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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</span> <span class="kw">crate</span> <span class="ident">core</span>; |
| <span class="kw">use</span> <span class="kw">super</span>::{<span class="ident">exp</span>, <span class="ident">floor</span>, <span class="ident">k_cos</span>, <span class="ident">k_sin</span>, <span class="ident">pow</span>}; |
| |
| <span class="kw">const</span> <span class="ident">PI</span>: <span class="ident">f64</span> <span class="op">=</span> <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> <span class="ident">sinpi</span>(<span class="kw-2">mut</span> <span class="ident">x</span>: <span class="ident">f64</span>) -> <span class="ident">f64</span> { |
| <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">n</span>: <span class="ident">isize</span>; |
| |
| <span class="comment">/* argument reduction: x = |x| mod 2 */</span> |
| <span class="comment">/* spurious inexact when x is odd int */</span> |
| <span class="ident">x</span> <span class="op">=</span> <span class="ident">x</span> <span class="op">*</span> <span class="number">0.5</span>; |
| <span class="ident">x</span> <span class="op">=</span> <span class="number">2.0</span> <span class="op">*</span> (<span class="ident">x</span> <span class="op">-</span> <span class="ident">floor</span>(<span class="ident">x</span>)); |
| |
| <span class="comment">/* reduce x into [-.25,.25] */</span> |
| <span class="ident">n</span> <span class="op">=</span> (<span class="number">4.0</span> <span class="op">*</span> <span class="ident">x</span>) <span class="kw">as</span> <span class="ident">isize</span>; |
| <span class="ident">n</span> <span class="op">=</span> (<span class="ident">n</span> <span class="op">+</span> <span class="number">1</span>) <span class="op">/</span> <span class="number">2</span>; |
| <span class="ident">x</span> <span class="op">-</span><span class="op">=</span> (<span class="ident">n</span> <span class="kw">as</span> <span class="ident">f64</span>) <span class="op">*</span> <span class="number">0.5</span>; |
| |
| <span class="ident">x</span> <span class="kw-2">*</span><span class="op">=</span> <span class="ident">PI</span>; |
| <span class="kw">match</span> <span class="ident">n</span> { |
| <span class="number">1</span> => <span class="ident">k_cos</span>(<span class="ident">x</span>, <span class="number">0.0</span>), |
| <span class="number">2</span> => <span class="ident">k_sin</span>(<span class="op">-</span><span class="ident">x</span>, <span class="number">0.0</span>, <span class="number">0</span>), |
| <span class="number">3</span> => <span class="op">-</span><span class="ident">k_cos</span>(<span class="ident">x</span>, <span class="number">0.0</span>), |
| <span class="number">0</span> <span class="op">|</span> <span class="kw">_</span> => <span class="ident">k_sin</span>(<span class="ident">x</span>, <span class="number">0.0</span>, <span class="number">0</span>), |
| } |
| } |
| |
| <span class="kw">const</span> <span class="ident">N</span>: <span class="ident">usize</span> <span class="op">=</span> <span class="number">12</span>; |
| <span class="comment">//static const double g = 6.024680040776729583740234375;</span> |
| <span class="kw">const</span> <span class="ident">GMHALF</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">5.524680040776729583740234375</span>; |
| <span class="kw">const</span> <span class="ident">SNUM</span>: [<span class="ident">f64</span>; <span class="ident">N</span> <span class="op">+</span> <span class="number">1</span>] <span class="op">=</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> <span class="ident">SDEN</span>: [<span class="ident">f64</span>; <span class="ident">N</span> <span class="op">+</span> <span class="number">1</span>] <span class="op">=</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> <span class="ident">FACT</span>: [<span class="ident">f64</span>; <span class="number">23</span>] <span class="op">=</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> <span class="ident">s</span>(<span class="ident">x</span>: <span class="ident">f64</span>) -> <span class="ident">f64</span> { |
| <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">num</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">0.0</span>; |
| <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">den</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">0.0</span>; |
| |
| <span class="comment">/* to avoid overflow handle large x differently */</span> |
| <span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="number">8.0</span> { |
| <span class="kw">for</span> <span class="ident">i</span> <span class="kw">in</span> (<span class="number">0</span>..<span class="op">=</span><span class="ident">N</span>).<span class="ident">rev</span>() { |
| <span class="ident">num</span> <span class="op">=</span> <span class="ident">num</span> <span class="op">*</span> <span class="ident">x</span> <span class="op">+</span> <span class="ident">SNUM</span>[<span class="ident">i</span>]; |
| <span class="ident">den</span> <span class="op">=</span> <span class="ident">den</span> <span class="op">*</span> <span class="ident">x</span> <span class="op">+</span> <span class="ident">SDEN</span>[<span class="ident">i</span>]; |
| } |
| } <span class="kw">else</span> { |
| <span class="kw">for</span> <span class="ident">i</span> <span class="kw">in</span> <span class="number">0</span>..<span class="op">=</span><span class="ident">N</span> { |
| <span class="ident">num</span> <span class="op">=</span> <span class="ident">num</span> <span class="op">/</span> <span class="ident">x</span> <span class="op">+</span> <span class="ident">SNUM</span>[<span class="ident">i</span>]; |
| <span class="ident">den</span> <span class="op">=</span> <span class="ident">den</span> <span class="op">/</span> <span class="ident">x</span> <span class="op">+</span> <span class="ident">SDEN</span>[<span class="ident">i</span>]; |
| } |
| } |
| <span class="kw">return</span> <span class="ident">num</span> <span class="op">/</span> <span class="ident">den</span>; |
| } |
| |
| <span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">tgamma</span>(<span class="kw-2">mut</span> <span class="ident">x</span>: <span class="ident">f64</span>) -> <span class="ident">f64</span> { |
| <span class="kw">let</span> <span class="ident">u</span>: <span class="ident">u64</span> <span class="op">=</span> <span class="ident">x</span>.<span class="ident">to_bits</span>(); |
| <span class="kw">let</span> <span class="ident">absx</span>: <span class="ident">f64</span>; |
| <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">y</span>: <span class="ident">f64</span>; |
| <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">dy</span>: <span class="ident">f64</span>; |
| <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">z</span>: <span class="ident">f64</span>; |
| <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">r</span>: <span class="ident">f64</span>; |
| <span class="kw">let</span> <span class="ident">ix</span>: <span class="ident">u32</span> <span class="op">=</span> ((<span class="ident">u</span> <span class="op">></span><span class="op">></span> <span class="number">32</span>) <span class="kw">as</span> <span class="ident">u32</span>) <span class="op">&</span> <span class="number">0x7fffffff</span>; |
| <span class="kw">let</span> <span class="ident">sign</span>: <span class="ident">bool</span> <span class="op">=</span> (<span class="ident">u</span> <span class="op">></span><span class="op">></span> <span class="number">63</span>) <span class="op">!</span><span class="op">=</span> <span class="number">0</span>; |
| |
| <span class="comment">/* special cases */</span> |
| <span class="kw">if</span> <span class="ident">ix</span> <span class="op">></span><span class="op">=</span> <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> <span class="ident">x</span> <span class="op">+</span> <span class="ident">core::f64::INFINITY</span>; |
| } |
| <span class="kw">if</span> <span class="ident">ix</span> <span class="op"><</span> ((<span class="number">0x3ff</span> <span class="op">-</span> <span class="number">54</span>) <span class="op"><</span><span class="op"><</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> <span class="op">/</span> <span class="ident">x</span>; |
| } |
| |
| <span class="comment">/* integer arguments */</span> |
| <span class="comment">/* raise inexact when non-integer */</span> |
| <span class="kw">if</span> <span class="ident">x</span> <span class="op">==</span> <span class="ident">floor</span>(<span class="ident">x</span>) { |
| <span class="kw">if</span> <span class="ident">sign</span> { |
| <span class="kw">return</span> <span class="number">0.0</span> <span class="op">/</span> <span class="number">0.0</span>; |
| } |
| <span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span><span class="op">=</span> <span class="ident">FACT</span>.<span class="ident">len</span>() <span class="kw">as</span> <span class="ident">f64</span> { |
| <span class="kw">return</span> <span class="ident">FACT</span>[(<span class="ident">x</span> <span class="kw">as</span> <span class="ident">usize</span>) <span class="op">-</span> <span class="number">1</span>]; |
| } |
| } |
| |
| <span class="comment">/* x >= 172: tgamma(x)=inf with overflow */</span> |
| <span class="comment">/* x =< -184: tgamma(x)=+-0 with underflow */</span> |
| <span class="kw">if</span> <span class="ident">ix</span> <span class="op">></span><span class="op">=</span> <span class="number">0x40670000</span> { |
| <span class="comment">/* |x| >= 184 */</span> |
| <span class="kw">if</span> <span class="ident">sign</span> { |
| <span class="kw">let</span> <span class="ident">x1p_126</span> <span class="op">=</span> <span class="ident">f64::from_bits</span>(<span class="number">0x3810000000000000</span>); <span class="comment">// 0x1p-126 == 2^-126</span> |
| <span class="macro">force_eval!</span>((<span class="ident">x1p_126</span> <span class="op">/</span> <span class="ident">x</span>) <span class="kw">as</span> <span class="ident">f32</span>); |
| <span class="kw">if</span> <span class="ident">floor</span>(<span class="ident">x</span>) <span class="op">*</span> <span class="number">0.5</span> <span class="op">==</span> <span class="ident">floor</span>(<span class="ident">x</span> <span class="op">*</span> <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="op">-</span><span class="number">0.0</span>; |
| } |
| } |
| <span class="kw">let</span> <span class="ident">x1p1023</span> <span class="op">=</span> <span class="ident">f64::from_bits</span>(<span class="number">0x7fe0000000000000</span>); <span class="comment">// 0x1p1023 == 2^1023</span> |
| <span class="ident">x</span> <span class="kw-2">*</span><span class="op">=</span> <span class="ident">x1p1023</span>; |
| <span class="kw">return</span> <span class="ident">x</span>; |
| } |
| |
| <span class="ident">absx</span> <span class="op">=</span> <span class="kw">if</span> <span class="ident">sign</span> { <span class="op">-</span><span class="ident">x</span> } <span class="kw">else</span> { <span class="ident">x</span> }; |
| |
| <span class="comment">/* handle the error of x + g - 0.5 */</span> |
| <span class="ident">y</span> <span class="op">=</span> <span class="ident">absx</span> <span class="op">+</span> <span class="ident">GMHALF</span>; |
| <span class="kw">if</span> <span class="ident">absx</span> <span class="op">></span> <span class="ident">GMHALF</span> { |
| <span class="ident">dy</span> <span class="op">=</span> <span class="ident">y</span> <span class="op">-</span> <span class="ident">absx</span>; |
| <span class="ident">dy</span> <span class="op">-</span><span class="op">=</span> <span class="ident">GMHALF</span>; |
| } <span class="kw">else</span> { |
| <span class="ident">dy</span> <span class="op">=</span> <span class="ident">y</span> <span class="op">-</span> <span class="ident">GMHALF</span>; |
| <span class="ident">dy</span> <span class="op">-</span><span class="op">=</span> <span class="ident">absx</span>; |
| } |
| |
| <span class="ident">z</span> <span class="op">=</span> <span class="ident">absx</span> <span class="op">-</span> <span class="number">0.5</span>; |
| <span class="ident">r</span> <span class="op">=</span> <span class="ident">s</span>(<span class="ident">absx</span>) <span class="op">*</span> <span class="ident">exp</span>(<span class="op">-</span><span class="ident">y</span>); |
| <span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="number">0.0</span> { |
| <span class="comment">/* reflection formula for negative x */</span> |
| <span class="comment">/* sinpi(absx) is not 0, integers are already handled */</span> |
| <span class="ident">r</span> <span class="op">=</span> <span class="op">-</span><span class="ident">PI</span> <span class="op">/</span> (<span class="ident">sinpi</span>(<span class="ident">absx</span>) <span class="op">*</span> <span class="ident">absx</span> <span class="op">*</span> <span class="ident">r</span>); |
| <span class="ident">dy</span> <span class="op">=</span> <span class="op">-</span><span class="ident">dy</span>; |
| <span class="ident">z</span> <span class="op">=</span> <span class="op">-</span><span class="ident">z</span>; |
| } |
| <span class="ident">r</span> <span class="op">+</span><span class="op">=</span> <span class="ident">dy</span> <span class="op">*</span> (<span class="ident">GMHALF</span> <span class="op">+</span> <span class="number">0.5</span>) <span class="op">*</span> <span class="ident">r</span> <span class="op">/</span> <span class="ident">y</span>; |
| <span class="ident">z</span> <span class="op">=</span> <span class="ident">pow</span>(<span class="ident">y</span>, <span class="number">0.5</span> <span class="op">*</span> <span class="ident">z</span>); |
| <span class="ident">y</span> <span class="op">=</span> <span class="ident">r</span> <span class="op">*</span> <span class="ident">z</span> <span class="op">*</span> <span class="ident">z</span>; |
| <span class="kw">return</span> <span class="ident">y</span>; |
| } |
| </code></pre></div> |
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