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

extern crate rand;
pub use self::rand::prelude::*;
pub use self::rand::distributions::uniform::SampleUniform;

use std::convert;
use std::u32;

#[allow(unused_imports)]
use structure::matrix::*;

/// Simple uniform random number generator with ThreadRng
///
/// # Examples
/// ```
/// extern crate peroxide;
/// use peroxide::*;
///
/// let mut rng = thread_rng();
/// println!("{}", rand_num(&mut rng, 1, 7));       // Roll a dice
/// println!("{}", rand_num(&mut rng, 0f64, 1f64)); // Uniform [0,1)
/// ```
pub fn rand_num<T>(rng: &mut ThreadRng, start: T, end: T) -> T where T: PartialOrd + SampleUniform + Copy {
    rng.gen_range(start, end)
}

// =============================================================================
// Back end utils
// =============================================================================

/// Gaussian random number generator using Marsaglia polar form
pub fn marsaglia_polar(rng: &mut ThreadRng, m: f64, s: f64) -> f64 {
    let mut x1 = 0f64;
    let mut x2 = 0f64;
    let mut _y2 = 0f64;
    let mut w = 0f64;

    while w == 0. || w >= 1. {
        x1 = 2.0 * rng.gen_range(0f64, 1f64) - 1.0;
        x2 = 2.0 * rng.gen_range(0f64, 1f64) - 1.0;
        w = x1 * x1 + x2 * x2;
    }

    w = (-2.0 * w.ln() / w).sqrt();
    let y1 = x1 * w;
    _y2 = x2 * w;

    return m + y1 * s;
}

// =============================================================================
// Ziggurat Table
// =============================================================================

/// Position of Right-most step
const PARAM_R: f64 = 3.44428647676;

/// Tabulated values for the height of the Ziggurat levels
const YTAB: [f64; 128] = [
    1., 0.963598623011, 0.936280813353, 0.913041104253,
    0.892278506696, 0.873239356919, 0.855496407634, 0.838778928349,
    0.822902083699, 0.807732738234, 0.793171045519, 0.779139726505,
    0.765577436082, 0.752434456248, 0.739669787677, 0.727249120285,
    0.715143377413, 0.703327646455, 0.691780377035, 0.68048276891,
    0.669418297233, 0.65857233912, 0.647931876189, 0.637485254896,
    0.62722199145, 0.617132611532, 0.607208517467, 0.597441877296,
    0.587825531465, 0.578352913803, 0.569017984198, 0.559815170911,
    0.550739320877, 0.541785656682, 0.532949739145, 0.524227434628,
    0.515614886373, 0.507108489253, 0.498704867478, 0.490400854812,
    0.482193476986, 0.47407993601, 0.466057596125, 0.458123971214,
    0.450276713467, 0.442513603171, 0.434832539473, 0.427231532022,
    0.419708693379, 0.41226223212, 0.404890446548, 0.397591718955,
    0.390364510382, 0.383207355816, 0.376118859788, 0.369097692334,
    0.362142585282, 0.355252328834, 0.348425768415, 0.341661801776,
    0.334959376311, 0.328317486588, 0.321735172063, 0.31521151497,
    0.308745638367, 0.302336704338, 0.29598391232, 0.289686497571,
    0.283443729739, 0.27725491156, 0.271119377649, 0.265036493387,
    0.259005653912, 0.253026283183, 0.247097833139, 0.241219782932,
    0.235391638239, 0.229612930649, 0.223883217122, 0.218202079518,
    0.212569124201, 0.206983981709, 0.201446306496, 0.195955776745,
    0.190512094256, 0.185114984406, 0.179764196185, 0.174459502324,
    0.169200699492, 0.1639876086, 0.158820075195, 0.153697969964,
    0.148621189348, 0.143589656295, 0.138603321143, 0.133662162669,
    0.128766189309, 0.123915440582, 0.119109988745, 0.114349940703,
    0.10963544023, 0.104966670533, 0.100343857232, 0.0957672718266,
    0.0912372357329, 0.0867541250127, 0.082318375932, 0.0779304915295,
    0.0735910494266, 0.0693007111742, 0.065060233529, 0.0608704821745,
    0.056732448584, 0.05264727098, 0.0486162607163, 0.0446409359769,
    0.0407230655415, 0.0368647267386, 0.0330683839378, 0.0293369977411,
    0.0256741818288, 0.0220844372634, 0.0185735200577, 0.0151490552854,
    0.0118216532614, 0.00860719483079, 0.00553245272614, 0.00265435214565
];

/// Tabulated values for 2^24 times `x[i] / x[i+1]`
/// Used to accept for `U*x[i+1] <= x[i]` without any floating point operations
const KTAB: [u32; 128] = [
    0, 12590644, 14272653, 14988939,
    15384584, 15635009, 15807561, 15933577,
    16029594, 16105155, 16166147, 16216399,
    16258508, 16294295, 16325078, 16351831,
    16375291, 16396026, 16414479, 16431002,
    16445880, 16459343, 16471578, 16482744,
    16492970, 16502368, 16511031, 16519039,
    16526459, 16533352, 16539769, 16545755,
    16551348, 16556584, 16561493, 16566101,
    16570433, 16574511, 16578353, 16581977,
    16585398, 16588629, 16591685, 16594575,
    16597311, 16599901, 16602354, 16604679,
    16606881, 16608968, 16610945, 16612818,
    16614592, 16616272, 16617861, 16619363,
    16620782, 16622121, 16623383, 16624570,
    16625685, 16626730, 16627708, 16628619,
    16629465, 16630248, 16630969, 16631628,
    16632228, 16632768, 16633248, 16633671,
    16634034, 16634340, 16634586, 16634774,
    16634903, 16634972, 16634980, 16634926,
    16634810, 16634628, 16634381, 16634066,
    16633680, 16633222, 16632688, 16632075,
    16631380, 16630598, 16629726, 16628757,
    16627686, 16626507, 16625212, 16623794,
    16622243, 16620548, 16618698, 16616679,
    16614476, 16612071, 16609444, 16606571,
    16603425, 16599973, 16596178, 16591995,
    16587369, 16582237, 16576520, 16570120,
    16562917, 16554758, 16545450, 16534739,
    16522287, 16507638, 16490152, 16468907,
    16442518, 16408804, 16364095, 16301683,
    16207738, 16047994, 15704248, 15472926
];

/// Tabulated values of `2^{-24} * x[i]`
const WTAB: [f64; 128] = [
    1.62318314817e-08, 2.16291505214e-08, 2.54246305087e-08, 2.84579525938e-08,
    3.10340022482e-08, 3.33011726243e-08, 3.53439060345e-08, 3.72152672658e-08,
    3.8950989572e-08, 4.05763964764e-08, 4.21101548915e-08, 4.35664624904e-08,
    4.49563968336e-08, 4.62887864029e-08, 4.75707945735e-08, 4.88083237257e-08,
    5.00063025384e-08, 5.11688950428e-08, 5.22996558616e-08, 5.34016475624e-08,
    5.44775307871e-08, 5.55296344581e-08, 5.65600111659e-08, 5.75704813695e-08,
    5.85626690412e-08, 5.95380306862e-08, 6.04978791776e-08, 6.14434034901e-08,
    6.23756851626e-08, 6.32957121259e-08, 6.42043903937e-08, 6.51025540077e-08,
    6.59909735447e-08, 6.68703634341e-08, 6.77413882848e-08, 6.8604668381e-08,
    6.94607844804e-08, 7.03102820203e-08, 7.11536748229e-08, 7.1991448372e-08,
    7.2824062723e-08, 7.36519550992e-08, 7.44755422158e-08, 7.52952223703e-08,
    7.61113773308e-08, 7.69243740467e-08, 7.77345662086e-08, 7.85422956743e-08,
    7.93478937793e-08, 8.01516825471e-08, 8.09539758128e-08, 8.17550802699e-08,
    8.25552964535e-08, 8.33549196661e-08, 8.41542408569e-08, 8.49535474601e-08,
    8.57531242006e-08, 8.65532538723e-08, 8.73542180955e-08, 8.8156298059e-08,
    8.89597752521e-08, 8.97649321908e-08, 9.05720531451e-08, 9.138142487e-08,
    9.21933373471e-08, 9.30080845407e-08, 9.38259651738e-08, 9.46472835298e-08,
    9.54723502847e-08, 9.63014833769e-08, 9.71350089201e-08, 9.79732621669e-08,
    9.88165885297e-08, 9.96653446693e-08, 1.00519899658e-07, 1.0138063623e-07,
    1.02247952126e-07, 1.03122261554e-07, 1.04003996769e-07, 1.04893609795e-07,
    1.05791574313e-07, 1.06698387725e-07, 1.07614573423e-07, 1.08540683296e-07,
    1.09477300508e-07, 1.1042504257e-07, 1.11384564771e-07, 1.12356564007e-07,
    1.13341783071e-07, 1.14341015475e-07, 1.15355110887e-07, 1.16384981291e-07,
    1.17431607977e-07, 1.18496049514e-07, 1.19579450872e-07, 1.20683053909e-07,
    1.21808209468e-07, 1.2295639141e-07, 1.24129212952e-07, 1.25328445797e-07,
    1.26556042658e-07, 1.27814163916e-07, 1.29105209375e-07, 1.30431856341e-07,
    1.31797105598e-07, 1.3320433736e-07, 1.34657379914e-07, 1.36160594606e-07,
    1.37718982103e-07, 1.39338316679e-07, 1.41025317971e-07, 1.42787873535e-07,
    1.44635331499e-07, 1.4657889173e-07, 1.48632138436e-07, 1.50811780719e-07,
    1.53138707402e-07, 1.55639532047e-07, 1.58348931426e-07, 1.61313325908e-07,
    1.64596952856e-07, 1.68292495203e-07, 1.72541128694e-07, 1.77574279496e-07,
    1.83813550477e-07, 1.92166040885e-07, 2.05295471952e-07, 2.22600839893e-07
];

/// Gaussian random numbers using the Ziggurat Method
///
/// The code is based on a [C implementation][1] by Jochen Voss.
///
/// [1]: https://www.seehuhn.de/pages/ziggurat.html
#[allow(unused_assignments)]
pub fn ziggurat(rng: &mut ThreadRng, sigma: f64) -> f64 {
    let (mut u, mut i, mut sign, mut j) = (0u32, 0usize, 0u32, 0u32);
    let mut x = 0f64;
    let mut y = 0f64;

    loop {
        u = rand_num(rng, u32::MIN, u32::MAX);
        i = (u & 0x0000007F) as usize;     // 7 bit to choose the step
        sign = u & 0x00000080;  // 1 bit for the sign
        j = u >> 8;             // 24 bit for the x-value

        x = j as f64 * WTAB[i];
        if j < KTAB[i] { break; }

        if i < 127 {
            let y0 = YTAB[i];
            let y1 = YTAB[i+1];
            y = y1 + (y0 - y1)*rand_num(rng, 0f64, 1f64);
        } else {
            x = PARAM_R - (1.0 - rand_num(rng, 0f64, 1f64).ln()) / PARAM_R;
            y = (-PARAM_R * (x - 0.5 * PARAM_R)).exp() * rand_num(rng, 0f64, 1f64);
        }

        if y < (-0.5 * x * x).exp() { break; }
    }

    if sign != 0 {
        sigma * x
    } else {
        -sigma * x
    }
}