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_r(x, signgamp)
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
90
91 static const long double
92 half = 0.5L,
93 one = 1.0L,
94 pi = 3.14159265358979323846264L,
95 two63 = 9.223372036854775808e18L,
96
97 /* lgam(1+x) = 0.5 x + x a(x)/b(x)
98 -0.268402099609375 <= x <= 0
99 peak relative error 6.6e-22 */
100 a0 = -6.343246574721079391729402781192128239938E2L,
101 a1 = 1.856560238672465796768677717168371401378E3L,
102 a2 = 2.404733102163746263689288466865843408429E3L,
103 a3 = 8.804188795790383497379532868917517596322E2L,
104 a4 = 1.135361354097447729740103745999661157426E2L,
105 a5 = 3.766956539107615557608581581190400021285E0L,
106
107 b0 = 8.214973713960928795704317259806842490498E3L,
108 b1 = 1.026343508841367384879065363925870888012E4L,
109 b2 = 4.553337477045763320522762343132210919277E3L,
110 b3 = 8.506975785032585797446253359230031874803E2L,
111 b4 = 6.042447899703295436820744186992189445813E1L,
112 /* b5 = 1.000000000000000000000000000000000000000E0 */
113
114
115 tc = 1.4616321449683623412626595423257213284682E0L,
116 tf = -1.2148629053584961146050602565082954242826E-1L,/* double precision */
117 /* tt = (tail of tf), i.e. tf + tt has extended precision. */
118 tt = 3.3649914684731379602768989080467587736363E-18L,
119 /* lgam ( 1.4616321449683623412626595423257213284682E0 ) =
120 -1.2148629053584960809551455717769158215135617312999903886372437313313530E-1 */
121
122 /* lgam (x + tc) = tf + tt + x g(x)/h(x)
123 - 0.230003726999612341262659542325721328468 <= x
124 <= 0.2699962730003876587373404576742786715318
125 peak relative error 2.1e-21 */
126 g0 = 3.645529916721223331888305293534095553827E-18L,
127 g1 = 5.126654642791082497002594216163574795690E3L,
128 g2 = 8.828603575854624811911631336122070070327E3L,
129 g3 = 5.464186426932117031234820886525701595203E3L,
130 g4 = 1.455427403530884193180776558102868592293E3L,
131 g5 = 1.541735456969245924860307497029155838446E2L,
132 g6 = 4.335498275274822298341872707453445815118E0L,
133
134 h0 = 1.059584930106085509696730443974495979641E4L,
135 h1 = 2.147921653490043010629481226937850618860E4L,
136 h2 = 1.643014770044524804175197151958100656728E4L,
137 h3 = 5.869021995186925517228323497501767586078E3L,
138 h4 = 9.764244777714344488787381271643502742293E2L,
139 h5 = 6.442485441570592541741092969581997002349E1L,
140 /* h6 = 1.000000000000000000000000000000000000000E0 */
141
142
143 /* lgam (x+1) = -0.5 x + x u(x)/v(x)
144 -0.100006103515625 <= x <= 0.231639862060546875
145 peak relative error 1.3e-21 */
146 u0 = -8.886217500092090678492242071879342025627E1L,
147 u1 = 6.840109978129177639438792958320783599310E2L,
148 u2 = 2.042626104514127267855588786511809932433E3L,
149 u3 = 1.911723903442667422201651063009856064275E3L,
150 u4 = 7.447065275665887457628865263491667767695E2L,
151 u5 = 1.132256494121790736268471016493103952637E2L,
152 u6 = 4.484398885516614191003094714505960972894E0L,
153
154 v0 = 1.150830924194461522996462401210374632929E3L,
155 v1 = 3.399692260848747447377972081399737098610E3L,
156 v2 = 3.786631705644460255229513563657226008015E3L,
157 v3 = 1.966450123004478374557778781564114347876E3L,
158 v4 = 4.741359068914069299837355438370682773122E2L,
159 v5 = 4.508989649747184050907206782117647852364E1L,
160 /* v6 = 1.000000000000000000000000000000000000000E0 */
161
162
163 /* lgam (x+2) = .5 x + x s(x)/r(x)
164 0 <= x <= 1
165 peak relative error 7.2e-22 */
166 s0 = 1.454726263410661942989109455292824853344E6L,
167 s1 = -3.901428390086348447890408306153378922752E6L,
168 s2 = -6.573568698209374121847873064292963089438E6L,
169 s3 = -3.319055881485044417245964508099095984643E6L,
170 s4 = -7.094891568758439227560184618114707107977E5L,
171 s5 = -6.263426646464505837422314539808112478303E4L,
172 s6 = -1.684926520999477529949915657519454051529E3L,
173
174 r0 = -1.883978160734303518163008696712983134698E7L,
175 r1 = -2.815206082812062064902202753264922306830E7L,
176 r2 = -1.600245495251915899081846093343626358398E7L,
177 r3 = -4.310526301881305003489257052083370058799E6L,
178 r4 = -5.563807682263923279438235987186184968542E5L,
179 r5 = -3.027734654434169996032905158145259713083E4L,
180 r6 = -4.501995652861105629217250715790764371267E2L,
181 /* r6 = 1.000000000000000000000000000000000000000E0 */
182
183
184 /* lgam(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x w(1/x^2)
185 x >= 8
186 Peak relative error 1.51e-21
187 w0 = LS2PI - 0.5 */
188 w0 = 4.189385332046727417803e-1L,
189 w1 = 8.333333333333331447505E-2L,
190 w2 = -2.777777777750349603440E-3L,
191 w3 = 7.936507795855070755671E-4L,
192 w4 = -5.952345851765688514613E-4L,
193 w5 = 8.412723297322498080632E-4L,
194 w6 = -1.880801938119376907179E-3L,
195 w7 = 4.885026142432270781165E-3L;
196
197 static const long double zero = 0.0L;
198
199 static long double
sin_pi(long double x)200 sin_pi(long double x)
201 {
202 long double y, z;
203 int n, ix;
204 u_int32_t se, i0, i1;
205
206 GET_LDOUBLE_WORDS (se, i0, i1, x);
207 ix = se & 0x7fff;
208 ix = (ix << 16) | (i0 >> 16);
209 if (ix < 0x3ffd8000) /* 0.25 */
210 return sinl (pi * x);
211 y = -x; /* x is assume negative */
212
213 /*
214 * argument reduction, make sure inexact flag not raised if input
215 * is an integer
216 */
217 z = floorl (y);
218 if (z != y)
219 { /* inexact anyway */
220 y *= 0.5L;
221 y = 2.0L*(y - floorl(y)); /* y = |x| mod 2.0 */
222 n = (int) (y*4.0L);
223 }
224 else
225 {
226 if (ix >= 0x403f8000) /* 2^64 */
227 {
228 y = zero; n = 0; /* y must be even */
229 }
230 else
231 {
232 if (ix < 0x403e8000) /* 2^63 */
233 z = y + two63; /* exact */
234 GET_LDOUBLE_WORDS (se, i0, i1, z);
235 n = i1 & 1;
236 y = n;
237 n <<= 2;
238 }
239 }
240
241 switch (n)
242 {
243 case 0:
244 y = sinl (pi * y);
245 break;
246 case 1:
247 case 2:
248 y = cosl (pi * (half - y));
249 break;
250 case 3:
251 case 4:
252 y = sinl (pi * (one - y));
253 break;
254 case 5:
255 case 6:
256 y = -cosl (pi * (y - 1.5L));
257 break;
258 default:
259 y = sinl (pi * (y - 2.0L));
260 break;
261 }
262 return -y;
263 }
264
265
266 long double
lgammal_r(long double x,int * signgamp)267 lgammal_r(long double x, int *signgamp)
268 {
269 long double t, y, z, nadj = 0, p, p1, p2, q, r, w;
270 int i, ix;
271 u_int32_t se, i0, i1;
272
273 *signgamp = 1;
274 GET_LDOUBLE_WORDS (se, i0, i1, x);
275 ix = se & 0x7fff;
276
277 if ((ix | i0 | i1) == 0)
278 {
279 if (se & 0x8000)
280 *signgamp = -1;
281 return __math_divzerol(0);
282 }
283
284 ix = (ix << 16) | (i0 >> 16);
285
286 /* purge off +-inf, NaN, +-0, and negative arguments */
287 if (ix >= 0x7fff0000)
288 return x * x;
289
290 if (ix < 0x3fc08000) /* 2^-63 */
291 { /* |x|<2**-63, return -log(|x|) */
292 if (se & 0x8000)
293 {
294 *signgamp = -1;
295 return -logl (-x);
296 }
297 else
298 return -logl (x);
299 }
300 if (se & 0x8000)
301 {
302 t = sin_pi (x);
303 if (t == zero)
304 return __math_divzerol(0); /* -integer */
305 nadj = logl (pi / fabsl (t * x));
306 if (t < zero)
307 *signgamp = -1;
308 x = -x;
309 }
310
311 /* purge off 1 and 2 */
312 if ((((ix - 0x3fff8000) | i0 | i1) == 0)
313 || (((ix - 0x40008000) | i0 | i1) == 0))
314 r = 0;
315 else if (ix < 0x40008000) /* 2.0 */
316 {
317 /* x < 2.0 */
318 if (ix <= 0x3ffee666) /* 8.99993896484375e-1 */
319 {
320 /* lgamma(x) = lgamma(x+1) - log(x) */
321 r = -logl (x);
322 if (ix >= 0x3ffebb4a) /* 7.31597900390625e-1 */
323 {
324 y = x - one;
325 i = 0;
326 }
327 else if (ix >= 0x3ffced33)/* 2.31639862060546875e-1 */
328 {
329 y = x - (tc - one);
330 i = 1;
331 }
332 else
333 {
334 /* x < 0.23 */
335 y = x;
336 i = 2;
337 }
338 }
339 else
340 {
341 r = zero;
342 if (ix >= 0x3fffdda6) /* 1.73162841796875 */
343 {
344 /* [1.7316,2] */
345 y = x - 2.0L;
346 i = 0;
347 }
348 else if (ix >= 0x3fff9da6)/* 1.23162841796875 */
349 {
350 /* [1.23,1.73] */
351 y = x - tc;
352 i = 1;
353 }
354 else
355 {
356 /* [0.9, 1.23] */
357 y = x - one;
358 i = 2;
359 }
360 }
361 switch (i)
362 {
363 case 0:
364 p1 = a0 + y * (a1 + y * (a2 + y * (a3 + y * (a4 + y * a5))));
365 p2 = b0 + y * (b1 + y * (b2 + y * (b3 + y * (b4 + y))));
366 r += half * y + y * p1/p2;
367 break;
368 case 1:
369 p1 = g0 + y * (g1 + y * (g2 + y * (g3 + y * (g4 + y * (g5 + y * g6)))));
370 p2 = h0 + y * (h1 + y * (h2 + y * (h3 + y * (h4 + y * (h5 + y)))));
371 p = tt + y * p1/p2;
372 r += (tf + p);
373 break;
374 case 2:
375 p1 = y * (u0 + y * (u1 + y * (u2 + y * (u3 + y * (u4 + y * (u5 + y * u6))))));
376 p2 = v0 + y * (v1 + y * (v2 + y * (v3 + y * (v4 + y * (v5 + y)))));
377 r += (-half * y + p1 / p2);
378 }
379 }
380 else if (ix < 0x40028000) /* 8.0 */
381 {
382 /* x < 8.0 */
383 i = (int) x;
384 t = zero;
385 y = x - (long double) i;
386 p = y *
387 (s0 + y * (s1 + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6))))));
388 q = r0 + y * (r1 + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * (r6 + y))))));
389 r = half * y + p / q;
390 z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
391 switch (i)
392 {
393 case 7:
394 z *= (y + 6.0L); /* FALLTHRU */
395 case 6:
396 z *= (y + 5.0L); /* FALLTHRU */
397 case 5:
398 z *= (y + 4.0L); /* FALLTHRU */
399 case 4:
400 z *= (y + 3.0L); /* FALLTHRU */
401 case 3:
402 z *= (y + 2.0L); /* FALLTHRU */
403 r += logl (z);
404 break;
405 }
406 }
407 else if (ix < 0x40418000) /* 2^66 */
408 {
409 /* 8.0 <= x < 2**66 */
410 t = logl (x);
411 z = one / x;
412 y = z * z;
413 w = w0 + z * (w1
414 + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * (w6 + y * w7))))));
415 r = (x - half) * (t - one) + w;
416 }
417 else
418 /* 2**66 <= x <= inf */
419 r = check_oflowl(x * (logl (x) - one));
420 if (se & 0x8000)
421 r = nadj - r;
422 return r;
423 }
424
