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
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
/*
"A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964)
"Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001)
"An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004)
approximation method:
(x - 0.5) S(x)
Gamma(x) = (x + g - 0.5) * ----------------
exp(x + g - 0.5)
with
a1 a2 a3 aN
S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ]
x + 1 x + 2 x + 3 x + N
with a0, a1, a2, a3,.. aN constants which depend on g.
for x < 0 the following reflection formula is used:
Gamma(x)*Gamma(-x) = -pi/(x sin(pi x))
most ideas and constants are from boost and python
*/
use super::{exp, floor, k_cos, k_sin, pow};
const PI: f64 = 3.141592653589793238462643383279502884;
/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */
fn sinpi(mut x: f64) -> f64 {
let mut n: isize;
/* argument reduction: x = |x| mod 2 */
/* spurious inexact when x is odd int */
x = x * 0.5;
x = 2.0 * (x - floor(x));
/* reduce x into [-.25,.25] */
n = (4.0 * x) as isize;
n = div!(n + 1, 2);
x -= (n as f64) * 0.5;
x *= PI;
match n {
1 => k_cos(x, 0.0),
2 => k_sin(-x, 0.0, 0),
3 => -k_cos(x, 0.0),
// 0
_ => k_sin(x, 0.0, 0),
}
}
const N: usize = 12;
//static const double g = 6.024680040776729583740234375;
const GMHALF: f64 = 5.524680040776729583740234375;
const SNUM: [f64; N + 1] = [
23531376880.410759688572007674451636754734846804940,
42919803642.649098768957899047001988850926355848959,
35711959237.355668049440185451547166705960488635843,
17921034426.037209699919755754458931112671403265390,
6039542586.3520280050642916443072979210699388420708,
1439720407.3117216736632230727949123939715485786772,
248874557.86205415651146038641322942321632125127801,
31426415.585400194380614231628318205362874684987640,
2876370.6289353724412254090516208496135991145378768,
186056.26539522349504029498971604569928220784236328,
8071.6720023658162106380029022722506138218516325024,
210.82427775157934587250973392071336271166969580291,
2.5066282746310002701649081771338373386264310793408,
];
const SDEN: [f64; N + 1] = [
0.0,
39916800.0,
120543840.0,
150917976.0,
105258076.0,
45995730.0,
13339535.0,
2637558.0,
357423.0,
32670.0,
1925.0,
66.0,
1.0,
];
/* n! for small integer n */
const FACT: [f64; 23] = [
1.0,
1.0,
2.0,
6.0,
24.0,
120.0,
720.0,
5040.0,
40320.0,
362880.0,
3628800.0,
39916800.0,
479001600.0,
6227020800.0,
87178291200.0,
1307674368000.0,
20922789888000.0,
355687428096000.0,
6402373705728000.0,
121645100408832000.0,
2432902008176640000.0,
51090942171709440000.0,
1124000727777607680000.0,
];
/* S(x) rational function for positive x */
fn s(x: f64) -> f64 {
let mut num: f64 = 0.0;
let mut den: f64 = 0.0;
/* to avoid overflow handle large x differently */
if x < 8.0 {
for i in (0..=N).rev() {
num = num * x + i!(SNUM, i);
den = den * x + i!(SDEN, i);
}
} else {
for i in 0..=N {
num = num / x + i!(SNUM, i);
den = den / x + i!(SDEN, i);
}
}
return num / den;
}
/// The [Gamma function](https://en.wikipedia.org/wiki/Gamma_function) (f64).
#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
pub fn tgamma(mut x: f64) -> f64 {
let u: u64 = x.to_bits();
let absx: f64;
let mut y: f64;
let mut dy: f64;
let mut z: f64;
let mut r: f64;
let ix: u32 = ((u >> 32) as u32) & 0x7fffffff;
let sign: bool = (u >> 63) != 0;
/* special cases */
if ix >= 0x7ff00000 {
/* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */
return x + f64::INFINITY;
}
if ix < ((0x3ff - 54) << 20) {
/* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */
return 1.0 / x;
}
/* integer arguments */
/* raise inexact when non-integer */
if x == floor(x) {
if sign {
return 0.0 / 0.0;
}
if x <= FACT.len() as f64 {
return i!(FACT, (x as usize) - 1);
}
}
/* x >= 172: tgamma(x)=inf with overflow */
/* x =< -184: tgamma(x)=+-0 with underflow */
if ix >= 0x40670000 {
/* |x| >= 184 */
if sign {
let x1p_126 = f64::from_bits(0x3810000000000000); // 0x1p-126 == 2^-126
force_eval!((x1p_126 / x) as f32);
if floor(x) * 0.5 == floor(x * 0.5) {
return 0.0;
} else {
return -0.0;
}
}
let x1p1023 = f64::from_bits(0x7fe0000000000000); // 0x1p1023 == 2^1023
x *= x1p1023;
return x;
}
absx = if sign { -x } else { x };
/* handle the error of x + g - 0.5 */
y = absx + GMHALF;
if absx > GMHALF {
dy = y - absx;
dy -= GMHALF;
} else {
dy = y - GMHALF;
dy -= absx;
}
z = absx - 0.5;
r = s(absx) * exp(-y);
if x < 0.0 {
/* reflection formula for negative x */
/* sinpi(absx) is not 0, integers are already handled */
r = -PI / (sinpi(absx) * absx * r);
dy = -dy;
z = -z;
}
r += dy * (GMHALF + 0.5) * r / y;
z = pow(y, 0.5 * z);
y = r * z * z;
return y;
}