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| </pre><pre class="rust"><code><span class="comment">/* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.c */</span> |
| <span class="comment">/* |
| * ==================================================== |
| * 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. |
| */</span> |
| <span class="comment">/* cbrt(x) |
| * Return cube root of x |
| */</span> |
| |
| <span class="kw">use</span> <span class="ident">core::f64</span>; |
| |
| <span class="kw">const</span> <span class="ident">B1</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">715094163</span>; <span class="comment">/* B1 = (1023-1023/3-0.03306235651)*2**20 */</span> |
| <span class="kw">const</span> <span class="ident">B2</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">696219795</span>; <span class="comment">/* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */</span> |
| |
| <span class="comment">/* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */</span> |
| <span class="kw">const</span> <span class="ident">P0</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">1.87595182427177009643</span>; <span class="comment">/* 0x3ffe03e6, 0x0f61e692 */</span> |
| <span class="kw">const</span> <span class="ident">P1</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="op">-</span><span class="number">1.88497979543377169875</span>; <span class="comment">/* 0xbffe28e0, 0x92f02420 */</span> |
| <span class="kw">const</span> <span class="ident">P2</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">1.621429720105354466140</span>; <span class="comment">/* 0x3ff9f160, 0x4a49d6c2 */</span> |
| <span class="kw">const</span> <span class="ident">P3</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="op">-</span><span class="number">0.758397934778766047437</span>; <span class="comment">/* 0xbfe844cb, 0xbee751d9 */</span> |
| <span class="kw">const</span> <span class="ident">P4</span>: <span class="ident">f64</span> <span class="op">=</span> <span class="number">0.145996192886612446982</span>; <span class="comment">/* 0x3fc2b000, 0xd4e4edd7 */</span> |
| |
| <span class="comment">// Cube root (f64)</span> |
| <span class="doccomment">///</span> |
| <span class="doccomment">/// Computes the cube root of the argument.</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">cbrt</span>(<span class="ident">x</span>: <span class="ident">f64</span>) -> <span class="ident">f64</span> { |
| <span class="kw">let</span> <span class="ident">x1p54</span> <span class="op">=</span> <span class="ident">f64::from_bits</span>(<span class="number">0x4350000000000000</span>); <span class="comment">// 0x1p54 === 2 ^ 54</span> |
| |
| <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">ui</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="kw-2">mut</span> <span class="ident">r</span>: <span class="ident">f64</span>; |
| <span class="kw">let</span> <span class="ident">s</span>: <span class="ident">f64</span>; |
| <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">t</span>: <span class="ident">f64</span>; |
| <span class="kw">let</span> <span class="ident">w</span>: <span class="ident">f64</span>; |
| <span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">hx</span>: <span class="ident">u32</span> <span class="op">=</span> (<span class="ident">ui</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">if</span> <span class="ident">hx</span> <span class="op">></span><span class="op">=</span> <span class="number">0x7ff00000</span> { |
| <span class="comment">/* cbrt(NaN,INF) is itself */</span> |
| <span class="kw">return</span> <span class="ident">x</span> <span class="op">+</span> <span class="ident">x</span>; |
| } |
| |
| <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> <span class="ident">hx</span> <span class="op"><</span> <span class="number">0x00100000</span> { |
| <span class="comment">/* zero or subnormal? */</span> |
| <span class="ident">ui</span> <span class="op">=</span> (<span class="ident">x</span> <span class="op">*</span> <span class="ident">x1p54</span>).<span class="ident">to_bits</span>(); |
| <span class="ident">hx</span> <span class="op">=</span> (<span class="ident">ui</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">if</span> <span class="ident">hx</span> <span class="op">==</span> <span class="number">0</span> { |
| <span class="kw">return</span> <span class="ident">x</span>; <span class="comment">/* cbrt(0) is itself */</span> |
| } |
| <span class="ident">hx</span> <span class="op">=</span> <span class="ident">hx</span> <span class="op">/</span> <span class="number">3</span> <span class="op">+</span> <span class="ident">B2</span>; |
| } <span class="kw">else</span> { |
| <span class="ident">hx</span> <span class="op">=</span> <span class="ident">hx</span> <span class="op">/</span> <span class="number">3</span> <span class="op">+</span> <span class="ident">B1</span>; |
| } |
| <span class="ident">ui</span> <span class="op">&=</span> <span class="number">1</span> <span class="op"><</span><span class="op"><</span> <span class="number">63</span>; |
| <span class="ident">ui</span> <span class="op">|</span><span class="op">=</span> (<span class="ident">hx</span> <span class="kw">as</span> <span class="ident">u64</span>) <span class="op"><</span><span class="op"><</span> <span class="number">32</span>; |
| <span class="ident">t</span> <span class="op">=</span> <span class="ident">f64::from_bits</span>(<span class="ident">ui</span>); |
| |
| <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> |
| <span class="ident">r</span> <span class="op">=</span> (<span class="ident">t</span> <span class="op">*</span> <span class="ident">t</span>) <span class="op">*</span> (<span class="ident">t</span> <span class="op">/</span> <span class="ident">x</span>); |
| <span class="ident">t</span> <span class="op">=</span> <span class="ident">t</span> <span class="op">*</span> ((<span class="ident">P0</span> <span class="op">+</span> <span class="ident">r</span> <span class="op">*</span> (<span class="ident">P1</span> <span class="op">+</span> <span class="ident">r</span> <span class="op">*</span> <span class="ident">P2</span>)) <span class="op">+</span> ((<span class="ident">r</span> <span class="op">*</span> <span class="ident">r</span>) <span class="op">*</span> <span class="ident">r</span>) <span class="op">*</span> (<span class="ident">P3</span> <span class="op">+</span> <span class="ident">r</span> <span class="op">*</span> <span class="ident">P4</span>)); |
| |
| <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> |
| <span class="ident">ui</span> <span class="op">=</span> <span class="ident">t</span>.<span class="ident">to_bits</span>(); |
| <span class="ident">ui</span> <span class="op">=</span> (<span class="ident">ui</span> <span class="op">+</span> <span class="number">0x80000000</span>) <span class="op">&</span> <span class="number">0xffffffffc0000000</span>; |
| <span class="ident">t</span> <span class="op">=</span> <span class="ident">f64::from_bits</span>(<span class="ident">ui</span>); |
| |
| <span class="comment">/* one step Newton iteration to 53 bits with error < 0.667 ulps */</span> |
| <span class="ident">s</span> <span class="op">=</span> <span class="ident">t</span> <span class="op">*</span> <span class="ident">t</span>; <span class="comment">/* t*t is exact */</span> |
| <span class="ident">r</span> <span class="op">=</span> <span class="ident">x</span> <span class="op">/</span> <span class="ident">s</span>; <span class="comment">/* error <= 0.5 ulps; |r| < |t| */</span> |
| <span class="ident">w</span> <span class="op">=</span> <span class="ident">t</span> <span class="op">+</span> <span class="ident">t</span>; <span class="comment">/* t+t is exact */</span> |
| <span class="ident">r</span> <span class="op">=</span> (<span class="ident">r</span> <span class="op">-</span> <span class="ident">t</span>) <span class="op">/</span> (<span class="ident">w</span> <span class="op">+</span> <span class="ident">r</span>); <span class="comment">/* r-t is exact; w+r ~= 3*t */</span> |
| <span class="ident">t</span> <span class="op">=</span> <span class="ident">t</span> <span class="op">+</span> <span class="ident">t</span> <span class="op">*</span> <span class="ident">r</span>; <span class="comment">/* error <= 0.5 + 0.5/3 + epsilon */</span> |
| <span class="ident">t</span> |
| } |
| </code></pre></div> |
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