1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
|
/*
* Copyright (c) 1985 Regents of the University of California.
* All rights reserved. The Berkeley software License Agreement
* specifies the terms and conditions for redistribution.
*/
/* Coefficients for long double precision by Stephen L. Moshier
(moshier@world.std.com). This routine was spot chaecked at
10,000 random arguments between -40 and +40. The largest error
observed was 2.2e-19. Error criterion was absolute for function
value < 1, relative otherwise. */
#if 0
#ifndef lint
static char sccsid[] = "@(#)lgamma.c 5.3 (Berkeley) 9/22/88";
#endif /* not lint */
#endif
/*
C program for floating point log Gamma function
lgamma(x) computes the log of the absolute
value of the Gamma function.
The sign of the Gamma function is returned in the
external quantity signgaml.
The coefficients for expansion around zero
are #5243 from Hart & Cheney; for expansion
around infinity they are #5404.
Calls log, floor and sin.
*/
#include <math.h>
#if 0
#include "mathimpl.h"
#else
#include <math.h>
#endif
#if defined(vax)||defined(tahoe)
#include <errno.h>
#endif /* defined(vax)||defined(tahoe) */
int signgaml;
/* log(2*pi)/2 */
static const long double goobie = 0.9189385332046727417803297L;
static const long double pi = 3.1415926535897932384626434L;
/*
log gamma(x) = ( x - 0.5 ) * log(x) - x + LS2PI + 1/x P(1/x^2)
Relative error
n=6, d=0
Peak error = 1.51e-21
Relative error spread = 5.7e-21
*/
#define M 7
static const long double p1[] = {
8.33333333333333144750e-2L,
-2.77777777775034960344e-3L,
7.93650779585507075567e-4L,
-5.95234585176568851461e-4L,
8.41272329732249808063e-4L,
-1.88080193811937690718e-3L,
4.88502614243227078116e-3L
};
/*
gamma(x+2) = P(x)/Q(x)
Relative error
n=7, d=8
Peak error = 1.84e-20
Relative error spread = 8.4e-23
*/
#define N 9
static const long double p2[] = {
-7.15743521530849602425e4L,
-5.99650230220855581642e4L,
-2.59780216007146401957e4L,
-7.96667499622741999770e3L,
-1.70730828800510297666e3L,
-2.92929976820724030353e2L,
-3.25157411956062339893e1L,
-3.01525602666895735709e0L,
0.0L
};
static const long double q2[] = {
-7.15743521530849602412e4L,
-2.97045081369399940529e4L,
1.60577839621734713377e4L,
3.31667508019495079814e3L,
-1.98526250512761318471e3L,
5.69440799097468430177e1L,
8.85946791747759881659e1L,
-1.67955233807178858919e1L,
1.00000000000000000000e0L,
};
static long double pos(long double);
static long double neg(long double);
static long double asym(long double);
long double
lgammal(long double arg)
{
signgaml = 1;
if(arg <= 0.L) return(neg(arg));
if(arg > 8.L) return(asym(arg));
return(logl(pos(arg)));
}
static long double
asym(long double arg)
{
long double n, argsq;
int i;
n = 0.0L;
/* Avoid overflow of arg*arg. */
if(arg > 1.0e10L)
goto noexpan;
argsq = 1./(arg*arg);
for(i=M-1; i>=0; i--){
n = n*argsq + p1[i];
}
noexpan:
return((arg-.5L)*logl(arg) - arg + goobie + n/arg);
}
static long double
neg(long double arg)
{
long double t;
arg = -arg;
/*
* to see if arg were a true integer, the old code used the
* mathematically correct observation:
* sin(n*pi) = 0 <=> n is an integer.
* but in finite precision arithmetic, sin(n*PI) will NEVER
* be zero simply because n*PI is a rational number. hence
* it failed to work with our newer, more accurate sin()
* which uses true pi to do the argument reduction...
* temp = sin(pi*arg);
*/
t = floorl(arg);
if (arg - t > 0.5e0L)
t += 1.e0L; /* t := integer nearest arg */
#if defined(vax)||defined(tahoe)
if (arg == t) {
return(__infnan(ERANGE)); /* +INF */
}
#endif /* defined(vax)||defined(tahoe) */
signgaml = (int) (t - 2*floorl(t/2)); /* signgam = 1 if t was odd, */
/* 0 if t was even */
signgaml = signgam - 1 + signgam; /* signgam = 1 if t was odd, */
/* -1 if t was even */
t = arg - t; /* -0.5 <= t <= 0.5 */
if (t < 0.e0L) {
t = -t;
signgaml = -signgaml;
}
return(-logl(arg*pos(arg)*sinl(pi*t)/pi));
}
static long double
pos(long double arg)
{
long double n, d, s;
register i;
if(arg < 2.L) return(pos(arg+1.L)/arg);
if(arg > 3.L) return((arg-1.0L)*pos(arg-1.0L));
s = arg - 2.0L;
/* for(n=0,d=0,i=N-1; i>=0; i--){*/
/*
n = 0.0L;
d = 0.0L;
*/
for(n=0,d=0,i=N-1; i>=0; i--){
n = n*s + p2[i];
d = d*s + q2[i];
}
return(n/d);
}
|
