<!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 to the Rust file `libcore/num/dec2flt/algorithm.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>algorithm.rs.html -- source</title><link rel="stylesheet" type="text/css" href="../../../../normalize.css"><link rel="stylesheet" type="text/css" href="../../../../rustdoc.css" id="mainThemeStyle"><link rel="stylesheet" type="text/css" href="../../../../dark.css"><link rel="stylesheet" type="text/css" href="../../../../light.css" id="themeStyle"><script src="../../../../storage.js"></script><link rel="shortcut icon" href="https://doc.rust-lang.org/favicon.ico"></head><body class="rustdoc source"><!--[if lte IE 8]><div class="warning">This old browser is unsupported and will most likely display funky things.</div><![endif]--><nav class="sidebar"><div class="sidebar-menu">☰</div><a href='../../../../core/index.html'><img src='https://www.rust-lang.org/logos/rust-logo-128x128-blk-v2.png' alt='logo' width='100'></a></nav><div class="theme-picker"><button id="theme-picker" aria-label="Pick another theme!"><img src="../../../../brush.svg" width="18" alt="Pick another theme!"></button><div id="theme-choices"></div></div><script src="../../../../theme.js"></script><nav class="sub"><form class="search-form js-only"><div class="search-container"><input class="search-input" name="search" autocomplete="off" placeholder="Click or press ‘S’ to search, ‘?’ for more options…" type="search"><a id="settings-menu" href="../../../../settings.html"><img src="../../../../wheel.svg" width="18" alt="Change settings"></a></div></form></nav><section id="main" class="content"><pre class="line-numbers"><span id="1"> 1</span>
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</pre><pre class="rust ">
<span class="comment">// Copyright 2015 The Rust Project Developers. See the COPYRIGHT</span>
<span class="comment">// file at the top-level directory of this distribution and at</span>
<span class="comment">// http://rust-lang.org/COPYRIGHT.</span>
<span class="comment">//</span>
<span class="comment">// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or</span>
<span class="comment">// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license</span>
<span class="comment">// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your</span>
<span class="comment">// option. This file may not be copied, modified, or distributed</span>
<span class="comment">// except according to those terms.</span>
<span class="doccomment">//! The various algorithms from the paper.</span>
<span class="kw">use</span> <span class="ident">cmp</span>::<span class="ident">min</span>;
<span class="kw">use</span> <span class="ident">cmp</span>::<span class="ident">Ordering</span>::{<span class="ident">Less</span>, <span class="ident">Equal</span>, <span class="ident">Greater</span>};
<span class="kw">use</span> <span class="ident">num</span>::<span class="ident">diy_float</span>::<span class="ident">Fp</span>;
<span class="kw">use</span> <span class="ident">num</span>::<span class="ident">dec2flt</span>::<span class="ident">table</span>;
<span class="kw">use</span> <span class="ident">num</span>::<span class="ident">dec2flt</span>::<span class="ident">rawfp</span>::{<span class="self">self</span>, <span class="ident">Unpacked</span>, <span class="ident">RawFloat</span>, <span class="ident">fp_to_float</span>, <span class="ident">next_float</span>, <span class="ident">prev_float</span>};
<span class="kw">use</span> <span class="ident">num</span>::<span class="ident">dec2flt</span>::<span class="ident">num</span>::{<span class="self">self</span>, <span class="ident">Big</span>};
<span class="doccomment">/// Number of significand bits in Fp</span>
<span class="kw">const</span> <span class="ident">P</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">64</span>;
<span class="comment">// We simply store the best approximation for *all* exponents, so the variable "h" and the</span>
<span class="comment">// associated conditions can be omitted. This trades performance for a couple kilobytes of space.</span>
<span class="kw">fn</span> <span class="ident">power_of_ten</span>(<span class="ident">e</span>: <span class="ident">i16</span>) <span class="op">-></span> <span class="ident">Fp</span> {
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">e</span> <span class="op">>=</span> <span class="ident">table</span>::<span class="ident">MIN_E</span>);
<span class="kw">let</span> <span class="ident">i</span> <span class="op">=</span> <span class="ident">e</span> <span class="op">-</span> <span class="ident">table</span>::<span class="ident">MIN_E</span>;
<span class="kw">let</span> <span class="ident">sig</span> <span class="op">=</span> <span class="ident">table</span>::<span class="ident">POWERS</span>.<span class="number">0</span>[<span class="ident">i</span> <span class="kw">as</span> <span class="ident">usize</span>];
<span class="kw">let</span> <span class="ident">exp</span> <span class="op">=</span> <span class="ident">table</span>::<span class="ident">POWERS</span>.<span class="number">1</span>[<span class="ident">i</span> <span class="kw">as</span> <span class="ident">usize</span>];
<span class="ident">Fp</span> { <span class="ident">f</span>: <span class="ident">sig</span>, <span class="ident">e</span>: <span class="ident">exp</span> }
}
<span class="comment">// In most architectures, floating point operations have an explicit bit size, therefore the</span>
<span class="comment">// precision of the computation is determined on a per-operation basis.</span>
<span class="attribute">#[<span class="ident">cfg</span>(<span class="ident">any</span>(<span class="ident">not</span>(<span class="ident">target_arch</span><span class="op">=</span><span class="string">"x86"</span>), <span class="ident">target_feature</span><span class="op">=</span><span class="string">"sse2"</span>))]</span>
<span class="kw">mod</span> <span class="ident">fpu_precision</span> {
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">set_precision</span><span class="op"><</span><span class="ident">T</span><span class="op">></span>() { }
}
<span class="comment">// On x86, the x87 FPU is used for float operations if the SSE/SSE2 extensions are not available.</span>
<span class="comment">// The x87 FPU operates with 80 bits of precision by default, which means that operations will</span>
<span class="comment">// round to 80 bits causing double rounding to happen when values are eventually represented as</span>
<span class="comment">// 32/64 bit float values. To overcome this, the FPU control word can be set so that the</span>
<span class="comment">// computations are performed in the desired precision.</span>
<span class="attribute">#[<span class="ident">cfg</span>(<span class="ident">all</span>(<span class="ident">target_arch</span><span class="op">=</span><span class="string">"x86"</span>, <span class="ident">not</span>(<span class="ident">target_feature</span><span class="op">=</span><span class="string">"sse2"</span>)))]</span>
<span class="kw">mod</span> <span class="ident">fpu_precision</span> {
<span class="kw">use</span> <span class="ident">mem</span>::<span class="ident">size_of</span>;
<span class="doccomment">/// A structure used to preserve the original value of the FPU control word, so that it can be</span>
<span class="doccomment">/// restored when the structure is dropped.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// The x87 FPU is a 16-bits register whose fields are as follows:</span>
<span class="doccomment">///</span>
<span class="doccomment">/// | 12-15 | 10-11 | 8-9 | 6-7 | 5 | 4 | 3 | 2 | 1 | 0 |</span>
<span class="doccomment">/// |------:|------:|----:|----:|---:|---:|---:|---:|---:|---:|</span>
<span class="doccomment">/// | | RC | PC | | PM | UM | OM | ZM | DM | IM |</span>
<span class="doccomment">///</span>
<span class="doccomment">/// The documentation for all of the fields is available in the IA-32 Architectures Software</span>
<span class="doccomment">/// Developer's Manual (Volume 1).</span>
<span class="doccomment">///</span>
<span class="doccomment">/// The only field which is relevant for the following code is PC, Precision Control. This</span>
<span class="doccomment">/// field determines the precision of the operations performed by the FPU. It can be set to:</span>
<span class="doccomment">/// - 0b00, single precision i.e. 32-bits</span>
<span class="doccomment">/// - 0b10, double precision i.e. 64-bits</span>
<span class="doccomment">/// - 0b11, double extended precision i.e. 80-bits (default state)</span>
<span class="doccomment">/// The 0b01 value is reserved and should not be used.</span>
<span class="kw">pub</span> <span class="kw">struct</span> <span class="ident">FPUControlWord</span>(<span class="ident">u16</span>);
<span class="kw">fn</span> <span class="ident">set_cw</span>(<span class="ident">cw</span>: <span class="ident">u16</span>) {
<span class="kw">unsafe</span> { <span class="macro">asm</span><span class="macro">!</span>(<span class="string">"fldcw $0"</span> :: <span class="string">"m"</span> (<span class="ident">cw</span>) :: <span class="string">"volatile"</span>) }
}
<span class="doccomment">/// Set the precision field of the FPU to `T` and return a `FPUControlWord`</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">set_precision</span><span class="op"><</span><span class="ident">T</span><span class="op">></span>() <span class="op">-></span> <span class="ident">FPUControlWord</span> {
<span class="kw">let</span> <span class="ident">cw</span> <span class="op">=</span> <span class="number">0u16</span>;
<span class="comment">// Compute the value for the Precision Control field that is appropriate for `T`.</span>
<span class="kw">let</span> <span class="ident">cw_precision</span> <span class="op">=</span> <span class="kw">match</span> <span class="ident">size_of</span>::<span class="op"><</span><span class="ident">T</span><span class="op">></span>() {
<span class="number">4</span> <span class="op">=></span> <span class="number">0x0000</span>, <span class="comment">// 32 bits</span>
<span class="number">8</span> <span class="op">=></span> <span class="number">0x0200</span>, <span class="comment">// 64 bits</span>
<span class="kw">_</span> <span class="op">=></span> <span class="number">0x0300</span>, <span class="comment">// default, 80 bits</span>
};
<span class="comment">// Get the original value of the control word to restore it later, when the</span>
<span class="comment">// `FPUControlWord` structure is dropped</span>
<span class="kw">unsafe</span> { <span class="macro">asm</span><span class="macro">!</span>(<span class="string">"fnstcw $0"</span> : <span class="string">"=*m"</span> (<span class="kw-2">&</span><span class="ident">cw</span>) ::: <span class="string">"volatile"</span>) }
<span class="comment">// Set the control word to the desired precision. This is achieved by masking away the old</span>
<span class="comment">// precision (bits 8 and 9, 0x300) and replacing it with the precision flag computed above.</span>
<span class="ident">set_cw</span>((<span class="ident">cw</span> <span class="op">&</span> <span class="number">0xFCFF</span>) <span class="op">|</span> <span class="ident">cw_precision</span>);
<span class="ident">FPUControlWord</span>(<span class="ident">cw</span>)
}
<span class="kw">impl</span> <span class="ident">Drop</span> <span class="kw">for</span> <span class="ident">FPUControlWord</span> {
<span class="kw">fn</span> <span class="ident">drop</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="self">self</span>) {
<span class="ident">set_cw</span>(<span class="self">self</span>.<span class="number">0</span>)
}
}
}
<span class="doccomment">/// The fast path of Bellerophon using machine-sized integers and floats.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// This is extracted into a separate function so that it can be attempted before constructing</span>
<span class="doccomment">/// a bignum.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">fast_path</span><span class="op"><</span><span class="ident">T</span>: <span class="ident">RawFloat</span><span class="op">></span>(<span class="ident">integral</span>: <span class="kw-2">&</span>[<span class="ident">u8</span>], <span class="ident">fractional</span>: <span class="kw-2">&</span>[<span class="ident">u8</span>], <span class="ident">e</span>: <span class="ident">i64</span>) <span class="op">-></span> <span class="prelude-ty">Option</span><span class="op"><</span><span class="ident">T</span><span class="op">></span> {
<span class="kw">let</span> <span class="ident">num_digits</span> <span class="op">=</span> <span class="ident">integral</span>.<span class="ident">len</span>() <span class="op">+</span> <span class="ident">fractional</span>.<span class="ident">len</span>();
<span class="comment">// log_10(f64::MAX_SIG) ~ 15.95. We compare the exact value to MAX_SIG near the end,</span>
<span class="comment">// this is just a quick, cheap rejection (and also frees the rest of the code from</span>
<span class="comment">// worrying about underflow).</span>
<span class="kw">if</span> <span class="ident">num_digits</span> <span class="op">></span> <span class="number">16</span> {
<span class="kw">return</span> <span class="prelude-val">None</span>;
}
<span class="kw">if</span> <span class="ident">e</span>.<span class="ident">abs</span>() <span class="op">>=</span> <span class="ident">T</span>::<span class="ident">CEIL_LOG5_OF_MAX_SIG</span> <span class="kw">as</span> <span class="ident">i64</span> {
<span class="kw">return</span> <span class="prelude-val">None</span>;
}
<span class="kw">let</span> <span class="ident">f</span> <span class="op">=</span> <span class="ident">num</span>::<span class="ident">from_str_unchecked</span>(<span class="ident">integral</span>.<span class="ident">iter</span>().<span class="ident">chain</span>(<span class="ident">fractional</span>.<span class="ident">iter</span>()));
<span class="kw">if</span> <span class="ident">f</span> <span class="op">></span> <span class="ident">T</span>::<span class="ident">MAX_SIG</span> {
<span class="kw">return</span> <span class="prelude-val">None</span>;
}
<span class="comment">// The fast path crucially depends on arithmetic being rounded to the correct number of bits</span>
<span class="comment">// without any intermediate rounding. On x86 (without SSE or SSE2) this requires the precision</span>
<span class="comment">// of the x87 FPU stack to be changed so that it directly rounds to 64/32 bit.</span>
<span class="comment">// The `set_precision` function takes care of setting the precision on architectures which</span>
<span class="comment">// require setting it by changing the global state (like the control word of the x87 FPU).</span>
<span class="kw">let</span> <span class="ident">_cw</span> <span class="op">=</span> <span class="ident">fpu_precision</span>::<span class="ident">set_precision</span>::<span class="op"><</span><span class="ident">T</span><span class="op">></span>();
<span class="comment">// The case e < 0 cannot be folded into the other branch. Negative powers result in</span>
<span class="comment">// a repeating fractional part in binary, which are rounded, which causes real</span>
<span class="comment">// (and occasionally quite significant!) errors in the final result.</span>
<span class="kw">if</span> <span class="ident">e</span> <span class="op">>=</span> <span class="number">0</span> {
<span class="prelude-val">Some</span>(<span class="ident">T</span>::<span class="ident">from_int</span>(<span class="ident">f</span>) <span class="op">*</span> <span class="ident">T</span>::<span class="ident">short_fast_pow10</span>(<span class="ident">e</span> <span class="kw">as</span> <span class="ident">usize</span>))
} <span class="kw">else</span> {
<span class="prelude-val">Some</span>(<span class="ident">T</span>::<span class="ident">from_int</span>(<span class="ident">f</span>) <span class="op">/</span> <span class="ident">T</span>::<span class="ident">short_fast_pow10</span>(<span class="ident">e</span>.<span class="ident">abs</span>() <span class="kw">as</span> <span class="ident">usize</span>))
}
}
<span class="doccomment">/// Algorithm Bellerophon is trivial code justified by non-trivial numeric analysis.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// It rounds ``f`` to a float with 64 bit significand and multiplies it by the best approximation</span>
<span class="doccomment">/// of `10^e` (in the same floating point format). This is often enough to get the correct result.</span>
<span class="doccomment">/// However, when the result is close to halfway between two adjacent (ordinary) floats, the</span>
<span class="doccomment">/// compound rounding error from multiplying two approximation means the result may be off by a</span>
<span class="doccomment">/// few bits. When this happens, the iterative Algorithm R fixes things up.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// The hand-wavy "close to halfway" is made precise by the numeric analysis in the paper.</span>
<span class="doccomment">/// In the words of Clinger:</span>
<span class="doccomment">///</span>
<span class="doccomment">/// > Slop, expressed in units of the least significant bit, is an inclusive bound for the error</span>
<span class="doccomment">/// > accumulated during the floating point calculation of the approximation to f * 10^e. (Slop is</span>
<span class="doccomment">/// > not a bound for the true error, but bounds the difference between the approximation z and</span>
<span class="doccomment">/// > the best possible approximation that uses p bits of significand.)</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">bellerophon</span><span class="op"><</span><span class="ident">T</span>: <span class="ident">RawFloat</span><span class="op">></span>(<span class="ident">f</span>: <span class="kw-2">&</span><span class="ident">Big</span>, <span class="ident">e</span>: <span class="ident">i16</span>) <span class="op">-></span> <span class="ident">T</span> {
<span class="kw">let</span> <span class="ident">slop</span>;
<span class="kw">if</span> <span class="ident">f</span> <span class="op"><=</span> <span class="kw-2">&</span><span class="ident">Big</span>::<span class="ident">from_u64</span>(<span class="ident">T</span>::<span class="ident">MAX_SIG</span>) {
<span class="comment">// The cases abs(e) < log5(2^N) are in fast_path()</span>
<span class="ident">slop</span> <span class="op">=</span> <span class="kw">if</span> <span class="ident">e</span> <span class="op">>=</span> <span class="number">0</span> { <span class="number">0</span> } <span class="kw">else</span> { <span class="number">3</span> };
} <span class="kw">else</span> {
<span class="ident">slop</span> <span class="op">=</span> <span class="kw">if</span> <span class="ident">e</span> <span class="op">>=</span> <span class="number">0</span> { <span class="number">1</span> } <span class="kw">else</span> { <span class="number">4</span> };
}
<span class="kw">let</span> <span class="ident">z</span> <span class="op">=</span> <span class="ident">rawfp</span>::<span class="ident">big_to_fp</span>(<span class="ident">f</span>).<span class="ident">mul</span>(<span class="kw-2">&</span><span class="ident">power_of_ten</span>(<span class="ident">e</span>)).<span class="ident">normalize</span>();
<span class="kw">let</span> <span class="ident">exp_p_n</span> <span class="op">=</span> <span class="number">1</span> <span class="op"><<</span> (<span class="ident">P</span> <span class="op">-</span> <span class="ident">T</span>::<span class="ident">SIG_BITS</span> <span class="kw">as</span> <span class="ident">u32</span>);
<span class="kw">let</span> <span class="ident">lowbits</span>: <span class="ident">i64</span> <span class="op">=</span> (<span class="ident">z</span>.<span class="ident">f</span> <span class="op">%</span> <span class="ident">exp_p_n</span>) <span class="kw">as</span> <span class="ident">i64</span>;
<span class="comment">// Is the slop large enough to make a difference when</span>
<span class="comment">// rounding to n bits?</span>
<span class="kw">if</span> (<span class="ident">lowbits</span> <span class="op">-</span> <span class="ident">exp_p_n</span> <span class="kw">as</span> <span class="ident">i64</span> <span class="op">/</span> <span class="number">2</span>).<span class="ident">abs</span>() <span class="op"><=</span> <span class="ident">slop</span> {
<span class="ident">algorithm_r</span>(<span class="ident">f</span>, <span class="ident">e</span>, <span class="ident">fp_to_float</span>(<span class="ident">z</span>))
} <span class="kw">else</span> {
<span class="ident">fp_to_float</span>(<span class="ident">z</span>)
}
}
<span class="doccomment">/// An iterative algorithm that improves a floating point approximation of `f * 10^e`.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Each iteration gets one unit in the last place closer, which of course takes terribly long to</span>
<span class="doccomment">/// converge if `z0` is even mildly off. Luckily, when used as fallback for Bellerophon, the</span>
<span class="doccomment">/// starting approximation is off by at most one ULP.</span>
<span class="kw">fn</span> <span class="ident">algorithm_r</span><span class="op"><</span><span class="ident">T</span>: <span class="ident">RawFloat</span><span class="op">></span>(<span class="ident">f</span>: <span class="kw-2">&</span><span class="ident">Big</span>, <span class="ident">e</span>: <span class="ident">i16</span>, <span class="ident">z0</span>: <span class="ident">T</span>) <span class="op">-></span> <span class="ident">T</span> {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">z</span> <span class="op">=</span> <span class="ident">z0</span>;
<span class="kw">loop</span> {
<span class="kw">let</span> <span class="ident">raw</span> <span class="op">=</span> <span class="ident">z</span>.<span class="ident">unpack</span>();
<span class="kw">let</span> (<span class="ident">m</span>, <span class="ident">k</span>) <span class="op">=</span> (<span class="ident">raw</span>.<span class="ident">sig</span>, <span class="ident">raw</span>.<span class="ident">k</span>);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">x</span> <span class="op">=</span> <span class="ident">f</span>.<span class="ident">clone</span>();
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">y</span> <span class="op">=</span> <span class="ident">Big</span>::<span class="ident">from_u64</span>(<span class="ident">m</span>);
<span class="comment">// Find positive integers `x`, `y` such that `x / y` is exactly `(f * 10^e) / (m * 2^k)`.</span>
<span class="comment">// This not only avoids dealing with the signs of `e` and `k`, we also eliminate the</span>
<span class="comment">// power of two common to `10^e` and `2^k` to make the numbers smaller.</span>
<span class="ident">make_ratio</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">x</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">y</span>, <span class="ident">e</span>, <span class="ident">k</span>);
<span class="kw">let</span> <span class="ident">m_digits</span> <span class="op">=</span> [(<span class="ident">m</span> <span class="op">&</span> <span class="number">0xFF_FF_FF_FF</span>) <span class="kw">as</span> <span class="ident">u32</span>, (<span class="ident">m</span> <span class="op">>></span> <span class="number">32</span>) <span class="kw">as</span> <span class="ident">u32</span>];
<span class="comment">// This is written a bit awkwardly because our bignums don't support</span>
<span class="comment">// negative numbers, so we use the absolute value + sign information.</span>
<span class="comment">// The multiplication with m_digits can't overflow. If `x` or `y` are large enough that</span>
<span class="comment">// we need to worry about overflow, then they are also large enough that `make_ratio` has</span>
<span class="comment">// reduced the fraction by a factor of 2^64 or more.</span>
<span class="kw">let</span> (<span class="ident">d2</span>, <span class="ident">d_negative</span>) <span class="op">=</span> <span class="kw">if</span> <span class="ident">x</span> <span class="op">>=</span> <span class="ident">y</span> {
<span class="comment">// Don't need x any more, save a clone().</span>
<span class="ident">x</span>.<span class="ident">sub</span>(<span class="kw-2">&</span><span class="ident">y</span>).<span class="ident">mul_pow2</span>(<span class="number">1</span>).<span class="ident">mul_digits</span>(<span class="kw-2">&</span><span class="ident">m_digits</span>);
(<span class="ident">x</span>, <span class="bool-val">false</span>)
} <span class="kw">else</span> {
<span class="comment">// Still need y - make a copy.</span>
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">y</span> <span class="op">=</span> <span class="ident">y</span>.<span class="ident">clone</span>();
<span class="ident">y</span>.<span class="ident">sub</span>(<span class="kw-2">&</span><span class="ident">x</span>).<span class="ident">mul_pow2</span>(<span class="number">1</span>).<span class="ident">mul_digits</span>(<span class="kw-2">&</span><span class="ident">m_digits</span>);
(<span class="ident">y</span>, <span class="bool-val">true</span>)
};
<span class="kw">if</span> <span class="ident">d2</span> <span class="op"><</span> <span class="ident">y</span> {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">d2_double</span> <span class="op">=</span> <span class="ident">d2</span>;
<span class="ident">d2_double</span>.<span class="ident">mul_pow2</span>(<span class="number">1</span>);
<span class="kw">if</span> <span class="ident">m</span> <span class="op">==</span> <span class="ident">T</span>::<span class="ident">MIN_SIG</span> <span class="op">&&</span> <span class="ident">d_negative</span> <span class="op">&&</span> <span class="ident">d2_double</span> <span class="op">></span> <span class="ident">y</span> {
<span class="ident">z</span> <span class="op">=</span> <span class="ident">prev_float</span>(<span class="ident">z</span>);
} <span class="kw">else</span> {
<span class="kw">return</span> <span class="ident">z</span>;
}
} <span class="kw">else</span> <span class="kw">if</span> <span class="ident">d2</span> <span class="op">==</span> <span class="ident">y</span> {
<span class="kw">if</span> <span class="ident">m</span> <span class="op">%</span> <span class="number">2</span> <span class="op">==</span> <span class="number">0</span> {
<span class="kw">if</span> <span class="ident">m</span> <span class="op">==</span> <span class="ident">T</span>::<span class="ident">MIN_SIG</span> <span class="op">&&</span> <span class="ident">d_negative</span> {
<span class="ident">z</span> <span class="op">=</span> <span class="ident">prev_float</span>(<span class="ident">z</span>);
} <span class="kw">else</span> {
<span class="kw">return</span> <span class="ident">z</span>;
}
} <span class="kw">else</span> <span class="kw">if</span> <span class="ident">d_negative</span> {
<span class="ident">z</span> <span class="op">=</span> <span class="ident">prev_float</span>(<span class="ident">z</span>);
} <span class="kw">else</span> {
<span class="ident">z</span> <span class="op">=</span> <span class="ident">next_float</span>(<span class="ident">z</span>);
}
} <span class="kw">else</span> <span class="kw">if</span> <span class="ident">d_negative</span> {
<span class="ident">z</span> <span class="op">=</span> <span class="ident">prev_float</span>(<span class="ident">z</span>);
} <span class="kw">else</span> {
<span class="ident">z</span> <span class="op">=</span> <span class="ident">next_float</span>(<span class="ident">z</span>);
}
}
}
<span class="doccomment">/// Given `x = f` and `y = m` where `f` represent input decimal digits as usual and `m` is the</span>
<span class="doccomment">/// significand of a floating point approximation, make the ratio `x / y` equal to</span>
<span class="doccomment">/// `(f * 10^e) / (m * 2^k)`, possibly reduced by a power of two both have in common.</span>
<span class="kw">fn</span> <span class="ident">make_ratio</span>(<span class="ident">x</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">Big</span>, <span class="ident">y</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">Big</span>, <span class="ident">e</span>: <span class="ident">i16</span>, <span class="ident">k</span>: <span class="ident">i16</span>) {
<span class="kw">let</span> (<span class="ident">e_abs</span>, <span class="ident">k_abs</span>) <span class="op">=</span> (<span class="ident">e</span>.<span class="ident">abs</span>() <span class="kw">as</span> <span class="ident">usize</span>, <span class="ident">k</span>.<span class="ident">abs</span>() <span class="kw">as</span> <span class="ident">usize</span>);
<span class="kw">if</span> <span class="ident">e</span> <span class="op">>=</span> <span class="number">0</span> {
<span class="kw">if</span> <span class="ident">k</span> <span class="op">>=</span> <span class="number">0</span> {
<span class="comment">// x = f * 10^e, y = m * 2^k, except that we reduce the fraction by some power of two.</span>
<span class="kw">let</span> <span class="ident">common</span> <span class="op">=</span> <span class="ident">min</span>(<span class="ident">e_abs</span>, <span class="ident">k_abs</span>);
<span class="ident">x</span>.<span class="ident">mul_pow5</span>(<span class="ident">e_abs</span>).<span class="ident">mul_pow2</span>(<span class="ident">e_abs</span> <span class="op">-</span> <span class="ident">common</span>);
<span class="ident">y</span>.<span class="ident">mul_pow2</span>(<span class="ident">k_abs</span> <span class="op">-</span> <span class="ident">common</span>);
} <span class="kw">else</span> {
<span class="comment">// x = f * 10^e * 2^abs(k), y = m</span>
<span class="comment">// This can't overflow because it requires positive `e` and negative `k`, which can</span>
<span class="comment">// only happen for values extremely close to 1, which means that `e` and `k` will be</span>
<span class="comment">// comparatively tiny.</span>
<span class="ident">x</span>.<span class="ident">mul_pow5</span>(<span class="ident">e_abs</span>).<span class="ident">mul_pow2</span>(<span class="ident">e_abs</span> <span class="op">+</span> <span class="ident">k_abs</span>);
}
} <span class="kw">else</span> {
<span class="kw">if</span> <span class="ident">k</span> <span class="op">>=</span> <span class="number">0</span> {
<span class="comment">// x = f, y = m * 10^abs(e) * 2^k</span>
<span class="comment">// This can't overflow either, see above.</span>
<span class="ident">y</span>.<span class="ident">mul_pow5</span>(<span class="ident">e_abs</span>).<span class="ident">mul_pow2</span>(<span class="ident">k_abs</span> <span class="op">+</span> <span class="ident">e_abs</span>);
} <span class="kw">else</span> {
<span class="comment">// x = f * 2^abs(k), y = m * 10^abs(e), again reducing by a common power of two.</span>
<span class="kw">let</span> <span class="ident">common</span> <span class="op">=</span> <span class="ident">min</span>(<span class="ident">e_abs</span>, <span class="ident">k_abs</span>);
<span class="ident">x</span>.<span class="ident">mul_pow2</span>(<span class="ident">k_abs</span> <span class="op">-</span> <span class="ident">common</span>);
<span class="ident">y</span>.<span class="ident">mul_pow5</span>(<span class="ident">e_abs</span>).<span class="ident">mul_pow2</span>(<span class="ident">e_abs</span> <span class="op">-</span> <span class="ident">common</span>);
}
}
}
<span class="doccomment">/// Conceptually, Algorithm M is the simplest way to convert a decimal to a float.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// We form a ratio that is equal to `f * 10^e`, then throwing in powers of two until it gives</span>
<span class="doccomment">/// a valid float significand. The binary exponent `k` is the number of times we multiplied</span>
<span class="doccomment">/// numerator or denominator by two, i.e., at all times `f * 10^e` equals `(u / v) * 2^k`.</span>
<span class="doccomment">/// When we have found out significand, we only need to round by inspecting the remainder of the</span>
<span class="doccomment">/// division, which is done in helper functions further below.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// This algorithm is super slow, even with the optimization described in `quick_start()`.</span>
<span class="doccomment">/// However, it's the simplest of the algorithms to adapt for overflow, underflow, and subnormal</span>
<span class="doccomment">/// results. This implementation takes over when Bellerophon and Algorithm R are overwhelmed.</span>
<span class="doccomment">/// Detecting underflow and overflow is easy: The ratio still isn't an in-range significand,</span>
<span class="doccomment">/// yet the minimum/maximum exponent has been reached. In the case of overflow, we simply return</span>
<span class="doccomment">/// infinity.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Handling underflow and subnormals is trickier. One big problem is that, with the minimum</span>
<span class="doccomment">/// exponent, the ratio might still be too large for a significand. See underflow() for details.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">algorithm_m</span><span class="op"><</span><span class="ident">T</span>: <span class="ident">RawFloat</span><span class="op">></span>(<span class="ident">f</span>: <span class="kw-2">&</span><span class="ident">Big</span>, <span class="ident">e</span>: <span class="ident">i16</span>) <span class="op">-></span> <span class="ident">T</span> {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">u</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">v</span>;
<span class="kw">let</span> <span class="ident">e_abs</span> <span class="op">=</span> <span class="ident">e</span>.<span class="ident">abs</span>() <span class="kw">as</span> <span class="ident">usize</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">k</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">if</span> <span class="ident">e</span> <span class="op"><</span> <span class="number">0</span> {
<span class="ident">u</span> <span class="op">=</span> <span class="ident">f</span>.<span class="ident">clone</span>();
<span class="ident">v</span> <span class="op">=</span> <span class="ident">Big</span>::<span class="ident">from_small</span>(<span class="number">1</span>);
<span class="ident">v</span>.<span class="ident">mul_pow5</span>(<span class="ident">e_abs</span>).<span class="ident">mul_pow2</span>(<span class="ident">e_abs</span>);
} <span class="kw">else</span> {
<span class="comment">// FIXME possible optimization: generalize big_to_fp so that we can do the equivalent of</span>
<span class="comment">// fp_to_float(big_to_fp(u)) here, only without the double rounding.</span>
<span class="ident">u</span> <span class="op">=</span> <span class="ident">f</span>.<span class="ident">clone</span>();
<span class="ident">u</span>.<span class="ident">mul_pow5</span>(<span class="ident">e_abs</span>).<span class="ident">mul_pow2</span>(<span class="ident">e_abs</span>);
<span class="ident">v</span> <span class="op">=</span> <span class="ident">Big</span>::<span class="ident">from_small</span>(<span class="number">1</span>);
}
<span class="ident">quick_start</span>::<span class="op"><</span><span class="ident">T</span><span class="op">></span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">u</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">v</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">k</span>);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">rem</span> <span class="op">=</span> <span class="ident">Big</span>::<span class="ident">from_small</span>(<span class="number">0</span>);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">x</span> <span class="op">=</span> <span class="ident">Big</span>::<span class="ident">from_small</span>(<span class="number">0</span>);
<span class="kw">let</span> <span class="ident">min_sig</span> <span class="op">=</span> <span class="ident">Big</span>::<span class="ident">from_u64</span>(<span class="ident">T</span>::<span class="ident">MIN_SIG</span>);
<span class="kw">let</span> <span class="ident">max_sig</span> <span class="op">=</span> <span class="ident">Big</span>::<span class="ident">from_u64</span>(<span class="ident">T</span>::<span class="ident">MAX_SIG</span>);
<span class="kw">loop</span> {
<span class="ident">u</span>.<span class="ident">div_rem</span>(<span class="kw-2">&</span><span class="ident">v</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">x</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">rem</span>);
<span class="kw">if</span> <span class="ident">k</span> <span class="op">==</span> <span class="ident">T</span>::<span class="ident">MIN_EXP_INT</span> {
<span class="comment">// We have to stop at the minimum exponent, if we wait until `k < T::MIN_EXP_INT`,</span>
<span class="comment">// then we'd be off by a factor of two. Unfortunately this means we have to special-</span>
<span class="comment">// case normal numbers with the minimum exponent.</span>
<span class="comment">// FIXME find a more elegant formulation, but run the `tiny-pow10` test to make sure</span>
<span class="comment">// that it's actually correct!</span>
<span class="kw">if</span> <span class="ident">x</span> <span class="op">>=</span> <span class="ident">min_sig</span> <span class="op">&&</span> <span class="ident">x</span> <span class="op"><=</span> <span class="ident">max_sig</span> {
<span class="kw">break</span>;
}
<span class="kw">return</span> <span class="ident">underflow</span>(<span class="ident">x</span>, <span class="ident">v</span>, <span class="ident">rem</span>);
}
<span class="kw">if</span> <span class="ident">k</span> <span class="op">></span> <span class="ident">T</span>::<span class="ident">MAX_EXP_INT</span> {
<span class="kw">return</span> <span class="ident">T</span>::<span class="ident">INFINITY</span>;
}
<span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="ident">min_sig</span> {
<span class="ident">u</span>.<span class="ident">mul_pow2</span>(<span class="number">1</span>);
<span class="ident">k</span> <span class="op">-=</span> <span class="number">1</span>;
} <span class="kw">else</span> <span class="kw">if</span> <span class="ident">x</span> <span class="op">></span> <span class="ident">max_sig</span> {
<span class="ident">v</span>.<span class="ident">mul_pow2</span>(<span class="number">1</span>);
<span class="ident">k</span> <span class="op">+=</span> <span class="number">1</span>;
} <span class="kw">else</span> {
<span class="kw">break</span>;
}
}
<span class="kw">let</span> <span class="ident">q</span> <span class="op">=</span> <span class="ident">num</span>::<span class="ident">to_u64</span>(<span class="kw-2">&</span><span class="ident">x</span>);
<span class="kw">let</span> <span class="ident">z</span> <span class="op">=</span> <span class="ident">rawfp</span>::<span class="ident">encode_normal</span>(<span class="ident">Unpacked</span>::<span class="ident">new</span>(<span class="ident">q</span>, <span class="ident">k</span>));
<span class="ident">round_by_remainder</span>(<span class="ident">v</span>, <span class="ident">rem</span>, <span class="ident">q</span>, <span class="ident">z</span>)
}
<span class="doccomment">/// Skip over most Algorithm M iterations by checking the bit length.</span>
<span class="kw">fn</span> <span class="ident">quick_start</span><span class="op"><</span><span class="ident">T</span>: <span class="ident">RawFloat</span><span class="op">></span>(<span class="ident">u</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">Big</span>, <span class="ident">v</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">Big</span>, <span class="ident">k</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">i16</span>) {
<span class="comment">// The bit length is an estimate of the base two logarithm, and log(u / v) = log(u) - log(v).</span>
<span class="comment">// The estimate is off by at most 1, but always an under-estimate, so the error on log(u)</span>
<span class="comment">// and log(v) are of the same sign and cancel out (if both are large). Therefore the error</span>
<span class="comment">// for log(u / v) is at most one as well.</span>
<span class="comment">// The target ratio is one where u/v is in an in-range significand. Thus our termination</span>
<span class="comment">// condition is log2(u / v) being the significand bits, plus/minus one.</span>
<span class="comment">// FIXME Looking at the second bit could improve the estimate and avoid some more divisions.</span>
<span class="kw">let</span> <span class="ident">target_ratio</span> <span class="op">=</span> <span class="ident">T</span>::<span class="ident">SIG_BITS</span> <span class="kw">as</span> <span class="ident">i16</span>;
<span class="kw">let</span> <span class="ident">log2_u</span> <span class="op">=</span> <span class="ident">u</span>.<span class="ident">bit_length</span>() <span class="kw">as</span> <span class="ident">i16</span>;
<span class="kw">let</span> <span class="ident">log2_v</span> <span class="op">=</span> <span class="ident">v</span>.<span class="ident">bit_length</span>() <span class="kw">as</span> <span class="ident">i16</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">u_shift</span>: <span class="ident">i16</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">v_shift</span>: <span class="ident">i16</span> <span class="op">=</span> <span class="number">0</span>;
<span class="macro">assert</span><span class="macro">!</span>(<span class="kw-2">*</span><span class="ident">k</span> <span class="op">==</span> <span class="number">0</span>);
<span class="kw">loop</span> {
<span class="kw">if</span> <span class="kw-2">*</span><span class="ident">k</span> <span class="op">==</span> <span class="ident">T</span>::<span class="ident">MIN_EXP_INT</span> {
<span class="comment">// Underflow or subnormal. Leave it to the main function.</span>
<span class="kw">break</span>;
}
<span class="kw">if</span> <span class="kw-2">*</span><span class="ident">k</span> <span class="op">==</span> <span class="ident">T</span>::<span class="ident">MAX_EXP_INT</span> {
<span class="comment">// Overflow. Leave it to the main function.</span>
<span class="kw">break</span>;
}
<span class="kw">let</span> <span class="ident">log2_ratio</span> <span class="op">=</span> (<span class="ident">log2_u</span> <span class="op">+</span> <span class="ident">u_shift</span>) <span class="op">-</span> (<span class="ident">log2_v</span> <span class="op">+</span> <span class="ident">v_shift</span>);
<span class="kw">if</span> <span class="ident">log2_ratio</span> <span class="op"><</span> <span class="ident">target_ratio</span> <span class="op">-</span> <span class="number">1</span> {
<span class="ident">u_shift</span> <span class="op">+=</span> <span class="number">1</span>;
<span class="kw-2">*</span><span class="ident">k</span> <span class="op">-=</span> <span class="number">1</span>;
} <span class="kw">else</span> <span class="kw">if</span> <span class="ident">log2_ratio</span> <span class="op">></span> <span class="ident">target_ratio</span> <span class="op">+</span> <span class="number">1</span> {
<span class="ident">v_shift</span> <span class="op">+=</span> <span class="number">1</span>;
<span class="kw-2">*</span><span class="ident">k</span> <span class="op">+=</span> <span class="number">1</span>;
} <span class="kw">else</span> {
<span class="kw">break</span>;
}
}
<span class="ident">u</span>.<span class="ident">mul_pow2</span>(<span class="ident">u_shift</span> <span class="kw">as</span> <span class="ident">usize</span>);
<span class="ident">v</span>.<span class="ident">mul_pow2</span>(<span class="ident">v_shift</span> <span class="kw">as</span> <span class="ident">usize</span>);
}
<span class="kw">fn</span> <span class="ident">underflow</span><span class="op"><</span><span class="ident">T</span>: <span class="ident">RawFloat</span><span class="op">></span>(<span class="ident">x</span>: <span class="ident">Big</span>, <span class="ident">v</span>: <span class="ident">Big</span>, <span class="ident">rem</span>: <span class="ident">Big</span>) <span class="op">-></span> <span class="ident">T</span> {
<span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="ident">Big</span>::<span class="ident">from_u64</span>(<span class="ident">T</span>::<span class="ident">MIN_SIG</span>) {
<span class="kw">let</span> <span class="ident">q</span> <span class="op">=</span> <span class="ident">num</span>::<span class="ident">to_u64</span>(<span class="kw-2">&</span><span class="ident">x</span>);
<span class="kw">let</span> <span class="ident">z</span> <span class="op">=</span> <span class="ident">rawfp</span>::<span class="ident">encode_subnormal</span>(<span class="ident">q</span>);
<span class="kw">return</span> <span class="ident">round_by_remainder</span>(<span class="ident">v</span>, <span class="ident">rem</span>, <span class="ident">q</span>, <span class="ident">z</span>);
}
<span class="comment">// Ratio isn't an in-range significand with the minimum exponent, so we need to round off</span>
<span class="comment">// excess bits and adjust the exponent accordingly. The real value now looks like this:</span>
<span class="comment">//</span>
<span class="comment">// x lsb</span>
<span class="comment">// /--------------\/</span>
<span class="comment">// 1010101010101010.10101010101010 * 2^k</span>
<span class="comment">// \-----/\-------/ \------------/</span>
<span class="comment">// q trunc. (represented by rem)</span>
<span class="comment">//</span>
<span class="comment">// Therefore, when the rounded-off bits are != 0.5 ULP, they decide the rounding</span>
<span class="comment">// on their own. When they are equal and the remainder is non-zero, the value still</span>
<span class="comment">// needs to be rounded up. Only when the rounded off bits are 1/2 and the remainder</span>
<span class="comment">// is zero, we have a half-to-even situation.</span>
<span class="kw">let</span> <span class="ident">bits</span> <span class="op">=</span> <span class="ident">x</span>.<span class="ident">bit_length</span>();
<span class="kw">let</span> <span class="ident">lsb</span> <span class="op">=</span> <span class="ident">bits</span> <span class="op">-</span> <span class="ident">T</span>::<span class="ident">SIG_BITS</span> <span class="kw">as</span> <span class="ident">usize</span>;
<span class="kw">let</span> <span class="ident">q</span> <span class="op">=</span> <span class="ident">num</span>::<span class="ident">get_bits</span>(<span class="kw-2">&</span><span class="ident">x</span>, <span class="ident">lsb</span>, <span class="ident">bits</span>);
<span class="kw">let</span> <span class="ident">k</span> <span class="op">=</span> <span class="ident">T</span>::<span class="ident">MIN_EXP_INT</span> <span class="op">+</span> <span class="ident">lsb</span> <span class="kw">as</span> <span class="ident">i16</span>;
<span class="kw">let</span> <span class="ident">z</span> <span class="op">=</span> <span class="ident">rawfp</span>::<span class="ident">encode_normal</span>(<span class="ident">Unpacked</span>::<span class="ident">new</span>(<span class="ident">q</span>, <span class="ident">k</span>));
<span class="kw">let</span> <span class="ident">q_even</span> <span class="op">=</span> <span class="ident">q</span> <span class="op">%</span> <span class="number">2</span> <span class="op">==</span> <span class="number">0</span>;
<span class="kw">match</span> <span class="ident">num</span>::<span class="ident">compare_with_half_ulp</span>(<span class="kw-2">&</span><span class="ident">x</span>, <span class="ident">lsb</span>) {
<span class="ident">Greater</span> <span class="op">=></span> <span class="ident">next_float</span>(<span class="ident">z</span>),
<span class="ident">Less</span> <span class="op">=></span> <span class="ident">z</span>,
<span class="ident">Equal</span> <span class="kw">if</span> <span class="ident">rem</span>.<span class="ident">is_zero</span>() <span class="op">&&</span> <span class="ident">q_even</span> <span class="op">=></span> <span class="ident">z</span>,
<span class="ident">Equal</span> <span class="op">=></span> <span class="ident">next_float</span>(<span class="ident">z</span>),
}
}
<span class="doccomment">/// Ordinary round-to-even, obfuscated by having to round based on the remainder of a division.</span>
<span class="kw">fn</span> <span class="ident">round_by_remainder</span><span class="op"><</span><span class="ident">T</span>: <span class="ident">RawFloat</span><span class="op">></span>(<span class="ident">v</span>: <span class="ident">Big</span>, <span class="ident">r</span>: <span class="ident">Big</span>, <span class="ident">q</span>: <span class="ident">u64</span>, <span class="ident">z</span>: <span class="ident">T</span>) <span class="op">-></span> <span class="ident">T</span> {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">v_minus_r</span> <span class="op">=</span> <span class="ident">v</span>;
<span class="ident">v_minus_r</span>.<span class="ident">sub</span>(<span class="kw-2">&</span><span class="ident">r</span>);
<span class="kw">if</span> <span class="ident">r</span> <span class="op"><</span> <span class="ident">v_minus_r</span> {
<span class="ident">z</span>
} <span class="kw">else</span> <span class="kw">if</span> <span class="ident">r</span> <span class="op">></span> <span class="ident">v_minus_r</span> {
<span class="ident">next_float</span>(<span class="ident">z</span>)
} <span class="kw">else</span> <span class="kw">if</span> <span class="ident">q</span> <span class="op">%</span> <span class="number">2</span> <span class="op">==</span> <span class="number">0</span> {
<span class="ident">z</span>
} <span class="kw">else</span> {
<span class="ident">next_float</span>(<span class="ident">z</span>)
}
}
</pre>
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