2 * Copyright (c) 1992, 1993
3 * The Regents of the University of California. All rights reserved.
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
13 * 3. All advertising materials mentioning features or use of this software
14 * must display the following acknowledgement:
15 * This product includes software developed by the University of
16 * California, Berkeley and its contributors.
17 * 4. Neither the name of the University nor the names of its contributors
18 * may be used to endorse or promote products derived from this software
19 * without specific prior written permission.
21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
35 static char sccsid[] = "@(#)lgamma.c 8.2 (Berkeley) 11/30/93";
39 * Coded by Peter McIlroy, Nov 1992;
41 * The financial support of UUNET Communications Services is greatfully
50 /* Log gamma function.
51 * Error: x > 0 error < 1.3ulp.
52 * x > 4, error < 1ulp.
53 * x > 9, error < .6ulp.
54 * x < 0, all bets are off. (When G(x) ~ 1, log(G(x)) ~ 0)
57 * Use the asymptotic expansion (Stirling's Formula)
59 * Use gamma(x+1) = x*gamma(x) for argument reduction.
60 * Use rational approximation in
62 * Two approximations are used, one centered at the
63 * minimum to ensure monotonicity; one centered at 2
64 * to maintain small relative error.
66 * Use the reflection formula,
67 * G(1-x)G(x) = PI/sin(PI*x)
69 * non-positive integer returns +Inf.
73 #if defined(vax) || defined(tahoe)
75 /* double and float have same size exponent field */
76 #define TRUNC(x) x = (double) (float) (x)
79 #define TRUNC(x) *(((int *) &x) + endian) &= 0xf8000000
83 static double small_lgam(double);
84 static double large_lgam(double);
85 static double neg_lgam(double);
86 static double zero = 0.0, one = 1.0;
89 #define UNDERFL (1e-1020 * 1e-1020)
91 #define LEFT (1.0 - (x0 + .25))
92 #define RIGHT (x0 - .218)
94 * Constants for approximation in [1.244,1.712]
96 #define x0 0.461632144968362356785
97 #define x0_lo -.000000000000000015522348162858676890521
98 #define a0_hi -0.12148629128932952880859
99 #define a0_lo .0000000007534799204229502
100 #define r0 -2.771227512955130520e-002
101 #define r1 -2.980729795228150847e-001
102 #define r2 -3.257411333183093394e-001
103 #define r3 -1.126814387531706041e-001
104 #define r4 -1.129130057170225562e-002
105 #define r5 -2.259650588213369095e-005
106 #define s0 1.714457160001714442e+000
107 #define s1 2.786469504618194648e+000
108 #define s2 1.564546365519179805e+000
109 #define s3 3.485846389981109850e-001
110 #define s4 2.467759345363656348e-002
112 * Constants for approximation in [1.71, 2.5]
114 #define a1_hi 4.227843350984671344505727574870e-01
115 #define a1_lo 4.670126436531227189e-18
116 #define p0 3.224670334241133695662995251041e-01
117 #define p1 3.569659696950364669021382724168e-01
118 #define p2 1.342918716072560025853732668111e-01
119 #define p3 1.950702176409779831089963408886e-02
120 #define p4 8.546740251667538090796227834289e-04
121 #define q0 1.000000000000000444089209850062e+00
122 #define q1 1.315850076960161985084596381057e+00
123 #define q2 6.274644311862156431658377186977e-01
124 #define q3 1.304706631926259297049597307705e-01
125 #define q4 1.102815279606722369265536798366e-02
126 #define q5 2.512690594856678929537585620579e-04
127 #define q6 -1.003597548112371003358107325598e-06
129 * Stirling's Formula, adjusted for equal-ripple. x in [6,Inf].
131 #define lns2pi .418938533204672741780329736405
132 #define pb0 8.33333333333333148296162562474e-02
133 #define pb1 -2.77777777774548123579378966497e-03
134 #define pb2 7.93650778754435631476282786423e-04
135 #define pb3 -5.95235082566672847950717262222e-04
136 #define pb4 8.41428560346653702135821806252e-04
137 #define pb5 -1.89773526463879200348872089421e-03
138 #define pb6 5.69394463439411649408050664078e-03
139 #define pb7 -1.44705562421428915453880392761e-02
147 endian = ((*(int *) &one)) ? 1 : 0;
152 else return (infnan(EDOM));
157 } else if (x > 1e-16)
158 return (small_lgam(x));
159 else if (x > -1e-16) {
161 signgam = -1, x = -x;
164 return (neg_lgam(x));
172 struct Double t, u, v;
178 v.b = (x - v.a) - 0.5;
180 t.b = x*u.b + v.b*u.a;
181 if (_IEEE == 0 && !finite(t.a))
182 return(infnan(ERANGE));
187 p = pb0+z*(pb1+z*(pb2+z*(pb3+z*(pb4+z*(pb5+z*(pb6+z*pb7))))));
188 /* error in approximation = 2.8e-19 */
190 p = p*x1; /* error < 2.3e-18 absolute */
191 /* 0 < p < 1/64 (at x = 5.5) */
193 TRUNC(v.a); /* truncate v.a to 26 bits. */
195 t.a = v.a*u.a; /* t = (x-.5)*(log(x)-1) */
196 t.b = v.b*u.a + x*u.b;
197 t.b += p; t.b += lns2pi; /* return t + lns2pi + p */
205 double y, z, t, r = 0, p, q, hi, lo;
209 if (x_int <= 2 && y > RIGHT) {
213 } else if (y < -LEFT) {
217 p = r0+z*(r1+z*(r2+z*(r3+z*(r4+z*r5))));
218 q = s0+z*(s1+z*(s2+z*(s3+z*s4)));
219 r = t*(z*(p/q) - x0_lo);
224 case 5: z *= (y + 4);
225 case 4: z *= (y + 3);
226 case 3: z *= (y + 2);
228 rr.b += a0_lo; rr.a += a0_hi;
229 return(((r+rr.b)+t+rr.a));
230 case 2: return(((r+a0_lo)+t)+a0_hi);
231 case 0: r -= log1p(x);
232 default: rr = __log__D(x);
233 rr.a -= a0_hi; rr.b -= a0_lo;
234 return(((r - rr.b) + t) - rr.a);
237 p = p0+y*(p1+y*(p2+y*(p3+y*p4)));
238 q = q0+y*(q1+y*(q2+y*(q3+y*(q4+y*(q5+y*q6)))));
240 t = (double)(float) y;
242 hi = (double)(float) (p+a1_hi);
243 lo = a1_hi - hi; lo += p; lo += a1_lo;
244 r = lo*y + z*hi; /* q + r = y*(a0+p/q) */
249 case 5: z *= (y + 4);
250 case 4: z *= (y + 3);
251 case 3: z *= (y + 2);
255 case 2: return (q+ r);
256 case 0: rr = __log__D(x);
257 r -= rr.b; r -= log1p(x);
260 default: rr = __log__D(x);
272 double y, z, one = 1.0, zero = 0.0;
273 extern double gamma();
275 /* avoid destructive cancellation as much as possible */
282 return(infnan(ERANGE));
285 y = -y, signgam = -1;
289 if (z == x) { /* convention: G(-(integer)) -> +Inf */
293 return (infnan(ERANGE));
299 z = fabs(x + z); /* 0 < z <= .5 */
303 z = cos(M_PI*(0.5-z));