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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/num-bigint-0.2.6/src/algorithms.rs`."><meta name="keywords" content="rust, rustlang, rust-lang"><title>algorithms.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="../../num_bigint/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="../../num_bigint/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="kw">use </span>std::borrow::Cow;
<span class="kw">use </span>std::cmp;
<span class="kw">use </span>std::cmp::Ordering::{<span class="self">self</span>, Equal, Greater, Less};
<span class="kw">use </span>std::iter::repeat;
<span class="kw">use </span>std::mem;
<span class="kw">use </span>traits;
<span class="kw">use </span>traits::{One, Zero};
<span class="kw">use </span>biguint::BigUint;
<span class="kw">use </span>bigint::BigInt;
<span class="kw">use </span>bigint::Sign;
<span class="kw">use </span>bigint::Sign::{Minus, NoSign, Plus};
<span class="kw">use </span>big_digit::{<span class="self">self</span>, BigDigit, DoubleBigDigit, SignedDoubleBigDigit};
<span class="comment">// Generic functions for add/subtract/multiply with carry/borrow:
// Add with carry:
</span><span class="attribute">#[inline]
</span><span class="kw">fn </span>adc(a: BigDigit, b: BigDigit, acc: <span class="kw-2">&amp;mut </span>DoubleBigDigit) -&gt; BigDigit {
<span class="kw-2">*</span>acc += DoubleBigDigit::from(a);
<span class="kw-2">*</span>acc += DoubleBigDigit::from(b);
<span class="kw">let </span>lo = <span class="kw-2">*</span>acc <span class="kw">as </span>BigDigit;
<span class="kw-2">*</span>acc &gt;&gt;= big_digit::BITS;
lo
}
<span class="comment">// Subtract with borrow:
</span><span class="attribute">#[inline]
</span><span class="kw">fn </span>sbb(a: BigDigit, b: BigDigit, acc: <span class="kw-2">&amp;mut </span>SignedDoubleBigDigit) -&gt; BigDigit {
<span class="kw-2">*</span>acc += SignedDoubleBigDigit::from(a);
<span class="kw-2">*</span>acc -= SignedDoubleBigDigit::from(b);
<span class="kw">let </span>lo = <span class="kw-2">*</span>acc <span class="kw">as </span>BigDigit;
<span class="kw-2">*</span>acc &gt;&gt;= big_digit::BITS;
lo
}
<span class="attribute">#[inline]
</span><span class="kw">pub fn </span>mac_with_carry(a: BigDigit, b: BigDigit, c: BigDigit, acc: <span class="kw-2">&amp;mut </span>DoubleBigDigit) -&gt; BigDigit {
<span class="kw-2">*</span>acc += DoubleBigDigit::from(a);
<span class="kw-2">*</span>acc += DoubleBigDigit::from(b) * DoubleBigDigit::from(c);
<span class="kw">let </span>lo = <span class="kw-2">*</span>acc <span class="kw">as </span>BigDigit;
<span class="kw-2">*</span>acc &gt;&gt;= big_digit::BITS;
lo
}
<span class="attribute">#[inline]
</span><span class="kw">pub fn </span>mul_with_carry(a: BigDigit, b: BigDigit, acc: <span class="kw-2">&amp;mut </span>DoubleBigDigit) -&gt; BigDigit {
<span class="kw-2">*</span>acc += DoubleBigDigit::from(a) * DoubleBigDigit::from(b);
<span class="kw">let </span>lo = <span class="kw-2">*</span>acc <span class="kw">as </span>BigDigit;
<span class="kw-2">*</span>acc &gt;&gt;= big_digit::BITS;
lo
}
<span class="doccomment">/// Divide a two digit numerator by a one digit divisor, returns quotient and remainder:
///
/// Note: the caller must ensure that both the quotient and remainder will fit into a single digit.
/// This is _not_ true for an arbitrary numerator/denominator.
///
/// (This function also matches what the x86 divide instruction does).
</span><span class="attribute">#[inline]
</span><span class="kw">fn </span>div_wide(hi: BigDigit, lo: BigDigit, divisor: BigDigit) -&gt; (BigDigit, BigDigit) {
<span class="macro">debug_assert!</span>(hi &lt; divisor);
<span class="kw">let </span>lhs = big_digit::to_doublebigdigit(hi, lo);
<span class="kw">let </span>rhs = DoubleBigDigit::from(divisor);
((lhs / rhs) <span class="kw">as </span>BigDigit, (lhs % rhs) <span class="kw">as </span>BigDigit)
}
<span class="kw">pub fn </span>div_rem_digit(<span class="kw-2">mut </span>a: BigUint, b: BigDigit) -&gt; (BigUint, BigDigit) {
<span class="kw">let </span><span class="kw-2">mut </span>rem = <span class="number">0</span>;
<span class="kw">for </span>d <span class="kw">in </span>a.data.iter_mut().rev() {
<span class="kw">let </span>(q, r) = div_wide(rem, <span class="kw-2">*</span>d, b);
<span class="kw-2">*</span>d = q;
rem = r;
}
(a.normalized(), rem)
}
<span class="kw">pub fn </span>rem_digit(a: <span class="kw-2">&amp;</span>BigUint, b: BigDigit) -&gt; BigDigit {
<span class="kw">let </span><span class="kw-2">mut </span>rem: DoubleBigDigit = <span class="number">0</span>;
<span class="kw">for </span><span class="kw-2">&amp;</span>digit <span class="kw">in </span>a.data.iter().rev() {
rem = (rem &lt;&lt; big_digit::BITS) + DoubleBigDigit::from(digit);
rem %= DoubleBigDigit::from(b);
}
rem <span class="kw">as </span>BigDigit
}
<span class="comment">// Only for the Add impl:
</span><span class="attribute">#[inline]
</span><span class="kw">pub fn </span>__add2(a: <span class="kw-2">&amp;mut </span>[BigDigit], b: <span class="kw-2">&amp;</span>[BigDigit]) -&gt; BigDigit {
<span class="macro">debug_assert!</span>(a.len() &gt;= b.len());
<span class="kw">let </span><span class="kw-2">mut </span>carry = <span class="number">0</span>;
<span class="kw">let </span>(a_lo, a_hi) = a.split_at_mut(b.len());
<span class="kw">for </span>(a, b) <span class="kw">in </span>a_lo.iter_mut().zip(b) {
<span class="kw-2">*</span>a = adc(<span class="kw-2">*</span>a, <span class="kw-2">*</span>b, <span class="kw-2">&amp;mut </span>carry);
}
<span class="kw">if </span>carry != <span class="number">0 </span>{
<span class="kw">for </span>a <span class="kw">in </span>a_hi {
<span class="kw-2">*</span>a = adc(<span class="kw-2">*</span>a, <span class="number">0</span>, <span class="kw-2">&amp;mut </span>carry);
<span class="kw">if </span>carry == <span class="number">0 </span>{
<span class="kw">break</span>;
}
}
}
carry <span class="kw">as </span>BigDigit
}
<span class="doccomment">/// Two argument addition of raw slices:
/// a += b
///
/// The caller _must_ ensure that a is big enough to store the result - typically this means
/// resizing a to max(a.len(), b.len()) + 1, to fit a possible carry.
</span><span class="kw">pub fn </span>add2(a: <span class="kw-2">&amp;mut </span>[BigDigit], b: <span class="kw-2">&amp;</span>[BigDigit]) {
<span class="kw">let </span>carry = __add2(a, b);
<span class="macro">debug_assert!</span>(carry == <span class="number">0</span>);
}
<span class="kw">pub fn </span>sub2(a: <span class="kw-2">&amp;mut </span>[BigDigit], b: <span class="kw-2">&amp;</span>[BigDigit]) {
<span class="kw">let </span><span class="kw-2">mut </span>borrow = <span class="number">0</span>;
<span class="kw">let </span>len = cmp::min(a.len(), b.len());
<span class="kw">let </span>(a_lo, a_hi) = a.split_at_mut(len);
<span class="kw">let </span>(b_lo, b_hi) = b.split_at(len);
<span class="kw">for </span>(a, b) <span class="kw">in </span>a_lo.iter_mut().zip(b_lo) {
<span class="kw-2">*</span>a = sbb(<span class="kw-2">*</span>a, <span class="kw-2">*</span>b, <span class="kw-2">&amp;mut </span>borrow);
}
<span class="kw">if </span>borrow != <span class="number">0 </span>{
<span class="kw">for </span>a <span class="kw">in </span>a_hi {
<span class="kw-2">*</span>a = sbb(<span class="kw-2">*</span>a, <span class="number">0</span>, <span class="kw-2">&amp;mut </span>borrow);
<span class="kw">if </span>borrow == <span class="number">0 </span>{
<span class="kw">break</span>;
}
}
}
<span class="comment">// note: we&#39;re _required_ to fail on underflow
</span><span class="macro">assert!</span>(
borrow == <span class="number">0 </span>&amp;&amp; b_hi.iter().all(|x| <span class="kw-2">*</span>x == <span class="number">0</span>),
<span class="string">&quot;Cannot subtract b from a because b is larger than a.&quot;
</span>);
}
<span class="comment">// Only for the Sub impl. `a` and `b` must have same length.
</span><span class="attribute">#[inline]
</span><span class="kw">pub fn </span>__sub2rev(a: <span class="kw-2">&amp;</span>[BigDigit], b: <span class="kw-2">&amp;mut </span>[BigDigit]) -&gt; BigDigit {
<span class="macro">debug_assert!</span>(b.len() == a.len());
<span class="kw">let </span><span class="kw-2">mut </span>borrow = <span class="number">0</span>;
<span class="kw">for </span>(ai, bi) <span class="kw">in </span>a.iter().zip(b) {
<span class="kw-2">*</span>bi = sbb(<span class="kw-2">*</span>ai, <span class="kw-2">*</span>bi, <span class="kw-2">&amp;mut </span>borrow);
}
borrow <span class="kw">as </span>BigDigit
}
<span class="kw">pub fn </span>sub2rev(a: <span class="kw-2">&amp;</span>[BigDigit], b: <span class="kw-2">&amp;mut </span>[BigDigit]) {
<span class="macro">debug_assert!</span>(b.len() &gt;= a.len());
<span class="kw">let </span>len = cmp::min(a.len(), b.len());
<span class="kw">let </span>(a_lo, a_hi) = a.split_at(len);
<span class="kw">let </span>(b_lo, b_hi) = b.split_at_mut(len);
<span class="kw">let </span>borrow = __sub2rev(a_lo, b_lo);
<span class="macro">assert!</span>(a_hi.is_empty());
<span class="comment">// note: we&#39;re _required_ to fail on underflow
</span><span class="macro">assert!</span>(
borrow == <span class="number">0 </span>&amp;&amp; b_hi.iter().all(|x| <span class="kw-2">*</span>x == <span class="number">0</span>),
<span class="string">&quot;Cannot subtract b from a because b is larger than a.&quot;
</span>);
}
<span class="kw">pub fn </span>sub_sign(a: <span class="kw-2">&amp;</span>[BigDigit], b: <span class="kw-2">&amp;</span>[BigDigit]) -&gt; (Sign, BigUint) {
<span class="comment">// Normalize:
</span><span class="kw">let </span>a = <span class="kw-2">&amp;</span>a[..a.iter().rposition(|<span class="kw-2">&amp;</span>x| x != <span class="number">0</span>).map_or(<span class="number">0</span>, |i| i + <span class="number">1</span>)];
<span class="kw">let </span>b = <span class="kw-2">&amp;</span>b[..b.iter().rposition(|<span class="kw-2">&amp;</span>x| x != <span class="number">0</span>).map_or(<span class="number">0</span>, |i| i + <span class="number">1</span>)];
<span class="kw">match </span>cmp_slice(a, b) {
Greater =&gt; {
<span class="kw">let </span><span class="kw-2">mut </span>a = a.to_vec();
sub2(<span class="kw-2">&amp;mut </span>a, b);
(Plus, BigUint::new(a))
}
Less =&gt; {
<span class="kw">let </span><span class="kw-2">mut </span>b = b.to_vec();
sub2(<span class="kw-2">&amp;mut </span>b, a);
(Minus, BigUint::new(b))
}
<span class="kw">_ </span>=&gt; (NoSign, Zero::zero()),
}
}
<span class="doccomment">/// Three argument multiply accumulate:
/// acc += b * c
</span><span class="kw">pub fn </span>mac_digit(acc: <span class="kw-2">&amp;mut </span>[BigDigit], b: <span class="kw-2">&amp;</span>[BigDigit], c: BigDigit) {
<span class="kw">if </span>c == <span class="number">0 </span>{
<span class="kw">return</span>;
}
<span class="kw">let </span><span class="kw-2">mut </span>carry = <span class="number">0</span>;
<span class="kw">let </span>(a_lo, a_hi) = acc.split_at_mut(b.len());
<span class="kw">for </span>(a, <span class="kw-2">&amp;</span>b) <span class="kw">in </span>a_lo.iter_mut().zip(b) {
<span class="kw-2">*</span>a = mac_with_carry(<span class="kw-2">*</span>a, b, c, <span class="kw-2">&amp;mut </span>carry);
}
<span class="kw">let </span><span class="kw-2">mut </span>a = a_hi.iter_mut();
<span class="kw">while </span>carry != <span class="number">0 </span>{
<span class="kw">let </span>a = a.next().expect(<span class="string">&quot;carry overflow during multiplication!&quot;</span>);
<span class="kw-2">*</span>a = adc(<span class="kw-2">*</span>a, <span class="number">0</span>, <span class="kw-2">&amp;mut </span>carry);
}
}
<span class="doccomment">/// Three argument multiply accumulate:
/// acc += b * c
</span><span class="kw">fn </span>mac3(acc: <span class="kw-2">&amp;mut </span>[BigDigit], b: <span class="kw-2">&amp;</span>[BigDigit], c: <span class="kw-2">&amp;</span>[BigDigit]) {
<span class="kw">let </span>(x, y) = <span class="kw">if </span>b.len() &lt; c.len() { (b, c) } <span class="kw">else </span>{ (c, b) };
<span class="comment">// We use three algorithms for different input sizes.
//
// - For small inputs, long multiplication is fastest.
// - Next we use Karatsuba multiplication (Toom-2), which we have optimized
// to avoid unnecessary allocations for intermediate values.
// - For the largest inputs we use Toom-3, which better optimizes the
// number of operations, but uses more temporary allocations.
//
// The thresholds are somewhat arbitrary, chosen by evaluating the results
// of `cargo bench --bench bigint multiply`.
</span><span class="kw">if </span>x.len() &lt;= <span class="number">32 </span>{
<span class="comment">// Long multiplication:
</span><span class="kw">for </span>(i, xi) <span class="kw">in </span>x.iter().enumerate() {
mac_digit(<span class="kw-2">&amp;mut </span>acc[i..], y, <span class="kw-2">*</span>xi);
}
} <span class="kw">else if </span>x.len() &lt;= <span class="number">256 </span>{
<span class="comment">/*
* Karatsuba multiplication:
*
* The idea is that we break x and y up into two smaller numbers that each have about half
* as many digits, like so (note that multiplying by b is just a shift):
*
* x = x0 + x1 * b
* y = y0 + y1 * b
*
* With some algebra, we can compute x * y with three smaller products, where the inputs to
* each of the smaller products have only about half as many digits as x and y:
*
* x * y = (x0 + x1 * b) * (y0 + y1 * b)
*
* x * y = x0 * y0
* + x0 * y1 * b
* + x1 * y0 * b
* + x1 * y1 * b^2
*
* Let p0 = x0 * y0 and p2 = x1 * y1:
*
* x * y = p0
* + (x0 * y1 + x1 * y0) * b
* + p2 * b^2
*
* The real trick is that middle term:
*
* x0 * y1 + x1 * y0
*
* = x0 * y1 + x1 * y0 - p0 + p0 - p2 + p2
*
* = x0 * y1 + x1 * y0 - x0 * y0 - x1 * y1 + p0 + p2
*
* Now we complete the square:
*
* = -(x0 * y0 - x0 * y1 - x1 * y0 + x1 * y1) + p0 + p2
*
* = -((x1 - x0) * (y1 - y0)) + p0 + p2
*
* Let p1 = (x1 - x0) * (y1 - y0), and substitute back into our original formula:
*
* x * y = p0
* + (p0 + p2 - p1) * b
* + p2 * b^2
*
* Where the three intermediate products are:
*
* p0 = x0 * y0
* p1 = (x1 - x0) * (y1 - y0)
* p2 = x1 * y1
*
* In doing the computation, we take great care to avoid unnecessary temporary variables
* (since creating a BigUint requires a heap allocation): thus, we rearrange the formula a
* bit so we can use the same temporary variable for all the intermediate products:
*
* x * y = p2 * b^2 + p2 * b
* + p0 * b + p0
* - p1 * b
*
* The other trick we use is instead of doing explicit shifts, we slice acc at the
* appropriate offset when doing the add.
*/
/*
* When x is smaller than y, it&#39;s significantly faster to pick b such that x is split in
* half, not y:
*/
</span><span class="kw">let </span>b = x.len() / <span class="number">2</span>;
<span class="kw">let </span>(x0, x1) = x.split_at(b);
<span class="kw">let </span>(y0, y1) = y.split_at(b);
<span class="comment">/*
* We reuse the same BigUint for all the intermediate multiplies and have to size p
* appropriately here: x1.len() &gt;= x0.len and y1.len() &gt;= y0.len():
*/
</span><span class="kw">let </span>len = x1.len() + y1.len() + <span class="number">1</span>;
<span class="kw">let </span><span class="kw-2">mut </span>p = BigUint { data: <span class="macro">vec!</span>[<span class="number">0</span>; len] };
<span class="comment">// p2 = x1 * y1
</span>mac3(<span class="kw-2">&amp;mut </span>p.data[..], x1, y1);
<span class="comment">// Not required, but the adds go faster if we drop any unneeded 0s from the end:
</span>p.normalize();
add2(<span class="kw-2">&amp;mut </span>acc[b..], <span class="kw-2">&amp;</span>p.data[..]);
add2(<span class="kw-2">&amp;mut </span>acc[b * <span class="number">2</span>..], <span class="kw-2">&amp;</span>p.data[..]);
<span class="comment">// Zero out p before the next multiply:
</span>p.data.truncate(<span class="number">0</span>);
p.data.extend(repeat(<span class="number">0</span>).take(len));
<span class="comment">// p0 = x0 * y0
</span>mac3(<span class="kw-2">&amp;mut </span>p.data[..], x0, y0);
p.normalize();
add2(<span class="kw-2">&amp;mut </span>acc[..], <span class="kw-2">&amp;</span>p.data[..]);
add2(<span class="kw-2">&amp;mut </span>acc[b..], <span class="kw-2">&amp;</span>p.data[..]);
<span class="comment">// p1 = (x1 - x0) * (y1 - y0)
// We do this one last, since it may be negative and acc can&#39;t ever be negative:
</span><span class="kw">let </span>(j0_sign, j0) = sub_sign(x1, x0);
<span class="kw">let </span>(j1_sign, j1) = sub_sign(y1, y0);
<span class="kw">match </span>j0_sign * j1_sign {
Plus =&gt; {
p.data.truncate(<span class="number">0</span>);
p.data.extend(repeat(<span class="number">0</span>).take(len));
mac3(<span class="kw-2">&amp;mut </span>p.data[..], <span class="kw-2">&amp;</span>j0.data[..], <span class="kw-2">&amp;</span>j1.data[..]);
p.normalize();
sub2(<span class="kw-2">&amp;mut </span>acc[b..], <span class="kw-2">&amp;</span>p.data[..]);
}
Minus =&gt; {
mac3(<span class="kw-2">&amp;mut </span>acc[b..], <span class="kw-2">&amp;</span>j0.data[..], <span class="kw-2">&amp;</span>j1.data[..]);
}
NoSign =&gt; (),
}
} <span class="kw">else </span>{
<span class="comment">// Toom-3 multiplication:
//
// Toom-3 is like Karatsuba above, but dividing the inputs into three parts.
// Both are instances of Toom-Cook, using `k=3` and `k=2` respectively.
//
// The general idea is to treat the large integers digits as
// polynomials of a certain degree and determine the coefficients/digits
// of the product of the two via interpolation of the polynomial product.
</span><span class="kw">let </span>i = y.len() / <span class="number">3 </span>+ <span class="number">1</span>;
<span class="kw">let </span>x0_len = cmp::min(x.len(), i);
<span class="kw">let </span>x1_len = cmp::min(x.len() - x0_len, i);
<span class="kw">let </span>y0_len = i;
<span class="kw">let </span>y1_len = cmp::min(y.len() - y0_len, i);
<span class="comment">// Break x and y into three parts, representating an order two polynomial.
// t is chosen to be the size of a digit so we can use faster shifts
// in place of multiplications.
//
// x(t) = x2*t^2 + x1*t + x0
</span><span class="kw">let </span>x0 = BigInt::from_slice(Plus, <span class="kw-2">&amp;</span>x[..x0_len]);
<span class="kw">let </span>x1 = BigInt::from_slice(Plus, <span class="kw-2">&amp;</span>x[x0_len..x0_len + x1_len]);
<span class="kw">let </span>x2 = BigInt::from_slice(Plus, <span class="kw-2">&amp;</span>x[x0_len + x1_len..]);
<span class="comment">// y(t) = y2*t^2 + y1*t + y0
</span><span class="kw">let </span>y0 = BigInt::from_slice(Plus, <span class="kw-2">&amp;</span>y[..y0_len]);
<span class="kw">let </span>y1 = BigInt::from_slice(Plus, <span class="kw-2">&amp;</span>y[y0_len..y0_len + y1_len]);
<span class="kw">let </span>y2 = BigInt::from_slice(Plus, <span class="kw-2">&amp;</span>y[y0_len + y1_len..]);
<span class="comment">// Let w(t) = x(t) * y(t)
//
// This gives us the following order-4 polynomial.
//
// w(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
//
// We need to find the coefficients w4, w3, w2, w1 and w0. Instead
// of simply multiplying the x and y in total, we can evaluate w
// at 5 points. An n-degree polynomial is uniquely identified by (n + 1)
// points.
//
// It is arbitrary as to what points we evaluate w at but we use the
// following.
//
// w(t) at t = 0, 1, -1, -2 and inf
//
// The values for w(t) in terms of x(t)*y(t) at these points are:
//
// let a = w(0) = x0 * y0
// let b = w(1) = (x2 + x1 + x0) * (y2 + y1 + y0)
// let c = w(-1) = (x2 - x1 + x0) * (y2 - y1 + y0)
// let d = w(-2) = (4*x2 - 2*x1 + x0) * (4*y2 - 2*y1 + y0)
// let e = w(inf) = x2 * y2 as t -&gt; inf
// x0 + x2, avoiding temporaries
</span><span class="kw">let </span>p = <span class="kw-2">&amp;</span>x0 + <span class="kw-2">&amp;</span>x2;
<span class="comment">// y0 + y2, avoiding temporaries
</span><span class="kw">let </span>q = <span class="kw-2">&amp;</span>y0 + <span class="kw-2">&amp;</span>y2;
<span class="comment">// x2 - x1 + x0, avoiding temporaries
</span><span class="kw">let </span>p2 = <span class="kw-2">&amp;</span>p - <span class="kw-2">&amp;</span>x1;
<span class="comment">// y2 - y1 + y0, avoiding temporaries
</span><span class="kw">let </span>q2 = <span class="kw-2">&amp;</span>q - <span class="kw-2">&amp;</span>y1;
<span class="comment">// w(0)
</span><span class="kw">let </span>r0 = <span class="kw-2">&amp;</span>x0 * <span class="kw-2">&amp;</span>y0;
<span class="comment">// w(inf)
</span><span class="kw">let </span>r4 = <span class="kw-2">&amp;</span>x2 * <span class="kw-2">&amp;</span>y2;
<span class="comment">// w(1)
</span><span class="kw">let </span>r1 = (p + x1) * (q + y1);
<span class="comment">// w(-1)
</span><span class="kw">let </span>r2 = <span class="kw-2">&amp;</span>p2 * <span class="kw-2">&amp;</span>q2;
<span class="comment">// w(-2)
</span><span class="kw">let </span>r3 = ((p2 + x2) * <span class="number">2 </span>- x0) * ((q2 + y2) * <span class="number">2 </span>- y0);
<span class="comment">// Evaluating these points gives us the following system of linear equations.
//
// 0 0 0 0 1 | a
// 1 1 1 1 1 | b
// 1 -1 1 -1 1 | c
// 16 -8 4 -2 1 | d
// 1 0 0 0 0 | e
//
// The solved equation (after gaussian elimination or similar)
// in terms of its coefficients:
//
// w0 = w(0)
// w1 = w(0)/2 + w(1)/3 - w(-1) + w(2)/6 - 2*w(inf)
// w2 = -w(0) + w(1)/2 + w(-1)/2 - w(inf)
// w3 = -w(0)/2 + w(1)/6 + w(-1)/2 - w(1)/6
// w4 = w(inf)
//
// This particular sequence is given by Bodrato and is an interpolation
// of the above equations.
</span><span class="kw">let </span><span class="kw-2">mut </span>comp3: BigInt = (r3 - <span class="kw-2">&amp;</span>r1) / <span class="number">3</span>;
<span class="kw">let </span><span class="kw-2">mut </span>comp1: BigInt = (r1 - <span class="kw-2">&amp;</span>r2) / <span class="number">2</span>;
<span class="kw">let </span><span class="kw-2">mut </span>comp2: BigInt = r2 - <span class="kw-2">&amp;</span>r0;
comp3 = (<span class="kw-2">&amp;</span>comp2 - comp3) / <span class="number">2 </span>+ <span class="kw-2">&amp;</span>r4 * <span class="number">2</span>;
comp2 += <span class="kw-2">&amp;</span>comp1 - <span class="kw-2">&amp;</span>r4;
comp1 -= <span class="kw-2">&amp;</span>comp3;
<span class="comment">// Recomposition. The coefficients of the polynomial are now known.
//
// Evaluate at w(t) where t is our given base to get the result.
</span><span class="kw">let </span>result = r0
+ (comp1 &lt;&lt; (<span class="number">32 </span>* i))
+ (comp2 &lt;&lt; (<span class="number">2 </span>* <span class="number">32 </span>* i))
+ (comp3 &lt;&lt; (<span class="number">3 </span>* <span class="number">32 </span>* i))
+ (r4 &lt;&lt; (<span class="number">4 </span>* <span class="number">32 </span>* i));
<span class="kw">let </span>result_pos = result.to_biguint().unwrap();
add2(<span class="kw-2">&amp;mut </span>acc[..], <span class="kw-2">&amp;</span>result_pos.data);
}
}
<span class="kw">pub fn </span>mul3(x: <span class="kw-2">&amp;</span>[BigDigit], y: <span class="kw-2">&amp;</span>[BigDigit]) -&gt; BigUint {
<span class="kw">let </span>len = x.len() + y.len() + <span class="number">1</span>;
<span class="kw">let </span><span class="kw-2">mut </span>prod = BigUint { data: <span class="macro">vec!</span>[<span class="number">0</span>; len] };
mac3(<span class="kw-2">&amp;mut </span>prod.data[..], x, y);
prod.normalized()
}
<span class="kw">pub fn </span>scalar_mul(a: <span class="kw-2">&amp;mut </span>[BigDigit], b: BigDigit) -&gt; BigDigit {
<span class="kw">let </span><span class="kw-2">mut </span>carry = <span class="number">0</span>;
<span class="kw">for </span>a <span class="kw">in </span>a.iter_mut() {
<span class="kw-2">*</span>a = mul_with_carry(<span class="kw-2">*</span>a, b, <span class="kw-2">&amp;mut </span>carry);
}
carry <span class="kw">as </span>BigDigit
}
<span class="kw">pub fn </span>div_rem(<span class="kw-2">mut </span>u: BigUint, <span class="kw-2">mut </span>d: BigUint) -&gt; (BigUint, BigUint) {
<span class="kw">if </span>d.is_zero() {
<span class="macro">panic!</span>()
}
<span class="kw">if </span>u.is_zero() {
<span class="kw">return </span>(Zero::zero(), Zero::zero());
}
<span class="kw">if </span>d.data.len() == <span class="number">1 </span>{
<span class="kw">if </span>d.data == [<span class="number">1</span>] {
<span class="kw">return </span>(u, Zero::zero());
}
<span class="kw">let </span>(div, rem) = div_rem_digit(u, d.data[<span class="number">0</span>]);
<span class="comment">// reuse d
</span>d.data.clear();
d += rem;
<span class="kw">return </span>(div, d);
}
<span class="comment">// Required or the q_len calculation below can underflow:
</span><span class="kw">match </span>u.cmp(<span class="kw-2">&amp;</span>d) {
Less =&gt; <span class="kw">return </span>(Zero::zero(), u),
Equal =&gt; {
u.set_one();
<span class="kw">return </span>(u, Zero::zero());
}
Greater =&gt; {} <span class="comment">// Do nothing
</span>}
<span class="comment">// This algorithm is from Knuth, TAOCP vol 2 section 4.3, algorithm D:
//
// First, normalize the arguments so the highest bit in the highest digit of the divisor is
// set: the main loop uses the highest digit of the divisor for generating guesses, so we
// want it to be the largest number we can efficiently divide by.
//
</span><span class="kw">let </span>shift = d.data.last().unwrap().leading_zeros() <span class="kw">as </span>usize;
<span class="kw">let </span>(q, r) = <span class="kw">if </span>shift == <span class="number">0 </span>{
<span class="comment">// no need to clone d
</span>div_rem_core(u, <span class="kw-2">&amp;</span>d)
} <span class="kw">else </span>{
div_rem_core(u &lt;&lt; shift, <span class="kw-2">&amp;</span>(d &lt;&lt; shift))
};
<span class="comment">// renormalize the remainder
</span>(q, r &gt;&gt; shift)
}
<span class="kw">pub fn </span>div_rem_ref(u: <span class="kw-2">&amp;</span>BigUint, d: <span class="kw-2">&amp;</span>BigUint) -&gt; (BigUint, BigUint) {
<span class="kw">if </span>d.is_zero() {
<span class="macro">panic!</span>()
}
<span class="kw">if </span>u.is_zero() {
<span class="kw">return </span>(Zero::zero(), Zero::zero());
}
<span class="kw">if </span>d.data.len() == <span class="number">1 </span>{
<span class="kw">if </span>d.data == [<span class="number">1</span>] {
<span class="kw">return </span>(u.clone(), Zero::zero());
}
<span class="kw">let </span>(div, rem) = div_rem_digit(u.clone(), d.data[<span class="number">0</span>]);
<span class="kw">return </span>(div, rem.into());
}
<span class="comment">// Required or the q_len calculation below can underflow:
</span><span class="kw">match </span>u.cmp(d) {
Less =&gt; <span class="kw">return </span>(Zero::zero(), u.clone()),
Equal =&gt; <span class="kw">return </span>(One::one(), Zero::zero()),
Greater =&gt; {} <span class="comment">// Do nothing
</span>}
<span class="comment">// This algorithm is from Knuth, TAOCP vol 2 section 4.3, algorithm D:
//
// First, normalize the arguments so the highest bit in the highest digit of the divisor is
// set: the main loop uses the highest digit of the divisor for generating guesses, so we
// want it to be the largest number we can efficiently divide by.
//
</span><span class="kw">let </span>shift = d.data.last().unwrap().leading_zeros() <span class="kw">as </span>usize;
<span class="kw">let </span>(q, r) = <span class="kw">if </span>shift == <span class="number">0 </span>{
<span class="comment">// no need to clone d
</span>div_rem_core(u.clone(), d)
} <span class="kw">else </span>{
div_rem_core(u &lt;&lt; shift, <span class="kw-2">&amp;</span>(d &lt;&lt; shift))
};
<span class="comment">// renormalize the remainder
</span>(q, r &gt;&gt; shift)
}
<span class="doccomment">/// an implementation of Knuth, TAOCP vol 2 section 4.3, algorithm D
///
/// # Correctness
///
/// This function requires the following conditions to run correctly and/or effectively
///
/// - `a &gt; b`
/// - `d.data.len() &gt; 1`
/// - `d.data.last().unwrap().leading_zeros() == 0`
</span><span class="kw">fn </span>div_rem_core(<span class="kw-2">mut </span>a: BigUint, b: <span class="kw-2">&amp;</span>BigUint) -&gt; (BigUint, BigUint) {
<span class="comment">// The algorithm works by incrementally calculating &quot;guesses&quot;, q0, for part of the
// remainder. Once we have any number q0 such that q0 * b &lt;= a, we can set
//
// q += q0
// a -= q0 * b
//
// and then iterate until a &lt; b. Then, (q, a) will be our desired quotient and remainder.
//
// q0, our guess, is calculated by dividing the last few digits of a by the last digit of b
// - this should give us a guess that is &quot;close&quot; to the actual quotient, but is possibly
// greater than the actual quotient. If q0 * b &gt; a, we simply use iterated subtraction
// until we have a guess such that q0 * b &lt;= a.
//
</span><span class="kw">let </span>bn = <span class="kw-2">*</span>b.data.last().unwrap();
<span class="kw">let </span>q_len = a.data.len() - b.data.len() + <span class="number">1</span>;
<span class="kw">let </span><span class="kw-2">mut </span>q = BigUint {
data: <span class="macro">vec!</span>[<span class="number">0</span>; q_len],
};
<span class="comment">// We reuse the same temporary to avoid hitting the allocator in our inner loop - this is
// sized to hold a0 (in the common case; if a particular digit of the quotient is zero a0
// can be bigger).
//
</span><span class="kw">let </span><span class="kw-2">mut </span>tmp = BigUint {
data: Vec::with_capacity(<span class="number">2</span>),
};
<span class="kw">for </span>j <span class="kw">in </span>(<span class="number">0</span>..q_len).rev() {
<span class="comment">/*
* When calculating our next guess q0, we don&#39;t need to consider the digits below j
* + b.data.len() - 1: we&#39;re guessing digit j of the quotient (i.e. q0 &lt;&lt; j) from
* digit bn of the divisor (i.e. bn &lt;&lt; (b.data.len() - 1) - so the product of those
* two numbers will be zero in all digits up to (j + b.data.len() - 1).
*/
</span><span class="kw">let </span>offset = j + b.data.len() - <span class="number">1</span>;
<span class="kw">if </span>offset &gt;= a.data.len() {
<span class="kw">continue</span>;
}
<span class="comment">/* just avoiding a heap allocation: */
</span><span class="kw">let </span><span class="kw-2">mut </span>a0 = tmp;
a0.data.truncate(<span class="number">0</span>);
a0.data.extend(a.data[offset..].iter().cloned());
<span class="comment">/*
* q0 &lt;&lt; j * big_digit::BITS is our actual quotient estimate - we do the shifts
* implicitly at the end, when adding and subtracting to a and q. Not only do we
* save the cost of the shifts, the rest of the arithmetic gets to work with
* smaller numbers.
*/
</span><span class="kw">let </span>(<span class="kw-2">mut </span>q0, <span class="kw">_</span>) = div_rem_digit(a0, bn);
<span class="kw">let </span><span class="kw-2">mut </span>prod = b * <span class="kw-2">&amp;</span>q0;
<span class="kw">while </span>cmp_slice(<span class="kw-2">&amp;</span>prod.data[..], <span class="kw-2">&amp;</span>a.data[j..]) == Greater {
<span class="kw">let </span>one: BigUint = One::one();
q0 -= one;
prod -= b;
}
add2(<span class="kw-2">&amp;mut </span>q.data[j..], <span class="kw-2">&amp;</span>q0.data[..]);
sub2(<span class="kw-2">&amp;mut </span>a.data[j..], <span class="kw-2">&amp;</span>prod.data[..]);
a.normalize();
tmp = q0;
}
<span class="macro">debug_assert!</span>(a &lt; <span class="kw-2">*</span>b);
(q.normalized(), a)
}
<span class="doccomment">/// Find last set bit
/// fls(0) == 0, fls(u32::MAX) == 32
</span><span class="kw">pub fn </span>fls&lt;T: traits::PrimInt&gt;(v: T) -&gt; usize {
mem::size_of::&lt;T&gt;() * <span class="number">8 </span>- v.leading_zeros() <span class="kw">as </span>usize
}
<span class="kw">pub fn </span>ilog2&lt;T: traits::PrimInt&gt;(v: T) -&gt; usize {
fls(v) - <span class="number">1
</span>}
<span class="attribute">#[inline]
</span><span class="kw">pub fn </span>biguint_shl(n: Cow&lt;BigUint&gt;, bits: usize) -&gt; BigUint {
<span class="kw">let </span>n_unit = bits / big_digit::BITS;
<span class="kw">let </span><span class="kw-2">mut </span>data = <span class="kw">match </span>n_unit {
<span class="number">0 </span>=&gt; n.into_owned().data,
<span class="kw">_ </span>=&gt; {
<span class="kw">let </span>len = n_unit + n.data.len() + <span class="number">1</span>;
<span class="kw">let </span><span class="kw-2">mut </span>data = Vec::with_capacity(len);
data.extend(repeat(<span class="number">0</span>).take(n_unit));
data.extend(n.data.iter().cloned());
data
}
};
<span class="kw">let </span>n_bits = bits % big_digit::BITS;
<span class="kw">if </span>n_bits &gt; <span class="number">0 </span>{
<span class="kw">let </span><span class="kw-2">mut </span>carry = <span class="number">0</span>;
<span class="kw">for </span>elem <span class="kw">in </span>data[n_unit..].iter_mut() {
<span class="kw">let </span>new_carry = <span class="kw-2">*</span>elem &gt;&gt; (big_digit::BITS - n_bits);
<span class="kw-2">*</span>elem = (<span class="kw-2">*</span>elem &lt;&lt; n_bits) | carry;
carry = new_carry;
}
<span class="kw">if </span>carry != <span class="number">0 </span>{
data.push(carry);
}
}
BigUint::new(data)
}
<span class="attribute">#[inline]
</span><span class="kw">pub fn </span>biguint_shr(n: Cow&lt;BigUint&gt;, bits: usize) -&gt; BigUint {
<span class="kw">let </span>n_unit = bits / big_digit::BITS;
<span class="kw">if </span>n_unit &gt;= n.data.len() {
<span class="kw">return </span>Zero::zero();
}
<span class="kw">let </span><span class="kw-2">mut </span>data = <span class="kw">match </span>n {
Cow::Borrowed(n) =&gt; n.data[n_unit..].to_vec(),
Cow::Owned(<span class="kw-2">mut </span>n) =&gt; {
n.data.drain(..n_unit);
n.data
}
};
<span class="kw">let </span>n_bits = bits % big_digit::BITS;
<span class="kw">if </span>n_bits &gt; <span class="number">0 </span>{
<span class="kw">let </span><span class="kw-2">mut </span>borrow = <span class="number">0</span>;
<span class="kw">for </span>elem <span class="kw">in </span>data.iter_mut().rev() {
<span class="kw">let </span>new_borrow = <span class="kw-2">*</span>elem &lt;&lt; (big_digit::BITS - n_bits);
<span class="kw-2">*</span>elem = (<span class="kw-2">*</span>elem &gt;&gt; n_bits) | borrow;
borrow = new_borrow;
}
}
BigUint::new(data)
}
<span class="kw">pub fn </span>cmp_slice(a: <span class="kw-2">&amp;</span>[BigDigit], b: <span class="kw-2">&amp;</span>[BigDigit]) -&gt; Ordering {
<span class="macro">debug_assert!</span>(a.last() != <span class="prelude-val">Some</span>(<span class="kw-2">&amp;</span><span class="number">0</span>));
<span class="macro">debug_assert!</span>(b.last() != <span class="prelude-val">Some</span>(<span class="kw-2">&amp;</span><span class="number">0</span>));
<span class="kw">let </span>(a_len, b_len) = (a.len(), b.len());
<span class="kw">if </span>a_len &lt; b_len {
<span class="kw">return </span>Less;
}
<span class="kw">if </span>a_len &gt; b_len {
<span class="kw">return </span>Greater;
}
<span class="kw">for </span>(<span class="kw-2">&amp;</span>ai, <span class="kw-2">&amp;</span>bi) <span class="kw">in </span>a.iter().rev().zip(b.iter().rev()) {
<span class="kw">if </span>ai &lt; bi {
<span class="kw">return </span>Less;
}
<span class="kw">if </span>ai &gt; bi {
<span class="kw">return </span>Greater;
}
}
Equal
}
<span class="attribute">#[cfg(test)]
</span><span class="kw">mod </span>algorithm_tests {
<span class="kw">use </span>big_digit::BigDigit;
<span class="kw">use </span>traits::Num;
<span class="kw">use </span>Sign::Plus;
<span class="kw">use </span>{BigInt, BigUint};
<span class="attribute">#[test]
</span><span class="kw">fn </span>test_sub_sign() {
<span class="kw">use </span><span class="kw">super</span>::sub_sign;
<span class="kw">fn </span>sub_sign_i(a: <span class="kw-2">&amp;</span>[BigDigit], b: <span class="kw-2">&amp;</span>[BigDigit]) -&gt; BigInt {
<span class="kw">let </span>(sign, val) = sub_sign(a, b);
BigInt::from_biguint(sign, val)
}
<span class="kw">let </span>a = BigUint::from_str_radix(<span class="string">&quot;265252859812191058636308480000000&quot;</span>, <span class="number">10</span>).unwrap();
<span class="kw">let </span>b = BigUint::from_str_radix(<span class="string">&quot;26525285981219105863630848000000&quot;</span>, <span class="number">10</span>).unwrap();
<span class="kw">let </span>a_i = BigInt::from_biguint(Plus, a.clone());
<span class="kw">let </span>b_i = BigInt::from_biguint(Plus, b.clone());
<span class="macro">assert_eq!</span>(sub_sign_i(<span class="kw-2">&amp;</span>a.data[..], <span class="kw-2">&amp;</span>b.data[..]), <span class="kw-2">&amp;</span>a_i - <span class="kw-2">&amp;</span>b_i);
<span class="macro">assert_eq!</span>(sub_sign_i(<span class="kw-2">&amp;</span>b.data[..], <span class="kw-2">&amp;</span>a.data[..]), <span class="kw-2">&amp;</span>b_i - <span class="kw-2">&amp;</span>a_i);
}
}
</code></pre></div>
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