pxfm/gamma/
gamma_q.rs

1/*
2 * // Copyright (c) Radzivon Bartoshyk 9/2025. All rights reserved.
3 * //
4 * // Redistribution and use in source and binary forms, with or without modification,
5 * // are permitted provided that the following conditions are met:
6 * //
7 * // 1.  Redistributions of source code must retain the above copyright notice, this
8 * // list of conditions and the following disclaimer.
9 * //
10 * // 2.  Redistributions in binary form must reproduce the above copyright notice,
11 * // this list of conditions and the following disclaimer in the documentation
12 * // and/or other materials provided with the distribution.
13 * //
14 * // 3.  Neither the name of the copyright holder nor the names of its
15 * // contributors may be used to endorse or promote products derived from
16 * // this software without specific prior written permission.
17 * //
18 * // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
19 * // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
21 * // DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
22 * // FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * // DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
24 * // SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
25 * // CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
26 * // OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
27 * // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
28 */
29use crate::bessel::i0_exp;
30use crate::double_double::DoubleDouble;
31use crate::gamma::lgamma::lgamma_core;
32use crate::logs::fast_log_d_to_dd;
33
34fn core_gamma_p(a: f64, x: f64) -> (DoubleDouble, Option<f64>) {
35    const BIG: f64 = 4503599627370496.0;
36    const BIG_INV: f64 = 2.22044604925031308085e-16;
37
38    const EPS: f64 = f64::EPSILON;
39
40    let da = a;
41    let dx = x;
42
43    let r = DoubleDouble::full_add_f64(-lgamma_core(a).0, -dx);
44    let ax = DoubleDouble::mul_f64_add(fast_log_d_to_dd(x), da, r).to_f64();
45
46    if ax <= -709.78271289338399 {
47        if a < x {
48            return (DoubleDouble::default(), Some(1.0));
49        }
50        return (DoubleDouble::default(), Some(0.0));
51    }
52    if ax >= 709.783 {
53        return (DoubleDouble::default(), Some(f64::INFINITY));
54    }
55
56    if x <= 1.0 || x <= a {
57        let mut r2 = DoubleDouble::new(0., da);
58        let mut c2 = DoubleDouble::new(0., 1.0);
59        let mut ans2 = DoubleDouble::new(0., 1.0);
60        let v_e = i0_exp(ax);
61        for _ in 0..200 {
62            r2 = DoubleDouble::full_add_f64(r2, 1.0);
63            c2 = DoubleDouble::quick_mult(DoubleDouble::from_f64_div_dd(dx, r2), c2);
64            c2 = DoubleDouble::from_exact_add(c2.hi, c2.lo);
65            ans2 = DoubleDouble::add(ans2, c2);
66
67            if c2.hi / ans2.hi <= EPS {
68                break;
69            }
70        }
71        let v0 = DoubleDouble::quick_mult(v_e, ans2);
72        return (DoubleDouble::div_dd_f64(v0, da), None);
73    }
74
75    let v_e = i0_exp(ax);
76
77    let mut y = 1.0 - da;
78    let mut z = dx + y + 1.0;
79    let mut c = 0i32;
80
81    let mut p3 = 1.0;
82    let mut q3 = dx;
83    let mut p2 = dx + 1.0;
84    let mut q2 = z * dx;
85    let mut ans = p2 / q2;
86
87    for _ in 0..200 {
88        y += 1.0;
89        z += 2.0;
90        c += 1;
91        let yc = y * c as f64;
92
93        let p = p2 * z - p3 * yc;
94        let q = q2 * z - q3 * yc;
95
96        p3 = p2;
97        p2 = p;
98        q3 = q2;
99        q2 = q;
100
101        if p.abs() > BIG {
102            p3 *= BIG_INV;
103            p2 *= BIG_INV;
104            q3 *= BIG_INV;
105            q2 *= BIG_INV;
106        }
107
108        if q != 0.0 {
109            let nextans = p / q;
110            let error = ((ans - nextans) / nextans).abs();
111            ans = nextans;
112
113            if error <= EPS {
114                break;
115            }
116        }
117    }
118
119    (DoubleDouble::mul_f64_add_f64(-v_e, ans, 1.0), None)
120}
121
122/// Regularized upper incomplete gamma
123pub fn f_gamma_q(a: f64, x: f64) -> f64 {
124    let aa = a.to_bits();
125    let ax = x.to_bits();
126
127    if aa >= 0x7ffu64 << 52 || aa == 0 || ax >= 0x7ffu64 << 52 || ax == 0 {
128        if (aa >> 63) != 0 || (ax >> 63) != 0 {
129            // |a| < 0 or |b| < 0
130            return f64::NAN;
131        }
132        if aa.wrapping_shl(1) == 0 {
133            // |a| == 0
134            return 1.0;
135        }
136        if ax.wrapping_shl(1) == 0 {
137            // |x| == 0
138            return 0.;
139        }
140        if a.is_infinite() {
141            // |a| == infinity
142            return f64::INFINITY;
143        }
144        if x.is_infinite() {
145            // |x| == infinity
146            return f64::INFINITY;
147        }
148        return a + f64::NAN;
149    }
150
151    const EPS: f64 = f64::EPSILON;
152
153    const BIG: f64 = 4503599627370496.0;
154    const BIG_INV: f64 = 2.22044604925031308085e-16;
155
156    if x < 1.0 || x <= a {
157        let gamma_p = core_gamma_p(a, x);
158        return match gamma_p.1 {
159            None => {
160                let z = DoubleDouble::full_add_f64(-gamma_p.0, 1.);
161                z.to_f64()
162            }
163            Some(v) => v,
164        };
165    }
166
167    let da = a;
168    let dx = x;
169
170    let r = DoubleDouble::full_add_f64(-lgamma_core(a).0, -dx);
171    let ax = DoubleDouble::mul_f64_add(fast_log_d_to_dd(x), da, r).to_f64();
172
173    if ax <= -709.78271289338399 {
174        if a < x {
175            return 1.0;
176        }
177        return 0.0;
178    }
179    if ax >= 709.783 {
180        return f64::INFINITY;
181    }
182
183    let mut y = 1.0 - da;
184    let mut z = dx + y + 1.0;
185    let mut c = 0.0;
186    let mut pkm2 = 1.0;
187    let mut qkm2 = dx;
188    let mut pkm1 = dx + 1.0;
189    let mut qkm1 = z * dx;
190    let mut ans = pkm1 / qkm1;
191    for _ in 0..200 {
192        y += 1.0;
193        z += 2.0;
194        c += 1.0;
195        let yc = y * c;
196        let pk = pkm1 * z - pkm2 * yc;
197        let qk = qkm1 * z - qkm2 * yc;
198
199        pkm2 = pkm1;
200        pkm1 = pk;
201        qkm2 = qkm1;
202        qkm1 = qk;
203
204        if pk.abs() > BIG {
205            pkm2 *= BIG_INV;
206            pkm1 *= BIG_INV;
207            qkm2 *= BIG_INV;
208            qkm1 *= BIG_INV;
209        }
210
211        if qk != 0.0 {
212            let r = pk / qk;
213            let t = ((ans - r) / r).abs();
214            ans = r;
215
216            if t <= EPS {
217                break;
218            }
219        }
220    }
221    let v_exp = i0_exp(ax);
222    DoubleDouble::quick_mult_f64(v_exp, ans).to_f64()
223}
224
225#[cfg(test)]
226mod tests {
227    use super::*;
228    #[test]
229    fn test_f_beta_pf() {
230        assert_eq!(f_gamma_q(1., f64::INFINITY), f64::INFINITY);
231        assert_eq!(f_gamma_q(23.421, 41.), 0.0011305253882165434);
232        assert_eq!(f_gamma_q(0.764, 0.432123), 0.5224700360458718);
233        assert_eq!(f_gamma_q(0.421, 1.), 0.12721313819176905);
234        assert!(f_gamma_q(-1., 12.).is_nan());
235        assert!(f_gamma_q(1., -12.).is_nan());
236        assert!(f_gamma_q(f64::NAN, 12.).is_nan());
237        assert!(f_gamma_q(1., f64::NAN).is_nan());
238        assert_eq!(f_gamma_q(f64::INFINITY, f64::INFINITY), f64::INFINITY);
239        assert_eq!(f_gamma_q(f64::INFINITY, 5.32), f64::INFINITY);
240    }
241}
pxfm/gamma/gamma_q.rs

pxfm/gamma/
gamma_q.rs
