peroxide/special/
function.rs

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
use crate::special::lanczos::{gamma_approx, ln_gamma_approx};
use std::f64::consts::PI;

/// Gaussian function
///
/// `N(x|μ,σ) = 1/√(2πσ^2) exp(-(x-μ)^2/(2σ^2))`
pub fn gaussian(x: f64, mu: f64, sigma: f64) -> f64 {
    1f64 / ((2f64 * PI).sqrt() * sigma) * (-0.5 * ((x - mu) / sigma).powi(2)).exp()
}

/// Gamma function
///
/// # Description
/// Use Lanczos approximation to implement Gamma function ($g=5, n=7$)
///
/// # References
/// * [Robert Munafo, Coefficients for the Lanczos Approximation to the Gamma Function](https://mrob.com/pub/ries/lanczos-gamma.html)
/// * [Paul Godfrey, A note on the computation of the convergent Lanczos complex Gamma approximation (web page), 2001.](http://my.fit.edu/~gabdo/gamma.txt)
pub fn gamma(x: f64) -> f64 {
    gamma_approx(x)
}

/// Logarithm Gamma function
///
/// # Description
/// Use Lanczos approximation to implement Gamma function ($g=5, n=7$)
///
/// # References
/// * [Robert Munafo, Coefficients for the Lanczos Approximation to the Gamma Function](https://mrob.com/pub/ries/lanczos-gamma.html)
/// * [Paul Godfrey, A note on the computation of the convergent Lanczos complex Gamma approximation (web page), 2001.](http://my.fit.edu/~gabdo/gamma.txt)
pub fn ln_gamma(x: f64) -> f64 {
    ln_gamma_approx(x)
}

/// Pochhammer symbol
pub fn poch(x: f64, n: usize) -> f64 {
    let mut s = 1f64;
    for i in 0..n {
        s *= x + i as f64;
    }
    s
}

// /// Digamma function
// ///
// /// Wrapper of `digamma` function of `special` crate
// pub fn digamma(x: f64) -> f64 {
//     x.digamma()
// }

/// Regularized incomplete gamma integral (Lower)
///
/// Wrapper of `gammp` function of `puruspe` crate
pub fn inc_gamma(a: f64, x: f64) -> f64 {
    puruspe::gammp(a, x)
}

/// Inverse of regularized incomplete gamma integral (Lower)
///
/// Wrapper of `invgammp` function of `puruspe` crate
pub fn inv_inc_gamma(p: f64, a: f64) -> f64 {
    puruspe::invgammp(p, a)
}

/// Error function
///
/// Wrapper of `erf` function of `puruspe` crate
pub fn erf(x: f64) -> f64 {
    puruspe::erf(x)
}

/// Complement error function
///
/// Wrapper of `erfc` function of `puruspe` crate
pub fn erfc(x: f64) -> f64 {
    puruspe::erfc(x)
}

/// Inverse error function
///
/// Wrapper of `inverf` function of `puruspe` crate
pub fn inv_erf(x: f64) -> f64 {
    puruspe::inverf(x)
}

/// Inverse complementary error function
///
/// Wrapper of `inverfc` function of `puruspe` crate
pub fn inv_erfc(p: f64) -> f64 {
    puruspe::inverfc(p)
}

/// Beta function
///
/// Wrapper of `beta` function of `puruspe` crate
pub fn beta(a: f64, b: f64) -> f64 {
    puruspe::beta(a, b)
}

/// Regularized incomplete Beta function
///
/// Wrapper of `betai` function of `puruspe` crate
pub fn inc_beta(a: f64, b: f64, x: f64) -> f64 {
    puruspe::betai(a, b, x)
}

/// Inverse regularized incomplete beta function
///
/// Wrapper of `invbetai` function of `puruspe` crate
pub fn inv_inc_beta(p: f64, a: f64, b: f64) -> f64 {
    puruspe::invbetai(p, a, b)
}

/// Phi (CDF for Normal Dist)
///
/// $$\Phi(x) = \frac{1}{2}\left[1 + \text{erf}\left(\frac{x}{\sqrt{2}}\right) \right]$$
pub fn phi(x: f64) -> f64 {
    0.5 * (1f64 + erf(x / 2f64.sqrt()))
}

/// The principal branch of the Lambert W function, W_0(`z`).
///
/// Returns [`NAN`](f64::NAN) if the given input is smaller than -1/e (≈ -0.36787944117144233).
///
/// Use [`Precise`](LambertWAccuracyMode::Precise) for 50 bits of accuracy and the [`Simple`](LambertWAccuracyMode::Simple) mode
/// for only 24 bits, but with faster execution time.
///
/// Wrapper of the `lambert_w_0` and `sp_lambert_w_0` functions of the `puruspe` crate.
///
/// # Reference
///
/// [Toshio Fukushima, Precise and fast computation of Lambert W function by piecewise minimax rational function approximation with variable transformation](https://www.researchgate.net/publication/346309410_Precise_and_fast_computation_of_Lambert_W_function_by_piecewise_minimax_rational_function_approximation_with_variable_transformation)
pub fn lambert_w0(z: f64, mode: LambertWAccuracyMode) -> f64 {
    match mode {
        LambertWAccuracyMode::Precise => puruspe::lambert_w0(z),
        LambertWAccuracyMode::Simple => puruspe::sp_lambert_w0(z),
    }
}

/// The secondary branch of the Lambert W function, W_-1(`z`).
///
/// Returns [`NAN`](f64::NAN) if the given input is positive or smaller than -1/e (≈ -0.36787944117144233).
///
/// Use [`Precise`](LambertWAccuracyMode::Precise) for 50 bits of accuracy and the [`Simple`](LambertWAccuracyMode::Simple) mode
/// for only 24 bits, but with faster execution time.
///
/// Wrapper of the `lambert_w_m1` and `sp_lambert_w_m1` functions of the `puruspe` crate.
///
/// # Reference
///
/// [Toshio Fukushima, Precise and fast computation of Lambert W function by piecewise minimax rational function approximation with variable transformation](https://www.researchgate.net/publication/346309410_Precise_and_fast_computation_of_Lambert_W_function_by_piecewise_minimax_rational_function_approximation_with_variable_transformation)
pub fn lambert_wm1(z: f64, mode: LambertWAccuracyMode) -> f64 {
    match mode {
        LambertWAccuracyMode::Precise => puruspe::lambert_wm1(z),
        LambertWAccuracyMode::Simple => puruspe::sp_lambert_wm1(z),
    }
}

/// Decides the accuracy mode of the Lambert W functions.
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub enum LambertWAccuracyMode {
    /// Faster, 24 bits of accuracy.
    Simple,
    /// Slower, 50 bits of accuracy.
    Precise,
}