1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_lgammal.c */ 2 /* 3 * ==================================================== 4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 5 * 6 * Developed at SunPro, a Sun Microsystems, Inc. business. 7 * Permission to use, copy, modify, and distribute this 8 * software is freely granted, provided that this notice 9 * is preserved. 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 /* lgammal(x) 28 * Reentrant version of the logarithm of the Gamma function 29 * with user provide pointer for the sign of Gamma(x). 30 * 31 * Method: 32 * 1. Argument Reduction for 0 < x <= 8 33 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 34 * reduce x to a number in [1.5,2.5] by 35 * lgamma(1+s) = log(s) + lgamma(s) 36 * for example, 37 * lgamma(7.3) = log(6.3) + lgamma(6.3) 38 * = log(6.3*5.3) + lgamma(5.3) 39 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) 40 * 2. Polynomial approximation of lgamma around its 41 * minimun ymin=1.461632144968362245 to maintain monotonicity. 42 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use 43 * Let z = x-ymin; 44 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) 45 * 2. Rational approximation in the primary interval [2,3] 46 * We use the following approximation: 47 * s = x-2.0; 48 * lgamma(x) = 0.5*s + s*P(s)/Q(s) 49 * Our algorithms are based on the following observation 50 * 51 * zeta(2)-1 2 zeta(3)-1 3 52 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... 53 * 2 3 54 * 55 * where Euler = 0.5771... is the Euler constant, which is very 56 * close to 0.5. 57 * 58 * 3. For x>=8, we have 59 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... 60 * (better formula: 61 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) 62 * Let z = 1/x, then we approximation 63 * f(z) = lgamma(x) - (x-0.5)(log(x)-1) 64 * by 65 * 3 5 11 66 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z 67 * 68 * 4. For negative x, since (G is gamma function) 69 * -x*G(-x)*G(x) = pi/sin(pi*x), 70 * we have 71 * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) 72 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 73 * Hence, for x<0, signgam = sign(sin(pi*x)) and 74 * lgamma(x) = log(|Gamma(x)|) 75 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); 76 * Note: one should avoid compute pi*(-x) directly in the 77 * computation of sin(pi*(-x)). 78 * 79 * 5. Special Cases 80 * lgamma(2+s) ~ s*(1-Euler) for tiny s 81 * lgamma(1)=lgamma(2)=0 82 * lgamma(x) ~ -log(x) for tiny x 83 * lgamma(0) = lgamma(inf) = inf 84 * lgamma(-integer) = +-inf 85 * 86 */ 87 88 #define _GNU_SOURCE 89 #include "libm.h" 90 91 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 92 long double __lgammal_r(long double x, int *sg) 93 { 94 return __lgamma_r(x, sg); 95 } 96 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 97 static const long double 98 pi = 3.14159265358979323846264L, 99 100 /* lgam(1+x) = 0.5 x + x a(x)/b(x) 101 -0.268402099609375 <= x <= 0 102 peak relative error 6.6e-22 */ 103 a0 = -6.343246574721079391729402781192128239938E2L, 104 a1 = 1.856560238672465796768677717168371401378E3L, 105 a2 = 2.404733102163746263689288466865843408429E3L, 106 a3 = 8.804188795790383497379532868917517596322E2L, 107 a4 = 1.135361354097447729740103745999661157426E2L, 108 a5 = 3.766956539107615557608581581190400021285E0L, 109 110 b0 = 8.214973713960928795704317259806842490498E3L, 111 b1 = 1.026343508841367384879065363925870888012E4L, 112 b2 = 4.553337477045763320522762343132210919277E3L, 113 b3 = 8.506975785032585797446253359230031874803E2L, 114 b4 = 6.042447899703295436820744186992189445813E1L, 115 /* b5 = 1.000000000000000000000000000000000000000E0 */ 116 117 118 tc = 1.4616321449683623412626595423257213284682E0L, 119 tf = -1.2148629053584961146050602565082954242826E-1, /* double precision */ 120 /* tt = (tail of tf), i.e. tf + tt has extended precision. */ 121 tt = 3.3649914684731379602768989080467587736363E-18L, 122 /* lgam ( 1.4616321449683623412626595423257213284682E0 ) = 123 -1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */ 124 125 /* lgam (x + tc) = tf + tt + x g(x)/h(x) 126 -0.230003726999612341262659542325721328468 <= x 127 <= 0.2699962730003876587373404576742786715318 128 peak relative error 2.1e-21 */ 129 g0 = 3.645529916721223331888305293534095553827E-18L, 130 g1 = 5.126654642791082497002594216163574795690E3L, 131 g2 = 8.828603575854624811911631336122070070327E3L, 132 g3 = 5.464186426932117031234820886525701595203E3L, 133 g4 = 1.455427403530884193180776558102868592293E3L, 134 g5 = 1.541735456969245924860307497029155838446E2L, 135 g6 = 4.335498275274822298341872707453445815118E0L, 136 137 h0 = 1.059584930106085509696730443974495979641E4L, 138 h1 = 2.147921653490043010629481226937850618860E4L, 139 h2 = 1.643014770044524804175197151958100656728E4L, 140 h3 = 5.869021995186925517228323497501767586078E3L, 141 h4 = 9.764244777714344488787381271643502742293E2L, 142 h5 = 6.442485441570592541741092969581997002349E1L, 143 /* h6 = 1.000000000000000000000000000000000000000E0 */ 144 145 146 /* lgam (x+1) = -0.5 x + x u(x)/v(x) 147 -0.100006103515625 <= x <= 0.231639862060546875 148 peak relative error 1.3e-21 */ 149 u0 = -8.886217500092090678492242071879342025627E1L, 150 u1 = 6.840109978129177639438792958320783599310E2L, 151 u2 = 2.042626104514127267855588786511809932433E3L, 152 u3 = 1.911723903442667422201651063009856064275E3L, 153 u4 = 7.447065275665887457628865263491667767695E2L, 154 u5 = 1.132256494121790736268471016493103952637E2L, 155 u6 = 4.484398885516614191003094714505960972894E0L, 156 157 v0 = 1.150830924194461522996462401210374632929E3L, 158 v1 = 3.399692260848747447377972081399737098610E3L, 159 v2 = 3.786631705644460255229513563657226008015E3L, 160 v3 = 1.966450123004478374557778781564114347876E3L, 161 v4 = 4.741359068914069299837355438370682773122E2L, 162 v5 = 4.508989649747184050907206782117647852364E1L, 163 /* v6 = 1.000000000000000000000000000000000000000E0 */ 164 165 166 /* lgam (x+2) = .5 x + x s(x)/r(x) 167 0 <= x <= 1 168 peak relative error 7.2e-22 */ 169 s0 = 1.454726263410661942989109455292824853344E6L, 170 s1 = -3.901428390086348447890408306153378922752E6L, 171 s2 = -6.573568698209374121847873064292963089438E6L, 172 s3 = -3.319055881485044417245964508099095984643E6L, 173 s4 = -7.094891568758439227560184618114707107977E5L, 174 s5 = -6.263426646464505837422314539808112478303E4L, 175 s6 = -1.684926520999477529949915657519454051529E3L, 176 177 r0 = -1.883978160734303518163008696712983134698E7L, 178 r1 = -2.815206082812062064902202753264922306830E7L, 179 r2 = -1.600245495251915899081846093343626358398E7L, 180 r3 = -4.310526301881305003489257052083370058799E6L, 181 r4 = -5.563807682263923279438235987186184968542E5L, 182 r5 = -3.027734654434169996032905158145259713083E4L, 183 r6 = -4.501995652861105629217250715790764371267E2L, 184 /* r6 = 1.000000000000000000000000000000000000000E0 */ 185 186 187 /* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2) 188 x >= 8 189 Peak relative error 1.51e-21 190 w0 = LS2PI - 0.5 */ 191 w0 = 4.189385332046727417803e-1L, 192 w1 = 8.333333333333331447505E-2L, 193 w2 = -2.777777777750349603440E-3L, 194 w3 = 7.936507795855070755671E-4L, 195 w4 = -5.952345851765688514613E-4L, 196 w5 = 8.412723297322498080632E-4L, 197 w6 = -1.880801938119376907179E-3L, 198 w7 = 4.885026142432270781165E-3L; 199 200 /* sin(pi*x) assuming x > 2^-1000, if sin(pi*x)==0 the sign is arbitrary */ 201 static long double sin_pi(long double x) 202 { 203 int n; 204 205 /* spurious inexact if odd int */ 206 x *= 0.5; 207 x = 2.0*(x - floorl(x)); /* x mod 2.0 */ 208 209 n = (int)(x*4.0); 210 n = (n+1)/2; 211 x -= n*0.5f; 212 x *= pi; 213 214 switch (n) { 215 default: /* case 4: */ 216 case 0: return __sinl(x, 0.0, 0); 217 case 1: return __cosl(x, 0.0); 218 case 2: return __sinl(-x, 0.0, 0); 219 case 3: return -__cosl(x, 0.0); 220 } 221 } 222 223 long double __lgammal_r(long double x, int *sg) { 224 long double t, y, z, nadj, p, p1, p2, q, r, w; 225 union ldshape u = {x}; 226 uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48; 227 int sign = u.i.se >> 15; 228 int i; 229 230 *sg = 1; 231 232 /* purge off +-inf, NaN, +-0, tiny and negative arguments */ 233 if (ix >= 0x7fff0000) 234 return x * x; 235 if (ix < 0x3fc08000) { /* |x|<2**-63, return -log(|x|) */ 236 if (sign) { 237 *sg = -1; 238 x = -x; 239 } 240 return -logl(x); 241 } 242 if (sign) { 243 x = -x; 244 t = sin_pi(x); 245 if (t == 0.0) 246 return 1.0 / (x-x); /* -integer */ 247 if (t > 0.0) 248 *sg = -1; 249 else 250 t = -t; 251 nadj = logl(pi / (t * x)); 252 } 253 254 /* purge off 1 and 2 (so the sign is ok with downward rounding) */ 255 if ((ix == 0x3fff8000 || ix == 0x40008000) && u.i.m == 0) { 256 r = 0; 257 } else if (ix < 0x40008000) { /* x < 2.0 */ 258 if (ix <= 0x3ffee666) { /* 8.99993896484375e-1 */ 259 /* lgamma(x) = lgamma(x+1) - log(x) */ 260 r = -logl(x); 261 if (ix >= 0x3ffebb4a) { /* 7.31597900390625e-1 */ 262 y = x - 1.0; 263 i = 0; 264 } else if (ix >= 0x3ffced33) { /* 2.31639862060546875e-1 */ 265 y = x - (tc - 1.0); 266 i = 1; 267 } else { /* x < 0.23 */ 268 y = x; 269 i = 2; 270 } 271 } else { 272 r = 0.0; 273 if (ix >= 0x3fffdda6) { /* 1.73162841796875 */ 274 /* [1.7316,2] */ 275 y = x - 2.0; 276 i = 0; 277 } else if (ix >= 0x3fff9da6) { /* 1.23162841796875 */ 278 /* [1.23,1.73] */ 279 y = x - tc; 280 i = 1; 281 } else { 282 /* [0.9, 1.23] */ 283 y = x - 1.0; 284 i = 2; 285 } 286 } 287 switch (i) { 288 case 0: 289 p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5)))); 290 p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y)))); 291 r += 0.5 * y + y * p1/p2; 292 break; 293 case 1: 294 p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6))))); 295 p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y))))); 296 p = tt + y * p1/p2; 297 r += (tf + p); 298 break; 299 case 2: 300 p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6)))))); 301 p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y))))); 302 r += (-0.5 * y + p1 / p2); 303 } 304 } else if (ix < 0x40028000) { /* 8.0 */ 305 /* x < 8.0 */ 306 i = (int)x; 307 y = x - (double)i; 308 p = y * (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6)))))); 309 q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y)))))); 310 r = 0.5 * y + p / q; 311 z = 1.0; 312 /* lgamma(1+s) = log(s) + lgamma(s) */ 313 switch (i) { 314 case 7: 315 z *= (y + 6.0); /* FALLTHRU */ 316 case 6: 317 z *= (y + 5.0); /* FALLTHRU */ 318 case 5: 319 z *= (y + 4.0); /* FALLTHRU */ 320 case 4: 321 z *= (y + 3.0); /* FALLTHRU */ 322 case 3: 323 z *= (y + 2.0); /* FALLTHRU */ 324 r += logl(z); 325 break; 326 } 327 } else if (ix < 0x40418000) { /* 2^66 */ 328 /* 8.0 <= x < 2**66 */ 329 t = logl(x); 330 z = 1.0 / x; 331 y = z * z; 332 w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7)))))); 333 r = (x - 0.5) * (t - 1.0) + w; 334 } else /* 2**66 <= x <= inf */ 335 r = x * (logl(x) - 1.0); 336 if (sign) 337 r = nadj - r; 338 return r; 339 } 340 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 341 // TODO: broken implementation to make things compile 342 long double __lgammal_r(long double x, int *sg) 343 { 344 return __lgamma_r(x, sg); 345 } 346 #endif 347 348 long double lgammal(long double x) 349 { 350 return __lgammal_r(x, &__signgam); 351 } 352 353 weak_alias(__lgammal_r, lgammal_r); 354