1/*
2 * CDDL HEADER START
3 *
4 * The contents of this file are subject to the terms of the
5 * Common Development and Distribution License (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21/*
22 * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
23 */
24/*
25 * Copyright 2005 Sun Microsystems, Inc.  All rights reserved.
26 * Use is subject to license terms.
27 */
28
29/*
30 * double __k_lgamma(double x, int *signgamp);
31 *
32 * K.C. Ng, March, 1989.
33 *
34 * Part of the algorithm is based on W. Cody's lgamma function.
35 */
36
37#include "libm.h"
38
39static const double
40one	= 1.0,
41zero	= 0.0,
42hln2pi	= 0.9189385332046727417803297,	/* log(2*pi)/2 */
43pi	= 3.1415926535897932384626434,
44two52	= 4503599627370496.0,		/* 43300000,00000000 (used by sin_pi) */
45/*
46 * Numerator and denominator coefficients for rational minimax Approximation
47 * P/Q over (0.5,1.5).
48 */
49D1 = 	-5.772156649015328605195174e-1,
50p7 =	 4.945235359296727046734888e0,
51p6 =	 2.018112620856775083915565e2,
52p5 =	 2.290838373831346393026739e3,
53p4 =	 1.131967205903380828685045e4,
54p3 =	 2.855724635671635335736389e4,
55p2 =	 3.848496228443793359990269e4,
56p1 =	 2.637748787624195437963534e4,
57p0 =	 7.225813979700288197698961e3,
58q7 =	 6.748212550303777196073036e1,
59q6 =	 1.113332393857199323513008e3,
60q5 =	 7.738757056935398733233834e3,
61q4 =	 2.763987074403340708898585e4,
62q3 =	 5.499310206226157329794414e4,
63q2 =	 6.161122180066002127833352e4,
64q1 =	 3.635127591501940507276287e4,
65q0 =	 8.785536302431013170870835e3,
66/*
67 * Numerator and denominator coefficients for rational minimax Approximation
68 * G/H over (1.5,4.0).
69 */
70D2 =	 4.227843350984671393993777e-1,
71g7 =	 4.974607845568932035012064e0,
72g6 =	 5.424138599891070494101986e2,
73g5 =	 1.550693864978364947665077e4,
74g4 =	 1.847932904445632425417223e5,
75g3 =	 1.088204769468828767498470e6,
76g2 =	 3.338152967987029735917223e6,
77g1 =	 5.106661678927352456275255e6,
78g0 =	 3.074109054850539556250927e6,
79h7 =	 1.830328399370592604055942e2,
80h6 =	 7.765049321445005871323047e3,
81h5 =	 1.331903827966074194402448e5,
82h4 =	 1.136705821321969608938755e6,
83h3 =	 5.267964117437946917577538e6,
84h2 =	 1.346701454311101692290052e7,
85h1 =	 1.782736530353274213975932e7,
86h0 =	 9.533095591844353613395747e6,
87/*
88 * Numerator and denominator coefficients for rational minimax Approximation
89 * U/V over (4.0,12.0).
90 */
91D4 =	 1.791759469228055000094023e0,
92u7 =	 1.474502166059939948905062e4,
93u6 =	 2.426813369486704502836312e6,
94u5 =	 1.214755574045093227939592e8,
95u4 =	 2.663432449630976949898078e9,
96u3 =	 2.940378956634553899906876e10,
97u2 =	 1.702665737765398868392998e11,
98u1 =	 4.926125793377430887588120e11,
99u0 =	 5.606251856223951465078242e11,
100v7 =	 2.690530175870899333379843e3,
101v6 =	 6.393885654300092398984238e5,
102v5 =	 4.135599930241388052042842e7,
103v4 =	 1.120872109616147941376570e9,
104v3 =	 1.488613728678813811542398e10,
105v2 =	 1.016803586272438228077304e11,
106v1 =	 3.417476345507377132798597e11,
107v0 =	 4.463158187419713286462081e11,
108/*
109 * Coefficients for minimax approximation over (12, INF).
110 */
111c5 =	-1.910444077728e-03,
112c4 =	 8.4171387781295e-04,
113c3 =	-5.952379913043012e-04,
114c2 =	 7.93650793500350248e-04,
115c1 =	-2.777777777777681622553e-03,
116c0 =	 8.333333333333333331554247e-02,
117c6 =	 5.7083835261e-03;
118
119/*
120 * Return sin(pi*x).  We assume x is finite and negative, and if it
121 * is an integer, then the sign of the zero returned doesn't matter.
122 */
123static double
124sin_pi(double x) {
125	double	y, z;
126	int	n;
127
128	y = -x;
129	if (y <= 0.25)
130		return (__k_sin(pi * x, 0.0));
131	if (y >= two52)
132		return (zero);
133	z = floor(y);
134	if (y == z)
135		return (zero);
136
137	/* argument reduction: set y = |x| mod 2 */
138	y *= 0.5;
139	y = 2.0 * (y - floor(y));
140
141	/* now floor(y * 4) tells which octant y is in */
142	n = (int)(y * 4.0);
143	switch (n) {
144	case 0:
145		y = __k_sin(pi * y, 0.0);
146		break;
147	case 1:
148	case 2:
149		y = __k_cos(pi * (0.5 - y), 0.0);
150		break;
151	case 3:
152	case 4:
153		y = __k_sin(pi * (1.0 - y), 0.0);
154		break;
155	case 5:
156	case 6:
157		y = -__k_cos(pi * (y - 1.5), 0.0);
158		break;
159	default:
160		y = __k_sin(pi * (y - 2.0), 0.0);
161		break;
162	}
163	return (-y);
164}
165
166static double
167neg(double z, int *signgamp) {
168	double	t, p;
169
170	/*
171	 * written by K.C. Ng,  Feb 2, 1989.
172	 *
173	 * Since
174	 *		-z*G(-z)*G(z) = pi/sin(pi*z),
175	 * we have
176	 * 	G(-z) = -pi/(sin(pi*z)*G(z)*z)
177	 * 	      =  pi/(sin(pi*(-z))*G(z)*z)
178	 * Algorithm
179	 *		z = |z|
180	 *		t = sin_pi(z); ...note that when z>2**52, z is an int
181	 *		and hence t=0.
182	 *
183	 *		if (t == 0.0) return 1.0/0.0;
184	 *		if (t< 0.0) *signgamp = -1; else t= -t;
185	 *		if (z+1.0 == 1.0)	...tiny z
186	 *		    return -log(z);
187	 *		else
188	 *		    return log(pi/(t*z))-__k_lgamma(z, signgamp);
189	 */
190
191	t = sin_pi(z);			/* t := sin(pi*z) */
192	if (t == zero)			/* return 1.0/0.0 = +INF */
193		return (one / fabs(t));
194	z = -z;
195	p = z + one;
196	if (p == one)
197		p = -log(z);
198	else
199		p = log(pi / (fabs(t) * z)) - __k_lgamma(z, signgamp);
200	if (t < zero)
201		*signgamp = -1;
202	return (p);
203}
204
205double
206__k_lgamma(double x, int *signgamp) {
207	double	t, p, q, cr, y;
208
209	/* purge off +-inf, NaN and negative arguments */
210	if (!finite(x))
211		return (x * x);
212	*signgamp = 1;
213	if (signbit(x))
214		return (neg(x, signgamp));
215
216	/* lgamma(x) ~ log(1/x) for really tiny x */
217	t = one + x;
218	if (t == one) {
219		if (x == zero)
220			return (one / x);
221		return (-log(x));
222	}
223
224	/* for tiny < x < inf */
225	if (x <= 1.5) {
226		if (x < 0.6796875) {
227			cr = -log(x);
228			y = x;
229		} else {
230			cr = zero;
231			y = x - one;
232		}
233
234		if (x <= 0.5 || x >= 0.6796875) {
235			if (x == one)
236				return (zero);
237			p = p0+y*(p1+y*(p2+y*(p3+y*(p4+y*(p5+y*(p6+y*p7))))));
238			q = q0+y*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*(q6+y*
239			    (q7+y)))))));
240			return (cr+y*(D1+y*(p/q)));
241		} else {
242			y = x - one;
243			p = g0+y*(g1+y*(g2+y*(g3+y*(g4+y*(g5+y*(g6+y*g7))))));
244			q = h0+y*(h1+y*(h2+y*(h3+y*(h4+y*(h5+y*(h6+y*
245			    (h7+y)))))));
246			return (cr+y*(D2+y*(p/q)));
247		}
248	} else if (x <= 4.0) {
249		if (x == 2.0)
250			return (zero);
251		y = x - 2.0;
252		p = g0+y*(g1+y*(g2+y*(g3+y*(g4+y*(g5+y*(g6+y*g7))))));
253		q = h0+y*(h1+y*(h2+y*(h3+y*(h4+y*(h5+y*(h6+y*(h7+y)))))));
254		return (y*(D2+y*(p/q)));
255	} else if (x <= 12.0) {
256		y = x - 4.0;
257		p = u0+y*(u1+y*(u2+y*(u3+y*(u4+y*(u5+y*(u6+y*u7))))));
258		q = v0+y*(v1+y*(v2+y*(v3+y*(v4+y*(v5+y*(v6+y*(v7-y)))))));
259		return (D4+y*(p/q));
260	} else if (x <= 1.0e17) {		/* x ~< 2**(prec+3) */
261		t = one / x;
262		y = t * t;
263		p = hln2pi+t*(c0+y*(c1+y*(c2+y*(c3+y*(c4+y*(c5+y*c6))))));
264		q = log(x);
265		return (x*(q-one)-(0.5*q-p));
266	} else {			/* may overflow */
267		return (x * (log(x) - 1.0));
268	}
269}
270