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</pre><pre class="rust"><code><span class="comment">// Copyright 2018 Developers of the Rand project.</span>
<span class="comment">//</span>
<span class="comment">// Licensed under the Apache License, Version 2.0 &lt;LICENSE-APACHE or</span>
<span class="comment">// https://www.apache.org/licenses/LICENSE-2.0&gt; or the MIT license</span>
<span class="comment">// &lt;LICENSE-MIT or https://opensource.org/licenses/MIT&gt;, at your</span>
<span class="comment">// option. This file may not be copied, modified, or distributed</span>
<span class="comment">// except according to those terms.</span>
<span class="doccomment">//! Math helper functions</span>
<span class="kw">use</span> <span class="ident"><span class="kw">crate</span>::ziggurat_tables</span>;
<span class="kw">use</span> <span class="ident">rand::distributions::hidden_export::IntoFloat</span>;
<span class="kw">use</span> <span class="ident">rand::Rng</span>;
<span class="kw">use</span> <span class="ident">num_traits::Float</span>;
<span class="doccomment">/// Calculates ln(gamma(x)) (natural logarithm of the gamma</span>
<span class="doccomment">/// function) using the Lanczos approximation.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// The approximation expresses the gamma function as:</span>
<span class="doccomment">/// `gamma(z+1) = sqrt(2*pi)*(z+g+0.5)^(z+0.5)*exp(-z-g-0.5)*Ag(z)`</span>
<span class="doccomment">/// `g` is an arbitrary constant; we use the approximation with `g=5`.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Noting that `gamma(z+1) = z*gamma(z)` and applying `ln` to both sides:</span>
<span class="doccomment">/// `ln(gamma(z)) = (z+0.5)*ln(z+g+0.5)-(z+g+0.5) + ln(sqrt(2*pi)*Ag(z)/z)`</span>
<span class="doccomment">///</span>
<span class="doccomment">/// `Ag(z)` is an infinite series with coefficients that can be calculated</span>
<span class="doccomment">/// ahead of time - we use just the first 6 terms, which is good enough</span>
<span class="doccomment">/// for most purposes.</span>
<span class="kw">pub</span>(<span class="kw">crate</span>) <span class="kw">fn</span> <span class="ident">log_gamma</span><span class="op">&lt;</span><span class="ident">F</span>: <span class="ident">Float</span><span class="op">&gt;</span>(<span class="ident">x</span>: <span class="ident">F</span>) -&gt; <span class="ident">F</span> {
<span class="comment">// precalculated 6 coefficients for the first 6 terms of the series</span>
<span class="kw">let</span> <span class="ident">coefficients</span>: [<span class="ident">F</span>; <span class="number">6</span>] <span class="op">=</span> [
<span class="ident">F::from</span>(<span class="number">76.18009172947146</span>).<span class="ident">unwrap</span>(),
<span class="ident">F::from</span>(<span class="op">-</span><span class="number">86.50532032941677</span>).<span class="ident">unwrap</span>(),
<span class="ident">F::from</span>(<span class="number">24.01409824083091</span>).<span class="ident">unwrap</span>(),
<span class="ident">F::from</span>(<span class="op">-</span><span class="number">1.231739572450155</span>).<span class="ident">unwrap</span>(),
<span class="ident">F::from</span>(<span class="number">0.1208650973866179e-2</span>).<span class="ident">unwrap</span>(),
<span class="ident">F::from</span>(<span class="op">-</span><span class="number">0.5395239384953e-5</span>).<span class="ident">unwrap</span>(),
];
<span class="comment">// (x+0.5)*ln(x+g+0.5)-(x+g+0.5)</span>
<span class="kw">let</span> <span class="ident">tmp</span> <span class="op">=</span> <span class="ident">x</span> <span class="op">+</span> <span class="ident">F::from</span>(<span class="number">5.5</span>).<span class="ident">unwrap</span>();
<span class="kw">let</span> <span class="ident">log</span> <span class="op">=</span> (<span class="ident">x</span> <span class="op">+</span> <span class="ident">F::from</span>(<span class="number">0.5</span>).<span class="ident">unwrap</span>()) <span class="op">*</span> <span class="ident">tmp</span>.<span class="ident">ln</span>() <span class="op">-</span> <span class="ident">tmp</span>;
<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> <span class="ident">a</span> <span class="op">=</span> <span class="ident">F::from</span>(<span class="number">1.000000000190015</span>).<span class="ident">unwrap</span>();
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">denom</span> <span class="op">=</span> <span class="ident">x</span>;
<span class="kw">for</span> <span class="kw-2">&amp;</span><span class="ident">coeff</span> <span class="kw">in</span> <span class="kw-2">&amp;</span><span class="ident">coefficients</span> {
<span class="ident">denom</span> <span class="op">=</span> <span class="ident">denom</span> <span class="op">+</span> <span class="ident">F::one</span>();
<span class="ident">a</span> <span class="op">=</span> <span class="ident">a</span> <span class="op">+</span> (<span class="ident">coeff</span> <span class="op">/</span> <span class="ident">denom</span>);
}
<span class="comment">// get everything together</span>
<span class="comment">// a is Ag(x)</span>
<span class="comment">// 2.5066... is sqrt(2pi)</span>
<span class="ident">log</span> <span class="op">+</span> (<span class="ident">F::from</span>(<span class="number">2.5066282746310005</span>).<span class="ident">unwrap</span>() <span class="op">*</span> <span class="ident">a</span> <span class="op">/</span> <span class="ident">x</span>).<span class="ident">ln</span>()
}
<span class="doccomment">/// Sample a random number using the Ziggurat method (specifically the</span>
<span class="doccomment">/// ZIGNOR variant from Doornik 2005). Most of the arguments are</span>
<span class="doccomment">/// directly from the paper:</span>
<span class="doccomment">///</span>
<span class="doccomment">/// * `rng`: source of randomness</span>
<span class="doccomment">/// * `symmetric`: whether this is a symmetric distribution, or one-sided with P(x &lt; 0) = 0.</span>
<span class="doccomment">/// * `X`: the $x_i$ abscissae.</span>
<span class="doccomment">/// * `F`: precomputed values of the PDF at the $x_i$, (i.e. $f(x_i)$)</span>
<span class="doccomment">/// * `F_DIFF`: precomputed values of $f(x_i) - f(x_{i+1})$</span>
<span class="doccomment">/// * `pdf`: the probability density function</span>
<span class="doccomment">/// * `zero_case`: manual sampling from the tail when we chose the</span>
<span class="doccomment">/// bottom box (i.e. i == 0)</span>
<span class="comment">// the perf improvement (25-50%) is definitely worth the extra code</span>
<span class="comment">// size from force-inlining.</span>
<span class="attribute">#[<span class="ident">inline</span>(<span class="ident">always</span>)]</span>
<span class="kw">pub</span>(<span class="kw">crate</span>) <span class="kw">fn</span> <span class="ident">ziggurat</span><span class="op">&lt;</span><span class="ident">R</span>: <span class="ident">Rng</span> <span class="op">+</span> <span class="question-mark">?</span><span class="ident">Sized</span>, <span class="ident">P</span>, <span class="ident">Z</span><span class="op">&gt;</span>(
<span class="ident">rng</span>: <span class="kw-2">&amp;mut</span> <span class="ident">R</span>,
<span class="ident">symmetric</span>: <span class="ident">bool</span>,
<span class="ident">x_tab</span>: <span class="ident">ziggurat_tables::ZigTable</span>,
<span class="ident">f_tab</span>: <span class="ident">ziggurat_tables::ZigTable</span>,
<span class="kw-2">mut</span> <span class="ident">pdf</span>: <span class="ident">P</span>,
<span class="kw-2">mut</span> <span class="ident">zero_case</span>: <span class="ident">Z</span>
) -&gt; <span class="ident">f64</span>
<span class="kw">where</span>
<span class="ident">P</span>: <span class="ident">FnMut</span>(<span class="ident">f64</span>) -&gt; <span class="ident">f64</span>,
<span class="ident">Z</span>: <span class="ident">FnMut</span>(<span class="kw-2">&amp;mut</span> <span class="ident">R</span>, <span class="ident">f64</span>) -&gt; <span class="ident">f64</span>,
{
<span class="kw">loop</span> {
<span class="comment">// As an optimisation we re-implement the conversion to a f64.</span>
<span class="comment">// From the remaining 12 most significant bits we use 8 to construct `i`.</span>
<span class="comment">// This saves us generating a whole extra random number, while the added</span>
<span class="comment">// precision of using 64 bits for f64 does not buy us much.</span>
<span class="kw">let</span> <span class="ident">bits</span> <span class="op">=</span> <span class="ident">rng</span>.<span class="ident">next_u64</span>();
<span class="kw">let</span> <span class="ident">i</span> <span class="op">=</span> <span class="ident">bits</span> <span class="kw">as</span> <span class="ident">usize</span> <span class="op">&amp;</span> <span class="number">0xff</span>;
<span class="kw">let</span> <span class="ident">u</span> <span class="op">=</span> <span class="kw">if</span> <span class="ident">symmetric</span> {
<span class="comment">// Convert to a value in the range [2,4) and subtract to get [-1,1)</span>
<span class="comment">// We can&#39;t convert to an open range directly, that would require</span>
<span class="comment">// subtracting `3.0 - EPSILON`, which is not representable.</span>
<span class="comment">// It is possible with an extra step, but an open range does not</span>
<span class="comment">// seem necessary for the ziggurat algorithm anyway.</span>
(<span class="ident">bits</span> <span class="op">&gt;</span><span class="op">&gt;</span> <span class="number">12</span>).<span class="ident">into_float_with_exponent</span>(<span class="number">1</span>) <span class="op">-</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>
(<span class="ident">bits</span> <span class="op">&gt;</span><span class="op">&gt;</span> <span class="number">12</span>).<span class="ident">into_float_with_exponent</span>(<span class="number">0</span>) <span class="op">-</span> (<span class="number">1.0</span> <span class="op">-</span> <span class="ident">core::f64::EPSILON</span> <span class="op">/</span> <span class="number">2.0</span>)
};
<span class="kw">let</span> <span class="ident">x</span> <span class="op">=</span> <span class="ident">u</span> <span class="op">*</span> <span class="ident">x_tab</span>[<span class="ident">i</span>];
<span class="kw">let</span> <span class="ident">test_x</span> <span class="op">=</span> <span class="kw">if</span> <span class="ident">symmetric</span> { <span class="ident">x</span>.<span class="ident">abs</span>() } <span class="kw">else</span> { <span class="ident">x</span> };
<span class="comment">// algebraically equivalent to |u| &lt; x_tab[i+1]/x_tab[i] (or u &lt; x_tab[i+1]/x_tab[i])</span>
<span class="kw">if</span> <span class="ident">test_x</span> <span class="op">&lt;</span> <span class="ident">x_tab</span>[<span class="ident">i</span> <span class="op">+</span> <span class="number">1</span>] {
<span class="kw">return</span> <span class="ident">x</span>;
}
<span class="kw">if</span> <span class="ident">i</span> <span class="op">==</span> <span class="number">0</span> {
<span class="kw">return</span> <span class="ident">zero_case</span>(<span class="ident">rng</span>, <span class="ident">u</span>);
}
<span class="comment">// algebraically equivalent to f1 + DRanU()*(f0 - f1) &lt; 1</span>
<span class="kw">if</span> <span class="ident">f_tab</span>[<span class="ident">i</span> <span class="op">+</span> <span class="number">1</span>] <span class="op">+</span> (<span class="ident">f_tab</span>[<span class="ident">i</span>] <span class="op">-</span> <span class="ident">f_tab</span>[<span class="ident">i</span> <span class="op">+</span> <span class="number">1</span>]) <span class="op">*</span> <span class="ident">rng</span>.<span class="ident">gen</span>::<span class="op">&lt;</span><span class="ident">f64</span><span class="op">&gt;</span>() <span class="op">&lt;</span> <span class="ident">pdf</span>(<span class="ident">x</span>) {
<span class="kw">return</span> <span class="ident">x</span>;
}
}
}
</code></pre></div>
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