-3L, 172 9.621911155035976733706E-3L, 173 -4.219773343731191721664E-2L, 174 -1.665386113944413519335E-1L, 175 -4.200263503402112910504E-2L, 176 6.558780715202536547116E-1L, 177 5.772156649015328608727E-1L, 178 -1.000000000000000000000E0L, 179 }; 180 181 static const long double PIL = 3.1415926535897932384626L; 182 183 static long double stirf ( long double ); 184 185 /* Gamma function computed by Stirling's formula. 186 */ 187 static long double stirf(long double x) 188 { 189 long double y, w, v; 190 191 w = 1.0L/x; 192 /* For large x, use rational coefficients from the analytical expansion. */ 193 if( x > 1024.0L ) 194 w = (((((6.97281375836585777429E-5L * w 195 + 7.84039221720066627474E-4L) * w 196 - 2.29472093621399176955E-4L) * w 197 - 2.68132716049382716049E-3L) * w 198 + 3.47222222222222222222E-3L) * w 199 + 8.33333333333333333333E-2L) * w 200 + 1.0L; 201 else 202 w = 1.0L + w * __polevll( w, STIR, 8 ); 203 y = expl(x); 204 if( x > MAXSTIR ) 205 { /* Avoid overflow in pow() */ 206 v = powl( x, 0.5L * x - 0.25L ); 207 y = v * (v / y); 208 } 209 else 210 { 211 y = powl( x, x - 0.5L ) / y; 212 } 213 y = SQTPI * y * w; 214 return( y ); 215 } 216 217 long double 218 tgammal(long double x) 219 { 220 long double p, q, z; 221 int i; 222 223 if( isnan(x) ) 224 return(NAN); 225 if(x == INFINITY) 226 return(INFINITY); 227 if(x == -INFINITY) 228 return(x - x); 229 if( x == 0.0L ) 230 return( 1.0L / x ); 231 q = fabsl(x); 232 233 if( q > 13.0L ) 234 { 235 int sign = 1; 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 sign = -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( sign * INFINITY); 258 } 259 z = PIL/z; 260 } 261 else 262 { 263 z = stirf(x); 264 } 265 return( sign * 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 return z; 298 299 small: 300 if( x == 0.0L ) 301 return (x - x) / (x - x); 302 else 303 { 304 if( x < 0.0L ) 305 { 306 x = -x; 307 q = z / (x * __polevll( x, SN, 8 )); 308 } 309 else 310 q = z / (x * __polevll( x, S, 8 )); 311 } 312 return q; 313 } 314