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
92long 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
97static const long double
98pi = 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 */
103a0 = -6.343246574721079391729402781192128239938E2L,
104a1 =  1.856560238672465796768677717168371401378E3L,
105a2 =  2.404733102163746263689288466865843408429E3L,
106a3 =  8.804188795790383497379532868917517596322E2L,
107a4 =  1.135361354097447729740103745999661157426E2L,
108a5 =  3.766956539107615557608581581190400021285E0L,
109
110b0 =  8.214973713960928795704317259806842490498E3L,
111b1 =  1.026343508841367384879065363925870888012E4L,
112b2 =  4.553337477045763320522762343132210919277E3L,
113b3 =  8.506975785032585797446253359230031874803E2L,
114b4 =  6.042447899703295436820744186992189445813E1L,
115/* b5 =  1.000000000000000000000000000000000000000E0 */
116
117
118tc =  1.4616321449683623412626595423257213284682E0L,
119tf = -1.2148629053584961146050602565082954242826E-1, /* double precision */
120/* tt = (tail of tf), i.e. tf + tt has extended precision. */
121tt = 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 */
129g0 = 3.645529916721223331888305293534095553827E-18L,
130g1 = 5.126654642791082497002594216163574795690E3L,
131g2 = 8.828603575854624811911631336122070070327E3L,
132g3 = 5.464186426932117031234820886525701595203E3L,
133g4 = 1.455427403530884193180776558102868592293E3L,
134g5 = 1.541735456969245924860307497029155838446E2L,
135g6 = 4.335498275274822298341872707453445815118E0L,
136
137h0 = 1.059584930106085509696730443974495979641E4L,
138h1 = 2.147921653490043010629481226937850618860E4L,
139h2 = 1.643014770044524804175197151958100656728E4L,
140h3 = 5.869021995186925517228323497501767586078E3L,
141h4 = 9.764244777714344488787381271643502742293E2L,
142h5 = 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 */
149u0 = -8.886217500092090678492242071879342025627E1L,
150u1 =  6.840109978129177639438792958320783599310E2L,
151u2 =  2.042626104514127267855588786511809932433E3L,
152u3 =  1.911723903442667422201651063009856064275E3L,
153u4 =  7.447065275665887457628865263491667767695E2L,
154u5 =  1.132256494121790736268471016493103952637E2L,
155u6 =  4.484398885516614191003094714505960972894E0L,
156
157v0 =  1.150830924194461522996462401210374632929E3L,
158v1 =  3.399692260848747447377972081399737098610E3L,
159v2 =  3.786631705644460255229513563657226008015E3L,
160v3 =  1.966450123004478374557778781564114347876E3L,
161v4 =  4.741359068914069299837355438370682773122E2L,
162v5 =  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 */
169s0 =  1.454726263410661942989109455292824853344E6L,
170s1 = -3.901428390086348447890408306153378922752E6L,
171s2 = -6.573568698209374121847873064292963089438E6L,
172s3 = -3.319055881485044417245964508099095984643E6L,
173s4 = -7.094891568758439227560184618114707107977E5L,
174s5 = -6.263426646464505837422314539808112478303E4L,
175s6 = -1.684926520999477529949915657519454051529E3L,
176
177r0 = -1.883978160734303518163008696712983134698E7L,
178r1 = -2.815206082812062064902202753264922306830E7L,
179r2 = -1.600245495251915899081846093343626358398E7L,
180r3 = -4.310526301881305003489257052083370058799E6L,
181r4 = -5.563807682263923279438235987186184968542E5L,
182r5 = -3.027734654434169996032905158145259713083E4L,
183r6 = -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
190w0 = LS2PI - 0.5 */
191w0 =  4.189385332046727417803e-1L,
192w1 =  8.333333333333331447505E-2L,
193w2 = -2.777777777750349603440E-3L,
194w3 =  7.936507795855070755671E-4L,
195w4 = -5.952345851765688514613E-4L,
196w5 =  8.412723297322498080632E-4L,
197w6 = -1.880801938119376907179E-3L,
198w7 =  4.885026142432270781165E-3L;
199
200/* sin(pi*x) assuming x > 2^-1000, if sin(pi*x)==0 the sign is arbitrary */
201static 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
223long 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
342long double __lgammal_r(long double x, int *sg)
343{
344	return __lgamma_r(x, sg);
345}
346#endif
347
348long double lgammal(long double x)
349{
350	return __lgammal_r(x, &__signgam);
351}
352
353weak_alias(__lgammal_r, lgammal_r);
354