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</pre><pre class="rust"><code><span class="comment">/* origin: FreeBSD /usr/src/lib/msun/src/e_exp.c */</span>
<span class="comment">/*
* ====================================================
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/</span>
<span class="comment">/* exp(x)
* Returns the exponential of x.
*
* Method
* 1. Argument reduction:
* Reduce x to an r so that |r| &lt;= 0.5*ln2 ~ 0.34658.
* Given x, find r and integer k such that
*
* x = k*ln2 + r, |r| &lt;= 0.5*ln2.
*
* Here r will be represented as r = hi-lo for better
* accuracy.
*
* 2. Approximation of exp(r) by a special rational function on
* the interval [0,0.34658]:
* Write
* R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
* We use a special Remez algorithm on [0,0.34658] to generate
* a polynomial of degree 5 to approximate R. The maximum error
* of this polynomial approximation is bounded by 2**-59. In
* other words,
* R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
* (where z=r*r, and the values of P1 to P5 are listed below)
* and
* | 5 | -59
* | 2.0+P1*z+...+P5*z - R(z) | &lt;= 2
* | |
* The computation of exp(r) thus becomes
* 2*r
* exp(r) = 1 + ----------
* R(r) - r
* r*c(r)
* = 1 + r + ----------- (for better accuracy)
* 2 - c(r)
* where
* 2 4 10
* c(r) = r - (P1*r + P2*r + ... + P5*r ).
*
* 3. Scale back to obtain exp(x):
* From step 1, we have
* exp(x) = 2^k * exp(r)
*
* Special cases:
* exp(INF) is INF, exp(NaN) is NaN;
* exp(-INF) is 0, and
* for finite argument, only exp(0)=1 is exact.
*
* Accuracy:
* according to an error analysis, the error is always less than
* 1 ulp (unit in the last place).
*
* Misc. info.
* For IEEE double
* if x &gt; 709.782712893383973096 then exp(x) overflows
* if x &lt; -745.133219101941108420 then exp(x) underflows
*/</span>
<span class="kw">use</span> <span class="ident"><span class="kw">super</span>::scalbn</span>;
<span class="kw">const</span> <span class="ident">HALF</span>: [<span class="ident">f64</span>; <span class="number">2</span>] <span class="op">=</span> [<span class="number">0.5</span>, <span class="op">-</span><span class="number">0.5</span>];
<span class="kw">const</span> <span class="ident">LN2HI</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">6.93147180369123816490e-01</span>; <span class="comment">/* 0x3fe62e42, 0xfee00000 */</span>
<span class="kw">const</span> <span class="ident">LN2LO</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">1.90821492927058770002e-10</span>; <span class="comment">/* 0x3dea39ef, 0x35793c76 */</span>
<span class="kw">const</span> <span class="ident">INVLN2</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">1.44269504088896338700e+00</span>; <span class="comment">/* 0x3ff71547, 0x652b82fe */</span>
<span class="kw">const</span> <span class="ident">P1</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">1.66666666666666019037e-01</span>; <span class="comment">/* 0x3FC55555, 0x5555553E */</span>
<span class="kw">const</span> <span class="ident">P2</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="op">-</span><span class="number">2.77777777770155933842e-03</span>; <span class="comment">/* 0xBF66C16C, 0x16BEBD93 */</span>
<span class="kw">const</span> <span class="ident">P3</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">6.61375632143793436117e-05</span>; <span class="comment">/* 0x3F11566A, 0xAF25DE2C */</span>
<span class="kw">const</span> <span class="ident">P4</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="op">-</span><span class="number">1.65339022054652515390e-06</span>; <span class="comment">/* 0xBEBBBD41, 0xC5D26BF1 */</span>
<span class="kw">const</span> <span class="ident">P5</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">4.13813679705723846039e-08</span>; <span class="comment">/* 0x3E663769, 0x72BEA4D0 */</span>
<span class="doccomment">/// Exponential, base *e* (f64)</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Calculate the exponential of `x`, that is, *e* raised to the power `x`</span>
<span class="doccomment">/// (where *e* is the base of the natural system of logarithms, approximately 2.71828).</span>
<span class="attribute">#[<span class="ident">cfg_attr</span>(<span class="ident">all</span>(<span class="ident">test</span>, <span class="ident">assert_no_panic</span>), <span class="ident">no_panic::no_panic</span>)]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">exp</span>(<span class="kw-2">mut</span> <span class="ident">x</span>: <span class="ident">f64</span>) -&gt; <span class="ident">f64</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="kw">let</span> <span class="ident">x1p_149</span> <span class="op">=</span> <span class="ident">f64::from_bits</span>(<span class="number">0x36a0000000000000</span>); <span class="comment">// 0x1p-149 === 2 ^ -149</span>
<span class="kw">let</span> <span class="ident">hi</span>: <span class="ident">f64</span>;
<span class="kw">let</span> <span class="ident">lo</span>: <span class="ident">f64</span>;
<span class="kw">let</span> <span class="ident">c</span>: <span class="ident">f64</span>;
<span class="kw">let</span> <span class="ident">xx</span>: <span class="ident">f64</span>;
<span class="kw">let</span> <span class="ident">y</span>: <span class="ident">f64</span>;
<span class="kw">let</span> <span class="ident">k</span>: <span class="ident">i32</span>;
<span class="kw">let</span> <span class="ident">sign</span>: <span class="ident">i32</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">hx</span>: <span class="ident">u32</span>;
<span class="ident">hx</span> <span class="op">=</span> (<span class="ident">x</span>.<span class="ident">to_bits</span>() <span class="op">&gt;</span><span class="op">&gt;</span> <span class="number">32</span>) <span class="kw">as</span> <span class="ident">u32</span>;
<span class="ident">sign</span> <span class="op">=</span> (<span class="ident">hx</span> <span class="op">&gt;</span><span class="op">&gt;</span> <span class="number">31</span>) <span class="kw">as</span> <span class="ident">i32</span>;
<span class="ident">hx</span> <span class="op">&amp;=</span> <span class="number">0x7fffffff</span>; <span class="comment">/* high word of |x| */</span>
<span class="comment">/* special cases */</span>
<span class="kw">if</span> <span class="ident">hx</span> <span class="op">&gt;</span><span class="op">=</span> <span class="number">0x4086232b</span> {
<span class="comment">/* if |x| &gt;= 708.39... */</span>
<span class="kw">if</span> <span class="ident">x</span>.<span class="ident">is_nan</span>() {
<span class="kw">return</span> <span class="ident">x</span>;
}
<span class="kw">if</span> <span class="ident">x</span> <span class="op">&gt;</span> <span class="number">709.782712893383973096</span> {
<span class="comment">/* overflow if x!=inf */</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="kw">if</span> <span class="ident">x</span> <span class="op">&lt;</span> <span class="op">-</span><span class="number">708.39641853226410622</span> {
<span class="comment">/* underflow if x!=-inf */</span>
<span class="macro">force_eval!</span>((<span class="op">-</span><span class="ident">x1p_149</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">x</span> <span class="op">&lt;</span> <span class="op">-</span><span class="number">745.13321910194110842</span> {
<span class="kw">return</span> <span class="number">0.</span>;
}
}
}
<span class="comment">/* argument reduction */</span>
<span class="kw">if</span> <span class="ident">hx</span> <span class="op">&gt;</span> <span class="number">0x3fd62e42</span> {
<span class="comment">/* if |x| &gt; 0.5 ln2 */</span>
<span class="kw">if</span> <span class="ident">hx</span> <span class="op">&gt;</span><span class="op">=</span> <span class="number">0x3ff0a2b2</span> {
<span class="comment">/* if |x| &gt;= 1.5 ln2 */</span>
<span class="ident">k</span> <span class="op">=</span> (<span class="ident">INVLN2</span> <span class="op">*</span> <span class="ident">x</span> <span class="op">+</span> <span class="ident">HALF</span>[<span class="ident">sign</span> <span class="kw">as</span> <span class="ident">usize</span>]) <span class="kw">as</span> <span class="ident">i32</span>;
} <span class="kw">else</span> {
<span class="ident">k</span> <span class="op">=</span> <span class="number">1</span> <span class="op">-</span> <span class="ident">sign</span> <span class="op">-</span> <span class="ident">sign</span>;
}
<span class="ident">hi</span> <span class="op">=</span> <span class="ident">x</span> <span class="op">-</span> <span class="ident">k</span> <span class="kw">as</span> <span class="ident">f64</span> <span class="op">*</span> <span class="ident">LN2HI</span>; <span class="comment">/* k*ln2hi is exact here */</span>
<span class="ident">lo</span> <span class="op">=</span> <span class="ident">k</span> <span class="kw">as</span> <span class="ident">f64</span> <span class="op">*</span> <span class="ident">LN2LO</span>;
<span class="ident">x</span> <span class="op">=</span> <span class="ident">hi</span> <span class="op">-</span> <span class="ident">lo</span>;
} <span class="kw">else</span> <span class="kw">if</span> <span class="ident">hx</span> <span class="op">&gt;</span> <span class="number">0x3e300000</span> {
<span class="comment">/* if |x| &gt; 2**-28 */</span>
<span class="ident">k</span> <span class="op">=</span> <span class="number">0</span>;
<span class="ident">hi</span> <span class="op">=</span> <span class="ident">x</span>;
<span class="ident">lo</span> <span class="op">=</span> <span class="number">0.</span>;
} <span class="kw">else</span> {
<span class="comment">/* inexact if x!=0 */</span>
<span class="macro">force_eval!</span>(<span class="ident">x1p1023</span> <span class="op">+</span> <span class="ident">x</span>);
<span class="kw">return</span> <span class="number">1.</span> <span class="op">+</span> <span class="ident">x</span>;
}
<span class="comment">/* x is now in primary range */</span>
<span class="ident">xx</span> <span class="op">=</span> <span class="ident">x</span> <span class="op">*</span> <span class="ident">x</span>;
<span class="ident">c</span> <span class="op">=</span> <span class="ident">x</span> <span class="op">-</span> <span class="ident">xx</span> <span class="op">*</span> (<span class="ident">P1</span> <span class="op">+</span> <span class="ident">xx</span> <span class="op">*</span> (<span class="ident">P2</span> <span class="op">+</span> <span class="ident">xx</span> <span class="op">*</span> (<span class="ident">P3</span> <span class="op">+</span> <span class="ident">xx</span> <span class="op">*</span> (<span class="ident">P4</span> <span class="op">+</span> <span class="ident">xx</span> <span class="op">*</span> <span class="ident">P5</span>))));
<span class="ident">y</span> <span class="op">=</span> <span class="number">1.</span> <span class="op">+</span> (<span class="ident">x</span> <span class="op">*</span> <span class="ident">c</span> <span class="op">/</span> (<span class="number">2.</span> <span class="op">-</span> <span class="ident">c</span>) <span class="op">-</span> <span class="ident">lo</span> <span class="op">+</span> <span class="ident">hi</span>);
<span class="kw">if</span> <span class="ident">k</span> <span class="op">==</span> <span class="number">0</span> {
<span class="ident">y</span>
} <span class="kw">else</span> {
<span class="ident">scalbn</span>(<span class="ident">y</span>, <span class="ident">k</span>)
}
}
</code></pre></div>
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