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| 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 #include "libc.h" |
| 91 |
| 92 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 |
| 93 double __lgamma_r(double x, int *sg); |
| 94 |
| 95 long double __lgammal_r(long double x, int *sg) |
| 96 { |
| 97 return __lgamma_r(x, sg); |
| 98 } |
| 99 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 |
| 100 static const long double |
| 101 pi = 3.14159265358979323846264L, |
| 102 |
| 103 /* lgam(1+x) = 0.5 x + x a(x)/b(x) |
| 104 -0.268402099609375 <= x <= 0 |
| 105 peak relative error 6.6e-22 */ |
| 106 a0 = -6.343246574721079391729402781192128239938E2L, |
| 107 a1 = 1.856560238672465796768677717168371401378E3L, |
| 108 a2 = 2.404733102163746263689288466865843408429E3L, |
| 109 a3 = 8.804188795790383497379532868917517596322E2L, |
| 110 a4 = 1.135361354097447729740103745999661157426E2L, |
| 111 a5 = 3.766956539107615557608581581190400021285E0L, |
| 112 |
| 113 b0 = 8.214973713960928795704317259806842490498E3L, |
| 114 b1 = 1.026343508841367384879065363925870888012E4L, |
| 115 b2 = 4.553337477045763320522762343132210919277E3L, |
| 116 b3 = 8.506975785032585797446253359230031874803E2L, |
| 117 b4 = 6.042447899703295436820744186992189445813E1L, |
| 118 /* b5 = 1.000000000000000000000000000000000000000E0 */ |
| 119 |
| 120 |
| 121 tc = 1.4616321449683623412626595423257213284682E0L, |
| 122 tf = -1.2148629053584961146050602565082954242826E-1, /* double precision */ |
| 123 /* tt = (tail of tf), i.e. tf + tt has extended precision. */ |
| 124 tt = 3.3649914684731379602768989080467587736363E-18L, |
| 125 /* lgam ( 1.4616321449683623412626595423257213284682E0 ) = |
| 126 -1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */ |
| 127 |
| 128 /* lgam (x + tc) = tf + tt + x g(x)/h(x) |
| 129 -0.230003726999612341262659542325721328468 <= x |
| 130 <= 0.2699962730003876587373404576742786715318 |
| 131 peak relative error 2.1e-21 */ |
| 132 g0 = 3.645529916721223331888305293534095553827E-18L, |
| 133 g1 = 5.126654642791082497002594216163574795690E3L, |
| 134 g2 = 8.828603575854624811911631336122070070327E3L, |
| 135 g3 = 5.464186426932117031234820886525701595203E3L, |
| 136 g4 = 1.455427403530884193180776558102868592293E3L, |
| 137 g5 = 1.541735456969245924860307497029155838446E2L, |
| 138 g6 = 4.335498275274822298341872707453445815118E0L, |
| 139 |
| 140 h0 = 1.059584930106085509696730443974495979641E4L, |
| 141 h1 = 2.147921653490043010629481226937850618860E4L, |
| 142 h2 = 1.643014770044524804175197151958100656728E4L, |
| 143 h3 = 5.869021995186925517228323497501767586078E3L, |
| 144 h4 = 9.764244777714344488787381271643502742293E2L, |
| 145 h5 = 6.442485441570592541741092969581997002349E1L, |
| 146 /* h6 = 1.000000000000000000000000000000000000000E0 */ |
| 147 |
| 148 |
| 149 /* lgam (x+1) = -0.5 x + x u(x)/v(x) |
| 150 -0.100006103515625 <= x <= 0.231639862060546875 |
| 151 peak relative error 1.3e-21 */ |
| 152 u0 = -8.886217500092090678492242071879342025627E1L, |
| 153 u1 = 6.840109978129177639438792958320783599310E2L, |
| 154 u2 = 2.042626104514127267855588786511809932433E3L, |
| 155 u3 = 1.911723903442667422201651063009856064275E3L, |
| 156 u4 = 7.447065275665887457628865263491667767695E2L, |
| 157 u5 = 1.132256494121790736268471016493103952637E2L, |
| 158 u6 = 4.484398885516614191003094714505960972894E0L, |
| 159 |
| 160 v0 = 1.150830924194461522996462401210374632929E3L, |
| 161 v1 = 3.399692260848747447377972081399737098610E3L, |
| 162 v2 = 3.786631705644460255229513563657226008015E3L, |
| 163 v3 = 1.966450123004478374557778781564114347876E3L, |
| 164 v4 = 4.741359068914069299837355438370682773122E2L, |
| 165 v5 = 4.508989649747184050907206782117647852364E1L, |
| 166 /* v6 = 1.000000000000000000000000000000000000000E0 */ |
| 167 |
| 168 |
| 169 /* lgam (x+2) = .5 x + x s(x)/r(x) |
| 170 0 <= x <= 1 |
| 171 peak relative error 7.2e-22 */ |
| 172 s0 = 1.454726263410661942989109455292824853344E6L, |
| 173 s1 = -3.901428390086348447890408306153378922752E6L, |
| 174 s2 = -6.573568698209374121847873064292963089438E6L, |
| 175 s3 = -3.319055881485044417245964508099095984643E6L, |
| 176 s4 = -7.094891568758439227560184618114707107977E5L, |
| 177 s5 = -6.263426646464505837422314539808112478303E4L, |
| 178 s6 = -1.684926520999477529949915657519454051529E3L, |
| 179 |
| 180 r0 = -1.883978160734303518163008696712983134698E7L, |
| 181 r1 = -2.815206082812062064902202753264922306830E7L, |
| 182 r2 = -1.600245495251915899081846093343626358398E7L, |
| 183 r3 = -4.310526301881305003489257052083370058799E6L, |
| 184 r4 = -5.563807682263923279438235987186184968542E5L, |
| 185 r5 = -3.027734654434169996032905158145259713083E4L, |
| 186 r6 = -4.501995652861105629217250715790764371267E2L, |
| 187 /* r6 = 1.000000000000000000000000000000000000000E0 */ |
| 188 |
| 189 |
| 190 /* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2) |
| 191 x >= 8 |
| 192 Peak relative error 1.51e-21 |
| 193 w0 = LS2PI - 0.5 */ |
| 194 w0 = 4.189385332046727417803e-1L, |
| 195 w1 = 8.333333333333331447505E-2L, |
| 196 w2 = -2.777777777750349603440E-3L, |
| 197 w3 = 7.936507795855070755671E-4L, |
| 198 w4 = -5.952345851765688514613E-4L, |
| 199 w5 = 8.412723297322498080632E-4L, |
| 200 w6 = -1.880801938119376907179E-3L, |
| 201 w7 = 4.885026142432270781165E-3L; |
| 202 |
| 203 /* sin(pi*x) assuming x > 2^-1000, if sin(pi*x)==0 the sign is arbitrary */ |
| 204 static long double sin_pi(long double x) |
| 205 { |
| 206 int n; |
| 207 |
| 208 /* spurious inexact if odd int */ |
| 209 x *= 0.5; |
| 210 x = 2.0*(x - floorl(x)); /* x mod 2.0 */ |
| 211 |
| 212 n = (int)(x*4.0); |
| 213 n = (n+1)/2; |
| 214 x -= n*0.5f; |
| 215 x *= pi; |
| 216 |
| 217 switch (n) { |
| 218 default: /* case 4: */ |
| 219 case 0: return __sinl(x, 0.0, 0); |
| 220 case 1: return __cosl(x, 0.0); |
| 221 case 2: return __sinl(-x, 0.0, 0); |
| 222 case 3: return -__cosl(x, 0.0); |
| 223 } |
| 224 } |
| 225 |
| 226 long double __lgammal_r(long double x, int *sg) { |
| 227 long double t, y, z, nadj, p, p1, p2, q, r, w; |
| 228 union ldshape u = {x}; |
| 229 uint32_t ix = (u.i.se & 0x7fffU)<<16 | u.i.m>>48; |
| 230 int sign = u.i.se >> 15; |
| 231 int i; |
| 232 |
| 233 *sg = 1; |
| 234 |
| 235 /* purge off +-inf, NaN, +-0, tiny and negative arguments */ |
| 236 if (ix >= 0x7fff0000) |
| 237 return x * x; |
| 238 if (ix < 0x3fc08000) { /* |x|<2**-63, return -log(|x|) */ |
| 239 if (sign) { |
| 240 *sg = -1; |
| 241 x = -x; |
| 242 } |
| 243 return -logl(x); |
| 244 } |
| 245 if (sign) { |
| 246 x = -x; |
| 247 t = sin_pi(x); |
| 248 if (t == 0.0) |
| 249 return 1.0 / (x-x); /* -integer */ |
| 250 if (t > 0.0) |
| 251 *sg = -1; |
| 252 else |
| 253 t = -t; |
| 254 nadj = logl(pi / (t * x)); |
| 255 } |
| 256 |
| 257 /* purge off 1 and 2 (so the sign is ok with downward rounding) */ |
| 258 if ((ix == 0x3fff8000 || ix == 0x40008000) && u.i.m == 0) { |
| 259 r = 0; |
| 260 } else if (ix < 0x40008000) { /* x < 2.0 */ |
| 261 if (ix <= 0x3ffee666) { /* 8.99993896484375e-1 */ |
| 262 /* lgamma(x) = lgamma(x+1) - log(x) */ |
| 263 r = -logl(x); |
| 264 if (ix >= 0x3ffebb4a) { /* 7.31597900390625e-1 */ |
| 265 y = x - 1.0; |
| 266 i = 0; |
| 267 } else if (ix >= 0x3ffced33) { /* 2.31639862060546875e-
1 */ |
| 268 y = x - (tc - 1.0); |
| 269 i = 1; |
| 270 } else { /* x < 0.23 */ |
| 271 y = x; |
| 272 i = 2; |
| 273 } |
| 274 } else { |
| 275 r = 0.0; |
| 276 if (ix >= 0x3fffdda6) { /* 1.73162841796875 */ |
| 277 /* [1.7316,2] */ |
| 278 y = x - 2.0; |
| 279 i = 0; |
| 280 } else if (ix >= 0x3fff9da6) { /* 1.23162841796875 */ |
| 281 /* [1.23,1.73] */ |
| 282 y = x - tc; |
| 283 i = 1; |
| 284 } else { |
| 285 /* [0.9, 1.23] */ |
| 286 y = x - 1.0; |
| 287 i = 2; |
| 288 } |
| 289 } |
| 290 switch (i) { |
| 291 case 0: |
| 292 p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5
)))); |
| 293 p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y)))); |
| 294 r += 0.5 * y + y * p1/p2; |
| 295 break; |
| 296 case 1: |
| 297 p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g
5 + y * g6))))); |
| 298 p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h
5 + y))))); |
| 299 p = tt + y * p1/p2; |
| 300 r += (tf + p); |
| 301 break; |
| 302 case 2: |
| 303 p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y
* (u5 + y * u6)))))); |
| 304 p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v
5 + y))))); |
| 305 r += (-0.5 * y + p1 / p2); |
| 306 } |
| 307 } else if (ix < 0x40028000) { /* 8.0 */ |
| 308 /* x < 8.0 */ |
| 309 i = (int)x; |
| 310 y = x - (double)i; |
| 311 p = y * (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 +
y * s6)))))); |
| 312 q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (
r6 + y)))))); |
| 313 r = 0.5 * y + p / q; |
| 314 z = 1.0; |
| 315 /* lgamma(1+s) = log(s) + lgamma(s) */ |
| 316 switch (i) { |
| 317 case 7: |
| 318 z *= (y + 6.0); /* FALLTHRU */ |
| 319 case 6: |
| 320 z *= (y + 5.0); /* FALLTHRU */ |
| 321 case 5: |
| 322 z *= (y + 4.0); /* FALLTHRU */ |
| 323 case 4: |
| 324 z *= (y + 3.0); /* FALLTHRU */ |
| 325 case 3: |
| 326 z *= (y + 2.0); /* FALLTHRU */ |
| 327 r += logl(z); |
| 328 break; |
| 329 } |
| 330 } else if (ix < 0x40418000) { /* 2^66 */ |
| 331 /* 8.0 <= x < 2**66 */ |
| 332 t = logl(x); |
| 333 z = 1.0 / x; |
| 334 y = z * z; |
| 335 w = w0 + z * (w1 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (
w6 + y * w7)))))); |
| 336 r = (x - 0.5) * (t - 1.0) + w; |
| 337 } else /* 2**66 <= x <= inf */ |
| 338 r = x * (logl(x) - 1.0); |
| 339 if (sign) |
| 340 r = nadj - r; |
| 341 return r; |
| 342 } |
| 343 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 |
| 344 // TODO: broken implementation to make things compile |
| 345 double __lgamma_r(double x, int *sg); |
| 346 |
| 347 long double __lgammal_r(long double x, int *sg) |
| 348 { |
| 349 return __lgamma_r(x, sg); |
| 350 } |
| 351 #endif |
| 352 |
| 353 extern int __signgam; |
| 354 |
| 355 long double lgammal(long double x) |
| 356 { |
| 357 return __lgammal_r(x, &__signgam); |
| 358 } |
| 359 |
| 360 weak_alias(__lgammal_r, lgammal_r); |
| OLD | NEW |
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Add a "fork" of musl as //fusl. (Closed)
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