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</pre><pre class="rust"><code><span class="comment">/* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.c */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*
* Optimized by Bruce D. Evans.
*/
/* cbrt(x)
* Return cube root of x
*/
</span><span class="kw">use </span>core::f64;
<span class="kw">const </span>B1: u32 = <span class="number">715094163</span>; <span class="comment">/* B1 = (1023-1023/3-0.03306235651)*2**20 */
</span><span class="kw">const </span>B2: u32 = <span class="number">696219795</span>; <span class="comment">/* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
/* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
</span><span class="kw">const </span>P0: f64 = <span class="number">1.87595182427177009643</span>; <span class="comment">/* 0x3ffe03e6, 0x0f61e692 */
</span><span class="kw">const </span>P1: f64 = -<span class="number">1.88497979543377169875</span>; <span class="comment">/* 0xbffe28e0, 0x92f02420 */
</span><span class="kw">const </span>P2: f64 = <span class="number">1.621429720105354466140</span>; <span class="comment">/* 0x3ff9f160, 0x4a49d6c2 */
</span><span class="kw">const </span>P3: f64 = -<span class="number">0.758397934778766047437</span>; <span class="comment">/* 0xbfe844cb, 0xbee751d9 */
</span><span class="kw">const </span>P4: f64 = <span class="number">0.145996192886612446982</span>; <span class="comment">/* 0x3fc2b000, 0xd4e4edd7 */
// Cube root (f64)
</span><span class="doccomment">///
/// Computes the cube root of the argument.
</span><span class="attr">#[cfg_attr(all(test, assert_no_panic), no_panic::no_panic)]
</span><span class="kw">pub fn </span>cbrt(x: f64) -> f64 {
<span class="kw">let </span>x1p54 = f64::from_bits(<span class="number">0x4350000000000000</span>); <span class="comment">// 0x1p54 === 2 ^ 54
</span><span class="kw">let </span><span class="kw-2">mut </span>ui: u64 = x.to_bits();
<span class="kw">let </span><span class="kw-2">mut </span>r: f64;
<span class="kw">let </span>s: f64;
<span class="kw">let </span><span class="kw-2">mut </span>t: f64;
<span class="kw">let </span>w: f64;
<span class="kw">let </span><span class="kw-2">mut </span>hx: u32 = (ui >> <span class="number">32</span>) <span class="kw">as </span>u32 & <span class="number">0x7fffffff</span>;
<span class="kw">if </span>hx >= <span class="number">0x7ff00000 </span>{
<span class="comment">/* cbrt(NaN,INF) is itself */
</span><span class="kw">return </span>x + x;
}
<span class="comment">/*
* Rough cbrt to 5 bits:
* cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
* where e is integral and >= 0, m is real and in [0, 1), and "/" and
* "%" are integer division and modulus with rounding towards minus
* infinity. The RHS is always >= the LHS and has a maximum relative
* error of about 1 in 16. Adding a bias of -0.03306235651 to the
* (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
* floating point representation, for finite positive normal values,
* ordinary integer divison of the value in bits magically gives
* almost exactly the RHS of the above provided we first subtract the
* exponent bias (1023 for doubles) and later add it back. We do the
* subtraction virtually to keep e >= 0 so that ordinary integer
* division rounds towards minus infinity; this is also efficient.
*/
</span><span class="kw">if </span>hx < <span class="number">0x00100000 </span>{
<span class="comment">/* zero or subnormal? */
</span>ui = (x * x1p54).to_bits();
hx = (ui >> <span class="number">32</span>) <span class="kw">as </span>u32 & <span class="number">0x7fffffff</span>;
<span class="kw">if </span>hx == <span class="number">0 </span>{
<span class="kw">return </span>x; <span class="comment">/* cbrt(0) is itself */
</span>}
hx = hx / <span class="number">3 </span>+ B2;
} <span class="kw">else </span>{
hx = hx / <span class="number">3 </span>+ B1;
}
ui &= <span class="number">1 </span><< <span class="number">63</span>;
ui |= (hx <span class="kw">as </span>u64) << <span class="number">32</span>;
t = f64::from_bits(ui);
<span class="comment">/*
* New cbrt to 23 bits:
* cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
* where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
* to within 2**-23.5 when |r - 1| < 1/10. The rough approximation
* has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
* gives us bounds for r = t**3/x.
*
* Try to optimize for parallel evaluation as in __tanf.c.
*/
</span>r = (t * t) * (t / x);
t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4));
<span class="comment">/*
* Round t away from zero to 23 bits (sloppily except for ensuring that
* the result is larger in magnitude than cbrt(x) but not much more than
* 2 23-bit ulps larger). With rounding towards zero, the error bound
* would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps
* in the rounded t, the infinite-precision error in the Newton
* approximation barely affects third digit in the final error
* 0.667; the error in the rounded t can be up to about 3 23-bit ulps
* before the final error is larger than 0.667 ulps.
*/
</span>ui = t.to_bits();
ui = (ui + <span class="number">0x80000000</span>) & <span class="number">0xffffffffc0000000</span>;
t = f64::from_bits(ui);
<span class="comment">/* one step Newton iteration to 53 bits with error < 0.667 ulps */
</span>s = t * t; <span class="comment">/* t*t is exact */
</span>r = x / s; <span class="comment">/* error <= 0.5 ulps; |r| < |t| */
</span>w = t + t; <span class="comment">/* t+t is exact */
</span>r = (r - t) / (w + r); <span class="comment">/* r-t is exact; w+r ~= 3*t */
</span>t = t + t * r; <span class="comment">/* error <= 0.5 + 0.5/3 + epsilon */
</span>t
}
</code></pre></div>
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