1 /* 2 * ==================================================== 3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 4 * 5 * Developed at SunPro, a Sun Microsystems, Inc. business. 6 * Permission to use, copy, modify, and distribute this 7 * software is freely granted, provided that this notice 8 * is preserved. 9 * ==================================================== 10 */ 11 12 /* 13 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 14 * 15 * Permission to use, copy, modify, and distribute this software for any 16 * purpose with or without fee is hereby granted, provided that the above 17 * copyright notice and this permission notice appear in all copies. 18 * 19 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 20 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 21 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 22 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 23 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 24 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 25 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 26 */ 27 28 /* lgammal(x) 29 * Reentrant version of the logarithm of the Gamma function 30 * with user provide pointer for the sign of Gamma(x). 31 * 32 * Method: 33 * 1. Argument Reduction for 0 < x <= 8 34 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 35 * reduce x to a number in [1.5,2.5] by 36 * lgamma(1+s) = log(s) + lgamma(s) 37 * for example, 38 * lgamma(7.3) = log(6.3) + lgamma(6.3) 39 * = log(6.3*5.3) + lgamma(5.3) 40 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) 41 * 2. Polynomial approximation of lgamma around its 42 * minimun ymin=1.461632144968362245 to maintain monotonicity. 43 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use 44 * Let z = x-ymin; 45 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) 46 * 2. Rational approximation in the primary interval [2,3] 47 * We use the following approximation: 48 * s = x-2.0; 49 * lgamma(x) = 0.5*s + s*P(s)/Q(s) 50 * Our algorithms are based on the following observation 51 * 52 * zeta(2)-1 2 zeta(3)-1 3 53 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... 54 * 2 3 55 * 56 * where Euler = 0.5771... is the Euler constant, which is very 57 * close to 0.5. 58 * 59 * 3. For x>=8, we have 60 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... 61 * (better formula: 62 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) 63 * Let z = 1/x, then we approximation 64 * f(z) = lgamma(x) - (x-0.5)(log(x)-1) 65 * by 66 * 3 5 11 67 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z 68 * 69 * 4. For negative x, since (G is gamma function) 70 * -x*G(-x)*G(x) = pi/sin(pi*x), 71 * we have 72 * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) 73 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 74 * Hence, for x<0, signgam = sign(sin(pi*x)) and 75 * lgamma(x) = log(|Gamma(x)|) 76 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); 77 * Note: one should avoid compute pi*(-x) directly in the 78 * computation of sin(pi*(-x)). 79 * 80 * 5. Special Cases 81 * lgamma(2+s) ~ s*(1-Euler) for tiny s 82 * lgamma(1)=lgamma(2)=0 83 * lgamma(x) ~ -log(x) for tiny x 84 * lgamma(0) = lgamma(inf) = inf 85 * lgamma(-integer) = +-inf 86 * 87 */ 88 89 #include <math.h> 90 91 #include "math_private.h" 92 93 static const long double 94 half = 0.5L, 95 one = 1.0L, 96 pi = 3.14159265358979323846264L, 97 two63 = 9.223372036854775808e18L, 98 99 /* lgam(1+x) = 0.5 x + x a(x)/b(x) 100 -0.268402099609375 <= x <= 0 101 peak relative error 6.6e-22 */ 102 a0 = -6.343246574721079391729402781192128239938E2L, 103 a1 = 1.856560238672465796768677717168371401378E3L, 104 a2 = 2.404733102163746263689288466865843408429E3L, 105 a3 = 8.804188795790383497379532868917517596322E2L, 106 a4 = 1.135361354097447729740103745999661157426E2L, 107 a5 = 3.766956539107615557608581581190400021285E0L, 108 109 b0 = 8.214973713960928795704317259806842490498E3L, 110 b1 = 1.026343508841367384879065363925870888012E4L, 111 b2 = 4.553337477045763320522762343132210919277E3L, 112 b3 = 8.506975785032585797446253359230031874803E2L, 113 b4 = 6.042447899703295436820744186992189445813E1L, 114 /* b5 = 1.000000000000000000000000000000000000000E0 */ 115 116 117 tc = 1.4616321449683623412626595423257213284682E0L, 118 tf = -1.2148629053584961146050602565082954242826E-1,/* double precision */ 119 /* tt = (tail of tf), i.e. tf + tt has extended precision. */ 120 tt = 3.3649914684731379602768989080467587736363E-18L, 121 /* lgam ( 1.4616321449683623412626595423257213284682E0 ) = 122 -1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */ 123 124 /* lgam (x + tc) = tf + tt + x g(x)/h(x) 125 - 0.230003726999612341262659542325721328468 <= x 126 <= 0.2699962730003876587373404576742786715318 127 peak relative error 2.1e-21 */ 128 g0 = 3.645529916721223331888305293534095553827E-18L, 129 g1 = 5.126654642791082497002594216163574795690E3L, 130 g2 = 8.828603575854624811911631336122070070327E3L, 131 g3 = 5.464186426932117031234820886525701595203E3L, 132 g4 = 1.455427403530884193180776558102868592293E3L, 133 g5 = 1.541735456969245924860307497029155838446E2L, 134 g6 = 4.335498275274822298341872707453445815118E0L, 135 136 h0 = 1.059584930106085509696730443974495979641E4L, 137 h1 = 2.147921653490043010629481226937850618860E4L, 138 h2 = 1.643014770044524804175197151958100656728E4L, 139 h3 = 5.869021995186925517228323497501767586078E3L, 140 h4 = 9.764244777714344488787381271643502742293E2L, 141 h5 = 6.442485441570592541741092969581997002349E1L, 142 /* h6 = 1.000000000000000000000000000000000000000E0 */ 143 144 145 /* lgam (x+1) = -0.5 x + x u(x)/v(x) 146 -0.100006103515625 <= x <= 0.231639862060546875 147 peak relative error 1.3e-21 */ 148 u0 = -8.886217500092090678492242071879342025627E1L, 149 u1 = 6.840109978129177639438792958320783599310E2L, 150 u2 = 2.042626104514127267855588786511809932433E3L, 151 u3 = 1.911723903442667422201651063009856064275E3L, 152 u4 = 7.447065275665887457628865263491667767695E2L, 153 u5 = 1.132256494121790736268471016493103952637E2L, 154 u6 = 4.484398885516614191003094714505960972894E0L, 155 156 v0 = 1.150830924194461522996462401210374632929E3L, 157 v1 = 3.399692260848747447377972081399737098610E3L, 158 v2 = 3.786631705644460255229513563657226008015E3L, 159 v3 = 1.966450123004478374557778781564114347876E3L, 160 v4 = 4.741359068914069299837355438370682773122E2L, 161 v5 = 4.508989649747184050907206782117647852364E1L, 162 /* v6 = 1.000000000000000000000000000000000000000E0 */ 163 164 165 /* lgam (x+2) = .5 x + x s(x)/r(x) 166 0 <= x <= 1 167 peak relative error 7.2e-22 */ 168 s0 = 1.454726263410661942989109455292824853344E6L, 169 s1 = -3.901428390086348447890408306153378922752E6L, 170 s2 = -6.573568698209374121847873064292963089438E6L, 171 s3 = -3.319055881485044417245964508099095984643E6L, 172 s4 = -7.094891568758439227560184618114707107977E5L, 173 s5 = -6.263426646464505837422314539808112478303E4L, 174 s6 = -1.684926520999477529949915657519454051529E3L, 175 176 r0 = -1.883978160734303518163008696712983134698E7L, 177 r1 = -2.815206082812062064902202753264922306830E7L, 178 r2 = -1.600245495251915899081846093343626358398E7L, 179 r3 = -4.310526301881305003489257052083370058799E6L, 180 r4 = -5.563807682263923279438235987186184968542E5L, 181 r5 = -3.027734654434169996032905158145259713083E4L, 182 r6 = -4.501995652861105629217250715790764371267E2L, 183 /* r6 = 1.000000000000000000000000000000000000000E0 */ 184 185 186 /* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2) 187 x >= 8 188 Peak relative error 1.51e-21 189 w0 = LS2PI - 0.5 */ 190 w0 = 4.189385332046727417803e-1L, 191 w1 = 8.333333333333331447505E-2L, 192 w2 = -2.777777777750349603440E-3L, 193 w3 = 7.936507795855070755671E-4L, 194 w4 = -5.952345851765688514613E-4L, 195 w5 = 8.412723297322498080632E-4L, 196 w6 = -1.880801938119376907179E-3L, 197 w7 = 4.885026142432270781165E-3L; 198 199 static const long double zero = 0.0L; 200 201 static long double 202 sin_pi(long double x) 203 { 204 long double y, z; 205 int n, ix; 206 u_int32_t se, i0, i1; 207 208 GET_LDOUBLE_WORDS (se, i0, i1, x); 209 ix = se & 0x7fff; 210 ix = (ix << 16) | (i0 >> 16); 211 if (ix < 0x3ffd8000) /* 0.25 */ 212 return sinl (pi * x); 213 y = -x; /* x is assume negative */ 214 215 /* 216 * argument reduction, make sure inexact flag not raised if input 217 * is an integer 218 */ 219 z = floorl (y); 220 if (z != y) 221 { /* inexact anyway */ 222 y *= 0.5; 223 y = 2.0*(y - floorl(y)); /* y = |x| mod 2.0 */ 224 n = (int) (y*4.0); 225 } 226 else 227 { 228 if (ix >= 0x403f8000) /* 2^64 */ 229 { 230 y = zero; n = 0; /* y must be even */ 231 } 232 else 233 { 234 if (ix < 0x403e8000) /* 2^63 */ 235 z = y + two63; /* exact */ 236 GET_LDOUBLE_WORDS (se, i0, i1, z); 237 n = i1 & 1; 238 y = n; 239 n <<= 2; 240 } 241 } 242 243 switch (n) 244 { 245 case 0: 246 y = sinl (pi * y); 247 break; 248 case 1: 249 case 2: 250 y = cosl (pi * (half - y)); 251 break; 252 case 3: 253 case 4: 254 y = sinl (pi * (one - y)); 255 break; 256 case 5: 257 case 6: 258 y = -cosl (pi * (y - 1.5)); 259 break; 260 default: 261 y = sinl (pi * (y - 2.0)); 262 break; 263 } 264 return -y; 265 } 266 267 268 long double 269 lgammal(long double x) 270 { 271 long double t, y, z, nadj, p, p1, p2, q, r, w; 272 int i, ix; 273 u_int32_t se, i0, i1; 274 275 signgam = 1; 276 GET_LDOUBLE_WORDS (se, i0, i1, x); 277 ix = se & 0x7fff; 278 279 if ((ix | i0 | i1) == 0) 280 { 281 if (se & 0x8000) 282 signgam = -1; 283 return one / fabsl (x); 284 } 285 286 ix = (ix << 16) | (i0 >> 16); 287 288 /* purge off +-inf, NaN, +-0, and negative arguments */ 289 if (ix >= 0x7fff0000) 290 return x * x; 291 292 if (ix < 0x3fc08000) /* 2^-63 */ 293 { /* |x|<2**-63, return -log(|x|) */ 294 if (se & 0x8000) 295 { 296 signgam = -1; 297 return -logl (-x); 298 } 299 else 300 return -logl (x); 301 } 302 if (se & 0x8000) 303 { 304 t = sin_pi (x); 305 if (t == zero) 306 return one / fabsl (t); /* -integer */ 307 nadj = logl (pi / fabsl (t * x)); 308 if (t < zero) 309 signgam = -1; 310 x = -x; 311 } 312 313 /* purge off 1 and 2 */ 314 if ((((ix - 0x3fff8000) | i0 | i1) == 0) 315 || (((ix - 0x40008000) | i0 | i1) == 0)) 316 r = 0; 317 else if (ix < 0x40008000) /* 2.0 */ 318 { 319 /* x < 2.0 */ 320 if (ix <= 0x3ffee666) /* 8.99993896484375e-1 */ 321 { 322 /* lgamma(x) = lgamma(x+1) - log(x) */ 323 r = -logl (x); 324 if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1 */ 325 { 326 y = x - one; 327 i = 0; 328 } 329 else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1 */ 330 { 331 y = x - (tc - one); 332 i = 1; 333 } 334 else 335 { 336 /* x < 0.23 */ 337 y = x; 338 i = 2; 339 } 340 } 341 else 342 { 343 r = zero; 344 if (ix >= 0x3fffdda6) /* 1.73162841796875 */ 345 { 346 /* [1.7316,2] */ 347 y = x - 2.0; 348 i = 0; 349 } 350 else if (ix >= 0x3fff9da6)/* 1.23162841796875 */ 351 { 352 /* [1.23,1.73] */ 353 y = x - tc; 354 i = 1; 355 } 356 else 357 { 358 /* [0.9, 1.23] */ 359 y = x - one; 360 i = 2; 361 } 362 } 363 switch (i) 364 { 365 case 0: 366 p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5)))); 367 p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y)))); 368 r += half * y + y * p1/p2; 369 break; 370 case 1: 371 p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6))))); 372 p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y))))); 373 p = tt + y * p1/p2; 374 r += (tf + p); 375 break; 376 case 2: 377 p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6)))))); 378 p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y))))); 379 r += (-half * y + p1 / p2); 380 } 381 } 382 else if (ix < 0x40028000) /* 8.0 */ 383 { 384 /* x < 8.0 */ 385 i = (int) x; 386 t = zero; 387 y = x - (double) i; 388 p = y * 389 (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6)))))); 390 q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y)))))); 391 r = half * y + p / q; 392 z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ 393 switch (i) 394 { 395 case 7: 396 z *= (y + 6.0); /* FALLTHRU */ 397 case 6: 398 z *= (y + 5.0); /* FALLTHRU */ 399 case 5: 400 z *= (y + 4.0); /* FALLTHRU */ 401 case 4: 402 z *= (y + 3.0); /* FALLTHRU */ 403 case 3: 404 z *= (y + 2.0); /* FALLTHRU */ 405 r += logl (z); 406 break; 407 } 408 } 409 else if (ix < 0x40418000) /* 2^66 */ 410 { 411 /* 8.0 <= x < 2**66 */ 412 t = logl (x); 413 z = one / x; 414 y = z * z; 415 w = w0 + z * (w1 416 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7)))))); 417 r = (x - half) * (t - one) + w; 418 } 419 else 420 /* 2**66 <= x <= inf */ 421 r = x * (logl (x) - one); 422 if (se & 0x8000) 423 r = nadj - r; 424 return r; 425 } 426