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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="attr">#[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>
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