| <!DOCTYPE html><html lang="en"><head><meta charset="utf-8"><meta name="viewport" content="width=device-width, initial-scale=1.0"><meta name="generator" content="rustdoc"><meta name="description" content="Source of the Rust file `/root/.cargo/registry/src/github.com-1ecc6299db9ec823/libm-0.2.7/src/math/tgamma.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>tgamma.rs - source</title><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../SourceSerif4-Regular.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../FiraSans-Regular.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../FiraSans-Medium.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../SourceCodePro-Regular.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../SourceSerif4-Bold.ttf.woff2"><link rel="preload" as="font" type="font/woff2" crossorigin href="../../../SourceCodePro-Semibold.ttf.woff2"><link rel="stylesheet" href="../../../normalize.css"><link rel="stylesheet" href="../../../rustdoc.css" id="mainThemeStyle"><link rel="stylesheet" href="../../../ayu.css" disabled><link rel="stylesheet" href="../../../dark.css" disabled><link rel="stylesheet" href="../../../light.css" id="themeStyle"><script id="default-settings" ></script><script src="../../../storage.js"></script><script defer src="../../../source-script.js"></script><script defer src="../../../source-files.js"></script><script defer src="../../../main.js"></script><noscript><link rel="stylesheet" href="../../../noscript.css"></noscript><link rel="alternate icon" type="image/png" href="../../../favicon-16x16.png"><link rel="alternate icon" type="image/png" href="../../../favicon-32x32.png"><link rel="icon" type="image/svg+xml" href="../../../favicon.svg"></head><body class="rustdoc source"><!--[if lte IE 11]><div class="warning">This old browser is unsupported and will most likely display funky things.</div><![endif]--><nav class="sidebar"><a class="sidebar-logo" href="../../../libm/index.html"><div class="logo-container"><img class="rust-logo" src="../../../rust-logo.svg" alt="logo"></div></a></nav><main><div class="width-limiter"><nav class="sub"><a class="sub-logo-container" href="../../../libm/index.html"><img class="rust-logo" src="../../../rust-logo.svg" alt="logo"></a><form class="search-form"><div class="search-container"><span></span><input class="search-input" name="search" autocomplete="off" spellcheck="false" placeholder="Click or press ‘S’ to search, ‘?’ for more options…" type="search"><div id="help-button" title="help" tabindex="-1"><a href="../../../help.html">?</a></div><div id="settings-menu" tabindex="-1"><a href="../../../settings.html" title="settings"><img width="22" height="22" alt="Change settings" src="../../../wheel.svg"></a></div></div></form></nav><section id="main-content" class="content"><div class="example-wrap"><pre class="src-line-numbers"><span id="1">1</span> |
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| </pre><pre class="rust"><code><span class="comment">/* |
| "A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964) |
| "Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001) |
| "An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004) |
| |
| approximation method: |
| |
| (x - 0.5) S(x) |
| Gamma(x) = (x + g - 0.5) * ---------------- |
| exp(x + g - 0.5) |
| |
| with |
| a1 a2 a3 aN |
| S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ] |
| x + 1 x + 2 x + 3 x + N |
| |
| with a0, a1, a2, a3,.. aN constants which depend on g. |
| |
| for x < 0 the following reflection formula is used: |
| |
| Gamma(x)*Gamma(-x) = -pi/(x sin(pi x)) |
| |
| most ideas and constants are from boost and python |
| */ |
| </span><span class="kw">extern crate </span>core; |
| <span class="kw">use super</span>::{exp, floor, k_cos, k_sin, pow}; |
| |
| <span class="kw">const </span>PI: f64 = <span class="number">3.141592653589793238462643383279502884</span>; |
| |
| <span class="comment">/* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */ |
| </span><span class="kw">fn </span>sinpi(<span class="kw-2">mut </span>x: f64) -> f64 { |
| <span class="kw">let </span><span class="kw-2">mut </span>n: isize; |
| |
| <span class="comment">/* argument reduction: x = |x| mod 2 */ |
| /* spurious inexact when x is odd int */ |
| </span>x = x * <span class="number">0.5</span>; |
| x = <span class="number">2.0 </span>* (x - floor(x)); |
| |
| <span class="comment">/* reduce x into [-.25,.25] */ |
| </span>n = (<span class="number">4.0 </span>* x) <span class="kw">as </span>isize; |
| n = <span class="macro">div!</span>(n + <span class="number">1</span>, <span class="number">2</span>); |
| x -= (n <span class="kw">as </span>f64) * <span class="number">0.5</span>; |
| |
| x <span class="kw-2">*</span>= PI; |
| <span class="kw">match </span>n { |
| <span class="number">1 </span>=> k_cos(x, <span class="number">0.0</span>), |
| <span class="number">2 </span>=> k_sin(-x, <span class="number">0.0</span>, <span class="number">0</span>), |
| <span class="number">3 </span>=> -k_cos(x, <span class="number">0.0</span>), |
| <span class="number">0 </span>| <span class="kw">_ </span>=> k_sin(x, <span class="number">0.0</span>, <span class="number">0</span>), |
| } |
| } |
| |
| <span class="kw">const </span>N: usize = <span class="number">12</span>; |
| <span class="comment">//static const double g = 6.024680040776729583740234375; |
| </span><span class="kw">const </span>GMHALF: f64 = <span class="number">5.524680040776729583740234375</span>; |
| <span class="kw">const </span>SNUM: [f64; N + <span class="number">1</span>] = [ |
| <span class="number">23531376880.410759688572007674451636754734846804940</span>, |
| <span class="number">42919803642.649098768957899047001988850926355848959</span>, |
| <span class="number">35711959237.355668049440185451547166705960488635843</span>, |
| <span class="number">17921034426.037209699919755754458931112671403265390</span>, |
| <span class="number">6039542586.3520280050642916443072979210699388420708</span>, |
| <span class="number">1439720407.3117216736632230727949123939715485786772</span>, |
| <span class="number">248874557.86205415651146038641322942321632125127801</span>, |
| <span class="number">31426415.585400194380614231628318205362874684987640</span>, |
| <span class="number">2876370.6289353724412254090516208496135991145378768</span>, |
| <span class="number">186056.26539522349504029498971604569928220784236328</span>, |
| <span class="number">8071.6720023658162106380029022722506138218516325024</span>, |
| <span class="number">210.82427775157934587250973392071336271166969580291</span>, |
| <span class="number">2.5066282746310002701649081771338373386264310793408</span>, |
| ]; |
| <span class="kw">const </span>SDEN: [f64; N + <span class="number">1</span>] = [ |
| <span class="number">0.0</span>, |
| <span class="number">39916800.0</span>, |
| <span class="number">120543840.0</span>, |
| <span class="number">150917976.0</span>, |
| <span class="number">105258076.0</span>, |
| <span class="number">45995730.0</span>, |
| <span class="number">13339535.0</span>, |
| <span class="number">2637558.0</span>, |
| <span class="number">357423.0</span>, |
| <span class="number">32670.0</span>, |
| <span class="number">1925.0</span>, |
| <span class="number">66.0</span>, |
| <span class="number">1.0</span>, |
| ]; |
| <span class="comment">/* n! for small integer n */ |
| </span><span class="kw">const </span>FACT: [f64; <span class="number">23</span>] = [ |
| <span class="number">1.0</span>, |
| <span class="number">1.0</span>, |
| <span class="number">2.0</span>, |
| <span class="number">6.0</span>, |
| <span class="number">24.0</span>, |
| <span class="number">120.0</span>, |
| <span class="number">720.0</span>, |
| <span class="number">5040.0</span>, |
| <span class="number">40320.0</span>, |
| <span class="number">362880.0</span>, |
| <span class="number">3628800.0</span>, |
| <span class="number">39916800.0</span>, |
| <span class="number">479001600.0</span>, |
| <span class="number">6227020800.0</span>, |
| <span class="number">87178291200.0</span>, |
| <span class="number">1307674368000.0</span>, |
| <span class="number">20922789888000.0</span>, |
| <span class="number">355687428096000.0</span>, |
| <span class="number">6402373705728000.0</span>, |
| <span class="number">121645100408832000.0</span>, |
| <span class="number">2432902008176640000.0</span>, |
| <span class="number">51090942171709440000.0</span>, |
| <span class="number">1124000727777607680000.0</span>, |
| ]; |
| |
| <span class="comment">/* S(x) rational function for positive x */ |
| </span><span class="kw">fn </span>s(x: f64) -> f64 { |
| <span class="kw">let </span><span class="kw-2">mut </span>num: f64 = <span class="number">0.0</span>; |
| <span class="kw">let </span><span class="kw-2">mut </span>den: f64 = <span class="number">0.0</span>; |
| |
| <span class="comment">/* to avoid overflow handle large x differently */ |
| </span><span class="kw">if </span>x < <span class="number">8.0 </span>{ |
| <span class="kw">for </span>i <span class="kw">in </span>(<span class="number">0</span>..=N).rev() { |
| num = num * x + <span class="macro">i!</span>(SNUM, i); |
| den = den * x + <span class="macro">i!</span>(SDEN, i); |
| } |
| } <span class="kw">else </span>{ |
| <span class="kw">for </span>i <span class="kw">in </span><span class="number">0</span>..=N { |
| num = num / x + <span class="macro">i!</span>(SNUM, i); |
| den = den / x + <span class="macro">i!</span>(SDEN, i); |
| } |
| } |
| <span class="kw">return </span>num / den; |
| } |
| |
| <span class="attribute">#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)] |
| </span><span class="kw">pub fn </span>tgamma(<span class="kw-2">mut </span>x: f64) -> f64 { |
| <span class="kw">let </span>u: u64 = x.to_bits(); |
| <span class="kw">let </span>absx: f64; |
| <span class="kw">let </span><span class="kw-2">mut </span>y: f64; |
| <span class="kw">let </span><span class="kw-2">mut </span>dy: f64; |
| <span class="kw">let </span><span class="kw-2">mut </span>z: f64; |
| <span class="kw">let </span><span class="kw-2">mut </span>r: f64; |
| <span class="kw">let </span>ix: u32 = ((u >> <span class="number">32</span>) <span class="kw">as </span>u32) & <span class="number">0x7fffffff</span>; |
| <span class="kw">let </span>sign: bool = (u >> <span class="number">63</span>) != <span class="number">0</span>; |
| |
| <span class="comment">/* special cases */ |
| </span><span class="kw">if </span>ix >= <span class="number">0x7ff00000 </span>{ |
| <span class="comment">/* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */ |
| </span><span class="kw">return </span>x + core::f64::INFINITY; |
| } |
| <span class="kw">if </span>ix < ((<span class="number">0x3ff </span>- <span class="number">54</span>) << <span class="number">20</span>) { |
| <span class="comment">/* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */ |
| </span><span class="kw">return </span><span class="number">1.0 </span>/ x; |
| } |
| |
| <span class="comment">/* integer arguments */ |
| /* raise inexact when non-integer */ |
| </span><span class="kw">if </span>x == floor(x) { |
| <span class="kw">if </span>sign { |
| <span class="kw">return </span><span class="number">0.0 </span>/ <span class="number">0.0</span>; |
| } |
| <span class="kw">if </span>x <= FACT.len() <span class="kw">as </span>f64 { |
| <span class="kw">return </span><span class="macro">i!</span>(FACT, (x <span class="kw">as </span>usize) - <span class="number">1</span>); |
| } |
| } |
| |
| <span class="comment">/* x >= 172: tgamma(x)=inf with overflow */ |
| /* x =< -184: tgamma(x)=+-0 with underflow */ |
| </span><span class="kw">if </span>ix >= <span class="number">0x40670000 </span>{ |
| <span class="comment">/* |x| >= 184 */ |
| </span><span class="kw">if </span>sign { |
| <span class="kw">let </span>x1p_126 = f64::from_bits(<span class="number">0x3810000000000000</span>); <span class="comment">// 0x1p-126 == 2^-126 |
| </span><span class="macro">force_eval!</span>((x1p_126 / x) <span class="kw">as </span>f32); |
| <span class="kw">if </span>floor(x) * <span class="number">0.5 </span>== floor(x * <span class="number">0.5</span>) { |
| <span class="kw">return </span><span class="number">0.0</span>; |
| } <span class="kw">else </span>{ |
| <span class="kw">return </span>-<span class="number">0.0</span>; |
| } |
| } |
| <span class="kw">let </span>x1p1023 = f64::from_bits(<span class="number">0x7fe0000000000000</span>); <span class="comment">// 0x1p1023 == 2^1023 |
| </span>x <span class="kw-2">*</span>= x1p1023; |
| <span class="kw">return </span>x; |
| } |
| |
| absx = <span class="kw">if </span>sign { -x } <span class="kw">else </span>{ x }; |
| |
| <span class="comment">/* handle the error of x + g - 0.5 */ |
| </span>y = absx + GMHALF; |
| <span class="kw">if </span>absx > GMHALF { |
| dy = y - absx; |
| dy -= GMHALF; |
| } <span class="kw">else </span>{ |
| dy = y - GMHALF; |
| dy -= absx; |
| } |
| |
| z = absx - <span class="number">0.5</span>; |
| r = s(absx) * exp(-y); |
| <span class="kw">if </span>x < <span class="number">0.0 </span>{ |
| <span class="comment">/* reflection formula for negative x */ |
| /* sinpi(absx) is not 0, integers are already handled */ |
| </span>r = -PI / (sinpi(absx) * absx * r); |
| dy = -dy; |
| z = -z; |
| } |
| r += dy * (GMHALF + <span class="number">0.5</span>) * r / y; |
| z = pow(y, <span class="number">0.5 </span>* z); |
| y = r * z * z; |
| <span class="kw">return </span>y; |
| } |
| </code></pre></div> |
| </section></div></main><div id="rustdoc-vars" data-root-path="../../../" data-current-crate="libm" data-themes="ayu,dark,light" data-resource-suffix="" data-rustdoc-version="1.66.0-nightly (5c8bff74b 2022-10-21)" ></div></body></html> |
