xref: /illumos-gate/usr/src/lib/libm/common/C/__lgamma.c (revision 25c28e83)
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 
39 static const double
40 one	= 1.0,
41 zero	= 0.0,
42 hln2pi	= 0.9189385332046727417803297,	/* log(2*pi)/2 */
43 pi	= 3.1415926535897932384626434,
44 two52	= 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  */
49 D1 = 	-5.772156649015328605195174e-1,
50 p7 =	 4.945235359296727046734888e0,
51 p6 =	 2.018112620856775083915565e2,
52 p5 =	 2.290838373831346393026739e3,
53 p4 =	 1.131967205903380828685045e4,
54 p3 =	 2.855724635671635335736389e4,
55 p2 =	 3.848496228443793359990269e4,
56 p1 =	 2.637748787624195437963534e4,
57 p0 =	 7.225813979700288197698961e3,
58 q7 =	 6.748212550303777196073036e1,
59 q6 =	 1.113332393857199323513008e3,
60 q5 =	 7.738757056935398733233834e3,
61 q4 =	 2.763987074403340708898585e4,
62 q3 =	 5.499310206226157329794414e4,
63 q2 =	 6.161122180066002127833352e4,
64 q1 =	 3.635127591501940507276287e4,
65 q0 =	 8.785536302431013170870835e3,
66 /*
67  * Numerator and denominator coefficients for rational minimax Approximation
68  * G/H over (1.5,4.0).
69  */
70 D2 =	 4.227843350984671393993777e-1,
71 g7 =	 4.974607845568932035012064e0,
72 g6 =	 5.424138599891070494101986e2,
73 g5 =	 1.550693864978364947665077e4,
74 g4 =	 1.847932904445632425417223e5,
75 g3 =	 1.088204769468828767498470e6,
76 g2 =	 3.338152967987029735917223e6,
77 g1 =	 5.106661678927352456275255e6,
78 g0 =	 3.074109054850539556250927e6,
79 h7 =	 1.830328399370592604055942e2,
80 h6 =	 7.765049321445005871323047e3,
81 h5 =	 1.331903827966074194402448e5,
82 h4 =	 1.136705821321969608938755e6,
83 h3 =	 5.267964117437946917577538e6,
84 h2 =	 1.346701454311101692290052e7,
85 h1 =	 1.782736530353274213975932e7,
86 h0 =	 9.533095591844353613395747e6,
87 /*
88  * Numerator and denominator coefficients for rational minimax Approximation
89  * U/V over (4.0,12.0).
90  */
91 D4 =	 1.791759469228055000094023e0,
92 u7 =	 1.474502166059939948905062e4,
93 u6 =	 2.426813369486704502836312e6,
94 u5 =	 1.214755574045093227939592e8,
95 u4 =	 2.663432449630976949898078e9,
96 u3 =	 2.940378956634553899906876e10,
97 u2 =	 1.702665737765398868392998e11,
98 u1 =	 4.926125793377430887588120e11,
99 u0 =	 5.606251856223951465078242e11,
100 v7 =	 2.690530175870899333379843e3,
101 v6 =	 6.393885654300092398984238e5,
102 v5 =	 4.135599930241388052042842e7,
103 v4 =	 1.120872109616147941376570e9,
104 v3 =	 1.488613728678813811542398e10,
105 v2 =	 1.016803586272438228077304e11,
106 v1 =	 3.417476345507377132798597e11,
107 v0 =	 4.463158187419713286462081e11,
108 /*
109  * Coefficients for minimax approximation over (12, INF).
110  */
111 c5 =	-1.910444077728e-03,
112 c4 =	 8.4171387781295e-04,
113 c3 =	-5.952379913043012e-04,
114 c2 =	 7.93650793500350248e-04,
115 c1 =	-2.777777777777681622553e-03,
116 c0 =	 8.333333333333333331554247e-02,
117 c6 =	 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  */
123 static double
sin_pi(double x)124 sin_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 
166 static double
neg(double z,int * signgamp)167 neg(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 
205 double
__k_lgamma(double x,int * signgamp)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