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 </pre><pre class="rust"><code><span class="comment">// Copyright 2018 Developers of the Rand project.
 // Copyright 2013 The Rust Project Developers.
 //
 // Licensed under the Apache License, Version 2.0 &lt;LICENSE-APACHE or
 // https://www.apache.org/licenses/LICENSE-2.0&gt; or the MIT license
 // &lt;LICENSE-MIT or https://opensource.org/licenses/MIT&gt;, at your
 // option. This file may not be copied, modified, or distributed
 // except according to those terms.

 </span><span class="doccomment">//! The normal and derived distributions.

 </span><span class="kw">use </span><span class="kw">crate</span>::utils::ziggurat;
 <span class="kw">use </span>num_traits::Float;
 <span class="kw">use crate</span>::{ziggurat_tables, Distribution, Open01};
 <span class="kw">use </span>rand::Rng;
 <span class="kw">use </span>core::fmt;

 <span class="doccomment">/// Samples floating-point numbers according to the normal distribution
 /// `N(0, 1)` (a.k.a. a standard normal, or Gaussian). This is equivalent to
 /// `Normal::new(0.0, 1.0)` but faster.
 ///
 /// See `Normal` for the general normal distribution.
 ///
 /// Implemented via the ZIGNOR variant[^1] of the Ziggurat method.
 ///
 /// [^1]: Jurgen A. Doornik (2005). [*An Improved Ziggurat Method to
 ///       Generate Normal Random Samples*](
 ///       https://www.doornik.com/research/ziggurat.pdf).
 ///       Nuffield College, Oxford
 ///
 /// # Example
 /// ```
 /// use rand::prelude::*;
 /// use rand_distr::StandardNormal;
 ///
 /// let val: f64 = thread_rng().sample(StandardNormal);
 /// println!(&quot;{}&quot;, val);
 /// ```
 </span><span class="attribute">#[derive(Clone, Copy, Debug)]
 #[cfg_attr(feature = <span class="string">&quot;serde1&quot;</span>, derive(serde::Serialize, serde::Deserialize))]
 </span><span class="kw">pub struct </span>StandardNormal;

 <span class="kw">impl </span>Distribution&lt;f32&gt; <span class="kw">for </span>StandardNormal {
     <span class="attribute">#[inline]
     </span><span class="kw">fn </span>sample&lt;R: Rng + <span class="question-mark">?</span>Sized&gt;(<span class="kw-2">&amp;</span><span class="self">self</span>, rng: <span class="kw-2">&amp;mut </span>R) -&gt; f32 {
         <span class="comment">// TODO: use optimal 32-bit implementation
         </span><span class="kw">let </span>x: f64 = <span class="self">self</span>.sample(rng);
         x <span class="kw">as </span>f32
     }
 }

 <span class="kw">impl </span>Distribution&lt;f64&gt; <span class="kw">for </span>StandardNormal {
     <span class="kw">fn </span>sample&lt;R: Rng + <span class="question-mark">?</span>Sized&gt;(<span class="kw-2">&amp;</span><span class="self">self</span>, rng: <span class="kw-2">&amp;mut </span>R) -&gt; f64 {
         <span class="attribute">#[inline]
         </span><span class="kw">fn </span>pdf(x: f64) -&gt; f64 {
             (-x * x / <span class="number">2.0</span>).exp()
         }
         <span class="attribute">#[inline]
         </span><span class="kw">fn </span>zero_case&lt;R: Rng + <span class="question-mark">?</span>Sized&gt;(rng: <span class="kw-2">&amp;mut </span>R, u: f64) -&gt; f64 {
             <span class="comment">// compute a random number in the tail by hand

             // strange initial conditions, because the loop is not
             // do-while, so the condition should be true on the first
             // run, they get overwritten anyway (0 &lt; 1, so these are
             // good).
             </span><span class="kw">let </span><span class="kw-2">mut </span>x = <span class="number">1.0f64</span>;
             <span class="kw">let </span><span class="kw-2">mut </span>y = <span class="number">0.0f64</span>;

             <span class="kw">while </span>-<span class="number">2.0 </span>* y &lt; x * x {
                 <span class="kw">let </span>x_: f64 = rng.sample(Open01);
                 <span class="kw">let </span>y_: f64 = rng.sample(Open01);

                 x = x_.ln() / ziggurat_tables::ZIG_NORM_R;
                 y = y_.ln();
             }

             <span class="kw">if </span>u &lt; <span class="number">0.0 </span>{
                 x - ziggurat_tables::ZIG_NORM_R
             } <span class="kw">else </span>{
                 ziggurat_tables::ZIG_NORM_R - x
             }
         }

         ziggurat(
             rng,
             <span class="bool-val">true</span>, <span class="comment">// this is symmetric
             </span><span class="kw-2">&amp;</span>ziggurat_tables::ZIG_NORM_X,
             <span class="kw-2">&amp;</span>ziggurat_tables::ZIG_NORM_F,
             pdf,
             zero_case,
         )
     }
 }

 <span class="doccomment">/// The normal distribution `N(mean, std_dev**2)`.
 ///
 /// This uses the ZIGNOR variant of the Ziggurat method, see [`StandardNormal`]
 /// for more details.
 ///
 /// Note that [`StandardNormal`] is an optimised implementation for mean 0, and
 /// standard deviation 1.
 ///
 /// # Example
 ///
 /// ```
 /// use rand_distr::{Normal, Distribution};
 ///
 /// // mean 2, standard deviation 3
 /// let normal = Normal::new(2.0, 3.0).unwrap();
 /// let v = normal.sample(&amp;mut rand::thread_rng());
 /// println!(&quot;{} is from a N(2, 9) distribution&quot;, v)
 /// ```
 ///
 /// [`StandardNormal`]: crate::StandardNormal
 </span><span class="attribute">#[derive(Clone, Copy, Debug)]
 #[cfg_attr(feature = <span class="string">&quot;serde1&quot;</span>, derive(serde::Serialize, serde::Deserialize))]
 </span><span class="kw">pub struct </span>Normal&lt;F&gt;
 <span class="kw">where </span>F: Float, StandardNormal: Distribution&lt;F&gt;
 {
     mean: F,
     std_dev: F,
 }

 <span class="doccomment">/// Error type returned from `Normal::new` and `LogNormal::new`.
 </span><span class="attribute">#[derive(Clone, Copy, Debug, PartialEq, Eq)]
 </span><span class="kw">pub enum </span>Error {
     <span class="doccomment">/// The mean value is too small (log-normal samples must be positive)
     </span>MeanTooSmall,
     <span class="doccomment">/// The standard deviation or other dispersion parameter is not finite.
     </span>BadVariance,
 }

 <span class="kw">impl </span>fmt::Display <span class="kw">for </span>Error {
     <span class="kw">fn </span>fmt(<span class="kw-2">&amp;</span><span class="self">self</span>, f: <span class="kw-2">&amp;mut </span>fmt::Formatter&lt;<span class="lifetime">&#39;_</span>&gt;) -&gt; fmt::Result {
         f.write_str(<span class="kw">match </span><span class="self">self </span>{
             Error::MeanTooSmall =&gt; <span class="string">&quot;mean &lt; 0 or NaN in log-normal distribution&quot;</span>,
             Error::BadVariance =&gt; <span class="string">&quot;variation parameter is non-finite in (log)normal distribution&quot;</span>,
         })
     }
 }

 <span class="attribute">#[cfg(feature = <span class="string">&quot;std&quot;</span>)]
 #[cfg_attr(doc_cfg, doc(cfg(feature = <span class="string">&quot;std&quot;</span>)))]
 </span><span class="kw">impl </span>std::error::Error <span class="kw">for </span>Error {}

 <span class="kw">impl</span>&lt;F&gt; Normal&lt;F&gt;
 <span class="kw">where </span>F: Float, StandardNormal: Distribution&lt;F&gt;
 {
     <span class="doccomment">/// Construct, from mean and standard deviation
     ///
     /// Parameters:
     ///
     /// -   mean (`μ`, unrestricted)
     /// -   standard deviation (`σ`, must be finite)
     </span><span class="attribute">#[inline]
     </span><span class="kw">pub fn </span>new(mean: F, std_dev: F) -&gt; <span class="prelude-ty">Result</span>&lt;Normal&lt;F&gt;, Error&gt; {
         <span class="kw">if </span>!std_dev.is_finite() {
             <span class="kw">return </span><span class="prelude-val">Err</span>(Error::BadVariance);
         }
         <span class="prelude-val">Ok</span>(Normal { mean, std_dev })
     }

     <span class="doccomment">/// Construct, from mean and coefficient of variation
     ///
     /// Parameters:
     ///
     /// -   mean (`μ`, unrestricted)
     /// -   coefficient of variation (`cv = abs(σ / μ)`)
     </span><span class="attribute">#[inline]
     </span><span class="kw">pub fn </span>from_mean_cv(mean: F, cv: F) -&gt; <span class="prelude-ty">Result</span>&lt;Normal&lt;F&gt;, Error&gt; {
         <span class="kw">if </span>!cv.is_finite() || cv &lt; F::zero() {
             <span class="kw">return </span><span class="prelude-val">Err</span>(Error::BadVariance);
         }
         <span class="kw">let </span>std_dev = cv * mean;
         <span class="prelude-val">Ok</span>(Normal { mean, std_dev })
     }

     <span class="doccomment">/// Sample from a z-score
     ///
     /// This may be useful for generating correlated samples `x1` and `x2`
     /// from two different distributions, as follows.
     /// ```
     /// # use rand::prelude::*;
     /// # use rand_distr::{Normal, StandardNormal};
     /// let mut rng = thread_rng();
     /// let z = StandardNormal.sample(&amp;mut rng);
     /// let x1 = Normal::new(0.0, 1.0).unwrap().from_zscore(z);
     /// let x2 = Normal::new(2.0, -3.0).unwrap().from_zscore(z);
     /// ```
     </span><span class="attribute">#[inline]
     </span><span class="kw">pub fn </span>from_zscore(<span class="kw-2">&amp;</span><span class="self">self</span>, zscore: F) -&gt; F {
         <span class="self">self</span>.mean + <span class="self">self</span>.std_dev * zscore
     }

     <span class="doccomment">/// Returns the mean (`μ`) of the distribution.
     </span><span class="kw">pub fn </span>mean(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; F {
         <span class="self">self</span>.mean
     }

     <span class="doccomment">/// Returns the standard deviation (`σ`) of the distribution.
     </span><span class="kw">pub fn </span>std_dev(<span class="kw-2">&amp;</span><span class="self">self</span>) -&gt; F {
         <span class="self">self</span>.std_dev
     }
 }

 <span class="kw">impl</span>&lt;F&gt; Distribution&lt;F&gt; <span class="kw">for </span>Normal&lt;F&gt;
 <span class="kw">where </span>F: Float, StandardNormal: Distribution&lt;F&gt;
 {
     <span class="kw">fn </span>sample&lt;R: Rng + <span class="question-mark">?</span>Sized&gt;(<span class="kw-2">&amp;</span><span class="self">self</span>, rng: <span class="kw-2">&amp;mut </span>R) -&gt; F {
         <span class="self">self</span>.from_zscore(rng.sample(StandardNormal))
     }
 }


 <span class="doccomment">/// The log-normal distribution `ln N(mean, std_dev**2)`.
 ///
 /// If `X` is log-normal distributed, then `ln(X)` is `N(mean, std_dev**2)`
 /// distributed.
 ///
 /// # Example
 ///
 /// ```
 /// use rand_distr::{LogNormal, Distribution};
 ///
 /// // mean 2, standard deviation 3
 /// let log_normal = LogNormal::new(2.0, 3.0).unwrap();
 /// let v = log_normal.sample(&amp;mut rand::thread_rng());
 /// println!(&quot;{} is from an ln N(2, 9) distribution&quot;, v)
 /// ```
 </span><span class="attribute">#[derive(Clone, Copy, Debug)]
 #[cfg_attr(feature = <span class="string">&quot;serde1&quot;</span>, derive(serde::Serialize, serde::Deserialize))]
 </span><span class="kw">pub struct </span>LogNormal&lt;F&gt;
 <span class="kw">where </span>F: Float, StandardNormal: Distribution&lt;F&gt;
 {
     norm: Normal&lt;F&gt;,
 }

 <span class="kw">impl</span>&lt;F&gt; LogNormal&lt;F&gt;
 <span class="kw">where </span>F: Float, StandardNormal: Distribution&lt;F&gt;
 {
     <span class="doccomment">/// Construct, from (log-space) mean and standard deviation
     ///
     /// Parameters are the &quot;standard&quot; log-space measures (these are the mean
     /// and standard deviation of the logarithm of samples):
     ///
     /// -   `mu` (`μ`, unrestricted) is the mean of the underlying distribution
     /// -   `sigma` (`σ`, must be finite) is the standard deviation of the
     ///     underlying Normal distribution
     </span><span class="attribute">#[inline]
     </span><span class="kw">pub fn </span>new(mu: F, sigma: F) -&gt; <span class="prelude-ty">Result</span>&lt;LogNormal&lt;F&gt;, Error&gt; {
         <span class="kw">let </span>norm = Normal::new(mu, sigma)<span class="question-mark">?</span>;
         <span class="prelude-val">Ok</span>(LogNormal { norm })
     }

     <span class="doccomment">/// Construct, from (linear-space) mean and coefficient of variation
     ///
     /// Parameters are linear-space measures:
     ///
     /// -   mean (`μ &gt; 0`) is the (real) mean of the distribution
     /// -   coefficient of variation (`cv = σ / μ`, requiring `cv ≥ 0`) is a
     ///     standardized measure of dispersion
     ///
     /// As a special exception, `μ = 0, cv = 0` is allowed (samples are `-inf`).
     </span><span class="attribute">#[inline]
     </span><span class="kw">pub fn </span>from_mean_cv(mean: F, cv: F) -&gt; <span class="prelude-ty">Result</span>&lt;LogNormal&lt;F&gt;, Error&gt; {
         <span class="kw">if </span>cv == F::zero() {
             <span class="kw">let </span>mu = mean.ln();
             <span class="kw">let </span>norm = Normal::new(mu, F::zero()).unwrap();
             <span class="kw">return </span><span class="prelude-val">Ok</span>(LogNormal { norm });
         }
         <span class="kw">if </span>!(mean &gt; F::zero()) {
             <span class="kw">return </span><span class="prelude-val">Err</span>(Error::MeanTooSmall);
         }
         <span class="kw">if </span>!(cv &gt;= F::zero()) {
             <span class="kw">return </span><span class="prelude-val">Err</span>(Error::BadVariance);
         }

         <span class="comment">// Using X ~ lognormal(μ, σ), CV² = Var(X) / E(X)²
         // E(X) = exp(μ + σ² / 2) = exp(μ) × exp(σ² / 2)
         // Var(X) = exp(2μ + σ²)(exp(σ²) - 1) = E(X)² × (exp(σ²) - 1)
         // but Var(X) = (CV × E(X))² so CV² = exp(σ²) - 1
         // thus σ² = log(CV² + 1)
         // and exp(μ) = E(X) / exp(σ² / 2) = E(X) / sqrt(CV² + 1)
         </span><span class="kw">let </span>a = F::one() + cv * cv; <span class="comment">// e
         </span><span class="kw">let </span>mu = F::from(<span class="number">0.5</span>).unwrap() * (mean * mean / a).ln();
         <span class="kw">let </span>sigma = a.ln().sqrt();
         <span class="kw">let </span>norm = Normal::new(mu, sigma)<span class="question-mark">?</span>;
         <span class="prelude-val">Ok</span>(LogNormal { norm })
     }

     <span class="doccomment">/// Sample from a z-score
     ///
     /// This may be useful for generating correlated samples `x1` and `x2`
     /// from two different distributions, as follows.
     /// ```
     /// # use rand::prelude::*;
     /// # use rand_distr::{LogNormal, StandardNormal};
     /// let mut rng = thread_rng();
     /// let z = StandardNormal.sample(&amp;mut rng);
     /// let x1 = LogNormal::from_mean_cv(3.0, 1.0).unwrap().from_zscore(z);
     /// let x2 = LogNormal::from_mean_cv(2.0, 4.0).unwrap().from_zscore(z);
     /// ```
     </span><span class="attribute">#[inline]
     </span><span class="kw">pub fn </span>from_zscore(<span class="kw-2">&amp;</span><span class="self">self</span>, zscore: F) -&gt; F {
         <span class="self">self</span>.norm.from_zscore(zscore).exp()
     }
 }

 <span class="kw">impl</span>&lt;F&gt; Distribution&lt;F&gt; <span class="kw">for </span>LogNormal&lt;F&gt;
 <span class="kw">where </span>F: Float, StandardNormal: Distribution&lt;F&gt;
 {
     <span class="attribute">#[inline]
     </span><span class="kw">fn </span>sample&lt;R: Rng + <span class="question-mark">?</span>Sized&gt;(<span class="kw-2">&amp;</span><span class="self">self</span>, rng: <span class="kw-2">&amp;mut </span>R) -&gt; F {
         <span class="self">self</span>.norm.sample(rng).exp()
     }
 }

 <span class="attribute">#[cfg(test)]
 </span><span class="kw">mod </span>tests {
     <span class="kw">use super</span>::<span class="kw-2">*</span>;

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>test_normal() {
         <span class="kw">let </span>norm = Normal::new(<span class="number">10.0</span>, <span class="number">10.0</span>).unwrap();
         <span class="kw">let </span><span class="kw-2">mut </span>rng = <span class="kw">crate</span>::test::rng(<span class="number">210</span>);
         <span class="kw">for _ in </span><span class="number">0</span>..<span class="number">1000 </span>{
             norm.sample(<span class="kw-2">&amp;mut </span>rng);
         }
     }
     <span class="attribute">#[test]
     </span><span class="kw">fn </span>test_normal_cv() {
         <span class="kw">let </span>norm = Normal::from_mean_cv(<span class="number">1024.0</span>, <span class="number">1.0 </span>/ <span class="number">256.0</span>).unwrap();
         <span class="macro">assert_eq!</span>((norm.mean, norm.std_dev), (<span class="number">1024.0</span>, <span class="number">4.0</span>));
     }
     <span class="attribute">#[test]
     </span><span class="kw">fn </span>test_normal_invalid_sd() {
         <span class="macro">assert!</span>(Normal::from_mean_cv(<span class="number">10.0</span>, -<span class="number">1.0</span>).is_err());
     }

     <span class="attribute">#[test]
     </span><span class="kw">fn </span>test_log_normal() {
         <span class="kw">let </span>lnorm = LogNormal::new(<span class="number">10.0</span>, <span class="number">10.0</span>).unwrap();
         <span class="kw">let </span><span class="kw-2">mut </span>rng = <span class="kw">crate</span>::test::rng(<span class="number">211</span>);
         <span class="kw">for _ in </span><span class="number">0</span>..<span class="number">1000 </span>{
             lnorm.sample(<span class="kw-2">&amp;mut </span>rng);
         }
     }
     <span class="attribute">#[test]
     </span><span class="kw">fn </span>test_log_normal_cv() {
         <span class="kw">let </span>lnorm = LogNormal::from_mean_cv(<span class="number">0.0</span>, <span class="number">0.0</span>).unwrap();
         <span class="macro">assert_eq!</span>((lnorm.norm.mean, lnorm.norm.std_dev), (-core::f64::INFINITY, <span class="number">0.0</span>));

         <span class="kw">let </span>lnorm = LogNormal::from_mean_cv(<span class="number">1.0</span>, <span class="number">0.0</span>).unwrap();
         <span class="macro">assert_eq!</span>((lnorm.norm.mean, lnorm.norm.std_dev), (<span class="number">0.0</span>, <span class="number">0.0</span>));

         <span class="kw">let </span>e = core::f64::consts::E;
         <span class="kw">let </span>lnorm = LogNormal::from_mean_cv(e.sqrt(), (e - <span class="number">1.0</span>).sqrt()).unwrap();
         <span class="macro">assert_almost_eq!</span>(lnorm.norm.mean, <span class="number">0.0</span>, <span class="number">2e-16</span>);
         <span class="macro">assert_almost_eq!</span>(lnorm.norm.std_dev, <span class="number">1.0</span>, <span class="number">2e-16</span>);

         <span class="kw">let </span>lnorm = LogNormal::from_mean_cv(e.powf(<span class="number">1.5</span>), (e - <span class="number">1.0</span>).sqrt()).unwrap();
         <span class="macro">assert_almost_eq!</span>(lnorm.norm.mean, <span class="number">1.0</span>, <span class="number">1e-15</span>);
         <span class="macro">assert_eq!</span>(lnorm.norm.std_dev, <span class="number">1.0</span>);
     }
     <span class="attribute">#[test]
     </span><span class="kw">fn </span>test_log_normal_invalid_sd() {
         <span class="macro">assert!</span>(LogNormal::from_mean_cv(-<span class="number">1.0</span>, <span class="number">1.0</span>).is_err());
         <span class="macro">assert!</span>(LogNormal::from_mean_cv(<span class="number">0.0</span>, <span class="number">1.0</span>).is_err());
         <span class="macro">assert!</span>(LogNormal::from_mean_cv(<span class="number">1.0</span>, -<span class="number">1.0</span>).is_err());
     }
 }
 </code></pre></div>
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