jaeckel 0.2.0

Rust port of Peter Jäckel's algorithms on http://www.jaeckel.org
Documentation 
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

use crate::erf_cody::erfc_cody;
use std::f64;

// Mathematical constants
pub const SQRT_TWO: f64 = 1.4142135623730950488016887242096980785696718753769;
pub const SQRT_TWO_PI: f64 = 2.5066282746310005024157652848110452530069867406099;

// Constants for norm_cdf asymptotic expansion
const NORM_CDF_ASYMPTOTIC_EXPANSION_FIRST_THRESHOLD: f64 = -10.0;
const NORM_CDF_ASYMPTOTIC_EXPANSION_SECOND_THRESHOLD: f64 = -67.4489750196082; // -1/sqrt(DBL_EPSILON)

// Constants for PJ-2024-Inverse-Normal algorithm
const U_MAX: f64 = 0.3413447460685429; // Phi(1) - 0.5

/// Standard normal probability density function
/// φ(x) = (1/√(2π))·exp(-x²/2)
pub fn norm_pdf(x: f64) -> f64 {
    (-0.5 * x * x).exp() / SQRT_TWO_PI
}

/// Standard normal cumulative distribution function
pub fn norm_cdf(z: f64) -> f64 {
    if z <= NORM_CDF_ASYMPTOTIC_EXPANSION_FIRST_THRESHOLD {
        // Asymptotic expansion for very negative z following (26.2.12) on page 408
        // in M. Abramowitz and A. Stegun, Pocketbook of Mathematical Functions
        let mut sum = 1.0;
        if z >= NORM_CDF_ASYMPTOTIC_EXPANSION_SECOND_THRESHOLD {
            let zsqr = z * z;
            let mut i = 1.0;
            let mut g = 1.0;
            let mut a = f64::MAX;
            let mut lasta;

            loop {
                lasta = a;
                let x = (4.0 * i - 3.0) / zsqr;
                let y = x * ((4.0 * i - 1.0) / zsqr);
                a = g * (x - y);
                sum -= a;
                g *= y;
                i += 1.0;
                a = a.abs();

                if !(lasta > a && a >= (sum * f64::EPSILON).abs()) {
                    break;
                }
            }
        }
        -norm_pdf(z) * sum / z
    } else {
        0.5 * erfc_cody(-z * (1.0 / SQRT_TWO))
    }
}

// PJ-2024-Inverse-Normal

/// Specialisation of x = Phi⁻¹(p) for x ≤ -1, p = Phi(x) ≤ 0.1586552539314570514148
fn inverse_norm_cdf_for_low_probabilities(p: f64) -> f64 {
    // Five branches based on r = sqrt(-ln(p))
    let r = (-p.ln()).sqrt();

    // All of the below are Remez-optimized minimax rational function approximations of order (5,5)
    if r < 6.7 {
        if r < 3.41 {
            if r < 2.05 {
                // Branch I. Accuracy better than 7.6E-17 in perfect arithmetic
                (3.691562302945566191
                    + r * (4.7170590600740689449e1
                        + r * (6.5451292110261454609e1
                            + r * (-7.4594687726045926821e1
                                + r * (-8.3383894003636969722e1 - 1.3054072340494093704e1 * r)))))
                    / (1.0
                        + r * (2.0837211328697753726e1
                            + r * (7.1813812182579255459e1
                                + r * (5.9270122556046077717e1
                                    + r * (9.2216887978737432303 + 1.8295174852053530579e-4 * r)))))
            } else {
                // Branch II. Accuracy better than 9.4E-17 in perfect arithmetic
                (3.2340179116317970288
                    + r * (1.449177828689122096e1
                        + r * (6.8397370256591532878e-1
                            + r * (-1.81254427791789183e1
                                + r * (-1.005916339568646151e1 - 1.2013147879435525574 * r)))))
                    / (1.0
                        + r * (8.8820931773304337525
                            + r * (1.4656370665176799712e1
                                + r * (7.1369811056109768745
                                    + r * (8.4884892199149255469e-1
                                        + 1.0957576098829595323e-5 * r)))))
            }
        } else {
            // Branch III. Accuracy better than 9.1E-17 in perfect arithmetic
            (3.1252235780087584807
                + r * (9.9483724317036560676
                    + r * (-5.1633929115525534628
                        + r * (-1.1070534689309368061e1
                            + r * (-2.8699061335882526744 - 1.5414319494013597492e-1 * r)))))
                / (1.0
                    + r * (7.076769154309171622
                        + r * (8.1086341122361532407
                            + r * (2.0307076064309043613
                                + r * (1.0897972234131828901e-1 + 1.3565983564441297634e-7 * r)))))
        }
    } else {
        if r < 12.9 {
            // Branch IV. Accuracy better than 9E-17 in perfect arithmetic
            (2.6161264950897283681
                + r * (2.250881388987032271
                    + r * (-3.688196041019692267
                        + r * (-2.9644251353150605663
                            + r * (-4.7595169546783216436e-1 - 1.612303318390145052e-2 * r)))))
                / (1.0
                    + r * (3.2517455169035921495
                        + r * (2.1282030272153188194
                            + r * (3.3663746405626400164e-1
                                + r * (1.1400087282177594359e-2 + 3.0848093570966787291e-9 * r)))))
        } else {
            // Branch V. Accuracy better than 9.5E-17 in perfect arithmetic
            (2.3226849047872302955
                + r * (-4.2799650734502094297e-2
                    + r * (-2.5894451568465728432
                        + r * (-8.6385181219213758847e-1
                            + r * (-6.5127593753781672404e-2 - 1.0566357727202585402e-3 * r)))))
                / (1.0
                    + r * (1.9361316119254412206
                        + r * (6.1320841329197493341e-1
                            + r * (4.6054974512474443189e-2
                                + r * (7.471447992167225483e-4 + 2.3135343206304887818e-11 * r)))))
        }
    }
}

/// Inverse of Phi(x)-0.5, i.e., for a given u ∈ [-u_max,u_max] find x such that u = Phi(x)-0.5
fn inverse_norm_cdfm_half_for_midrange_probabilities(u: f64) -> f64 {
    // Accuracy better than 9.8E-17 in perfect arithmetic within this branch
    let s = U_MAX * U_MAX - u * u;
    u * ((2.92958954698308805
        + s * (5.0260572167303103e1
            + s * (3.01870541922933937e2
                + s * (7.4997781456657924e2
                    + s * (6.90489242061408612e2
                        + s * (1.34233243502653864e2 - 7.58939881401259242 * s))))))
        / (1.0
            + s * (1.8918538074574598e1
                + s * (1.29404120448755281e2
                    + s * (3.86821208540417453e2
                        + s * (4.79123914509756757e2 + 1.79227008508102628e2 * s))))))
}

/// Inverse standard normal cumulative distribution function
/// Given p, return x such that p = Phi(x), i.e., x = Phi⁻¹(p)
pub fn inverse_norm_cdf(p: f64) -> f64 {
    let u = p - 0.5;
    if u.abs() < U_MAX {
        inverse_norm_cdfm_half_for_midrange_probabilities(u)
    } else {
        // Tail probability min(p,1-p) = 0.15866. r = √(-ln(min(p,1-p))) > 1.3568425277
        if u > 0.0 {
            -inverse_norm_cdf_for_low_probabilities(1.0 - p)
        } else {
            inverse_norm_cdf_for_low_probabilities(p)
        }
    }
}

/// Inverse error function
/// We can use the internal branches of Phi⁻¹(·) to implement erfinv()
/// avoiding catastrophic subtractive cancellation for small arguments
pub fn erfinv(e: f64) -> f64 {
    // Phi(x) = erfc(-x/√2)/2 = (1+erf(x/√2))/2 = erf(x/√2)/2 + 0.5
    // erf(z) = 2 · ( Phi(√2·z) - 0.5 )
    // Hence, if e = erf(z), and y is the solution to Phi(y) - 0.5 = e/2, then z = y/√2
    if e.abs() < 2.0 * U_MAX {
        inverse_norm_cdfm_half_for_midrange_probabilities(0.5 * e) * (1.0 / SQRT_TWO)
    } else {
        (if e < 0.0 {
            inverse_norm_cdf_for_low_probabilities(0.5 * e + 0.5)
        } else {
            -inverse_norm_cdf_for_low_probabilities(-0.5 * e + 0.5)
        }) * (1.0 / SQRT_TWO)
    }
}