| File: | src/lib/libm/src/ld80/e_lgammal.c |
| Warning: | line 386, column 7 Value stored to 't' is never read |
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| 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)do { ieee_extended_shape_type ew_u; ew_u.value = (x); (se) = ew_u .parts.exp; (i0) = ew_u.parts.msw; (i1) = ew_u.parts.lsw; } while (0); |
| 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)do { ieee_extended_shape_type ew_u; ew_u.value = (z); (se) = ew_u .parts.exp; (i0) = ew_u.parts.msw; (i1) = ew_u.parts.lsw; } while (0); |
| 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)do { ieee_extended_shape_type ew_u; ew_u.value = (x); (se) = ew_u .parts.exp; (i0) = ew_u.parts.msw; (i1) = ew_u.parts.lsw; } while (0); |
| 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; |
Value stored to 't' is never read | |
| 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 | DEF_STD(lgammal)__asm__(".global " "lgammal" " ; " "lgammal" " = " "_libm_lgammal" ); |