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
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
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
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 }