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| </pre><pre class="rust"><code><span class="comment">// Copyright 2018 Developers of the Rand project. |
| // |
| // Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or |
| // https://www.apache.org/licenses/LICENSE-2.0> or the MIT license |
| // <LICENSE-MIT or https://opensource.org/licenses/MIT>, at your |
| // option. This file may not be copied, modified, or distributed |
| // except according to those terms. |
| |
| </span><span class="doccomment">//! Math helper functions |
| |
| </span><span class="kw">use </span><span class="kw">crate</span>::ziggurat_tables; |
| <span class="kw">use </span>rand::distributions::hidden_export::IntoFloat; |
| <span class="kw">use </span>rand::Rng; |
| <span class="kw">use </span>num_traits::Float; |
| |
| <span class="doccomment">/// Calculates ln(gamma(x)) (natural logarithm of the gamma |
| /// function) using the Lanczos approximation. |
| /// |
| /// The approximation expresses the gamma function as: |
| /// `gamma(z+1) = sqrt(2*pi)*(z+g+0.5)^(z+0.5)*exp(-z-g-0.5)*Ag(z)` |
| /// `g` is an arbitrary constant; we use the approximation with `g=5`. |
| /// |
| /// Noting that `gamma(z+1) = z*gamma(z)` and applying `ln` to both sides: |
| /// `ln(gamma(z)) = (z+0.5)*ln(z+g+0.5)-(z+g+0.5) + ln(sqrt(2*pi)*Ag(z)/z)` |
| /// |
| /// `Ag(z)` is an infinite series with coefficients that can be calculated |
| /// ahead of time - we use just the first 6 terms, which is good enough |
| /// for most purposes. |
| </span><span class="kw">pub</span>(<span class="kw">crate</span>) <span class="kw">fn </span>log_gamma<F: Float>(x: F) -> F { |
| <span class="comment">// precalculated 6 coefficients for the first 6 terms of the series |
| </span><span class="kw">let </span>coefficients: [F; <span class="number">6</span>] = [ |
| F::from(<span class="number">76.18009172947146</span>).unwrap(), |
| F::from(-<span class="number">86.50532032941677</span>).unwrap(), |
| F::from(<span class="number">24.01409824083091</span>).unwrap(), |
| F::from(-<span class="number">1.231739572450155</span>).unwrap(), |
| F::from(<span class="number">0.1208650973866179e-2</span>).unwrap(), |
| F::from(-<span class="number">0.5395239384953e-5</span>).unwrap(), |
| ]; |
| |
| <span class="comment">// (x+0.5)*ln(x+g+0.5)-(x+g+0.5) |
| </span><span class="kw">let </span>tmp = x + F::from(<span class="number">5.5</span>).unwrap(); |
| <span class="kw">let </span>log = (x + F::from(<span class="number">0.5</span>).unwrap()) * tmp.ln() - tmp; |
| |
| <span class="comment">// the first few terms of the series for Ag(x) |
| </span><span class="kw">let </span><span class="kw-2">mut </span>a = F::from(<span class="number">1.000000000190015</span>).unwrap(); |
| <span class="kw">let </span><span class="kw-2">mut </span>denom = x; |
| <span class="kw">for </span><span class="kw-2">&</span>coeff <span class="kw">in </span><span class="kw-2">&</span>coefficients { |
| denom = denom + F::one(); |
| a = a + (coeff / denom); |
| } |
| |
| <span class="comment">// get everything together |
| // a is Ag(x) |
| // 2.5066... is sqrt(2pi) |
| </span>log + (F::from(<span class="number">2.5066282746310005</span>).unwrap() * a / x).ln() |
| } |
| |
| <span class="doccomment">/// Sample a random number using the Ziggurat method (specifically the |
| /// ZIGNOR variant from Doornik 2005). Most of the arguments are |
| /// directly from the paper: |
| /// |
| /// * `rng`: source of randomness |
| /// * `symmetric`: whether this is a symmetric distribution, or one-sided with P(x < 0) = 0. |
| /// * `X`: the $x_i$ abscissae. |
| /// * `F`: precomputed values of the PDF at the $x_i$, (i.e. $f(x_i)$) |
| /// * `F_DIFF`: precomputed values of $f(x_i) - f(x_{i+1})$ |
| /// * `pdf`: the probability density function |
| /// * `zero_case`: manual sampling from the tail when we chose the |
| /// bottom box (i.e. i == 0) |
| |
| </span><span class="comment">// the perf improvement (25-50%) is definitely worth the extra code |
| // size from force-inlining. |
| </span><span class="attribute">#[inline(always)] |
| </span><span class="kw">pub</span>(<span class="kw">crate</span>) <span class="kw">fn </span>ziggurat<R: Rng + <span class="question-mark">?</span>Sized, P, Z>( |
| rng: <span class="kw-2">&mut </span>R, |
| symmetric: bool, |
| x_tab: ziggurat_tables::ZigTable, |
| f_tab: ziggurat_tables::ZigTable, |
| <span class="kw-2">mut </span>pdf: P, |
| <span class="kw-2">mut </span>zero_case: Z |
| ) -> f64 |
| <span class="kw">where |
| </span>P: FnMut(f64) -> f64, |
| Z: FnMut(<span class="kw-2">&mut </span>R, f64) -> f64, |
| { |
| <span class="kw">loop </span>{ |
| <span class="comment">// As an optimisation we re-implement the conversion to a f64. |
| // From the remaining 12 most significant bits we use 8 to construct `i`. |
| // This saves us generating a whole extra random number, while the added |
| // precision of using 64 bits for f64 does not buy us much. |
| </span><span class="kw">let </span>bits = rng.next_u64(); |
| <span class="kw">let </span>i = bits <span class="kw">as </span>usize & <span class="number">0xff</span>; |
| |
| <span class="kw">let </span>u = <span class="kw">if </span>symmetric { |
| <span class="comment">// Convert to a value in the range [2,4) and subtract to get [-1,1) |
| // We can't convert to an open range directly, that would require |
| // subtracting `3.0 - EPSILON`, which is not representable. |
| // It is possible with an extra step, but an open range does not |
| // seem necessary for the ziggurat algorithm anyway. |
| </span>(bits >> <span class="number">12</span>).into_float_with_exponent(<span class="number">1</span>) - <span class="number">3.0 |
| </span>} <span class="kw">else </span>{ |
| <span class="comment">// Convert to a value in the range [1,2) and subtract to get (0,1) |
| </span>(bits >> <span class="number">12</span>).into_float_with_exponent(<span class="number">0</span>) - (<span class="number">1.0 </span>- core::f64::EPSILON / <span class="number">2.0</span>) |
| }; |
| <span class="kw">let </span>x = u * x_tab[i]; |
| |
| <span class="kw">let </span>test_x = <span class="kw">if </span>symmetric { x.abs() } <span class="kw">else </span>{ x }; |
| |
| <span class="comment">// algebraically equivalent to |u| < x_tab[i+1]/x_tab[i] (or u < x_tab[i+1]/x_tab[i]) |
| </span><span class="kw">if </span>test_x < x_tab[i + <span class="number">1</span>] { |
| <span class="kw">return </span>x; |
| } |
| <span class="kw">if </span>i == <span class="number">0 </span>{ |
| <span class="kw">return </span>zero_case(rng, u); |
| } |
| <span class="comment">// algebraically equivalent to f1 + DRanU()*(f0 - f1) < 1 |
| </span><span class="kw">if </span>f_tab[i + <span class="number">1</span>] + (f_tab[i] - f_tab[i + <span class="number">1</span>]) * rng.gen::<f64>() < pdf(x) { |
| <span class="kw">return </span>x; |
| } |
| } |
| } |
| </code></pre></div> |
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