lgammal.c raw
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