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<!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/jn.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>jn.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">/* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
/*
* jn(n, x), yn(n, x)
* floating point Bessel&#39;s function of the 1st and 2nd kind
* of order n
*
* Special cases:
* y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
* y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
* Note 2. About jn(n,x), yn(n,x)
* For n=0, j0(x) is called,
* for n=1, j1(x) is called,
* for n&lt;=x, forward recursion is used starting
* from values of j0(x) and j1(x).
* for n&gt;x, a continued fraction approximation to
* j(n,x)/j(n-1,x) is evaluated and then backward
* recursion is used starting from a supposed value
* for j(n,x). The resulting value of j(0,x) is
* compared with the actual value to correct the
* supposed value of j(n,x).
*
* yn(n,x) is similar in all respects, except
* that forward recursion is used for all
* values of n&gt;1.
*/
</span><span class="kw">use super</span>::{cos, fabs, get_high_word, get_low_word, j0, j1, log, sin, sqrt, y0, y1};
<span class="kw">const </span>INVSQRTPI: f64 = <span class="number">5.64189583547756279280e-01</span>; <span class="comment">/* 0x3FE20DD7, 0x50429B6D */
</span><span class="kw">pub fn </span>jn(n: i32, <span class="kw-2">mut </span>x: f64) -&gt; f64 {
<span class="kw">let </span><span class="kw-2">mut </span>ix: u32;
<span class="kw">let </span>lx: u32;
<span class="kw">let </span>nm1: i32;
<span class="kw">let </span><span class="kw-2">mut </span>i: i32;
<span class="kw">let </span><span class="kw-2">mut </span>sign: bool;
<span class="kw">let </span><span class="kw-2">mut </span>a: f64;
<span class="kw">let </span><span class="kw-2">mut </span>b: f64;
<span class="kw">let </span><span class="kw-2">mut </span>temp: f64;
ix = get_high_word(x);
lx = get_low_word(x);
sign = (ix &gt;&gt; <span class="number">31</span>) != <span class="number">0</span>;
ix &amp;= <span class="number">0x7fffffff</span>;
<span class="comment">// -lx == !lx + 1
</span><span class="kw">if </span>(ix | (lx | ((!lx).wrapping_add(<span class="number">1</span>))) &gt;&gt; <span class="number">31</span>) &gt; <span class="number">0x7ff00000 </span>{
<span class="comment">/* nan */
</span><span class="kw">return </span>x;
}
<span class="comment">/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
* Thus, J(-n,x) = J(n,-x)
*/
/* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */
</span><span class="kw">if </span>n == <span class="number">0 </span>{
<span class="kw">return </span>j0(x);
}
<span class="kw">if </span>n &lt; <span class="number">0 </span>{
nm1 = -(n + <span class="number">1</span>);
x = -x;
sign = !sign;
} <span class="kw">else </span>{
nm1 = n - <span class="number">1</span>;
}
<span class="kw">if </span>nm1 == <span class="number">0 </span>{
<span class="kw">return </span>j1(x);
}
sign &amp;= (n &amp; <span class="number">1</span>) != <span class="number">0</span>; <span class="comment">/* even n: 0, odd n: signbit(x) */
</span>x = fabs(x);
<span class="kw">if </span>(ix | lx) == <span class="number">0 </span>|| ix == <span class="number">0x7ff00000 </span>{
<span class="comment">/* if x is 0 or inf */
</span>b = <span class="number">0.0</span>;
} <span class="kw">else if </span>(nm1 <span class="kw">as </span>f64) &lt; x {
<span class="comment">/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
</span><span class="kw">if </span>ix &gt;= <span class="number">0x52d00000 </span>{
<span class="comment">/* x &gt; 2**302 */
/* (x &gt;&gt; n**2)
* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* Let s=sin(x), c=cos(x),
* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
*
* n sin(xn)*sqt2 cos(xn)*sqt2
* ----------------------------------
* 0 s-c c+s
* 1 -s-c -c+s
* 2 -s+c -c-s
* 3 s+c c-s
*/
</span>temp = <span class="kw">match </span>nm1 &amp; <span class="number">3 </span>{
<span class="number">0 </span>=&gt; -cos(x) + sin(x),
<span class="number">1 </span>=&gt; -cos(x) - sin(x),
<span class="number">2 </span>=&gt; cos(x) - sin(x),
<span class="number">3 </span>| <span class="kw">_ </span>=&gt; cos(x) + sin(x),
};
b = INVSQRTPI * temp / sqrt(x);
} <span class="kw">else </span>{
a = j0(x);
b = j1(x);
i = <span class="number">0</span>;
<span class="kw">while </span>i &lt; nm1 {
i += <span class="number">1</span>;
temp = b;
b = b * (<span class="number">2.0 </span>* (i <span class="kw">as </span>f64) / x) - a; <span class="comment">/* avoid underflow */
</span>a = temp;
}
}
} <span class="kw">else </span>{
<span class="kw">if </span>ix &lt; <span class="number">0x3e100000 </span>{
<span class="comment">/* x &lt; 2**-29 */
/* x is tiny, return the first Taylor expansion of J(n,x)
* J(n,x) = 1/n!*(x/2)^n - ...
*/
</span><span class="kw">if </span>nm1 &gt; <span class="number">32 </span>{
<span class="comment">/* underflow */
</span>b = <span class="number">0.0</span>;
} <span class="kw">else </span>{
temp = x * <span class="number">0.5</span>;
b = temp;
a = <span class="number">1.0</span>;
i = <span class="number">2</span>;
<span class="kw">while </span>i &lt;= nm1 + <span class="number">1 </span>{
a <span class="kw-2">*</span>= i <span class="kw">as </span>f64; <span class="comment">/* a = n! */
</span>b <span class="kw-2">*</span>= temp; <span class="comment">/* b = (x/2)^n */
</span>i += <span class="number">1</span>;
}
b = b / a;
}
} <span class="kw">else </span>{
<span class="comment">/* use backward recurrence */
/* x x^2 x^2
* J(n,x)/J(n-1,x) = ---- ------ ------ .....
* 2n - 2(n+1) - 2(n+2)
*
* 1 1 1
* (for large x) = ---- ------ ------ .....
* 2n 2(n+1) 2(n+2)
* -- - ------ - ------ -
* x x x
*
* Let w = 2n/x and h=2/x, then the above quotient
* is equal to the continued fraction:
* 1
* = -----------------------
* 1
* w - -----------------
* 1
* w+h - ---------
* w+2h - ...
*
* To determine how many terms needed, let
* Q(0) = w, Q(1) = w(w+h) - 1,
* Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
* When Q(k) &gt; 1e4 good for single
* When Q(k) &gt; 1e9 good for double
* When Q(k) &gt; 1e17 good for quadruple
*/
/* determine k */
</span><span class="kw">let </span><span class="kw-2">mut </span>t: f64;
<span class="kw">let </span><span class="kw-2">mut </span>q0: f64;
<span class="kw">let </span><span class="kw-2">mut </span>q1: f64;
<span class="kw">let </span><span class="kw-2">mut </span>w: f64;
<span class="kw">let </span>h: 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>tmp: f64;
<span class="kw">let </span>nf: f64;
<span class="kw">let </span><span class="kw-2">mut </span>k: i32;
nf = (nm1 <span class="kw">as </span>f64) + <span class="number">1.0</span>;
w = <span class="number">2.0 </span>* nf / x;
h = <span class="number">2.0 </span>/ x;
z = w + h;
q0 = w;
q1 = w * z - <span class="number">1.0</span>;
k = <span class="number">1</span>;
<span class="kw">while </span>q1 &lt; <span class="number">1.0e9 </span>{
k += <span class="number">1</span>;
z += h;
tmp = z * q1 - q0;
q0 = q1;
q1 = tmp;
}
t = <span class="number">0.0</span>;
i = k;
<span class="kw">while </span>i &gt;= <span class="number">0 </span>{
t = <span class="number">1.0 </span>/ (<span class="number">2.0 </span>* ((i <span class="kw">as </span>f64) + nf) / x - t);
i -= <span class="number">1</span>;
}
a = t;
b = <span class="number">1.0</span>;
<span class="comment">/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
* Hence, if n*(log(2n/x)) &gt; ...
* single 8.8722839355e+01
* double 7.09782712893383973096e+02
* long double 1.1356523406294143949491931077970765006170e+04
* then recurrent value may overflow and the result is
* likely underflow to zero
*/
</span>tmp = nf * log(fabs(w));
<span class="kw">if </span>tmp &lt; <span class="number">7.09782712893383973096e+02 </span>{
i = nm1;
<span class="kw">while </span>i &gt; <span class="number">0 </span>{
temp = b;
b = b * (<span class="number">2.0 </span>* (i <span class="kw">as </span>f64)) / x - a;
a = temp;
i -= <span class="number">1</span>;
}
} <span class="kw">else </span>{
i = nm1;
<span class="kw">while </span>i &gt; <span class="number">0 </span>{
temp = b;
b = b * (<span class="number">2.0 </span>* (i <span class="kw">as </span>f64)) / x - a;
a = temp;
<span class="comment">/* scale b to avoid spurious overflow */
</span><span class="kw">let </span>x1p500 = f64::from_bits(<span class="number">0x5f30000000000000</span>); <span class="comment">// 0x1p500 == 2^500
</span><span class="kw">if </span>b &gt; x1p500 {
a /= b;
t /= b;
b = <span class="number">1.0</span>;
}
i -= <span class="number">1</span>;
}
}
z = j0(x);
w = j1(x);
<span class="kw">if </span>fabs(z) &gt;= fabs(w) {
b = t * z / b;
} <span class="kw">else </span>{
b = t * w / a;
}
}
}
<span class="kw">if </span>sign {
-b
} <span class="kw">else </span>{
b
}
}
<span class="kw">pub fn </span>yn(n: i32, x: f64) -&gt; f64 {
<span class="kw">let </span><span class="kw-2">mut </span>ix: u32;
<span class="kw">let </span>lx: u32;
<span class="kw">let </span><span class="kw-2">mut </span>ib: u32;
<span class="kw">let </span>nm1: i32;
<span class="kw">let </span><span class="kw-2">mut </span>sign: bool;
<span class="kw">let </span><span class="kw-2">mut </span>i: i32;
<span class="kw">let </span><span class="kw-2">mut </span>a: f64;
<span class="kw">let </span><span class="kw-2">mut </span>b: f64;
<span class="kw">let </span><span class="kw-2">mut </span>temp: f64;
ix = get_high_word(x);
lx = get_low_word(x);
sign = (ix &gt;&gt; <span class="number">31</span>) != <span class="number">0</span>;
ix &amp;= <span class="number">0x7fffffff</span>;
<span class="comment">// -lx == !lx + 1
</span><span class="kw">if </span>(ix | (lx | ((!lx).wrapping_add(<span class="number">1</span>))) &gt;&gt; <span class="number">31</span>) &gt; <span class="number">0x7ff00000 </span>{
<span class="comment">/* nan */
</span><span class="kw">return </span>x;
}
<span class="kw">if </span>sign &amp;&amp; (ix | lx) != <span class="number">0 </span>{
<span class="comment">/* x &lt; 0 */
</span><span class="kw">return </span><span class="number">0.0 </span>/ <span class="number">0.0</span>;
}
<span class="kw">if </span>ix == <span class="number">0x7ff00000 </span>{
<span class="kw">return </span><span class="number">0.0</span>;
}
<span class="kw">if </span>n == <span class="number">0 </span>{
<span class="kw">return </span>y0(x);
}
<span class="kw">if </span>n &lt; <span class="number">0 </span>{
nm1 = -(n + <span class="number">1</span>);
sign = (n &amp; <span class="number">1</span>) != <span class="number">0</span>;
} <span class="kw">else </span>{
nm1 = n - <span class="number">1</span>;
sign = <span class="bool-val">false</span>;
}
<span class="kw">if </span>nm1 == <span class="number">0 </span>{
<span class="kw">if </span>sign {
<span class="kw">return </span>-y1(x);
} <span class="kw">else </span>{
<span class="kw">return </span>y1(x);
}
}
<span class="kw">if </span>ix &gt;= <span class="number">0x52d00000 </span>{
<span class="comment">/* x &gt; 2**302 */
/* (x &gt;&gt; n**2)
* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* Let s=sin(x), c=cos(x),
* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
*
* n sin(xn)*sqt2 cos(xn)*sqt2
* ----------------------------------
* 0 s-c c+s
* 1 -s-c -c+s
* 2 -s+c -c-s
* 3 s+c c-s
*/
</span>temp = <span class="kw">match </span>nm1 &amp; <span class="number">3 </span>{
<span class="number">0 </span>=&gt; -sin(x) - cos(x),
<span class="number">1 </span>=&gt; -sin(x) + cos(x),
<span class="number">2 </span>=&gt; sin(x) + cos(x),
<span class="number">3 </span>| <span class="kw">_ </span>=&gt; sin(x) - cos(x),
};
b = INVSQRTPI * temp / sqrt(x);
} <span class="kw">else </span>{
a = y0(x);
b = y1(x);
<span class="comment">/* quit if b is -inf */
</span>ib = get_high_word(b);
i = <span class="number">0</span>;
<span class="kw">while </span>i &lt; nm1 &amp;&amp; ib != <span class="number">0xfff00000 </span>{
i += <span class="number">1</span>;
temp = b;
b = (<span class="number">2.0 </span>* (i <span class="kw">as </span>f64) / x) * b - a;
ib = get_high_word(b);
a = temp;
}
}
<span class="kw">if </span>sign {
-b
} <span class="kw">else </span>{
b
}
}
</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>