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</pre><pre class="rust ">
<span class="comment">// Copyright 2015 The Rust Project Developers. See the COPYRIGHT</span>
<span class="comment">// file at the top-level directory of this distribution and at</span>
<span class="comment">// http://rust-lang.org/COPYRIGHT.</span>
<span class="comment">//</span>
<span class="comment">// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or</span>
<span class="comment">// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license</span>
<span class="comment">// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your</span>
<span class="comment">// option. This file may not be copied, modified, or distributed</span>
<span class="comment">// except according to those terms.</span>
<span class="doccomment">/*!
Rust adaptation of Grisu3 algorithm described in [1]. It uses about
1KB of precomputed table, and in turn, it's very quick for most inputs.
[1] Florian Loitsch. 2010. Printing floating-point numbers quickly and
accurately with integers. SIGPLAN Not. 45, 6 (June 2010), 233-243.
*/</span>
<span class="kw">use</span> <span class="ident">num</span>::<span class="ident">diy_float</span>::<span class="ident">Fp</span>;
<span class="kw">use</span> <span class="ident">num</span>::<span class="ident">flt2dec</span>::{<span class="ident">Decoded</span>, <span class="ident">MAX_SIG_DIGITS</span>, <span class="ident">round_up</span>};
<span class="comment">// see the comments in `format_shortest_opt` for the rationale.</span>
<span class="attribute">#[<span class="ident">doc</span>(<span class="ident">hidden</span>)]</span> <span class="kw">pub</span> <span class="kw">const</span> <span class="ident">ALPHA</span>: <span class="ident">i16</span> <span class="op">=</span> <span class="op">-</span><span class="number">60</span>;
<span class="attribute">#[<span class="ident">doc</span>(<span class="ident">hidden</span>)]</span> <span class="kw">pub</span> <span class="kw">const</span> <span class="ident">GAMMA</span>: <span class="ident">i16</span> <span class="op">=</span> <span class="op">-</span><span class="number">32</span>;
<span class="comment">/*
# the following Python code generates this table:
for i in xrange(-308, 333, 8):
if i >= 0: f = 10**i; e = 0
else: f = 2**(80-4*i) // 10**-i; e = 4 * i - 80
l = f.bit_length()
f = ((f << 64 >> (l-1)) + 1) >> 1; e += l - 64
print ' (%#018x, %5d, %4d),' % (f, e, i)
*/</span>
<span class="attribute">#[<span class="ident">doc</span>(<span class="ident">hidden</span>)]</span>
<span class="kw">pub</span> <span class="kw">static</span> <span class="ident">CACHED_POW10</span>: [(<span class="ident">u64</span>, <span class="ident">i16</span>, <span class="ident">i16</span>); <span class="number">81</span>] <span class="op">=</span> [ <span class="comment">// (f, e, k)</span>
(<span class="number">0xe61acf033d1a45df</span>, <span class="op">-</span><span class="number">1087</span>, <span class="op">-</span><span class="number">308</span>),
(<span class="number">0xab70fe17c79ac6ca</span>, <span class="op">-</span><span class="number">1060</span>, <span class="op">-</span><span class="number">300</span>),
(<span class="number">0xff77b1fcbebcdc4f</span>, <span class="op">-</span><span class="number">1034</span>, <span class="op">-</span><span class="number">292</span>),
(<span class="number">0xbe5691ef416bd60c</span>, <span class="op">-</span><span class="number">1007</span>, <span class="op">-</span><span class="number">284</span>),
(<span class="number">0x8dd01fad907ffc3c</span>, <span class="op">-</span><span class="number">980</span>, <span class="op">-</span><span class="number">276</span>),
(<span class="number">0xd3515c2831559a83</span>, <span class="op">-</span><span class="number">954</span>, <span class="op">-</span><span class="number">268</span>),
(<span class="number">0x9d71ac8fada6c9b5</span>, <span class="op">-</span><span class="number">927</span>, <span class="op">-</span><span class="number">260</span>),
(<span class="number">0xea9c227723ee8bcb</span>, <span class="op">-</span><span class="number">901</span>, <span class="op">-</span><span class="number">252</span>),
(<span class="number">0xaecc49914078536d</span>, <span class="op">-</span><span class="number">874</span>, <span class="op">-</span><span class="number">244</span>),
(<span class="number">0x823c12795db6ce57</span>, <span class="op">-</span><span class="number">847</span>, <span class="op">-</span><span class="number">236</span>),
(<span class="number">0xc21094364dfb5637</span>, <span class="op">-</span><span class="number">821</span>, <span class="op">-</span><span class="number">228</span>),
(<span class="number">0x9096ea6f3848984f</span>, <span class="op">-</span><span class="number">794</span>, <span class="op">-</span><span class="number">220</span>),
(<span class="number">0xd77485cb25823ac7</span>, <span class="op">-</span><span class="number">768</span>, <span class="op">-</span><span class="number">212</span>),
(<span class="number">0xa086cfcd97bf97f4</span>, <span class="op">-</span><span class="number">741</span>, <span class="op">-</span><span class="number">204</span>),
(<span class="number">0xef340a98172aace5</span>, <span class="op">-</span><span class="number">715</span>, <span class="op">-</span><span class="number">196</span>),
(<span class="number">0xb23867fb2a35b28e</span>, <span class="op">-</span><span class="number">688</span>, <span class="op">-</span><span class="number">188</span>),
(<span class="number">0x84c8d4dfd2c63f3b</span>, <span class="op">-</span><span class="number">661</span>, <span class="op">-</span><span class="number">180</span>),
(<span class="number">0xc5dd44271ad3cdba</span>, <span class="op">-</span><span class="number">635</span>, <span class="op">-</span><span class="number">172</span>),
(<span class="number">0x936b9fcebb25c996</span>, <span class="op">-</span><span class="number">608</span>, <span class="op">-</span><span class="number">164</span>),
(<span class="number">0xdbac6c247d62a584</span>, <span class="op">-</span><span class="number">582</span>, <span class="op">-</span><span class="number">156</span>),
(<span class="number">0xa3ab66580d5fdaf6</span>, <span class="op">-</span><span class="number">555</span>, <span class="op">-</span><span class="number">148</span>),
(<span class="number">0xf3e2f893dec3f126</span>, <span class="op">-</span><span class="number">529</span>, <span class="op">-</span><span class="number">140</span>),
(<span class="number">0xb5b5ada8aaff80b8</span>, <span class="op">-</span><span class="number">502</span>, <span class="op">-</span><span class="number">132</span>),
(<span class="number">0x87625f056c7c4a8b</span>, <span class="op">-</span><span class="number">475</span>, <span class="op">-</span><span class="number">124</span>),
(<span class="number">0xc9bcff6034c13053</span>, <span class="op">-</span><span class="number">449</span>, <span class="op">-</span><span class="number">116</span>),
(<span class="number">0x964e858c91ba2655</span>, <span class="op">-</span><span class="number">422</span>, <span class="op">-</span><span class="number">108</span>),
(<span class="number">0xdff9772470297ebd</span>, <span class="op">-</span><span class="number">396</span>, <span class="op">-</span><span class="number">100</span>),
(<span class="number">0xa6dfbd9fb8e5b88f</span>, <span class="op">-</span><span class="number">369</span>, <span class="op">-</span><span class="number">92</span>),
(<span class="number">0xf8a95fcf88747d94</span>, <span class="op">-</span><span class="number">343</span>, <span class="op">-</span><span class="number">84</span>),
(<span class="number">0xb94470938fa89bcf</span>, <span class="op">-</span><span class="number">316</span>, <span class="op">-</span><span class="number">76</span>),
(<span class="number">0x8a08f0f8bf0f156b</span>, <span class="op">-</span><span class="number">289</span>, <span class="op">-</span><span class="number">68</span>),
(<span class="number">0xcdb02555653131b6</span>, <span class="op">-</span><span class="number">263</span>, <span class="op">-</span><span class="number">60</span>),
(<span class="number">0x993fe2c6d07b7fac</span>, <span class="op">-</span><span class="number">236</span>, <span class="op">-</span><span class="number">52</span>),
(<span class="number">0xe45c10c42a2b3b06</span>, <span class="op">-</span><span class="number">210</span>, <span class="op">-</span><span class="number">44</span>),
(<span class="number">0xaa242499697392d3</span>, <span class="op">-</span><span class="number">183</span>, <span class="op">-</span><span class="number">36</span>),
(<span class="number">0xfd87b5f28300ca0e</span>, <span class="op">-</span><span class="number">157</span>, <span class="op">-</span><span class="number">28</span>),
(<span class="number">0xbce5086492111aeb</span>, <span class="op">-</span><span class="number">130</span>, <span class="op">-</span><span class="number">20</span>),
(<span class="number">0x8cbccc096f5088cc</span>, <span class="op">-</span><span class="number">103</span>, <span class="op">-</span><span class="number">12</span>),
(<span class="number">0xd1b71758e219652c</span>, <span class="op">-</span><span class="number">77</span>, <span class="op">-</span><span class="number">4</span>),
(<span class="number">0x9c40000000000000</span>, <span class="op">-</span><span class="number">50</span>, <span class="number">4</span>),
(<span class="number">0xe8d4a51000000000</span>, <span class="op">-</span><span class="number">24</span>, <span class="number">12</span>),
(<span class="number">0xad78ebc5ac620000</span>, <span class="number">3</span>, <span class="number">20</span>),
(<span class="number">0x813f3978f8940984</span>, <span class="number">30</span>, <span class="number">28</span>),
(<span class="number">0xc097ce7bc90715b3</span>, <span class="number">56</span>, <span class="number">36</span>),
(<span class="number">0x8f7e32ce7bea5c70</span>, <span class="number">83</span>, <span class="number">44</span>),
(<span class="number">0xd5d238a4abe98068</span>, <span class="number">109</span>, <span class="number">52</span>),
(<span class="number">0x9f4f2726179a2245</span>, <span class="number">136</span>, <span class="number">60</span>),
(<span class="number">0xed63a231d4c4fb27</span>, <span class="number">162</span>, <span class="number">68</span>),
(<span class="number">0xb0de65388cc8ada8</span>, <span class="number">189</span>, <span class="number">76</span>),
(<span class="number">0x83c7088e1aab65db</span>, <span class="number">216</span>, <span class="number">84</span>),
(<span class="number">0xc45d1df942711d9a</span>, <span class="number">242</span>, <span class="number">92</span>),
(<span class="number">0x924d692ca61be758</span>, <span class="number">269</span>, <span class="number">100</span>),
(<span class="number">0xda01ee641a708dea</span>, <span class="number">295</span>, <span class="number">108</span>),
(<span class="number">0xa26da3999aef774a</span>, <span class="number">322</span>, <span class="number">116</span>),
(<span class="number">0xf209787bb47d6b85</span>, <span class="number">348</span>, <span class="number">124</span>),
(<span class="number">0xb454e4a179dd1877</span>, <span class="number">375</span>, <span class="number">132</span>),
(<span class="number">0x865b86925b9bc5c2</span>, <span class="number">402</span>, <span class="number">140</span>),
(<span class="number">0xc83553c5c8965d3d</span>, <span class="number">428</span>, <span class="number">148</span>),
(<span class="number">0x952ab45cfa97a0b3</span>, <span class="number">455</span>, <span class="number">156</span>),
(<span class="number">0xde469fbd99a05fe3</span>, <span class="number">481</span>, <span class="number">164</span>),
(<span class="number">0xa59bc234db398c25</span>, <span class="number">508</span>, <span class="number">172</span>),
(<span class="number">0xf6c69a72a3989f5c</span>, <span class="number">534</span>, <span class="number">180</span>),
(<span class="number">0xb7dcbf5354e9bece</span>, <span class="number">561</span>, <span class="number">188</span>),
(<span class="number">0x88fcf317f22241e2</span>, <span class="number">588</span>, <span class="number">196</span>),
(<span class="number">0xcc20ce9bd35c78a5</span>, <span class="number">614</span>, <span class="number">204</span>),
(<span class="number">0x98165af37b2153df</span>, <span class="number">641</span>, <span class="number">212</span>),
(<span class="number">0xe2a0b5dc971f303a</span>, <span class="number">667</span>, <span class="number">220</span>),
(<span class="number">0xa8d9d1535ce3b396</span>, <span class="number">694</span>, <span class="number">228</span>),
(<span class="number">0xfb9b7cd9a4a7443c</span>, <span class="number">720</span>, <span class="number">236</span>),
(<span class="number">0xbb764c4ca7a44410</span>, <span class="number">747</span>, <span class="number">244</span>),
(<span class="number">0x8bab8eefb6409c1a</span>, <span class="number">774</span>, <span class="number">252</span>),
(<span class="number">0xd01fef10a657842c</span>, <span class="number">800</span>, <span class="number">260</span>),
(<span class="number">0x9b10a4e5e9913129</span>, <span class="number">827</span>, <span class="number">268</span>),
(<span class="number">0xe7109bfba19c0c9d</span>, <span class="number">853</span>, <span class="number">276</span>),
(<span class="number">0xac2820d9623bf429</span>, <span class="number">880</span>, <span class="number">284</span>),
(<span class="number">0x80444b5e7aa7cf85</span>, <span class="number">907</span>, <span class="number">292</span>),
(<span class="number">0xbf21e44003acdd2d</span>, <span class="number">933</span>, <span class="number">300</span>),
(<span class="number">0x8e679c2f5e44ff8f</span>, <span class="number">960</span>, <span class="number">308</span>),
(<span class="number">0xd433179d9c8cb841</span>, <span class="number">986</span>, <span class="number">316</span>),
(<span class="number">0x9e19db92b4e31ba9</span>, <span class="number">1013</span>, <span class="number">324</span>),
(<span class="number">0xeb96bf6ebadf77d9</span>, <span class="number">1039</span>, <span class="number">332</span>),
];
<span class="attribute">#[<span class="ident">doc</span>(<span class="ident">hidden</span>)]</span> <span class="kw">pub</span> <span class="kw">const</span> <span class="ident">CACHED_POW10_FIRST_E</span>: <span class="ident">i16</span> <span class="op">=</span> <span class="op">-</span><span class="number">1087</span>;
<span class="attribute">#[<span class="ident">doc</span>(<span class="ident">hidden</span>)]</span> <span class="kw">pub</span> <span class="kw">const</span> <span class="ident">CACHED_POW10_LAST_E</span>: <span class="ident">i16</span> <span class="op">=</span> <span class="number">1039</span>;
<span class="attribute">#[<span class="ident">doc</span>(<span class="ident">hidden</span>)]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">cached_power</span>(<span class="ident">alpha</span>: <span class="ident">i16</span>, <span class="ident">gamma</span>: <span class="ident">i16</span>) <span class="op">-></span> (<span class="ident">i16</span>, <span class="ident">Fp</span>) {
<span class="kw">let</span> <span class="ident">offset</span> <span class="op">=</span> <span class="ident">CACHED_POW10_FIRST_E</span> <span class="kw">as</span> <span class="ident">i32</span>;
<span class="kw">let</span> <span class="ident">range</span> <span class="op">=</span> (<span class="ident">CACHED_POW10</span>.<span class="ident">len</span>() <span class="kw">as</span> <span class="ident">i32</span>) <span class="op">-</span> <span class="number">1</span>;
<span class="kw">let</span> <span class="ident">domain</span> <span class="op">=</span> (<span class="ident">CACHED_POW10_LAST_E</span> <span class="op">-</span> <span class="ident">CACHED_POW10_FIRST_E</span>) <span class="kw">as</span> <span class="ident">i32</span>;
<span class="kw">let</span> <span class="ident">idx</span> <span class="op">=</span> ((<span class="ident">gamma</span> <span class="kw">as</span> <span class="ident">i32</span>) <span class="op">-</span> <span class="ident">offset</span>) <span class="op">*</span> <span class="ident">range</span> <span class="op">/</span> <span class="ident">domain</span>;
<span class="kw">let</span> (<span class="ident">f</span>, <span class="ident">e</span>, <span class="ident">k</span>) <span class="op">=</span> <span class="ident">CACHED_POW10</span>[<span class="ident">idx</span> <span class="kw">as</span> <span class="ident">usize</span>];
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">alpha</span> <span class="op"><=</span> <span class="ident">e</span> <span class="op">&&</span> <span class="ident">e</span> <span class="op"><=</span> <span class="ident">gamma</span>);
(<span class="ident">k</span>, <span class="ident">Fp</span> { <span class="ident">f</span>: <span class="ident">f</span>, <span class="ident">e</span>: <span class="ident">e</span> })
}
<span class="doccomment">/// Given `x > 0`, returns `(k, 10^k)` such that `10^k <= x < 10^(k+1)`.</span>
<span class="attribute">#[<span class="ident">doc</span>(<span class="ident">hidden</span>)]</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">max_pow10_no_more_than</span>(<span class="ident">x</span>: <span class="ident">u32</span>) <span class="op">-></span> (<span class="ident">u8</span>, <span class="ident">u32</span>) {
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">x</span> <span class="op">></span> <span class="number">0</span>);
<span class="kw">const</span> <span class="ident">X9</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">10_0000_0000</span>;
<span class="kw">const</span> <span class="ident">X8</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">1_0000_0000</span>;
<span class="kw">const</span> <span class="ident">X7</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">1000_0000</span>;
<span class="kw">const</span> <span class="ident">X6</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">100_0000</span>;
<span class="kw">const</span> <span class="ident">X5</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">10_0000</span>;
<span class="kw">const</span> <span class="ident">X4</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">1_0000</span>;
<span class="kw">const</span> <span class="ident">X3</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">1000</span>;
<span class="kw">const</span> <span class="ident">X2</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">100</span>;
<span class="kw">const</span> <span class="ident">X1</span>: <span class="ident">u32</span> <span class="op">=</span> <span class="number">10</span>;
<span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="ident">X4</span> {
<span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="ident">X2</span> { <span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="ident">X1</span> {(<span class="number">0</span>, <span class="number">1</span>)} <span class="kw">else</span> {(<span class="number">1</span>, <span class="ident">X1</span>)} }
<span class="kw">else</span> { <span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="ident">X3</span> {(<span class="number">2</span>, <span class="ident">X2</span>)} <span class="kw">else</span> {(<span class="number">3</span>, <span class="ident">X3</span>)} }
} <span class="kw">else</span> {
<span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="ident">X6</span> { <span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="ident">X5</span> {(<span class="number">4</span>, <span class="ident">X4</span>)} <span class="kw">else</span> {(<span class="number">5</span>, <span class="ident">X5</span>)} }
<span class="kw">else</span> <span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="ident">X8</span> { <span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="ident">X7</span> {(<span class="number">6</span>, <span class="ident">X6</span>)} <span class="kw">else</span> {(<span class="number">7</span>, <span class="ident">X7</span>)} }
<span class="kw">else</span> { <span class="kw">if</span> <span class="ident">x</span> <span class="op"><</span> <span class="ident">X9</span> {(<span class="number">8</span>, <span class="ident">X8</span>)} <span class="kw">else</span> {(<span class="number">9</span>, <span class="ident">X9</span>)} }
}
}
<span class="doccomment">/// The shortest mode implementation for Grisu.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// It returns `None` when it would return an inexact representation otherwise.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">format_shortest_opt</span>(<span class="ident">d</span>: <span class="kw-2">&</span><span class="ident">Decoded</span>,
<span class="ident">buf</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">u8</span>]) <span class="op">-></span> <span class="prelude-ty">Option</span><span class="op"><</span>(<span class="comment">/*#digits*/</span> <span class="ident">usize</span>, <span class="comment">/*exp*/</span> <span class="ident">i16</span>)<span class="op">></span> {
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">d</span>.<span class="ident">mant</span> <span class="op">></span> <span class="number">0</span>);
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">d</span>.<span class="ident">minus</span> <span class="op">></span> <span class="number">0</span>);
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">d</span>.<span class="ident">plus</span> <span class="op">></span> <span class="number">0</span>);
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">d</span>.<span class="ident">mant</span>.<span class="ident">checked_add</span>(<span class="ident">d</span>.<span class="ident">plus</span>).<span class="ident">is_some</span>());
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">d</span>.<span class="ident">mant</span>.<span class="ident">checked_sub</span>(<span class="ident">d</span>.<span class="ident">minus</span>).<span class="ident">is_some</span>());
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">buf</span>.<span class="ident">len</span>() <span class="op">>=</span> <span class="ident">MAX_SIG_DIGITS</span>);
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">d</span>.<span class="ident">mant</span> <span class="op">+</span> <span class="ident">d</span>.<span class="ident">plus</span> <span class="op"><</span> (<span class="number">1</span> <span class="op"><<</span> <span class="number">61</span>)); <span class="comment">// we need at least three bits of additional precision</span>
<span class="comment">// start with the normalized values with the shared exponent</span>
<span class="kw">let</span> <span class="ident">plus</span> <span class="op">=</span> <span class="ident">Fp</span> { <span class="ident">f</span>: <span class="ident">d</span>.<span class="ident">mant</span> <span class="op">+</span> <span class="ident">d</span>.<span class="ident">plus</span>, <span class="ident">e</span>: <span class="ident">d</span>.<span class="ident">exp</span> }.<span class="ident">normalize</span>();
<span class="kw">let</span> <span class="ident">minus</span> <span class="op">=</span> <span class="ident">Fp</span> { <span class="ident">f</span>: <span class="ident">d</span>.<span class="ident">mant</span> <span class="op">-</span> <span class="ident">d</span>.<span class="ident">minus</span>, <span class="ident">e</span>: <span class="ident">d</span>.<span class="ident">exp</span> }.<span class="ident">normalize_to</span>(<span class="ident">plus</span>.<span class="ident">e</span>);
<span class="kw">let</span> <span class="ident">v</span> <span class="op">=</span> <span class="ident">Fp</span> { <span class="ident">f</span>: <span class="ident">d</span>.<span class="ident">mant</span>, <span class="ident">e</span>: <span class="ident">d</span>.<span class="ident">exp</span> }.<span class="ident">normalize_to</span>(<span class="ident">plus</span>.<span class="ident">e</span>);
<span class="comment">// find any `cached = 10^minusk` such that `ALPHA <= minusk + plus.e + 64 <= GAMMA`.</span>
<span class="comment">// since `plus` is normalized, this means `2^(62 + ALPHA) <= plus * cached < 2^(64 + GAMMA)`;</span>
<span class="comment">// given our choices of `ALPHA` and `GAMMA`, this puts `plus * cached` into `[4, 2^32)`.</span>
<span class="comment">//</span>
<span class="comment">// it is obviously desirable to maximize `GAMMA - ALPHA`,</span>
<span class="comment">// so that we don't need many cached powers of 10, but there are some considerations:</span>
<span class="comment">//</span>
<span class="comment">// 1. we want to keep `floor(plus * cached)` within `u32` since it needs a costly division.</span>
<span class="comment">// (this is not really avoidable, remainder is required for accuracy estimation.)</span>
<span class="comment">// 2. the remainder of `floor(plus * cached)` repeatedly gets multiplied by 10,</span>
<span class="comment">// and it should not overflow.</span>
<span class="comment">//</span>
<span class="comment">// the first gives `64 + GAMMA <= 32`, while the second gives `10 * 2^-ALPHA <= 2^64`;</span>
<span class="comment">// -60 and -32 is the maximal range with this constraint, and V8 also uses them.</span>
<span class="kw">let</span> (<span class="ident">minusk</span>, <span class="ident">cached</span>) <span class="op">=</span> <span class="ident">cached_power</span>(<span class="ident">ALPHA</span> <span class="op">-</span> <span class="ident">plus</span>.<span class="ident">e</span> <span class="op">-</span> <span class="number">64</span>, <span class="ident">GAMMA</span> <span class="op">-</span> <span class="ident">plus</span>.<span class="ident">e</span> <span class="op">-</span> <span class="number">64</span>);
<span class="comment">// scale fps. this gives the maximal error of 1 ulp (proved from Theorem 5.1).</span>
<span class="kw">let</span> <span class="ident">plus</span> <span class="op">=</span> <span class="ident">plus</span>.<span class="ident">mul</span>(<span class="kw-2">&</span><span class="ident">cached</span>);
<span class="kw">let</span> <span class="ident">minus</span> <span class="op">=</span> <span class="ident">minus</span>.<span class="ident">mul</span>(<span class="kw-2">&</span><span class="ident">cached</span>);
<span class="kw">let</span> <span class="ident">v</span> <span class="op">=</span> <span class="ident">v</span>.<span class="ident">mul</span>(<span class="kw-2">&</span><span class="ident">cached</span>);
<span class="macro">debug_assert_eq</span><span class="macro">!</span>(<span class="ident">plus</span>.<span class="ident">e</span>, <span class="ident">minus</span>.<span class="ident">e</span>);
<span class="macro">debug_assert_eq</span><span class="macro">!</span>(<span class="ident">plus</span>.<span class="ident">e</span>, <span class="ident">v</span>.<span class="ident">e</span>);
<span class="comment">// +- actual range of minus</span>
<span class="comment">// | <---|---------------------- unsafe region --------------------------> |</span>
<span class="comment">// | | |</span>
<span class="comment">// | |<--->| | <--------------- safe region ---------------> | |</span>
<span class="comment">// | | | | | |</span>
<span class="comment">// |1 ulp|1 ulp| |1 ulp|1 ulp| |1 ulp|1 ulp|</span>
<span class="comment">// |<--->|<--->| |<--->|<--->| |<--->|<--->|</span>
<span class="comment">// |-----|-----|-------...-------|-----|-----|-------...-------|-----|-----|</span>
<span class="comment">// | minus | | v | | plus |</span>
<span class="comment">// minus1 minus0 v - 1 ulp v + 1 ulp plus0 plus1</span>
<span class="comment">//</span>
<span class="comment">// above `minus`, `v` and `plus` are *quantized* approximations (error < 1 ulp).</span>
<span class="comment">// as we don't know the error is positive or negative, we use two approximations spaced equally</span>
<span class="comment">// and have the maximal error of 2 ulps.</span>
<span class="comment">//</span>
<span class="comment">// the "unsafe region" is a liberal interval which we initially generate.</span>
<span class="comment">// the "safe region" is a conservative interval which we only accept.</span>
<span class="comment">// we start with the correct repr within the unsafe region, and try to find the closest repr</span>
<span class="comment">// to `v` which is also within the safe region. if we can't, we give up.</span>
<span class="kw">let</span> <span class="ident">plus1</span> <span class="op">=</span> <span class="ident">plus</span>.<span class="ident">f</span> <span class="op">+</span> <span class="number">1</span>;
<span class="comment">// let plus0 = plus.f - 1; // only for explanation</span>
<span class="comment">// let minus0 = minus.f + 1; // only for explanation</span>
<span class="kw">let</span> <span class="ident">minus1</span> <span class="op">=</span> <span class="ident">minus</span>.<span class="ident">f</span> <span class="op">-</span> <span class="number">1</span>;
<span class="kw">let</span> <span class="ident">e</span> <span class="op">=</span> <span class="op">-</span><span class="ident">plus</span>.<span class="ident">e</span> <span class="kw">as</span> <span class="ident">usize</span>; <span class="comment">// shared exponent</span>
<span class="comment">// divide `plus1` into integral and fractional parts.</span>
<span class="comment">// integral parts are guaranteed to fit in u32, since cached power guarantees `plus < 2^32`</span>
<span class="comment">// and normalized `plus.f` is always less than `2^64 - 2^4` due to the precision requirement.</span>
<span class="kw">let</span> <span class="ident">plus1int</span> <span class="op">=</span> (<span class="ident">plus1</span> <span class="op">>></span> <span class="ident">e</span>) <span class="kw">as</span> <span class="ident">u32</span>;
<span class="kw">let</span> <span class="ident">plus1frac</span> <span class="op">=</span> <span class="ident">plus1</span> <span class="op">&</span> ((<span class="number">1</span> <span class="op"><<</span> <span class="ident">e</span>) <span class="op">-</span> <span class="number">1</span>);
<span class="comment">// calculate the largest `10^max_kappa` no more than `plus1` (thus `plus1 < 10^(max_kappa+1)`).</span>
<span class="comment">// this is an upper bound of `kappa` below.</span>
<span class="kw">let</span> (<span class="ident">max_kappa</span>, <span class="ident">max_ten_kappa</span>) <span class="op">=</span> <span class="ident">max_pow10_no_more_than</span>(<span class="ident">plus1int</span>);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">i</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">let</span> <span class="ident">exp</span> <span class="op">=</span> <span class="ident">max_kappa</span> <span class="kw">as</span> <span class="ident">i16</span> <span class="op">-</span> <span class="ident">minusk</span> <span class="op">+</span> <span class="number">1</span>;
<span class="comment">// Theorem 6.2: if `k` is the greatest integer s.t. `0 <= y mod 10^k <= y - x`,</span>
<span class="comment">// then `V = floor(y / 10^k) * 10^k` is in `[x, y]` and one of the shortest</span>
<span class="comment">// representations (with the minimal number of significant digits) in that range.</span>
<span class="comment">//</span>
<span class="comment">// find the digit length `kappa` between `(minus1, plus1)` as per Theorem 6.2.</span>
<span class="comment">// Theorem 6.2 can be adopted to exclude `x` by requiring `y mod 10^k < y - x` instead.</span>
<span class="comment">// (e.g. `x` = 32000, `y` = 32777; `kappa` = 2 since `y mod 10^3 = 777 < y - x = 777`.)</span>
<span class="comment">// the algorithm relies on the later verification phase to exclude `y`.</span>
<span class="kw">let</span> <span class="ident">delta1</span> <span class="op">=</span> <span class="ident">plus1</span> <span class="op">-</span> <span class="ident">minus1</span>;
<span class="comment">// let delta1int = (delta1 >> e) as usize; // only for explanation</span>
<span class="kw">let</span> <span class="ident">delta1frac</span> <span class="op">=</span> <span class="ident">delta1</span> <span class="op">&</span> ((<span class="number">1</span> <span class="op"><<</span> <span class="ident">e</span>) <span class="op">-</span> <span class="number">1</span>);
<span class="comment">// render integral parts, while checking for the accuracy at each step.</span>
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">kappa</span> <span class="op">=</span> <span class="ident">max_kappa</span> <span class="kw">as</span> <span class="ident">i16</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">ten_kappa</span> <span class="op">=</span> <span class="ident">max_ten_kappa</span>; <span class="comment">// 10^kappa</span>
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">remainder</span> <span class="op">=</span> <span class="ident">plus1int</span>; <span class="comment">// digits yet to be rendered</span>
<span class="kw">loop</span> { <span class="comment">// we always have at least one digit to render, as `plus1 >= 10^kappa`</span>
<span class="comment">// invariants:</span>
<span class="comment">// - `delta1int <= remainder < 10^(kappa+1)`</span>
<span class="comment">// - `plus1int = d[0..n-1] * 10^(kappa+1) + remainder`</span>
<span class="comment">// (it follows that `remainder = plus1int % 10^(kappa+1)`)</span>
<span class="comment">// divide `remainder` by `10^kappa`. both are scaled by `2^-e`.</span>
<span class="kw">let</span> <span class="ident">q</span> <span class="op">=</span> <span class="ident">remainder</span> <span class="op">/</span> <span class="ident">ten_kappa</span>;
<span class="kw">let</span> <span class="ident">r</span> <span class="op">=</span> <span class="ident">remainder</span> <span class="op">%</span> <span class="ident">ten_kappa</span>;
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">q</span> <span class="op"><</span> <span class="number">10</span>);
<span class="ident">buf</span>[<span class="ident">i</span>] <span class="op">=</span> <span class="string">b'0'</span> <span class="op">+</span> <span class="ident">q</span> <span class="kw">as</span> <span class="ident">u8</span>;
<span class="ident">i</span> <span class="op">+=</span> <span class="number">1</span>;
<span class="kw">let</span> <span class="ident">plus1rem</span> <span class="op">=</span> ((<span class="ident">r</span> <span class="kw">as</span> <span class="ident">u64</span>) <span class="op"><<</span> <span class="ident">e</span>) <span class="op">+</span> <span class="ident">plus1frac</span>; <span class="comment">// == (plus1 % 10^kappa) * 2^e</span>
<span class="kw">if</span> <span class="ident">plus1rem</span> <span class="op"><</span> <span class="ident">delta1</span> {
<span class="comment">// `plus1 % 10^kappa < delta1 = plus1 - minus1`; we've found the correct `kappa`.</span>
<span class="kw">let</span> <span class="ident">ten_kappa</span> <span class="op">=</span> (<span class="ident">ten_kappa</span> <span class="kw">as</span> <span class="ident">u64</span>) <span class="op"><<</span> <span class="ident">e</span>; <span class="comment">// scale 10^kappa back to the shared exponent</span>
<span class="kw">return</span> <span class="ident">round_and_weed</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">buf</span>[..<span class="ident">i</span>], <span class="ident">exp</span>, <span class="ident">plus1rem</span>, <span class="ident">delta1</span>, <span class="ident">plus1</span> <span class="op">-</span> <span class="ident">v</span>.<span class="ident">f</span>, <span class="ident">ten_kappa</span>, <span class="number">1</span>);
}
<span class="comment">// break the loop when we have rendered all integral digits.</span>
<span class="comment">// the exact number of digits is `max_kappa + 1` as `plus1 < 10^(max_kappa+1)`.</span>
<span class="kw">if</span> <span class="ident">i</span> <span class="op">></span> <span class="ident">max_kappa</span> <span class="kw">as</span> <span class="ident">usize</span> {
<span class="macro">debug_assert_eq</span><span class="macro">!</span>(<span class="ident">ten_kappa</span>, <span class="number">1</span>);
<span class="macro">debug_assert_eq</span><span class="macro">!</span>(<span class="ident">kappa</span>, <span class="number">0</span>);
<span class="kw">break</span>;
}
<span class="comment">// restore invariants</span>
<span class="ident">kappa</span> <span class="op">-=</span> <span class="number">1</span>;
<span class="ident">ten_kappa</span> <span class="op">/=</span> <span class="number">10</span>;
<span class="ident">remainder</span> <span class="op">=</span> <span class="ident">r</span>;
}
<span class="comment">// render fractional parts, while checking for the accuracy at each step.</span>
<span class="comment">// this time we rely on repeated multiplications, as division will lose the precision.</span>
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">remainder</span> <span class="op">=</span> <span class="ident">plus1frac</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">threshold</span> <span class="op">=</span> <span class="ident">delta1frac</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">ulp</span> <span class="op">=</span> <span class="number">1</span>;
<span class="kw">loop</span> { <span class="comment">// the next digit should be significant as we've tested that before breaking out</span>
<span class="comment">// invariants, where `m = max_kappa + 1` (# of digits in the integral part):</span>
<span class="comment">// - `remainder < 2^e`</span>
<span class="comment">// - `plus1frac * 10^(n-m) = d[m..n-1] * 2^e + remainder`</span>
<span class="ident">remainder</span> <span class="op">*=</span> <span class="number">10</span>; <span class="comment">// won't overflow, `2^e * 10 < 2^64`</span>
<span class="ident">threshold</span> <span class="op">*=</span> <span class="number">10</span>;
<span class="ident">ulp</span> <span class="op">*=</span> <span class="number">10</span>;
<span class="comment">// divide `remainder` by `10^kappa`.</span>
<span class="comment">// both are scaled by `2^e / 10^kappa`, so the latter is implicit here.</span>
<span class="kw">let</span> <span class="ident">q</span> <span class="op">=</span> <span class="ident">remainder</span> <span class="op">>></span> <span class="ident">e</span>;
<span class="kw">let</span> <span class="ident">r</span> <span class="op">=</span> <span class="ident">remainder</span> <span class="op">&</span> ((<span class="number">1</span> <span class="op"><<</span> <span class="ident">e</span>) <span class="op">-</span> <span class="number">1</span>);
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">q</span> <span class="op"><</span> <span class="number">10</span>);
<span class="ident">buf</span>[<span class="ident">i</span>] <span class="op">=</span> <span class="string">b'0'</span> <span class="op">+</span> <span class="ident">q</span> <span class="kw">as</span> <span class="ident">u8</span>;
<span class="ident">i</span> <span class="op">+=</span> <span class="number">1</span>;
<span class="kw">if</span> <span class="ident">r</span> <span class="op"><</span> <span class="ident">threshold</span> {
<span class="kw">let</span> <span class="ident">ten_kappa</span> <span class="op">=</span> <span class="number">1</span> <span class="op"><<</span> <span class="ident">e</span>; <span class="comment">// implicit divisor</span>
<span class="kw">return</span> <span class="ident">round_and_weed</span>(<span class="kw-2">&</span><span class="kw-2">mut</span> <span class="ident">buf</span>[..<span class="ident">i</span>], <span class="ident">exp</span>, <span class="ident">r</span>, <span class="ident">threshold</span>,
(<span class="ident">plus1</span> <span class="op">-</span> <span class="ident">v</span>.<span class="ident">f</span>) <span class="op">*</span> <span class="ident">ulp</span>, <span class="ident">ten_kappa</span>, <span class="ident">ulp</span>);
}
<span class="comment">// restore invariants</span>
<span class="ident">kappa</span> <span class="op">-=</span> <span class="number">1</span>;
<span class="ident">remainder</span> <span class="op">=</span> <span class="ident">r</span>;
}
<span class="comment">// we've generated all significant digits of `plus1`, but not sure if it's the optimal one.</span>
<span class="comment">// for example, if `minus1` is 3.14153... and `plus1` is 3.14158..., there are 5 different</span>
<span class="comment">// shortest representation from 3.14154 to 3.14158 but we only have the greatest one.</span>
<span class="comment">// we have to successively decrease the last digit and check if this is the optimal repr.</span>
<span class="comment">// there are at most 9 candidates (..1 to ..9), so this is fairly quick. ("rounding" phase)</span>
<span class="comment">//</span>
<span class="comment">// the function checks if this "optimal" repr is actually within the ulp ranges,</span>
<span class="comment">// and also, it is possible that the "second-to-optimal" repr can actually be optimal</span>
<span class="comment">// due to the rounding error. in either cases this returns `None`. ("weeding" phase)</span>
<span class="comment">//</span>
<span class="comment">// all arguments here are scaled by the common (but implicit) value `k`, so that:</span>
<span class="comment">// - `remainder = (plus1 % 10^kappa) * k`</span>
<span class="comment">// - `threshold = (plus1 - minus1) * k` (and also, `remainder < threshold`)</span>
<span class="comment">// - `plus1v = (plus1 - v) * k` (and also, `threshold > plus1v` from prior invariants)</span>
<span class="comment">// - `ten_kappa = 10^kappa * k`</span>
<span class="comment">// - `ulp = 2^-e * k`</span>
<span class="kw">fn</span> <span class="ident">round_and_weed</span>(<span class="ident">buf</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">u8</span>], <span class="ident">exp</span>: <span class="ident">i16</span>, <span class="ident">remainder</span>: <span class="ident">u64</span>, <span class="ident">threshold</span>: <span class="ident">u64</span>, <span class="ident">plus1v</span>: <span class="ident">u64</span>,
<span class="ident">ten_kappa</span>: <span class="ident">u64</span>, <span class="ident">ulp</span>: <span class="ident">u64</span>) <span class="op">-></span> <span class="prelude-ty">Option</span><span class="op"><</span>(<span class="ident">usize</span>, <span class="ident">i16</span>)<span class="op">></span> {
<span class="macro">assert</span><span class="macro">!</span>(<span class="op">!</span><span class="ident">buf</span>.<span class="ident">is_empty</span>());
<span class="comment">// produce two approximations to `v` (actually `plus1 - v`) within 1.5 ulps.</span>
<span class="comment">// the resulting representation should be the closest representation to both.</span>
<span class="comment">//</span>
<span class="comment">// here `plus1 - v` is used since calculations are done with respect to `plus1`</span>
<span class="comment">// in order to avoid overflow/underflow (hence the seemingly swapped names).</span>
<span class="kw">let</span> <span class="ident">plus1v_down</span> <span class="op">=</span> <span class="ident">plus1v</span> <span class="op">+</span> <span class="ident">ulp</span>; <span class="comment">// plus1 - (v - 1 ulp)</span>
<span class="kw">let</span> <span class="ident">plus1v_up</span> <span class="op">=</span> <span class="ident">plus1v</span> <span class="op">-</span> <span class="ident">ulp</span>; <span class="comment">// plus1 - (v + 1 ulp)</span>
<span class="comment">// decrease the last digit and stop at the closest representation to `v + 1 ulp`.</span>
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">plus1w</span> <span class="op">=</span> <span class="ident">remainder</span>; <span class="comment">// plus1w(n) = plus1 - w(n)</span>
{
<span class="kw">let</span> <span class="ident">last</span> <span class="op">=</span> <span class="ident">buf</span>.<span class="ident">last_mut</span>().<span class="ident">unwrap</span>();
<span class="comment">// we work with the approximated digits `w(n)`, which is initially equal to `plus1 -</span>
<span class="comment">// plus1 % 10^kappa`. after running the loop body `n` times, `w(n) = plus1 -</span>
<span class="comment">// plus1 % 10^kappa - n * 10^kappa`. we set `plus1w(n) = plus1 - w(n) =</span>
<span class="comment">// plus1 % 10^kappa + n * 10^kappa` (thus `remainder = plus1w(0)`) to simplify checks.</span>
<span class="comment">// note that `plus1w(n)` is always increasing.</span>
<span class="comment">//</span>
<span class="comment">// we have three conditions to terminate. any of them will make the loop unable to</span>
<span class="comment">// proceed, but we then have at least one valid representation known to be closest to</span>
<span class="comment">// `v + 1 ulp` anyway. we will denote them as TC1 through TC3 for brevity.</span>
<span class="comment">//</span>
<span class="comment">// TC1: `w(n) <= v + 1 ulp`, i.e. this is the last repr that can be the closest one.</span>
<span class="comment">// this is equivalent to `plus1 - w(n) = plus1w(n) >= plus1 - (v + 1 ulp) = plus1v_up`.</span>
<span class="comment">// combined with TC2 (which checks if `w(n+1)` is valid), this prevents the possible</span>
<span class="comment">// overflow on the calculation of `plus1w(n)`.</span>
<span class="comment">//</span>
<span class="comment">// TC2: `w(n+1) < minus1`, i.e. the next repr definitely does not round to `v`.</span>
<span class="comment">// this is equivalent to `plus1 - w(n) + 10^kappa = plus1w(n) + 10^kappa ></span>
<span class="comment">// plus1 - minus1 = threshold`. the left hand side can overflow, but we know</span>
<span class="comment">// `threshold > plus1v`, so if TC1 is false, `threshold - plus1w(n) ></span>
<span class="comment">// threshold - (plus1v - 1 ulp) > 1 ulp` and we can safely test if</span>
<span class="comment">// `threshold - plus1w(n) < 10^kappa` instead.</span>
<span class="comment">//</span>
<span class="comment">// TC3: `abs(w(n) - (v + 1 ulp)) <= abs(w(n+1) - (v + 1 ulp))`, i.e. the next repr is</span>
<span class="comment">// no closer to `v + 1 ulp` than the current repr. given `z(n) = plus1v_up - plus1w(n)`,</span>
<span class="comment">// this becomes `abs(z(n)) <= abs(z(n+1))`. again assuming that TC1 is false, we have</span>
<span class="comment">// `z(n) > 0`. we have two cases to consider:</span>
<span class="comment">//</span>
<span class="comment">// - when `z(n+1) >= 0`: TC3 becomes `z(n) <= z(n+1)`. as `plus1w(n)` is increasing,</span>
<span class="comment">// `z(n)` should be decreasing and this is clearly false.</span>
<span class="comment">// - when `z(n+1) < 0`:</span>
<span class="comment">// - TC3a: the precondition is `plus1v_up < plus1w(n) + 10^kappa`. assuming TC2 is</span>
<span class="comment">// false, `threshold >= plus1w(n) + 10^kappa` so it cannot overflow.</span>
<span class="comment">// - TC3b: TC3 becomes `z(n) <= -z(n+1)`, i.e. `plus1v_up - plus1w(n) >=</span>
<span class="comment">// plus1w(n+1) - plus1v_up = plus1w(n) + 10^kappa - plus1v_up`. the negated TC1</span>
<span class="comment">// gives `plus1v_up > plus1w(n)`, so it cannot overflow or underflow when</span>
<span class="comment">// combined with TC3a.</span>
<span class="comment">//</span>
<span class="comment">// consequently, we should stop when `TC1 || TC2 || (TC3a && TC3b)`. the following is</span>
<span class="comment">// equal to its inverse, `!TC1 && !TC2 && (!TC3a || !TC3b)`.</span>
<span class="kw">while</span> <span class="ident">plus1w</span> <span class="op"><</span> <span class="ident">plus1v_up</span> <span class="op">&&</span>
<span class="ident">threshold</span> <span class="op">-</span> <span class="ident">plus1w</span> <span class="op">>=</span> <span class="ident">ten_kappa</span> <span class="op">&&</span>
(<span class="ident">plus1w</span> <span class="op">+</span> <span class="ident">ten_kappa</span> <span class="op"><</span> <span class="ident">plus1v_up</span> <span class="op">||</span>
<span class="ident">plus1v_up</span> <span class="op">-</span> <span class="ident">plus1w</span> <span class="op">>=</span> <span class="ident">plus1w</span> <span class="op">+</span> <span class="ident">ten_kappa</span> <span class="op">-</span> <span class="ident">plus1v_up</span>) {
<span class="kw-2">*</span><span class="ident">last</span> <span class="op">-=</span> <span class="number">1</span>;
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="kw-2">*</span><span class="ident">last</span> <span class="op">></span> <span class="string">b'0'</span>); <span class="comment">// the shortest repr cannot end with `0`</span>
<span class="ident">plus1w</span> <span class="op">+=</span> <span class="ident">ten_kappa</span>;
}
}
<span class="comment">// check if this representation is also the closest representation to `v - 1 ulp`.</span>
<span class="comment">//</span>
<span class="comment">// this is simply same to the terminating conditions for `v + 1 ulp`, with all `plus1v_up`</span>
<span class="comment">// replaced by `plus1v_down` instead. overflow analysis equally holds.</span>
<span class="kw">if</span> <span class="ident">plus1w</span> <span class="op"><</span> <span class="ident">plus1v_down</span> <span class="op">&&</span>
<span class="ident">threshold</span> <span class="op">-</span> <span class="ident">plus1w</span> <span class="op">>=</span> <span class="ident">ten_kappa</span> <span class="op">&&</span>
(<span class="ident">plus1w</span> <span class="op">+</span> <span class="ident">ten_kappa</span> <span class="op"><</span> <span class="ident">plus1v_down</span> <span class="op">||</span>
<span class="ident">plus1v_down</span> <span class="op">-</span> <span class="ident">plus1w</span> <span class="op">>=</span> <span class="ident">plus1w</span> <span class="op">+</span> <span class="ident">ten_kappa</span> <span class="op">-</span> <span class="ident">plus1v_down</span>) {
<span class="kw">return</span> <span class="prelude-val">None</span>;
}
<span class="comment">// now we have the closest representation to `v` between `plus1` and `minus1`.</span>
<span class="comment">// this is too liberal, though, so we reject any `w(n)` not between `plus0` and `minus0`,</span>
<span class="comment">// i.e. `plus1 - plus1w(n) <= minus0` or `plus1 - plus1w(n) >= plus0`. we utilize the facts</span>
<span class="comment">// that `threshold = plus1 - minus1` and `plus1 - plus0 = minus0 - minus1 = 2 ulp`.</span>
<span class="kw">if</span> <span class="number">2</span> <span class="op">*</span> <span class="ident">ulp</span> <span class="op"><=</span> <span class="ident">plus1w</span> <span class="op">&&</span> <span class="ident">plus1w</span> <span class="op"><=</span> <span class="ident">threshold</span> <span class="op">-</span> <span class="number">4</span> <span class="op">*</span> <span class="ident">ulp</span> {
<span class="prelude-val">Some</span>((<span class="ident">buf</span>.<span class="ident">len</span>(), <span class="ident">exp</span>))
} <span class="kw">else</span> {
<span class="prelude-val">None</span>
}
}
}
<span class="doccomment">/// The shortest mode implementation for Grisu with Dragon fallback.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// This should be used for most cases.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">format_shortest</span>(<span class="ident">d</span>: <span class="kw-2">&</span><span class="ident">Decoded</span>, <span class="ident">buf</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">u8</span>]) <span class="op">-></span> (<span class="comment">/*#digits*/</span> <span class="ident">usize</span>, <span class="comment">/*exp*/</span> <span class="ident">i16</span>) {
<span class="kw">use</span> <span class="ident">num</span>::<span class="ident">flt2dec</span>::<span class="ident">strategy</span>::<span class="ident">dragon</span>::<span class="ident">format_shortest</span> <span class="kw">as</span> <span class="ident">fallback</span>;
<span class="kw">match</span> <span class="ident">format_shortest_opt</span>(<span class="ident">d</span>, <span class="ident">buf</span>) {
<span class="prelude-val">Some</span>(<span class="ident">ret</span>) <span class="op">=></span> <span class="ident">ret</span>,
<span class="prelude-val">None</span> <span class="op">=></span> <span class="ident">fallback</span>(<span class="ident">d</span>, <span class="ident">buf</span>),
}
}
<span class="doccomment">/// The exact and fixed mode implementation for Grisu.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// It returns `None` when it would return an inexact representation otherwise.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">format_exact_opt</span>(<span class="ident">d</span>: <span class="kw-2">&</span><span class="ident">Decoded</span>, <span class="ident">buf</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">u8</span>], <span class="ident">limit</span>: <span class="ident">i16</span>)
<span class="op">-></span> <span class="prelude-ty">Option</span><span class="op"><</span>(<span class="comment">/*#digits*/</span> <span class="ident">usize</span>, <span class="comment">/*exp*/</span> <span class="ident">i16</span>)<span class="op">></span> {
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">d</span>.<span class="ident">mant</span> <span class="op">></span> <span class="number">0</span>);
<span class="macro">assert</span><span class="macro">!</span>(<span class="ident">d</span>.<span class="ident">mant</span> <span class="op"><</span> (<span class="number">1</span> <span class="op"><<</span> <span class="number">61</span>)); <span class="comment">// we need at least three bits of additional precision</span>
<span class="macro">assert</span><span class="macro">!</span>(<span class="op">!</span><span class="ident">buf</span>.<span class="ident">is_empty</span>());
<span class="comment">// normalize and scale `v`.</span>
<span class="kw">let</span> <span class="ident">v</span> <span class="op">=</span> <span class="ident">Fp</span> { <span class="ident">f</span>: <span class="ident">d</span>.<span class="ident">mant</span>, <span class="ident">e</span>: <span class="ident">d</span>.<span class="ident">exp</span> }.<span class="ident">normalize</span>();
<span class="kw">let</span> (<span class="ident">minusk</span>, <span class="ident">cached</span>) <span class="op">=</span> <span class="ident">cached_power</span>(<span class="ident">ALPHA</span> <span class="op">-</span> <span class="ident">v</span>.<span class="ident">e</span> <span class="op">-</span> <span class="number">64</span>, <span class="ident">GAMMA</span> <span class="op">-</span> <span class="ident">v</span>.<span class="ident">e</span> <span class="op">-</span> <span class="number">64</span>);
<span class="kw">let</span> <span class="ident">v</span> <span class="op">=</span> <span class="ident">v</span>.<span class="ident">mul</span>(<span class="kw-2">&</span><span class="ident">cached</span>);
<span class="comment">// divide `v` into integral and fractional parts.</span>
<span class="kw">let</span> <span class="ident">e</span> <span class="op">=</span> <span class="op">-</span><span class="ident">v</span>.<span class="ident">e</span> <span class="kw">as</span> <span class="ident">usize</span>;
<span class="kw">let</span> <span class="ident">vint</span> <span class="op">=</span> (<span class="ident">v</span>.<span class="ident">f</span> <span class="op">>></span> <span class="ident">e</span>) <span class="kw">as</span> <span class="ident">u32</span>;
<span class="kw">let</span> <span class="ident">vfrac</span> <span class="op">=</span> <span class="ident">v</span>.<span class="ident">f</span> <span class="op">&</span> ((<span class="number">1</span> <span class="op"><<</span> <span class="ident">e</span>) <span class="op">-</span> <span class="number">1</span>);
<span class="comment">// both old `v` and new `v` (scaled by `10^-k`) has an error of < 1 ulp (Theorem 5.1).</span>
<span class="comment">// as we don't know the error is positive or negative, we use two approximations</span>
<span class="comment">// spaced equally and have the maximal error of 2 ulps (same to the shortest case).</span>
<span class="comment">//</span>
<span class="comment">// the goal is to find the exactly rounded series of digits that are common to</span>
<span class="comment">// both `v - 1 ulp` and `v + 1 ulp`, so that we are maximally confident.</span>
<span class="comment">// if this is not possible, we don't know which one is the correct output for `v`,</span>
<span class="comment">// so we give up and fall back.</span>
<span class="comment">//</span>
<span class="comment">// `err` is defined as `1 ulp * 2^e` here (same to the ulp in `vfrac`),</span>
<span class="comment">// and we will scale it whenever `v` gets scaled.</span>
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">err</span> <span class="op">=</span> <span class="number">1</span>;
<span class="comment">// calculate the largest `10^max_kappa` no more than `v` (thus `v < 10^(max_kappa+1)`).</span>
<span class="comment">// this is an upper bound of `kappa` below.</span>
<span class="kw">let</span> (<span class="ident">max_kappa</span>, <span class="ident">max_ten_kappa</span>) <span class="op">=</span> <span class="ident">max_pow10_no_more_than</span>(<span class="ident">vint</span>);
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">i</span> <span class="op">=</span> <span class="number">0</span>;
<span class="kw">let</span> <span class="ident">exp</span> <span class="op">=</span> <span class="ident">max_kappa</span> <span class="kw">as</span> <span class="ident">i16</span> <span class="op">-</span> <span class="ident">minusk</span> <span class="op">+</span> <span class="number">1</span>;
<span class="comment">// if we are working with the last-digit limitation, we need to shorten the buffer</span>
<span class="comment">// before the actual rendering in order to avoid double rounding.</span>
<span class="comment">// note that we have to enlarge the buffer again when rounding up happens!</span>
<span class="kw">let</span> <span class="ident">len</span> <span class="op">=</span> <span class="kw">if</span> <span class="ident">exp</span> <span class="op"><=</span> <span class="ident">limit</span> {
<span class="comment">// oops, we cannot even produce *one* digit.</span>
<span class="comment">// this is possible when, say, we've got something like 9.5 and it's being rounded to 10.</span>
<span class="comment">//</span>
<span class="comment">// in principle we can immediately call `possibly_round` with an empty buffer,</span>
<span class="comment">// but scaling `max_ten_kappa << e` by 10 can result in overflow.</span>
<span class="comment">// thus we are being sloppy here and widen the error range by a factor of 10.</span>
<span class="comment">// this will increase the false negative rate, but only very, *very* slightly;</span>
<span class="comment">// it can only matter noticeably when the mantissa is bigger than 60 bits.</span>
<span class="kw">return</span> <span class="ident">possibly_round</span>(<span class="ident">buf</span>, <span class="number">0</span>, <span class="ident">exp</span>, <span class="ident">limit</span>, <span class="ident">v</span>.<span class="ident">f</span> <span class="op">/</span> <span class="number">10</span>, (<span class="ident">max_ten_kappa</span> <span class="kw">as</span> <span class="ident">u64</span>) <span class="op"><<</span> <span class="ident">e</span>, <span class="ident">err</span> <span class="op"><<</span> <span class="ident">e</span>);
} <span class="kw">else</span> <span class="kw">if</span> ((<span class="ident">exp</span> <span class="kw">as</span> <span class="ident">i32</span> <span class="op">-</span> <span class="ident">limit</span> <span class="kw">as</span> <span class="ident">i32</span>) <span class="kw">as</span> <span class="ident">usize</span>) <span class="op"><</span> <span class="ident">buf</span>.<span class="ident">len</span>() {
(<span class="ident">exp</span> <span class="op">-</span> <span class="ident">limit</span>) <span class="kw">as</span> <span class="ident">usize</span>
} <span class="kw">else</span> {
<span class="ident">buf</span>.<span class="ident">len</span>()
};
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">len</span> <span class="op">></span> <span class="number">0</span>);
<span class="comment">// render integral parts.</span>
<span class="comment">// the error is entirely fractional, so we don't need to check it in this part.</span>
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">kappa</span> <span class="op">=</span> <span class="ident">max_kappa</span> <span class="kw">as</span> <span class="ident">i16</span>;
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">ten_kappa</span> <span class="op">=</span> <span class="ident">max_ten_kappa</span>; <span class="comment">// 10^kappa</span>
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">remainder</span> <span class="op">=</span> <span class="ident">vint</span>; <span class="comment">// digits yet to be rendered</span>
<span class="kw">loop</span> { <span class="comment">// we always have at least one digit to render</span>
<span class="comment">// invariants:</span>
<span class="comment">// - `remainder < 10^(kappa+1)`</span>
<span class="comment">// - `vint = d[0..n-1] * 10^(kappa+1) + remainder`</span>
<span class="comment">// (it follows that `remainder = vint % 10^(kappa+1)`)</span>
<span class="comment">// divide `remainder` by `10^kappa`. both are scaled by `2^-e`.</span>
<span class="kw">let</span> <span class="ident">q</span> <span class="op">=</span> <span class="ident">remainder</span> <span class="op">/</span> <span class="ident">ten_kappa</span>;
<span class="kw">let</span> <span class="ident">r</span> <span class="op">=</span> <span class="ident">remainder</span> <span class="op">%</span> <span class="ident">ten_kappa</span>;
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">q</span> <span class="op"><</span> <span class="number">10</span>);
<span class="ident">buf</span>[<span class="ident">i</span>] <span class="op">=</span> <span class="string">b'0'</span> <span class="op">+</span> <span class="ident">q</span> <span class="kw">as</span> <span class="ident">u8</span>;
<span class="ident">i</span> <span class="op">+=</span> <span class="number">1</span>;
<span class="comment">// is the buffer full? run the rounding pass with the remainder.</span>
<span class="kw">if</span> <span class="ident">i</span> <span class="op">==</span> <span class="ident">len</span> {
<span class="kw">let</span> <span class="ident">vrem</span> <span class="op">=</span> ((<span class="ident">r</span> <span class="kw">as</span> <span class="ident">u64</span>) <span class="op"><<</span> <span class="ident">e</span>) <span class="op">+</span> <span class="ident">vfrac</span>; <span class="comment">// == (v % 10^kappa) * 2^e</span>
<span class="kw">return</span> <span class="ident">possibly_round</span>(<span class="ident">buf</span>, <span class="ident">len</span>, <span class="ident">exp</span>, <span class="ident">limit</span>, <span class="ident">vrem</span>, (<span class="ident">ten_kappa</span> <span class="kw">as</span> <span class="ident">u64</span>) <span class="op"><<</span> <span class="ident">e</span>, <span class="ident">err</span> <span class="op"><<</span> <span class="ident">e</span>);
}
<span class="comment">// break the loop when we have rendered all integral digits.</span>
<span class="comment">// the exact number of digits is `max_kappa + 1` as `plus1 < 10^(max_kappa+1)`.</span>
<span class="kw">if</span> <span class="ident">i</span> <span class="op">></span> <span class="ident">max_kappa</span> <span class="kw">as</span> <span class="ident">usize</span> {
<span class="macro">debug_assert_eq</span><span class="macro">!</span>(<span class="ident">ten_kappa</span>, <span class="number">1</span>);
<span class="macro">debug_assert_eq</span><span class="macro">!</span>(<span class="ident">kappa</span>, <span class="number">0</span>);
<span class="kw">break</span>;
}
<span class="comment">// restore invariants</span>
<span class="ident">kappa</span> <span class="op">-=</span> <span class="number">1</span>;
<span class="ident">ten_kappa</span> <span class="op">/=</span> <span class="number">10</span>;
<span class="ident">remainder</span> <span class="op">=</span> <span class="ident">r</span>;
}
<span class="comment">// render fractional parts.</span>
<span class="comment">//</span>
<span class="comment">// in principle we can continue to the last available digit and check for the accuracy.</span>
<span class="comment">// unfortunately we are working with the finite-sized integers, so we need some criterion</span>
<span class="comment">// to detect the overflow. V8 uses `remainder > err`, which becomes false when</span>
<span class="comment">// the first `i` significant digits of `v - 1 ulp` and `v` differ. however this rejects</span>
<span class="comment">// too many otherwise valid input.</span>
<span class="comment">//</span>
<span class="comment">// since the later phase has a correct overflow detection, we instead use tighter criterion:</span>
<span class="comment">// we continue til `err` exceeds `10^kappa / 2`, so that the range between `v - 1 ulp` and</span>
<span class="comment">// `v + 1 ulp` definitely contains two or more rounded representations. this is same to</span>
<span class="comment">// the first two comparisons from `possibly_round`, for the reference.</span>
<span class="kw">let</span> <span class="kw-2">mut</span> <span class="ident">remainder</span> <span class="op">=</span> <span class="ident">vfrac</span>;
<span class="kw">let</span> <span class="ident">maxerr</span> <span class="op">=</span> <span class="number">1</span> <span class="op"><<</span> (<span class="ident">e</span> <span class="op">-</span> <span class="number">1</span>);
<span class="kw">while</span> <span class="ident">err</span> <span class="op"><</span> <span class="ident">maxerr</span> {
<span class="comment">// invariants, where `m = max_kappa + 1` (# of digits in the integral part):</span>
<span class="comment">// - `remainder < 2^e`</span>
<span class="comment">// - `vfrac * 10^(n-m) = d[m..n-1] * 2^e + remainder`</span>
<span class="comment">// - `err = 10^(n-m)`</span>
<span class="ident">remainder</span> <span class="op">*=</span> <span class="number">10</span>; <span class="comment">// won't overflow, `2^e * 10 < 2^64`</span>
<span class="ident">err</span> <span class="op">*=</span> <span class="number">10</span>; <span class="comment">// won't overflow, `err * 10 < 2^e * 5 < 2^64`</span>
<span class="comment">// divide `remainder` by `10^kappa`.</span>
<span class="comment">// both are scaled by `2^e / 10^kappa`, so the latter is implicit here.</span>
<span class="kw">let</span> <span class="ident">q</span> <span class="op">=</span> <span class="ident">remainder</span> <span class="op">>></span> <span class="ident">e</span>;
<span class="kw">let</span> <span class="ident">r</span> <span class="op">=</span> <span class="ident">remainder</span> <span class="op">&</span> ((<span class="number">1</span> <span class="op"><<</span> <span class="ident">e</span>) <span class="op">-</span> <span class="number">1</span>);
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">q</span> <span class="op"><</span> <span class="number">10</span>);
<span class="ident">buf</span>[<span class="ident">i</span>] <span class="op">=</span> <span class="string">b'0'</span> <span class="op">+</span> <span class="ident">q</span> <span class="kw">as</span> <span class="ident">u8</span>;
<span class="ident">i</span> <span class="op">+=</span> <span class="number">1</span>;
<span class="comment">// is the buffer full? run the rounding pass with the remainder.</span>
<span class="kw">if</span> <span class="ident">i</span> <span class="op">==</span> <span class="ident">len</span> {
<span class="kw">return</span> <span class="ident">possibly_round</span>(<span class="ident">buf</span>, <span class="ident">len</span>, <span class="ident">exp</span>, <span class="ident">limit</span>, <span class="ident">r</span>, <span class="number">1</span> <span class="op"><<</span> <span class="ident">e</span>, <span class="ident">err</span>);
}
<span class="comment">// restore invariants</span>
<span class="ident">remainder</span> <span class="op">=</span> <span class="ident">r</span>;
}
<span class="comment">// further calculation is useless (`possibly_round` definitely fails), so we give up.</span>
<span class="kw">return</span> <span class="prelude-val">None</span>;
<span class="comment">// we've generated all requested digits of `v`, which should be also same to corresponding</span>
<span class="comment">// digits of `v - 1 ulp`. now we check if there is a unique representation shared by</span>
<span class="comment">// both `v - 1 ulp` and `v + 1 ulp`; this can be either same to generated digits, or</span>
<span class="comment">// to the rounded-up version of those digits. if the range contains multiple representations</span>
<span class="comment">// of the same length, we cannot be sure and should return `None` instead.</span>
<span class="comment">//</span>
<span class="comment">// all arguments here are scaled by the common (but implicit) value `k`, so that:</span>
<span class="comment">// - `remainder = (v % 10^kappa) * k`</span>
<span class="comment">// - `ten_kappa = 10^kappa * k`</span>
<span class="comment">// - `ulp = 2^-e * k`</span>
<span class="kw">fn</span> <span class="ident">possibly_round</span>(<span class="ident">buf</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">u8</span>], <span class="kw-2">mut</span> <span class="ident">len</span>: <span class="ident">usize</span>, <span class="kw-2">mut</span> <span class="ident">exp</span>: <span class="ident">i16</span>, <span class="ident">limit</span>: <span class="ident">i16</span>,
<span class="ident">remainder</span>: <span class="ident">u64</span>, <span class="ident">ten_kappa</span>: <span class="ident">u64</span>, <span class="ident">ulp</span>: <span class="ident">u64</span>) <span class="op">-></span> <span class="prelude-ty">Option</span><span class="op"><</span>(<span class="ident">usize</span>, <span class="ident">i16</span>)<span class="op">></span> {
<span class="macro">debug_assert</span><span class="macro">!</span>(<span class="ident">remainder</span> <span class="op"><</span> <span class="ident">ten_kappa</span>);
<span class="comment">// 10^kappa</span>
<span class="comment">// : : :<->: :</span>
<span class="comment">// : : : : :</span>
<span class="comment">// :|1 ulp|1 ulp| :</span>
<span class="comment">// :|<--->|<--->| :</span>
<span class="comment">// ----|-----|-----|----</span>
<span class="comment">// | v |</span>
<span class="comment">// v - 1 ulp v + 1 ulp</span>
<span class="comment">//</span>
<span class="comment">// (for the reference, the dotted line indicates the exact value for</span>
<span class="comment">// possible representations in given number of digits.)</span>
<span class="comment">//</span>
<span class="comment">// error is too large that there are at least three possible representations</span>
<span class="comment">// between `v - 1 ulp` and `v + 1 ulp`. we cannot determine which one is correct.</span>
<span class="kw">if</span> <span class="ident">ulp</span> <span class="op">>=</span> <span class="ident">ten_kappa</span> { <span class="kw">return</span> <span class="prelude-val">None</span>; }
<span class="comment">// 10^kappa</span>
<span class="comment">// :<------->:</span>
<span class="comment">// : :</span>
<span class="comment">// : |1 ulp|1 ulp|</span>
<span class="comment">// : |<--->|<--->|</span>
<span class="comment">// ----|-----|-----|----</span>
<span class="comment">// | v |</span>
<span class="comment">// v - 1 ulp v + 1 ulp</span>
<span class="comment">//</span>
<span class="comment">// in fact, 1/2 ulp is enough to introduce two possible representations.</span>
<span class="comment">// (remember that we need a unique representation for both `v - 1 ulp` and `v + 1 ulp`.)</span>
<span class="comment">// this won't overflow, as `ulp < ten_kappa` from the first check.</span>
<span class="kw">if</span> <span class="ident">ten_kappa</span> <span class="op">-</span> <span class="ident">ulp</span> <span class="op"><=</span> <span class="ident">ulp</span> { <span class="kw">return</span> <span class="prelude-val">None</span>; }
<span class="comment">// remainder</span>
<span class="comment">// :<->| :</span>
<span class="comment">// : | :</span>
<span class="comment">// :<--------- 10^kappa ---------->:</span>
<span class="comment">// | : | :</span>
<span class="comment">// |1 ulp|1 ulp| :</span>
<span class="comment">// |<--->|<--->| :</span>
<span class="comment">// ----|-----|-----|------------------------</span>
<span class="comment">// | v |</span>
<span class="comment">// v - 1 ulp v + 1 ulp</span>
<span class="comment">//</span>
<span class="comment">// if `v + 1 ulp` is closer to the rounded-down representation (which is already in `buf`),</span>
<span class="comment">// then we can safely return. note that `v - 1 ulp` *can* be less than the current</span>
<span class="comment">// representation, but as `1 ulp < 10^kappa / 2`, this condition is enough:</span>
<span class="comment">// the distance between `v - 1 ulp` and the current representation</span>
<span class="comment">// cannot exceed `10^kappa / 2`.</span>
<span class="comment">//</span>
<span class="comment">// the condition equals to `remainder + ulp < 10^kappa / 2`.</span>
<span class="comment">// since this can easily overflow, first check if `remainder < 10^kappa / 2`.</span>
<span class="comment">// we've already verified that `ulp < 10^kappa / 2`, so as long as</span>
<span class="comment">// `10^kappa` did not overflow after all, the second check is fine.</span>
<span class="kw">if</span> <span class="ident">ten_kappa</span> <span class="op">-</span> <span class="ident">remainder</span> <span class="op">></span> <span class="ident">remainder</span> <span class="op">&&</span> <span class="ident">ten_kappa</span> <span class="op">-</span> <span class="number">2</span> <span class="op">*</span> <span class="ident">remainder</span> <span class="op">>=</span> <span class="number">2</span> <span class="op">*</span> <span class="ident">ulp</span> {
<span class="kw">return</span> <span class="prelude-val">Some</span>((<span class="ident">len</span>, <span class="ident">exp</span>));
}
<span class="comment">// :<------- remainder ------>| :</span>
<span class="comment">// : | :</span>
<span class="comment">// :<--------- 10^kappa --------->:</span>
<span class="comment">// : | | : |</span>
<span class="comment">// : |1 ulp|1 ulp|</span>
<span class="comment">// : |<--->|<--->|</span>
<span class="comment">// -----------------------|-----|-----|-----</span>
<span class="comment">// | v |</span>
<span class="comment">// v - 1 ulp v + 1 ulp</span>
<span class="comment">//</span>
<span class="comment">// on the other hands, if `v - 1 ulp` is closer to the rounded-up representation,</span>
<span class="comment">// we should round up and return. for the same reason we don't need to check `v + 1 ulp`.</span>
<span class="comment">//</span>
<span class="comment">// the condition equals to `remainder - ulp >= 10^kappa / 2`.</span>
<span class="comment">// again we first check if `remainder > ulp` (note that this is not `remainder >= ulp`,</span>
<span class="comment">// as `10^kappa` is never zero). also note that `remainder - ulp <= 10^kappa`,</span>
<span class="comment">// so the second check does not overflow.</span>
<span class="kw">if</span> <span class="ident">remainder</span> <span class="op">></span> <span class="ident">ulp</span> <span class="op">&&</span> <span class="ident">ten_kappa</span> <span class="op">-</span> (<span class="ident">remainder</span> <span class="op">-</span> <span class="ident">ulp</span>) <span class="op"><=</span> <span class="ident">remainder</span> <span class="op">-</span> <span class="ident">ulp</span> {
<span class="kw">if</span> <span class="kw">let</span> <span class="prelude-val">Some</span>(<span class="ident">c</span>) <span class="op">=</span> <span class="ident">round_up</span>(<span class="ident">buf</span>, <span class="ident">len</span>) {
<span class="comment">// only add an additional digit when we've been requested the fixed precision.</span>
<span class="comment">// we also need to check that, if the original buffer was empty,</span>
<span class="comment">// the additional digit can only be added when `exp == limit` (edge case).</span>
<span class="ident">exp</span> <span class="op">+=</span> <span class="number">1</span>;
<span class="kw">if</span> <span class="ident">exp</span> <span class="op">></span> <span class="ident">limit</span> <span class="op">&&</span> <span class="ident">len</span> <span class="op"><</span> <span class="ident">buf</span>.<span class="ident">len</span>() {
<span class="ident">buf</span>[<span class="ident">len</span>] <span class="op">=</span> <span class="ident">c</span>;
<span class="ident">len</span> <span class="op">+=</span> <span class="number">1</span>;
}
}
<span class="kw">return</span> <span class="prelude-val">Some</span>((<span class="ident">len</span>, <span class="ident">exp</span>));
}
<span class="comment">// otherwise we are doomed (i.e. some values between `v - 1 ulp` and `v + 1 ulp` are</span>
<span class="comment">// rounding down and others are rounding up) and give up.</span>
<span class="prelude-val">None</span>
}
}
<span class="doccomment">/// The exact and fixed mode implementation for Grisu with Dragon fallback.</span>
<span class="doccomment">///</span>
<span class="doccomment">/// This should be used for most cases.</span>
<span class="kw">pub</span> <span class="kw">fn</span> <span class="ident">format_exact</span>(<span class="ident">d</span>: <span class="kw-2">&</span><span class="ident">Decoded</span>, <span class="ident">buf</span>: <span class="kw-2">&</span><span class="kw-2">mut</span> [<span class="ident">u8</span>], <span class="ident">limit</span>: <span class="ident">i16</span>) <span class="op">-></span> (<span class="comment">/*#digits*/</span> <span class="ident">usize</span>, <span class="comment">/*exp*/</span> <span class="ident">i16</span>) {
<span class="kw">use</span> <span class="ident">num</span>::<span class="ident">flt2dec</span>::<span class="ident">strategy</span>::<span class="ident">dragon</span>::<span class="ident">format_exact</span> <span class="kw">as</span> <span class="ident">fallback</span>;
<span class="kw">match</span> <span class="ident">format_exact_opt</span>(<span class="ident">d</span>, <span class="ident">buf</span>, <span class="ident">limit</span>) {
<span class="prelude-val">Some</span>(<span class="ident">ret</span>) <span class="op">=></span> <span class="ident">ret</span>,
<span class="prelude-val">None</span> <span class="op">=></span> <span class="ident">fallback</span>(<span class="ident">d</span>, <span class="ident">buf</span>, <span class="ident">limit</span>),
}
}
</pre>
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