libm/math/
tgamma.rs

Help
1/*
2"A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964)
3"Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001)
4"An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004)
5
6approximation method:
7
8                        (x - 0.5)         S(x)
9Gamma(x) = (x + g - 0.5)         *  ----------------
10                                    exp(x + g - 0.5)
11
12with
13                 a1      a2      a3            aN
14S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ]
15               x + 1   x + 2   x + 3         x + N
16
17with a0, a1, a2, a3,.. aN constants which depend on g.
18
19for x < 0 the following reflection formula is used:
20
21Gamma(x)*Gamma(-x) = -pi/(x sin(pi x))
22
23most ideas and constants are from boost and python
24*/
25extern crate core;
26use super::{exp, floor, k_cos, k_sin, pow};
27
28const PI: f64 = 3.141592653589793238462643383279502884;
29
30/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */
31fn sinpi(mut x: f64) -> f64 {
32    let mut n: isize;
33
34    /* argument reduction: x = |x| mod 2 */
35    /* spurious inexact when x is odd int */
36    x = x * 0.5;
37    x = 2.0 * (x - floor(x));
38
39    /* reduce x into [-.25,.25] */
40    n = (4.0 * x) as isize;
41    n = div!(n + 1, 2);
42    x -= (n as f64) * 0.5;
43
44    x *= PI;
45    match n {
46        1 => k_cos(x, 0.0),
47        2 => k_sin(-x, 0.0, 0),
48        3 => -k_cos(x, 0.0),
49        0 | _ => k_sin(x, 0.0, 0),
50    }
51}
52
53const N: usize = 12;
54//static const double g = 6.024680040776729583740234375;
55const GMHALF: f64 = 5.524680040776729583740234375;
56const SNUM: [f64; N + 1] = [
57    23531376880.410759688572007674451636754734846804940,
58    42919803642.649098768957899047001988850926355848959,
59    35711959237.355668049440185451547166705960488635843,
60    17921034426.037209699919755754458931112671403265390,
61    6039542586.3520280050642916443072979210699388420708,
62    1439720407.3117216736632230727949123939715485786772,
63    248874557.86205415651146038641322942321632125127801,
64    31426415.585400194380614231628318205362874684987640,
65    2876370.6289353724412254090516208496135991145378768,
66    186056.26539522349504029498971604569928220784236328,
67    8071.6720023658162106380029022722506138218516325024,
68    210.82427775157934587250973392071336271166969580291,
69    2.5066282746310002701649081771338373386264310793408,
70];
71const SDEN: [f64; N + 1] = [
72    0.0,
73    39916800.0,
74    120543840.0,
75    150917976.0,
76    105258076.0,
77    45995730.0,
78    13339535.0,
79    2637558.0,
80    357423.0,
81    32670.0,
82    1925.0,
83    66.0,
84    1.0,
85];
86/* n! for small integer n */
87const FACT: [f64; 23] = [
88    1.0,
89    1.0,
90    2.0,
91    6.0,
92    24.0,
93    120.0,
94    720.0,
95    5040.0,
96    40320.0,
97    362880.0,
98    3628800.0,
99    39916800.0,
100    479001600.0,
101    6227020800.0,
102    87178291200.0,
103    1307674368000.0,
104    20922789888000.0,
105    355687428096000.0,
106    6402373705728000.0,
107    121645100408832000.0,
108    2432902008176640000.0,
109    51090942171709440000.0,
110    1124000727777607680000.0,
111];
112
113/* S(x) rational function for positive x */
114fn s(x: f64) -> f64 {
115    let mut num: f64 = 0.0;
116    let mut den: f64 = 0.0;
117
118    /* to avoid overflow handle large x differently */
119    if x < 8.0 {
120        for i in (0..=N).rev() {
121            num = num * x + i!(SNUM, i);
122            den = den * x + i!(SDEN, i);
123        }
124    } else {
125        for i in 0..=N {
126            num = num / x + i!(SNUM, i);
127            den = den / x + i!(SDEN, i);
128        }
129    }
130    return num / den;
131}
132
133#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
134pub fn tgamma(mut x: f64) -> f64 {
135    let u: u64 = x.to_bits();
136    let absx: f64;
137    let mut y: f64;
138    let mut dy: f64;
139    let mut z: f64;
140    let mut r: f64;
141    let ix: u32 = ((u >> 32) as u32) & 0x7fffffff;
142    let sign: bool = (u >> 63) != 0;
143
144    /* special cases */
145    if ix >= 0x7ff00000 {
146        /* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */
147        return x + core::f64::INFINITY;
148    }
149    if ix < ((0x3ff - 54) << 20) {
150        /* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */
151        return 1.0 / x;
152    }
153
154    /* integer arguments */
155    /* raise inexact when non-integer */
156    if x == floor(x) {
157        if sign {
158            return 0.0 / 0.0;
159        }
160        if x <= FACT.len() as f64 {
161            return i!(FACT, (x as usize) - 1);
162        }
163    }
164
165    /* x >= 172: tgamma(x)=inf with overflow */
166    /* x =< -184: tgamma(x)=+-0 with underflow */
167    if ix >= 0x40670000 {
168        /* |x| >= 184 */
169        if sign {
170            let x1p_126 = f64::from_bits(0x3810000000000000); // 0x1p-126 == 2^-126
171            force_eval!((x1p_126 / x) as f32);
172            if floor(x) * 0.5 == floor(x * 0.5) {
173                return 0.0;
174            } else {
175                return -0.0;
176            }
177        }
178        let x1p1023 = f64::from_bits(0x7fe0000000000000); // 0x1p1023 == 2^1023
179        x *= x1p1023;
180        return x;
181    }
182
183    absx = if sign { -x } else { x };
184
185    /* handle the error of x + g - 0.5 */
186    y = absx + GMHALF;
187    if absx > GMHALF {
188        dy = y - absx;
189        dy -= GMHALF;
190    } else {
191        dy = y - GMHALF;
192        dy -= absx;
193    }
194
195    z = absx - 0.5;
196    r = s(absx) * exp(-y);
197    if x < 0.0 {
198        /* reflection formula for negative x */
199        /* sinpi(absx) is not 0, integers are already handled */
200        r = -PI / (sinpi(absx) * absx * r);
201        dy = -dy;
202        z = -z;
203    }
204    r += dy * (GMHALF + 0.5) * r / y;
205    z = pow(y, 0.5 * z);
206    y = r * z * z;
207    return y;
208}
libm/math/tgamma.rs

libm/math/
tgamma.rs
