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</pre><pre class="rust ">
<span class="kw">use</span> <span class="ident">std</span>::<span class="ident">borrow</span>::<span class="ident">Cow</span>;
<span class="kw">use</span> <span class="ident">std</span>::<span class="ident">cmp</span>;
<span class="kw">use</span> <span class="ident">std</span>::<span class="ident">cmp</span>::<span class="ident">Ordering</span>::{<span class="self">self</span>, <span class="ident">Less</span>, <span class="ident">Greater</span>, <span class="ident">Equal</span>};
<span class="kw">use</span> <span class="ident">std</span>::<span class="ident">iter</span>::<span class="ident">repeat</span>;
<span class="kw">use</span> <span class="ident">std</span>::<span class="ident">mem</span>;
<span class="kw">use</span> <span class="ident">traits</span>;
<span class="kw">use</span> <span class="ident">traits</span>::{<span class="ident">Zero</span>, <span class="ident">One</span>};
<span class="kw">use</span> <span class="ident">biguint</span>::<span class="ident">BigUint</span>;
<span class="kw">use</span> <span class="ident">bigint</span>::<span class="ident">BigInt</span>;
<span class="kw">use</span> <span class="ident">bigint</span>::<span class="ident">Sign</span>;
<span class="kw">use</span> <span class="ident">bigint</span>::<span class="ident">Sign</span>::{<span class="ident">Minus</span>, <span class="ident">NoSign</span>, <span class="ident">Plus</span>};
<span class="attribute">#[<span class="ident">allow</span>(<span class="ident">non_snake_case</span>)]</span>
<span class="kw">pub</span> <span class="kw">mod</span> <span class="ident">big_digit</span> {
<span class="doccomment">/// A `BigDigit` is a `BigUint`'s composing element.</span>
<span class="kw">pub</span> <span class="kw">type</span> <span class="ident">BigDigit</span> <span class="op">=</span> <span class="ident">u32</span>;
<span class="doccomment">/// A `DoubleBigDigit` is the internal type used to do the computations. Its</span>
<span class="doccomment">/// size is the double of the size of `BigDigit`.</span>
<span class="kw">pub</span> <span class="kw">type</span> <span class="ident">DoubleBigDigit</span> <span class="op">=</span> <span class="ident">u64</span>;
<span class="kw">pub</span> <span class="kw">const</span> <span class="ident">ZERO_BIG_DIGIT</span>: <span class="ident">BigDigit</span> <span class="op">=</span> <span class="number">0</span>;
<span class="comment">// `DoubleBigDigit` size dependent</span>
<span class="kw">pub</span> <span class="kw">const</span> <span class="ident">BITS</span>: <span class="ident">usize</span> <span class="op">=</span> <span class="number">32</span>;
<span class="kw">pub</span> <span class="kw">const</span> <span class="ident">BASE</span>: <span class="ident">DoubleBigDigit</span> <span class="op">=</span> <span class="number">1</span> <span class="op"><<</span> <span class="ident">BITS</span>;
<span class="kw">const</span> <span class="ident">LO_MASK</span>: <span class="ident">DoubleBigDigit</span> <span class="op">=</span> (<span class="op">-</span><span class="number">1i32</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op">>></span> <span class="ident">BITS</span>;
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">get_hi</span>(<span class="ident">n</span>: <span class="ident">DoubleBigDigit</span>) <span class="op">-></span> <span class="ident">BigDigit</span> {
(<span class="ident">n</span> <span class="op">>></span> <span class="ident">BITS</span>) <span class="kw">as</span> <span class="ident">BigDigit</span>
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">get_lo</span>(<span class="ident">n</span>: <span class="ident">DoubleBigDigit</span>) <span class="op">-></span> <span class="ident">BigDigit</span> {
(<span class="ident">n</span> <span class="op">&</span> <span class="ident">LO_MASK</span>) <span class="kw">as</span> <span class="ident">BigDigit</span>
}
<span class="doccomment">/// Split one `DoubleBigDigit` into two `BigDigit`s.</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">from_doublebigdigit</span>(<span class="ident">n</span>: <span class="ident">DoubleBigDigit</span>) <span class="op">-></span> (<span class="ident">BigDigit</span>, <span class="ident">BigDigit</span>) {
(<span class="ident">get_hi</span>(<span class="ident">n</span>), <span class="ident">get_lo</span>(<span class="ident">n</span>))
}
<span class="doccomment">/// Join two `BigDigit`s into one `DoubleBigDigit`</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">to_doublebigdigit</span>(<span class="ident">hi</span>: <span class="ident">BigDigit</span>, <span class="ident">lo</span>: <span class="ident">BigDigit</span>) <span class="op">-></span> <span class="ident">DoubleBigDigit</span> {
(<span class="ident">lo</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op">|</span> ((<span class="ident">hi</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op"><<</span> <span class="ident">BITS</span>)
}
}
<span class="kw">use</span> <span class="ident">big_digit</span>::{<span class="ident">BigDigit</span>, <span class="ident">DoubleBigDigit</span>};
<span class="comment">// Generic functions for add/subtract/multiply with carry/borrow:</span>
<span class="comment">// Add with carry:</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">adc</span>(<span class="ident">a</span>: <span class="ident">BigDigit</span>, <span class="ident">b</span>: <span class="ident">BigDigit</span>, <span class="ident">carry</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">BigDigit</span>) <span class="op">-></span> <span class="ident">BigDigit</span> {
<span class="kw">let</span> (<span class="ident">hi</span>, <span class="ident">lo</span>) <span class="op">=</span> <span class="ident">big_digit</span>::<span class="ident">from_doublebigdigit</span>((<span class="ident">a</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op">+</span> (<span class="ident">b</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op">+</span>
(<span class="kw-2">*</span><span class="ident">carry</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>));
<span class="kw-2">*</span><span class="ident">carry</span> <span class="op">=</span> <span class="ident">hi</span>;
<span class="ident">lo</span>
}
<span class="comment">// Subtract with borrow:</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">sbb</span>(<span class="ident">a</span>: <span class="ident">BigDigit</span>, <span class="ident">b</span>: <span class="ident">BigDigit</span>, <span class="ident">borrow</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">BigDigit</span>) <span class="op">-></span> <span class="ident">BigDigit</span> {
<span class="kw">let</span> (<span class="ident">hi</span>, <span class="ident">lo</span>) <span class="op">=</span> <span class="ident">big_digit</span>::<span class="ident">from_doublebigdigit</span>(<span class="ident">big_digit</span>::<span class="ident">BASE</span> <span class="op">+</span> (<span class="ident">a</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op">-</span>
(<span class="ident">b</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op">-</span>
(<span class="kw-2">*</span><span class="ident">borrow</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>));
<span class="comment">// hi * (base) + lo == 1*(base) + ai - bi - borrow</span>
<span class="comment">// => ai - bi - borrow < 0 <=> hi == 0</span>
<span class="kw-2">*</span><span class="ident">borrow</span> <span class="op">=</span> (<span class="ident">hi</span> <span class="op">==</span> <span class="number">0</span>) <span class="kw">as</span> <span class="ident">BigDigit</span>;
<span class="ident">lo</span>
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">mac_with_carry</span>(<span class="ident">a</span>: <span class="ident">BigDigit</span>, <span class="ident">b</span>: <span class="ident">BigDigit</span>, <span class="ident">c</span>: <span class="ident">BigDigit</span>, <span class="ident">carry</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">BigDigit</span>) <span class="op">-></span> <span class="ident">BigDigit</span> {
<span class="kw">let</span> (<span class="ident">hi</span>, <span class="ident">lo</span>) <span class="op">=</span> <span class="ident">big_digit</span>::<span class="ident">from_doublebigdigit</span>((<span class="ident">a</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op">+</span>
(<span class="ident">b</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op">*</span> (<span class="ident">c</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op">+</span>
(<span class="kw-2">*</span><span class="ident">carry</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>));
<span class="kw-2">*</span><span class="ident">carry</span> <span class="op">=</span> <span class="ident">hi</span>;
<span class="ident">lo</span>
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">mul_with_carry</span>(<span class="ident">a</span>: <span class="ident">BigDigit</span>, <span class="ident">b</span>: <span class="ident">BigDigit</span>, <span class="ident">carry</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">BigDigit</span>) <span class="op">-></span> <span class="ident">BigDigit</span> {
<span class="kw">let</span> (<span class="ident">hi</span>, <span class="ident">lo</span>) <span class="op">=</span> <span class="ident">big_digit</span>::<span class="ident">from_doublebigdigit</span>((<span class="ident">a</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op">*</span> (<span class="ident">b</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>) <span class="op">+</span>
(<span class="kw-2">*</span><span class="ident">carry</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>));
<span class="kw-2">*</span><span class="ident">carry</span> <span class="op">=</span> <span class="ident">hi</span>;
<span class="ident">lo</span>
}
<span class="doccomment">/// Divide a two digit numerator by a one digit divisor, returns quotient and remainder:</span>
<span class="doccomment">///</span>
<span class="doccomment">/// Note: the caller must ensure that both the quotient and remainder will fit into a single digit.</span>
<span class="doccomment">/// This is _not_ true for an arbitrary numerator/denominator.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// (This function also matches what the x86 divide instruction does).</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">fn</span> <span class="ident">div_wide</span>(<span class="ident">hi</span>: <span class="ident">BigDigit</span>, <span class="ident">lo</span>: <span class="ident">BigDigit</span>, <span class="ident">divisor</span>: <span class="ident">BigDigit</span>) <span class="op">-></span> (<span class="ident">BigDigit</span>, <span class="ident">BigDigit</span>) {
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">hi</span> <span class="op"><</span> <span class="ident">divisor</span>);
<span class="kw">let</span> <span class="ident">lhs</span> <span class="op">=</span> <span class="ident">big_digit</span>::<span class="ident">to_doublebigdigit</span>(<span class="ident">hi</span>, <span class="ident">lo</span>);
<span class="kw">let</span> <span class="ident">rhs</span> <span class="op">=</span> <span class="ident">divisor</span> <span class="kw">as</span> <span class="ident">DoubleBigDigit</span>;
((<span class="ident">lhs</span> <span class="op">/</span> <span class="ident">rhs</span>) <span class="kw">as</span> <span class="ident">BigDigit</span>, (<span class="ident">lhs</span> <span class="op">%</span> <span class="ident">rhs</span>) <span class="kw">as</span> <span class="ident">BigDigit</span>)
}
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">div_rem_digit</span>(<span class="kw-2">mut</span> <span class="ident">a</span>: <span class="ident">BigUint</span>, <span class="ident">b</span>: <span class="ident">BigDigit</span>) <span class="op">-></span> (<span class="ident">BigUint</span>, <span class="ident">BigDigit</span>) {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">rem</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">for</span> <span class="ident">d</span> <span class="kw">in</span> <span class="ident">a</span>.<span class="ident">data</span>.<span class="ident">iter_mut</span>().<span class="ident">rev</span>() {
<span class="kw">let</span> (<span class="ident">q</span>, <span class="ident">r</span>) <span class="op">=</span> <span class="ident">div_wide</span>(<span class="ident">rem</span>, <span class="kw-2">*</span><span class="ident">d</span>, <span class="ident">b</span>);
<span class="kw-2">*</span><span class="ident">d</span> <span class="op">=</span> <span class="ident">q</span>;
<span class="ident">rem</span> <span class="op">=</span> <span class="ident">r</span>;
}
(<span class="ident">a</span>.<span class="ident">normalized</span>(), <span class="ident">rem</span>)
}
<span class="comment">// Only for the Add impl:</span>
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">__add2</span>(<span class="ident">a</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">BigDigit</span>], <span class="ident">b</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>]) <span class="op">-></span> <span class="ident">BigDigit</span> {
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">a</span>.<span class="ident">len</span>() <span class="op">>=</span> <span class="ident">b</span>.<span class="ident">len</span>());
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">carry</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">let</span> (<span class="ident">a_lo</span>, <span class="ident">a_hi</span>) <span class="op">=</span> <span class="ident">a</span>.<span class="ident">split_at_mut</span>(<span class="ident">b</span>.<span class="ident">len</span>());
<span class="kw">for</span> (<span class="ident">a</span>, <span class="ident">b</span>) <span class="kw">in</span> <span class="ident">a_lo</span>.<span class="ident">iter_mut</span>().<span class="ident">zip</span>(<span class="ident">b</span>) {
<span class="kw-2">*</span><span class="ident">a</span> <span class="op">=</span> <span class="ident">adc</span>(<span class="kw-2">*</span><span class="ident">a</span>, <span class="kw-2">*</span><span class="ident">b</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">carry</span>);
}
<span class="kw">if</span> <span class="ident">carry</span> <span class="op">!=</span> <span class="number">0</span> {
<span class="kw">for</span> <span class="ident">a</span> <span class="kw">in</span> <span class="ident">a_hi</span> {
<span class="kw-2">*</span><span class="ident">a</span> <span class="op">=</span> <span class="ident">adc</span>(<span class="kw-2">*</span><span class="ident">a</span>, <span class="number">0</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">carry</span>);
<span class="kw">if</span> <span class="ident">carry</span> <span class="op">==</span> <span class="number">0</span> { <span class="kw">break</span> }
}
}
<span class="ident">carry</span>
}
<span class="doccomment">/// /Two argument addition of raw slices:</span>
<span class="doccomment">/// a += b</span>
<span class="doccomment">///</span>
<span class="doccomment">/// The caller _must_ ensure that a is big enough to store the result - typically this means</span>
<span class="doccomment">/// resizing a to max(a.len(), b.len()) + 1, to fit a possible carry.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">add2</span>(<span class="ident">a</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">BigDigit</span>], <span class="ident">b</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>]) {
<span class="kw">let</span> <span class="ident">carry</span> <span class="op">=</span> <span class="ident">__add2</span>(<span class="ident">a</span>, <span class="ident">b</span>);
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">carry</span> <span class="op">==</span> <span class="number">0</span>);
}
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">sub2</span>(<span class="ident">a</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">BigDigit</span>], <span class="ident">b</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>]) {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">borrow</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">let</span> <span class="ident">len</span> <span class="op">=</span> <span class="ident">cmp</span>::<span class="ident">min</span>(<span class="ident">a</span>.<span class="ident">len</span>(), <span class="ident">b</span>.<span class="ident">len</span>());
<span class="kw">let</span> (<span class="ident">a_lo</span>, <span class="ident">a_hi</span>) <span class="op">=</span> <span class="ident">a</span>.<span class="ident">split_at_mut</span>(<span class="ident">len</span>);
<span class="kw">let</span> (<span class="ident">b_lo</span>, <span class="ident">b_hi</span>) <span class="op">=</span> <span class="ident">b</span>.<span class="ident">split_at</span>(<span class="ident">len</span>);
<span class="kw">for</span> (<span class="ident">a</span>, <span class="ident">b</span>) <span class="kw">in</span> <span class="ident">a_lo</span>.<span class="ident">iter_mut</span>().<span class="ident">zip</span>(<span class="ident">b_lo</span>) {
<span class="kw-2">*</span><span class="ident">a</span> <span class="op">=</span> <span class="ident">sbb</span>(<span class="kw-2">*</span><span class="ident">a</span>, <span class="kw-2">*</span><span class="ident">b</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">borrow</span>);
}
<span class="kw">if</span> <span class="ident">borrow</span> <span class="op">!=</span> <span class="number">0</span> {
<span class="kw">for</span> <span class="ident">a</span> <span class="kw">in</span> <span class="ident">a_hi</span> {
<span class="kw-2">*</span><span class="ident">a</span> <span class="op">=</span> <span class="ident">sbb</span>(<span class="kw-2">*</span><span class="ident">a</span>, <span class="number">0</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">borrow</span>);
<span class="kw">if</span> <span class="ident">borrow</span> <span class="op">==</span> <span class="number">0</span> { <span class="kw">break</span> }
}
}
<span class="comment">// note: we're _required_ to fail on underflow</span>
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">borrow</span> <span class="op">==</span> <span class="number">0</span> <span class="op">&&</span> <span class="ident">b_hi</span>.<span class="ident">iter</span>().<span class="ident">all</span>(<span class="op">|</span><span class="ident">x</span><span class="op">|</span> <span class="kw-2">*</span><span class="ident">x</span> <span class="op">==</span> <span class="number">0</span>),
<span class="string">"Cannot subtract b from a because b is larger than a."</span>);
}
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">sub2rev</span>(<span class="ident">a</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>], <span class="ident">b</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">BigDigit</span>]) {
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">b</span>.<span class="ident">len</span>() <span class="op">>=</span> <span class="ident">a</span>.<span class="ident">len</span>());
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">borrow</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">let</span> <span class="ident">len</span> <span class="op">=</span> <span class="ident">cmp</span>::<span class="ident">min</span>(<span class="ident">a</span>.<span class="ident">len</span>(), <span class="ident">b</span>.<span class="ident">len</span>());
<span class="kw">let</span> (<span class="ident">a_lo</span>, <span class="ident">a_hi</span>) <span class="op">=</span> <span class="ident">a</span>.<span class="ident">split_at</span>(<span class="ident">len</span>);
<span class="kw">let</span> (<span class="ident">b_lo</span>, <span class="ident">b_hi</span>) <span class="op">=</span> <span class="ident">b</span>.<span class="ident">split_at_mut</span>(<span class="ident">len</span>);
<span class="kw">for</span> (<span class="ident">a</span>, <span class="ident">b</span>) <span class="kw">in</span> <span class="ident">a_lo</span>.<span class="ident">iter</span>().<span class="ident">zip</span>(<span class="ident">b_lo</span>) {
<span class="kw-2">*</span><span class="ident">b</span> <span class="op">=</span> <span class="ident">sbb</span>(<span class="kw-2">*</span><span class="ident">a</span>, <span class="kw-2">*</span><span class="ident">b</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">borrow</span>);
}
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">a_hi</span>.<span class="ident">is_empty</span>());
<span class="comment">// note: we're _required_ to fail on underflow</span>
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">borrow</span> <span class="op">==</span> <span class="number">0</span> <span class="op">&&</span> <span class="ident">b_hi</span>.<span class="ident">iter</span>().<span class="ident">all</span>(<span class="op">|</span><span class="ident">x</span><span class="op">|</span> <span class="kw-2">*</span><span class="ident">x</span> <span class="op">==</span> <span class="number">0</span>),
<span class="string">"Cannot subtract b from a because b is larger than a."</span>);
}
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">sub_sign</span>(<span class="ident">a</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>], <span class="ident">b</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>]) <span class="op">-></span> (<span class="ident">Sign</span>, <span class="ident">BigUint</span>) {
<span class="comment">// Normalize:</span>
<span class="kw">let</span> <span class="ident">a</span> <span class="op">=</span> <span class="kw-2">&</span><span class="ident">a</span>[..<span class="ident">a</span>.<span class="ident">iter</span>().<span class="ident">rposition</span>(<span class="op">|</span><span class="kw-2">&</span><span class="ident">x</span><span class="op">|</span> <span class="ident">x</span> <span class="op">!=</span> <span class="number">0</span>).<span class="ident">map_or</span>(<span class="number">0</span>, <span class="op">|</span><span class="ident">i</span><span class="op">|</span> <span class="ident">i</span> <span class="op">+</span> <span class="number">1</span>)];
<span class="kw">let</span> <span class="ident">b</span> <span class="op">=</span> <span class="kw-2">&</span><span class="ident">b</span>[..<span class="ident">b</span>.<span class="ident">iter</span>().<span class="ident">rposition</span>(<span class="op">|</span><span class="kw-2">&</span><span class="ident">x</span><span class="op">|</span> <span class="ident">x</span> <span class="op">!=</span> <span class="number">0</span>).<span class="ident">map_or</span>(<span class="number">0</span>, <span class="op">|</span><span class="ident">i</span><span class="op">|</span> <span class="ident">i</span> <span class="op">+</span> <span class="number">1</span>)];
<span class="kw">match</span> <span class="ident">cmp_slice</span>(<span class="ident">a</span>, <span class="ident">b</span>) {
<span class="ident">Greater</span> <span class="op">=></span> {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">a</span> <span class="op">=</span> <span class="ident">a</span>.<span class="ident">to_vec</span>();
<span class="ident">sub2</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">a</span>, <span class="ident">b</span>);
(<span class="ident">Plus</span>, <span class="ident">BigUint</span>::<span class="ident">new</span>(<span class="ident">a</span>))
}
<span class="ident">Less</span> <span class="op">=></span> {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">b</span> <span class="op">=</span> <span class="ident">b</span>.<span class="ident">to_vec</span>();
<span class="ident">sub2</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">b</span>, <span class="ident">a</span>);
(<span class="ident">Minus</span>, <span class="ident">BigUint</span>::<span class="ident">new</span>(<span class="ident">b</span>))
}
_ <span class="op">=></span> (<span class="ident">NoSign</span>, <span class="ident">Zero</span>::<span class="ident">zero</span>()),
}
}
<span class="doccomment">/// Three argument multiply accumulate:</span>
<span class="doccomment">/// acc += b * c</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">mac_digit</span>(<span class="ident">acc</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">BigDigit</span>], <span class="ident">b</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>], <span class="ident">c</span>: <span class="ident">BigDigit</span>) {
<span class="kw">if</span> <span class="ident">c</span> <span class="op">==</span> <span class="number">0</span> {
<span class="kw">return</span>;
}
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">carry</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">let</span> (<span class="ident">a_lo</span>, <span class="ident">a_hi</span>) <span class="op">=</span> <span class="ident">acc</span>.<span class="ident">split_at_mut</span>(<span class="ident">b</span>.<span class="ident">len</span>());
<span class="kw">for</span> (<span class="ident">a</span>, <span class="kw-2">&</span><span class="ident">b</span>) <span class="kw">in</span> <span class="ident">a_lo</span>.<span class="ident">iter_mut</span>().<span class="ident">zip</span>(<span class="ident">b</span>) {
<span class="kw-2">*</span><span class="ident">a</span> <span class="op">=</span> <span class="ident">mac_with_carry</span>(<span class="kw-2">*</span><span class="ident">a</span>, <span class="ident">b</span>, <span class="ident">c</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">carry</span>);
}
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">a</span> <span class="op">=</span> <span class="ident">a_hi</span>.<span class="ident">iter_mut</span>();
<span class="kw">while</span> <span class="ident">carry</span> <span class="op">!=</span> <span class="number">0</span> {
<span class="kw">let</span> <span class="ident">a</span> <span class="op">=</span> <span class="ident">a</span>.<span class="ident">next</span>().<span class="ident">expect</span>(<span class="string">"carry overflow during multiplication!"</span>);
<span class="kw-2">*</span><span class="ident">a</span> <span class="op">=</span> <span class="ident">adc</span>(<span class="kw-2">*</span><span class="ident">a</span>, <span class="number">0</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">carry</span>);
}
}
<span class="doccomment">/// Three argument multiply accumulate:</span>
<span class="doccomment">/// acc += b * c</span>
<span class="kw">fn</span> <span class="ident">mac3</span>(<span class="ident">acc</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">BigDigit</span>], <span class="ident">b</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>], <span class="ident">c</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>]) {
<span class="kw">let</span> (<span class="ident">x</span>, <span class="ident">y</span>) <span class="op">=</span> <span class="kw">if</span> <span class="ident">b</span>.<span class="ident">len</span>() <span class="op"><</span> <span class="ident">c</span>.<span class="ident">len</span>() {
(<span class="ident">b</span>, <span class="ident">c</span>)
} <span class="kw">else</span> {
(<span class="ident">c</span>, <span class="ident">b</span>)
};
<span class="comment">// We use three algorithms for different input sizes.</span>
<span class="comment">//</span>
<span class="comment">// - For small inputs, long multiplication is fastest.</span>
<span class="comment">// - Next we use Karatsuba multiplication (Toom-2), which we have optimized</span>
<span class="comment">// to avoid unnecessary allocations for intermediate values.</span>
<span class="comment">// - For the largest inputs we use Toom-3, which better optimizes the</span>
<span class="comment">// number of operations, but uses more temporary allocations.</span>
<span class="comment">//</span>
<span class="comment">// The thresholds are somewhat arbitrary, chosen by evaluating the results</span>
<span class="comment">// of `cargo bench --bench bigint multiply`.</span>
<span class="kw">if</span> <span class="ident">x</span>.<span class="ident">len</span>() <span class="op"><=</span> <span class="number">32</span> {
<span class="comment">// Long multiplication:</span>
<span class="kw">for</span> (<span class="ident">i</span>, <span class="ident">xi</span>) <span class="kw">in</span> <span class="ident">x</span>.<span class="ident">iter</span>().<span class="ident">enumerate</span>() {
<span class="ident">mac_digit</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">acc</span>[<span class="ident">i</span>..], <span class="ident">y</span>, <span class="kw-2">*</span><span class="ident">xi</span>);
}
} <span class="kw">else</span> <span class="kw">if</span> <span class="ident">x</span>.<span class="ident">len</span>() <span class="op"><=</span> <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.
*/</span>
<span class="comment">/*
* When x is smaller than y, it's significantly faster to pick b such that x is split in
* half, not y:
*/</span>
<span class="kw">let</span> <span class="ident">b</span> <span class="op">=</span> <span class="ident">x</span>.<span class="ident">len</span>() <span class="op">/</span> <span class="number">2</span>;
<span class="kw">let</span> (<span class="ident">x0</span>, <span class="ident">x1</span>) <span class="op">=</span> <span class="ident">x</span>.<span class="ident">split_at</span>(<span class="ident">b</span>);
<span class="kw">let</span> (<span class="ident">y0</span>, <span class="ident">y1</span>) <span class="op">=</span> <span class="ident">y</span>.<span class="ident">split_at</span>(<span class="ident">b</span>);
<span class="comment">/*
* We reuse the same BigUint for all the intermediate multiplies and have to size p
* appropriately here: x1.len() >= x0.len and y1.len() >= y0.len():
*/</span>
<span class="kw">let</span> <span class="ident">len</span> <span class="op">=</span> <span class="ident">x1</span>.<span class="ident">len</span>() <span class="op">+</span> <span class="ident">y1</span>.<span class="ident">len</span>() <span class="op">+</span> <span class="number">1</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">p</span> <span class="op">=</span> <span class="ident">BigUint</span> { <span class="ident">data</span>: <span class="macro">vec</span><span class="macro">!</span>[<span class="number">0</span>; <span class="ident">len</span>] };
<span class="comment">// p2 = x1 * y1</span>
<span class="ident">mac3</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">p</span>.<span class="ident">data</span>[..], <span class="ident">x1</span>, <span class="ident">y1</span>);
<span class="comment">// Not required, but the adds go faster if we drop any unneeded 0s from the end:</span>
<span class="ident">p</span>.<span class="ident">normalize</span>();
<span class="ident">add2</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">acc</span>[<span class="ident">b</span>..], <span class="kw-2">&</span><span class="ident">p</span>.<span class="ident">data</span>[..]);
<span class="ident">add2</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">acc</span>[<span class="ident">b</span> <span class="op">*</span> <span class="number">2</span>..], <span class="kw-2">&</span><span class="ident">p</span>.<span class="ident">data</span>[..]);
<span class="comment">// Zero out p before the next multiply:</span>
<span class="ident">p</span>.<span class="ident">data</span>.<span class="ident">truncate</span>(<span class="number">0</span>);
<span class="ident">p</span>.<span class="ident">data</span>.<span class="ident">extend</span>(<span class="ident">repeat</span>(<span class="number">0</span>).<span class="ident">take</span>(<span class="ident">len</span>));
<span class="comment">// p0 = x0 * y0</span>
<span class="ident">mac3</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">p</span>.<span class="ident">data</span>[..], <span class="ident">x0</span>, <span class="ident">y0</span>);
<span class="ident">p</span>.<span class="ident">normalize</span>();
<span class="ident">add2</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">acc</span>[..], <span class="kw-2">&</span><span class="ident">p</span>.<span class="ident">data</span>[..]);
<span class="ident">add2</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">acc</span>[<span class="ident">b</span>..], <span class="kw-2">&</span><span class="ident">p</span>.<span class="ident">data</span>[..]);
<span class="comment">// p1 = (x1 - x0) * (y1 - y0)</span>
<span class="comment">// We do this one last, since it may be negative and acc can't ever be negative:</span>
<span class="kw">let</span> (<span class="ident">j0_sign</span>, <span class="ident">j0</span>) <span class="op">=</span> <span class="ident">sub_sign</span>(<span class="ident">x1</span>, <span class="ident">x0</span>);
<span class="kw">let</span> (<span class="ident">j1_sign</span>, <span class="ident">j1</span>) <span class="op">=</span> <span class="ident">sub_sign</span>(<span class="ident">y1</span>, <span class="ident">y0</span>);
<span class="kw">match</span> <span class="ident">j0_sign</span> <span class="op">*</span> <span class="ident">j1_sign</span> {
<span class="ident">Plus</span> <span class="op">=></span> {
<span class="ident">p</span>.<span class="ident">data</span>.<span class="ident">truncate</span>(<span class="number">0</span>);
<span class="ident">p</span>.<span class="ident">data</span>.<span class="ident">extend</span>(<span class="ident">repeat</span>(<span class="number">0</span>).<span class="ident">take</span>(<span class="ident">len</span>));
<span class="ident">mac3</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">p</span>.<span class="ident">data</span>[..], <span class="kw-2">&</span><span class="ident">j0</span>.<span class="ident">data</span>[..], <span class="kw-2">&</span><span class="ident">j1</span>.<span class="ident">data</span>[..]);
<span class="ident">p</span>.<span class="ident">normalize</span>();
<span class="ident">sub2</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">acc</span>[<span class="ident">b</span>..], <span class="kw-2">&</span><span class="ident">p</span>.<span class="ident">data</span>[..]);
},
<span class="ident">Minus</span> <span class="op">=></span> {
<span class="ident">mac3</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">acc</span>[<span class="ident">b</span>..], <span class="kw-2">&</span><span class="ident">j0</span>.<span class="ident">data</span>[..], <span class="kw-2">&</span><span class="ident">j1</span>.<span class="ident">data</span>[..]);
},
<span class="ident">NoSign</span> <span class="op">=></span> (),
}
} <span class="kw">else</span> {
<span class="comment">// Toom-3 multiplication:</span>
<span class="comment">//</span>
<span class="comment">// Toom-3 is like Karatsuba above, but dividing the inputs into three parts.</span>
<span class="comment">// Both are instances of Toom-Cook, using `k=3` and `k=2` respectively.</span>
<span class="comment">//</span>
<span class="comment">// FIXME: It would be nice to have comments breaking down the operations below.</span>
<span class="kw">let</span> <span class="ident">i</span> <span class="op">=</span> <span class="ident">y</span>.<span class="ident">len</span>()<span class="op">/</span><span class="number">3</span> <span class="op">+</span> <span class="number">1</span>;
<span class="kw">let</span> <span class="ident">x0_len</span> <span class="op">=</span> <span class="ident">cmp</span>::<span class="ident">min</span>(<span class="ident">x</span>.<span class="ident">len</span>(), <span class="ident">i</span>);
<span class="kw">let</span> <span class="ident">x1_len</span> <span class="op">=</span> <span class="ident">cmp</span>::<span class="ident">min</span>(<span class="ident">x</span>.<span class="ident">len</span>() <span class="op">-</span> <span class="ident">x0_len</span>, <span class="ident">i</span>);
<span class="kw">let</span> <span class="ident">y0_len</span> <span class="op">=</span> <span class="ident">i</span>;
<span class="kw">let</span> <span class="ident">y1_len</span> <span class="op">=</span> <span class="ident">cmp</span>::<span class="ident">min</span>(<span class="ident">y</span>.<span class="ident">len</span>() <span class="op">-</span> <span class="ident">y0_len</span>, <span class="ident">i</span>);
<span class="kw">let</span> <span class="ident">x0</span> <span class="op">=</span> <span class="ident">BigInt</span>::<span class="ident">from_slice</span>(<span class="ident">Plus</span>, <span class="kw-2">&</span><span class="ident">x</span>[..<span class="ident">x0_len</span>]);
<span class="kw">let</span> <span class="ident">x1</span> <span class="op">=</span> <span class="ident">BigInt</span>::<span class="ident">from_slice</span>(<span class="ident">Plus</span>, <span class="kw-2">&</span><span class="ident">x</span>[<span class="ident">x0_len</span>..<span class="ident">x0_len</span> <span class="op">+</span> <span class="ident">x1_len</span>]);
<span class="kw">let</span> <span class="ident">x2</span> <span class="op">=</span> <span class="ident">BigInt</span>::<span class="ident">from_slice</span>(<span class="ident">Plus</span>, <span class="kw-2">&</span><span class="ident">x</span>[<span class="ident">x0_len</span> <span class="op">+</span> <span class="ident">x1_len</span>..]);
<span class="kw">let</span> <span class="ident">y0</span> <span class="op">=</span> <span class="ident">BigInt</span>::<span class="ident">from_slice</span>(<span class="ident">Plus</span>, <span class="kw-2">&</span><span class="ident">y</span>[..<span class="ident">y0_len</span>]);
<span class="kw">let</span> <span class="ident">y1</span> <span class="op">=</span> <span class="ident">BigInt</span>::<span class="ident">from_slice</span>(<span class="ident">Plus</span>, <span class="kw-2">&</span><span class="ident">y</span>[<span class="ident">y0_len</span>..<span class="ident">y0_len</span> <span class="op">+</span> <span class="ident">y1_len</span>]);
<span class="kw">let</span> <span class="ident">y2</span> <span class="op">=</span> <span class="ident">BigInt</span>::<span class="ident">from_slice</span>(<span class="ident">Plus</span>, <span class="kw-2">&</span><span class="ident">y</span>[<span class="ident">y0_len</span> <span class="op">+</span> <span class="ident">y1_len</span>..]);
<span class="kw">let</span> <span class="ident">p</span> <span class="op">=</span> <span class="kw-2">&</span><span class="ident">x0</span> <span class="op">+</span> <span class="kw-2">&</span><span class="ident">x2</span>;
<span class="kw">let</span> <span class="ident">q</span> <span class="op">=</span> <span class="kw-2">&</span><span class="ident">y0</span> <span class="op">+</span> <span class="kw-2">&</span><span class="ident">y2</span>;
<span class="kw">let</span> <span class="ident">p2</span> <span class="op">=</span> <span class="kw-2">&</span><span class="ident">p</span> <span class="op">-</span> <span class="kw-2">&</span><span class="ident">x1</span>;
<span class="kw">let</span> <span class="ident">q2</span> <span class="op">=</span> <span class="kw-2">&</span><span class="ident">q</span> <span class="op">-</span> <span class="kw-2">&</span><span class="ident">y1</span>;
<span class="kw">let</span> <span class="ident">r0</span> <span class="op">=</span> <span class="kw-2">&</span><span class="ident">x0</span> <span class="op">*</span> <span class="kw-2">&</span><span class="ident">y0</span>;
<span class="kw">let</span> <span class="ident">r4</span> <span class="op">=</span> <span class="kw-2">&</span><span class="ident">x2</span> <span class="op">*</span> <span class="kw-2">&</span><span class="ident">y2</span>;
<span class="kw">let</span> <span class="ident">r1</span> <span class="op">=</span> (<span class="ident">p</span> <span class="op">+</span> <span class="ident">x1</span>) <span class="op">*</span> (<span class="ident">q</span> <span class="op">+</span> <span class="ident">y1</span>);
<span class="kw">let</span> <span class="ident">r2</span> <span class="op">=</span> <span class="kw-2">&</span><span class="ident">p2</span> <span class="op">*</span> <span class="kw-2">&</span><span class="ident">q2</span>;
<span class="kw">let</span> <span class="ident">r3</span> <span class="op">=</span> ((<span class="ident">p2</span> <span class="op">+</span> <span class="ident">x2</span>)<span class="kw-2">*</span><span class="number">2</span> <span class="op">-</span> <span class="ident">x0</span>) <span class="op">*</span> ((<span class="ident">q2</span> <span class="op">+</span> <span class="ident">y2</span>)<span class="kw-2">*</span><span class="number">2</span> <span class="op">-</span> <span class="ident">y0</span>);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">comp3</span>: <span class="ident">BigInt</span> <span class="op">=</span> (<span class="ident">r3</span> <span class="op">-</span> <span class="kw-2">&</span><span class="ident">r1</span>) <span class="op">/</span> <span class="number">3</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">comp1</span>: <span class="ident">BigInt</span> <span class="op">=</span> (<span class="ident">r1</span> <span class="op">-</span> <span class="kw-2">&</span><span class="ident">r2</span>) <span class="op">/</span> <span class="number">2</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">comp2</span>: <span class="ident">BigInt</span> <span class="op">=</span> <span class="ident">r2</span> <span class="op">-</span> <span class="kw-2">&</span><span class="ident">r0</span>;
<span class="ident">comp3</span> <span class="op">=</span> (<span class="kw-2">&</span><span class="ident">comp2</span> <span class="op">-</span> <span class="ident">comp3</span>)<span class="op">/</span><span class="number">2</span> <span class="op">+</span> <span class="kw-2">&</span><span class="ident">r4</span><span class="kw-2">*</span><span class="number">2</span>;
<span class="ident">comp2</span> <span class="op">=</span> <span class="ident">comp2</span> <span class="op">+</span> <span class="kw-2">&</span><span class="ident">comp1</span> <span class="op">-</span> <span class="kw-2">&</span><span class="ident">r4</span>;
<span class="ident">comp1</span> <span class="op">=</span> <span class="ident">comp1</span> <span class="op">-</span> <span class="kw-2">&</span><span class="ident">comp3</span>;
<span class="kw">let</span> <span class="ident">result</span> <span class="op">=</span> <span class="ident">r0</span> <span class="op">+</span> (<span class="ident">comp1</span> <span class="op"><<</span> <span class="number">32</span><span class="kw-2">*</span><span class="ident">i</span>) <span class="op">+</span> (<span class="ident">comp2</span> <span class="op"><<</span> <span class="number">2</span><span class="kw-2">*</span><span class="number">32</span><span class="kw-2">*</span><span class="ident">i</span>) <span class="op">+</span> (<span class="ident">comp3</span> <span class="op"><<</span> <span class="number">3</span><span class="kw-2">*</span><span class="number">32</span><span class="kw-2">*</span><span class="ident">i</span>) <span class="op">+</span> (<span class="ident">r4</span> <span class="op"><<</span> <span class="number">4</span><span class="kw-2">*</span><span class="number">32</span><span class="kw-2">*</span><span class="ident">i</span>);
<span class="kw">let</span> <span class="ident">result_pos</span> <span class="op">=</span> <span class="ident">result</span>.<span class="ident">to_biguint</span>().<span class="ident">unwrap</span>();
<span class="ident">add2</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">acc</span>[..], <span class="kw-2">&</span><span class="ident">result_pos</span>.<span class="ident">data</span>);
}
}
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">mul3</span>(<span class="ident">x</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>], <span class="ident">y</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>]) <span class="op">-></span> <span class="ident">BigUint</span> {
<span class="kw">let</span> <span class="ident">len</span> <span class="op">=</span> <span class="ident">x</span>.<span class="ident">len</span>() <span class="op">+</span> <span class="ident">y</span>.<span class="ident">len</span>() <span class="op">+</span> <span class="number">1</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">prod</span> <span class="op">=</span> <span class="ident">BigUint</span> { <span class="ident">data</span>: <span class="macro">vec</span><span class="macro">!</span>[<span class="number">0</span>; <span class="ident">len</span>] };
<span class="ident">mac3</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">prod</span>.<span class="ident">data</span>[..], <span class="ident">x</span>, <span class="ident">y</span>);
<span class="ident">prod</span>.<span class="ident">normalized</span>()
}
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">scalar_mul</span>(<span class="ident">a</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">BigDigit</span>], <span class="ident">b</span>: <span class="ident">BigDigit</span>) <span class="op">-></span> <span class="ident">BigDigit</span> {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">carry</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">for</span> <span class="ident">a</span> <span class="kw">in</span> <span class="ident">a</span>.<span class="ident">iter_mut</span>() {
<span class="kw-2">*</span><span class="ident">a</span> <span class="op">=</span> <span class="ident">mul_with_carry</span>(<span class="kw-2">*</span><span class="ident">a</span>, <span class="ident">b</span>, <span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">carry</span>);
}
<span class="ident">carry</span>
}
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">div_rem</span>(<span class="ident">u</span>: <span class="kw-2">&</span><span class="ident">BigUint</span>, <span class="ident">d</span>: <span class="kw-2">&</span><span class="ident">BigUint</span>) <span class="op">-></span> (<span class="ident">BigUint</span>, <span class="ident">BigUint</span>) {
<span class="kw">if</span> <span class="ident">d</span>.<span class="ident">is_zero</span>() {
<span class="macro">panic</span><span class="macro">!</span>()
}
<span class="kw">if</span> <span class="ident">u</span>.<span class="ident">is_zero</span>() {
<span class="kw">return</span> (<span class="ident">Zero</span>::<span class="ident">zero</span>(), <span class="ident">Zero</span>::<span class="ident">zero</span>());
}
<span class="kw">if</span> <span class="kw-2">*</span><span class="ident">d</span> <span class="op">==</span> <span class="ident">One</span>::<span class="ident">one</span>() {
<span class="kw">return</span> (<span class="ident">u</span>.<span class="ident">clone</span>(), <span class="ident">Zero</span>::<span class="ident">zero</span>());
}
<span class="comment">// Required or the q_len calculation below can underflow:</span>
<span class="kw">match</span> <span class="ident">u</span>.<span class="ident">cmp</span>(<span class="ident">d</span>) {
<span class="ident">Less</span> <span class="op">=></span> <span class="kw">return</span> (<span class="ident">Zero</span>::<span class="ident">zero</span>(), <span class="ident">u</span>.<span class="ident">clone</span>()),
<span class="ident">Equal</span> <span class="op">=></span> <span class="kw">return</span> (<span class="ident">One</span>::<span class="ident">one</span>(), <span class="ident">Zero</span>::<span class="ident">zero</span>()),
<span class="ident">Greater</span> <span class="op">=></span> {} <span class="comment">// Do nothing</span>
}
<span class="comment">// This algorithm is from Knuth, TAOCP vol 2 section 4.3, algorithm D:</span>
<span class="comment">//</span>
<span class="comment">// First, normalize the arguments so the highest bit in the highest digit of the divisor is</span>
<span class="comment">// set: the main loop uses the highest digit of the divisor for generating guesses, so we</span>
<span class="comment">// want it to be the largest number we can efficiently divide by.</span>
<span class="comment">//</span>
<span class="kw">let</span> <span class="ident">shift</span> <span class="op">=</span> <span class="ident">d</span>.<span class="ident">data</span>.<span class="ident">last</span>().<span class="ident">unwrap</span>().<span class="ident">leading_zeros</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">a</span> <span class="op">=</span> <span class="ident">u</span> <span class="op"><<</span> <span class="ident">shift</span>;
<span class="kw">let</span> <span class="ident">b</span> <span class="op">=</span> <span class="ident">d</span> <span class="op"><<</span> <span class="ident">shift</span>;
<span class="comment">// The algorithm works by incrementally calculating "guesses", q0, for part of the</span>
<span class="comment">// remainder. Once we have any number q0 such that q0 * b <= a, we can set</span>
<span class="comment">//</span>
<span class="comment">// q += q0</span>
<span class="comment">// a -= q0 * b</span>
<span class="comment">//</span>
<span class="comment">// and then iterate until a < b. Then, (q, a) will be our desired quotient and remainder.</span>
<span class="comment">//</span>
<span class="comment">// q0, our guess, is calculated by dividing the last few digits of a by the last digit of b</span>
<span class="comment">// - this should give us a guess that is "close" to the actual quotient, but is possibly</span>
<span class="comment">// greater than the actual quotient. If q0 * b > a, we simply use iterated subtraction</span>
<span class="comment">// until we have a guess such that q0 * b <= a.</span>
<span class="comment">//</span>
<span class="kw">let</span> <span class="ident">bn</span> <span class="op">=</span> <span class="kw-2">*</span><span class="ident">b</span>.<span class="ident">data</span>.<span class="ident">last</span>().<span class="ident">unwrap</span>();
<span class="kw">let</span> <span class="ident">q_len</span> <span class="op">=</span> <span class="ident">a</span>.<span class="ident">data</span>.<span class="ident">len</span>() <span class="op">-</span> <span class="ident">b</span>.<span class="ident">data</span>.<span class="ident">len</span>() <span class="op">+</span> <span class="number">1</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">q</span> <span class="op">=</span> <span class="ident">BigUint</span> { <span class="ident">data</span>: <span class="macro">vec</span><span class="macro">!</span>[<span class="number">0</span>; <span class="ident">q_len</span>] };
<span class="comment">// We reuse the same temporary to avoid hitting the allocator in our inner loop - this is</span>
<span class="comment">// sized to hold a0 (in the common case; if a particular digit of the quotient is zero a0</span>
<span class="comment">// can be bigger).</span>
<span class="comment">//</span>
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">tmp</span> <span class="op">=</span> <span class="ident">BigUint</span> { <span class="ident">data</span>: <span class="ident">Vec</span>::<span class="ident">with_capacity</span>(<span class="number">2</span>) };
<span class="kw">for</span> <span class="ident">j</span> <span class="kw">in</span> (<span class="number">0</span>..<span class="ident">q_len</span>).<span class="ident">rev</span>() {
<span class="comment">/*
* When calculating our next guess q0, we don't need to consider the digits below j
* + b.data.len() - 1: we're guessing digit j of the quotient (i.e. q0 << j) from
* digit bn of the divisor (i.e. bn << (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> <span class="ident">offset</span> <span class="op">=</span> <span class="ident">j</span> <span class="op">+</span> <span class="ident">b</span>.<span class="ident">data</span>.<span class="ident">len</span>() <span class="op">-</span> <span class="number">1</span>;
<span class="kw">if</span> <span class="ident">offset</span> <span class="op">>=</span> <span class="ident">a</span>.<span class="ident">data</span>.<span class="ident">len</span>() {
<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> <span class="ident">a0</span> <span class="op">=</span> <span class="ident">tmp</span>;
<span class="ident">a0</span>.<span class="ident">data</span>.<span class="ident">truncate</span>(<span class="number">0</span>);
<span class="ident">a0</span>.<span class="ident">data</span>.<span class="ident">extend</span>(<span class="ident">a</span>.<span class="ident">data</span>[<span class="ident">offset</span>..].<span class="ident">iter</span>().<span class="ident">cloned</span>());
<span class="comment">/*
* q0 << 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> <span class="ident">q0</span>, _) <span class="op">=</span> <span class="ident">div_rem_digit</span>(<span class="ident">a0</span>, <span class="ident">bn</span>);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">prod</span> <span class="op">=</span> <span class="kw-2">&</span><span class="ident">b</span> <span class="op">*</span> <span class="kw-2">&</span><span class="ident">q0</span>;
<span class="kw">while</span> <span class="ident">cmp_slice</span>(<span class="kw-2">&</span><span class="ident">prod</span>.<span class="ident">data</span>[..], <span class="kw-2">&</span><span class="ident">a</span>.<span class="ident">data</span>[<span class="ident">j</span>..]) <span class="op">==</span> <span class="ident">Greater</span> {
<span class="kw">let</span> <span class="ident">one</span>: <span class="ident">BigUint</span> <span class="op">=</span> <span class="ident">One</span>::<span class="ident">one</span>();
<span class="ident">q0</span> <span class="op">=</span> <span class="ident">q0</span> <span class="op">-</span> <span class="ident">one</span>;
<span class="ident">prod</span> <span class="op">=</span> <span class="ident">prod</span> <span class="op">-</span> <span class="kw-2">&</span><span class="ident">b</span>;
}
<span class="ident">add2</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">q</span>.<span class="ident">data</span>[<span class="ident">j</span>..], <span class="kw-2">&</span><span class="ident">q0</span>.<span class="ident">data</span>[..]);
<span class="ident">sub2</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">a</span>.<span class="ident">data</span>[<span class="ident">j</span>..], <span class="kw-2">&</span><span class="ident">prod</span>.<span class="ident">data</span>[..]);
<span class="ident">a</span>.<span class="ident">normalize</span>();
<span class="ident">tmp</span> <span class="op">=</span> <span class="ident">q0</span>;
}
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">a</span> <span class="op"><</span> <span class="ident">b</span>);
(<span class="ident">q</span>.<span class="ident">normalized</span>(), <span class="ident">a</span> <span class="op">>></span> <span class="ident">shift</span>)
}
<span class="doccomment">/// Find last set bit</span>
<span class="doccomment">/// fls(0) == 0, fls(u32::MAX) == 32</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">fls</span><span class="op"><</span><span class="ident">T</span>: <span class="ident">traits</span>::<span class="ident">PrimInt</span><span class="op">></span>(<span class="ident">v</span>: <span class="ident">T</span>) <span class="op">-></span> <span class="ident">usize</span> {
<span class="ident">mem</span>::<span class="ident">size_of</span>::<span class="op"><</span><span class="ident">T</span><span class="op">></span>() <span class="op">*</span> <span class="number">8</span> <span class="op">-</span> <span class="ident">v</span>.<span class="ident">leading_zeros</span>() <span class="kw">as</span> <span class="ident">usize</span>
}
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">ilog2</span><span class="op"><</span><span class="ident">T</span>: <span class="ident">traits</span>::<span class="ident">PrimInt</span><span class="op">></span>(<span class="ident">v</span>: <span class="ident">T</span>) <span class="op">-></span> <span class="ident">usize</span> {
<span class="ident">fls</span>(<span class="ident">v</span>) <span class="op">-</span> <span class="number">1</span>
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">biguint_shl</span>(<span class="ident">n</span>: <span class="ident">Cow</span><span class="op"><</span><span class="ident">BigUint</span><span class="op">></span>, <span class="ident">bits</span>: <span class="ident">usize</span>) <span class="op">-></span> <span class="ident">BigUint</span> {
<span class="kw">let</span> <span class="ident">n_unit</span> <span class="op">=</span> <span class="ident">bits</span> <span class="op">/</span> <span class="ident">big_digit</span>::<span class="ident">BITS</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">data</span> <span class="op">=</span> <span class="kw">match</span> <span class="ident">n_unit</span> {
<span class="number">0</span> <span class="op">=></span> <span class="ident">n</span>.<span class="ident">into_owned</span>().<span class="ident">data</span>,
_ <span class="op">=></span> {
<span class="kw">let</span> <span class="ident">len</span> <span class="op">=</span> <span class="ident">n_unit</span> <span class="op">+</span> <span class="ident">n</span>.<span class="ident">data</span>.<span class="ident">len</span>() <span class="op">+</span> <span class="number">1</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">data</span> <span class="op">=</span> <span class="ident">Vec</span>::<span class="ident">with_capacity</span>(<span class="ident">len</span>);
<span class="ident">data</span>.<span class="ident">extend</span>(<span class="ident">repeat</span>(<span class="number">0</span>).<span class="ident">take</span>(<span class="ident">n_unit</span>));
<span class="ident">data</span>.<span class="ident">extend</span>(<span class="ident">n</span>.<span class="ident">data</span>.<span class="ident">iter</span>().<span class="ident">cloned</span>());
<span class="ident">data</span>
}
};
<span class="kw">let</span> <span class="ident">n_bits</span> <span class="op">=</span> <span class="ident">bits</span> <span class="op">%</span> <span class="ident">big_digit</span>::<span class="ident">BITS</span>;
<span class="kw">if</span> <span class="ident">n_bits</span> <span class="op">></span> <span class="number">0</span> {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">carry</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">for</span> <span class="ident">elem</span> <span class="kw">in</span> <span class="ident">data</span>[<span class="ident">n_unit</span>..].<span class="ident">iter_mut</span>() {
<span class="kw">let</span> <span class="ident">new_carry</span> <span class="op">=</span> <span class="kw-2">*</span><span class="ident">elem</span> <span class="op">>></span> (<span class="ident">big_digit</span>::<span class="ident">BITS</span> <span class="op">-</span> <span class="ident">n_bits</span>);
<span class="kw-2">*</span><span class="ident">elem</span> <span class="op">=</span> (<span class="kw-2">*</span><span class="ident">elem</span> <span class="op"><<</span> <span class="ident">n_bits</span>) <span class="op">|</span> <span class="ident">carry</span>;
<span class="ident">carry</span> <span class="op">=</span> <span class="ident">new_carry</span>;
}
<span class="kw">if</span> <span class="ident">carry</span> <span class="op">!=</span> <span class="number">0</span> {
<span class="ident">data</span>.<span class="ident">push</span>(<span class="ident">carry</span>);
}
}
<span class="ident">BigUint</span>::<span class="ident">new</span>(<span class="ident">data</span>)
}
<span class="attribute">#[<span class="ident">inline</span>]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">biguint_shr</span>(<span class="ident">n</span>: <span class="ident">Cow</span><span class="op"><</span><span class="ident">BigUint</span><span class="op">></span>, <span class="ident">bits</span>: <span class="ident">usize</span>) <span class="op">-></span> <span class="ident">BigUint</span> {
<span class="kw">let</span> <span class="ident">n_unit</span> <span class="op">=</span> <span class="ident">bits</span> <span class="op">/</span> <span class="ident">big_digit</span>::<span class="ident">BITS</span>;
<span class="kw">if</span> <span class="ident">n_unit</span> <span class="op">>=</span> <span class="ident">n</span>.<span class="ident">data</span>.<span class="ident">len</span>() {
<span class="kw">return</span> <span class="ident">Zero</span>::<span class="ident">zero</span>();
}
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">data</span> <span class="op">=</span> <span class="kw">match</span> <span class="ident">n</span> {
<span class="ident">Cow</span>::<span class="ident">Borrowed</span>(<span class="ident">n</span>) <span class="op">=></span> <span class="ident">n</span>.<span class="ident">data</span>[<span class="ident">n_unit</span>..].<span class="ident">to_vec</span>(),
<span class="ident">Cow</span>::<span class="ident">Owned</span>(<span class="kw-2">mut</span> <span class="ident">n</span>) <span class="op">=></span> {
<span class="ident">n</span>.<span class="ident">data</span>.<span class="ident">drain</span>(..<span class="ident">n_unit</span>);
<span class="ident">n</span>.<span class="ident">data</span>
}
};
<span class="kw">let</span> <span class="ident">n_bits</span> <span class="op">=</span> <span class="ident">bits</span> <span class="op">%</span> <span class="ident">big_digit</span>::<span class="ident">BITS</span>;
<span class="kw">if</span> <span class="ident">n_bits</span> <span class="op">></span> <span class="number">0</span> {
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">borrow</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">for</span> <span class="ident">elem</span> <span class="kw">in</span> <span class="ident">data</span>.<span class="ident">iter_mut</span>().<span class="ident">rev</span>() {
<span class="kw">let</span> <span class="ident">new_borrow</span> <span class="op">=</span> <span class="kw-2">*</span><span class="ident">elem</span> <span class="op"><<</span> (<span class="ident">big_digit</span>::<span class="ident">BITS</span> <span class="op">-</span> <span class="ident">n_bits</span>);
<span class="kw-2">*</span><span class="ident">elem</span> <span class="op">=</span> (<span class="kw-2">*</span><span class="ident">elem</span> <span class="op">>></span> <span class="ident">n_bits</span>) <span class="op">|</span> <span class="ident">borrow</span>;
<span class="ident">borrow</span> <span class="op">=</span> <span class="ident">new_borrow</span>;
}
}
<span class="ident">BigUint</span>::<span class="ident">new</span>(<span class="ident">data</span>)
}
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">cmp_slice</span>(<span class="ident">a</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>], <span class="ident">b</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>]) <span class="op">-></span> <span class="ident">Ordering</span> {
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">a</span>.<span class="ident">last</span>() <span class="op">!=</span> <span class="prelude-val">Some</span>(<span class="kw-2">&</span><span class="number">0</span>));
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">b</span>.<span class="ident">last</span>() <span class="op">!=</span> <span class="prelude-val">Some</span>(<span class="kw-2">&</span><span class="number">0</span>));
<span class="kw">let</span> (<span class="ident">a_len</span>, <span class="ident">b_len</span>) <span class="op">=</span> (<span class="ident">a</span>.<span class="ident">len</span>(), <span class="ident">b</span>.<span class="ident">len</span>());
<span class="kw">if</span> <span class="ident">a_len</span> <span class="op"><</span> <span class="ident">b_len</span> {
<span class="kw">return</span> <span class="ident">Less</span>;
}
<span class="kw">if</span> <span class="ident">a_len</span> <span class="op">></span> <span class="ident">b_len</span> {
<span class="kw">return</span> <span class="ident">Greater</span>;
}
<span class="kw">for</span> (<span class="kw-2">&</span><span class="ident">ai</span>, <span class="kw-2">&</span><span class="ident">bi</span>) <span class="kw">in</span> <span class="ident">a</span>.<span class="ident">iter</span>().<span class="ident">rev</span>().<span class="ident">zip</span>(<span class="ident">b</span>.<span class="ident">iter</span>().<span class="ident">rev</span>()) {
<span class="kw">if</span> <span class="ident">ai</span> <span class="op"><</span> <span class="ident">bi</span> {
<span class="kw">return</span> <span class="ident">Less</span>;
}
<span class="kw">if</span> <span class="ident">ai</span> <span class="op">></span> <span class="ident">bi</span> {
<span class="kw">return</span> <span class="ident">Greater</span>;
}
}
<span class="kw">return</span> <span class="ident">Equal</span>;
}
<span class="attribute">#[<span class="ident">cfg</span>(<span class="ident">test</span>)]</span>
<span class="kw">mod</span> <span class="ident">algorithm_tests</span> {
<span class="kw">use</span> {<span class="ident">BigDigit</span>, <span class="ident">BigUint</span>, <span class="ident">BigInt</span>};
<span class="kw">use</span> <span class="ident">Sign</span>::<span class="ident">Plus</span>;
<span class="kw">use</span> <span class="ident">traits</span>::<span class="ident">Num</span>;
<span class="attribute">#[<span class="ident">test</span>]</span>
<span class="kw">fn</span> <span class="ident">test_sub_sign</span>() {
<span class="kw">use</span> <span class="kw">super</span>::<span class="ident">sub_sign</span>;
<span class="kw">fn</span> <span class="ident">sub_sign_i</span>(<span class="ident">a</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>], <span class="ident">b</span>: <span class="kw-2">&</span>[<span class="ident">BigDigit</span>]) <span class="op">-></span> <span class="ident">BigInt</span> {
<span class="kw">let</span> (<span class="ident">sign</span>, <span class="ident">val</span>) <span class="op">=</span> <span class="ident">sub_sign</span>(<span class="ident">a</span>, <span class="ident">b</span>);
<span class="ident">BigInt</span>::<span class="ident">from_biguint</span>(<span class="ident">sign</span>, <span class="ident">val</span>)
}
<span class="kw">let</span> <span class="ident">a</span> <span class="op">=</span> <span class="ident">BigUint</span>::<span class="ident">from_str_radix</span>(<span class="string">"265252859812191058636308480000000"</span>, <span class="number">10</span>).<span class="ident">unwrap</span>();
<span class="kw">let</span> <span class="ident">b</span> <span class="op">=</span> <span class="ident">BigUint</span>::<span class="ident">from_str_radix</span>(<span class="string">"26525285981219105863630848000000"</span>, <span class="number">10</span>).<span class="ident">unwrap</span>();
<span class="kw">let</span> <span class="ident">a_i</span> <span class="op">=</span> <span class="ident">BigInt</span>::<span class="ident">from_biguint</span>(<span class="ident">Plus</span>, <span class="ident">a</span>.<span class="ident">clone</span>());
<span class="kw">let</span> <span class="ident">b_i</span> <span class="op">=</span> <span class="ident">BigInt</span>::<span class="ident">from_biguint</span>(<span class="ident">Plus</span>, <span class="ident">b</span>.<span class="ident">clone</span>());
<span class="macro">assert_eq</span><span class="macro">!</span>(<span class="ident">sub_sign_i</span>(<span class="kw-2">&</span><span class="ident">a</span>.<span class="ident">data</span>[..], <span class="kw-2">&</span><span class="ident">b</span>.<span class="ident">data</span>[..]), <span class="kw-2">&</span><span class="ident">a_i</span> <span class="op">-</span> <span class="kw-2">&</span><span class="ident">b_i</span>);
<span class="macro">assert_eq</span><span class="macro">!</span>(<span class="ident">sub_sign_i</span>(<span class="kw-2">&</span><span class="ident">b</span>.<span class="ident">data</span>[..], <span class="kw-2">&</span><span class="ident">a</span>.<span class="ident">data</span>[..]), <span class="kw-2">&</span><span class="ident">b_i</span> <span class="op">-</span> <span class="kw-2">&</span><span class="ident">a_i</span>);
}
}
</pre>
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