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" ); |