1 // Copyright 2015 The Rust Project Developers. See the COPYRIGHT
2 // file at the top-level directory of this distribution and at
3 // http://rust-lang.org/COPYRIGHT.
5 // Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
6 // http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
7 // <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
8 // option. This file may not be copied, modified, or distributed
9 // except according to those terms.
11 //! Rust adaptation of the Grisu3 algorithm described in "Printing Floating-Point Numbers Quickly
12 //! and Accurately with Integers"[^1]. It uses about 1KB of precomputed table, and in turn, it's
13 //! very quick for most inputs.
15 //! [^1]: Florian Loitsch. 2010. Printing floating-point numbers quickly and
16 //! accurately with integers. SIGPLAN Not. 45, 6 (June 2010), 233-243.
18 use num::diy_float::Fp;
19 use num::flt2dec::{Decoded, MAX_SIG_DIGITS, round_up};
22 // see the comments in `format_shortest_opt` for the rationale.
23 #[doc(hidden)] pub const ALPHA: i16 = -60;
24 #[doc(hidden)] pub const GAMMA: i16 = -32;
27 # the following Python code generates this table:
28 for i in xrange(-308, 333, 8):
29 if i >= 0: f = 10**i; e = 0
30 else: f = 2**(80-4*i) // 10**-i; e = 4 * i - 80
32 f = ((f << 64 >> (l-1)) + 1) >> 1; e += l - 64
33 print ' (%#018x, %5d, %4d),' % (f, e, i)
37 pub static CACHED_POW10: [(u64, i16, i16); 81] = [ // (f, e, k)
38 (0xe61acf033d1a45df, -1087, -308),
39 (0xab70fe17c79ac6ca, -1060, -300),
40 (0xff77b1fcbebcdc4f, -1034, -292),
41 (0xbe5691ef416bd60c, -1007, -284),
42 (0x8dd01fad907ffc3c, -980, -276),
43 (0xd3515c2831559a83, -954, -268),
44 (0x9d71ac8fada6c9b5, -927, -260),
45 (0xea9c227723ee8bcb, -901, -252),
46 (0xaecc49914078536d, -874, -244),
47 (0x823c12795db6ce57, -847, -236),
48 (0xc21094364dfb5637, -821, -228),
49 (0x9096ea6f3848984f, -794, -220),
50 (0xd77485cb25823ac7, -768, -212),
51 (0xa086cfcd97bf97f4, -741, -204),
52 (0xef340a98172aace5, -715, -196),
53 (0xb23867fb2a35b28e, -688, -188),
54 (0x84c8d4dfd2c63f3b, -661, -180),
55 (0xc5dd44271ad3cdba, -635, -172),
56 (0x936b9fcebb25c996, -608, -164),
57 (0xdbac6c247d62a584, -582, -156),
58 (0xa3ab66580d5fdaf6, -555, -148),
59 (0xf3e2f893dec3f126, -529, -140),
60 (0xb5b5ada8aaff80b8, -502, -132),
61 (0x87625f056c7c4a8b, -475, -124),
62 (0xc9bcff6034c13053, -449, -116),
63 (0x964e858c91ba2655, -422, -108),
64 (0xdff9772470297ebd, -396, -100),
65 (0xa6dfbd9fb8e5b88f, -369, -92),
66 (0xf8a95fcf88747d94, -343, -84),
67 (0xb94470938fa89bcf, -316, -76),
68 (0x8a08f0f8bf0f156b, -289, -68),
69 (0xcdb02555653131b6, -263, -60),
70 (0x993fe2c6d07b7fac, -236, -52),
71 (0xe45c10c42a2b3b06, -210, -44),
72 (0xaa242499697392d3, -183, -36),
73 (0xfd87b5f28300ca0e, -157, -28),
74 (0xbce5086492111aeb, -130, -20),
75 (0x8cbccc096f5088cc, -103, -12),
76 (0xd1b71758e219652c, -77, -4),
77 (0x9c40000000000000, -50, 4),
78 (0xe8d4a51000000000, -24, 12),
79 (0xad78ebc5ac620000, 3, 20),
80 (0x813f3978f8940984, 30, 28),
81 (0xc097ce7bc90715b3, 56, 36),
82 (0x8f7e32ce7bea5c70, 83, 44),
83 (0xd5d238a4abe98068, 109, 52),
84 (0x9f4f2726179a2245, 136, 60),
85 (0xed63a231d4c4fb27, 162, 68),
86 (0xb0de65388cc8ada8, 189, 76),
87 (0x83c7088e1aab65db, 216, 84),
88 (0xc45d1df942711d9a, 242, 92),
89 (0x924d692ca61be758, 269, 100),
90 (0xda01ee641a708dea, 295, 108),
91 (0xa26da3999aef774a, 322, 116),
92 (0xf209787bb47d6b85, 348, 124),
93 (0xb454e4a179dd1877, 375, 132),
94 (0x865b86925b9bc5c2, 402, 140),
95 (0xc83553c5c8965d3d, 428, 148),
96 (0x952ab45cfa97a0b3, 455, 156),
97 (0xde469fbd99a05fe3, 481, 164),
98 (0xa59bc234db398c25, 508, 172),
99 (0xf6c69a72a3989f5c, 534, 180),
100 (0xb7dcbf5354e9bece, 561, 188),
101 (0x88fcf317f22241e2, 588, 196),
102 (0xcc20ce9bd35c78a5, 614, 204),
103 (0x98165af37b2153df, 641, 212),
104 (0xe2a0b5dc971f303a, 667, 220),
105 (0xa8d9d1535ce3b396, 694, 228),
106 (0xfb9b7cd9a4a7443c, 720, 236),
107 (0xbb764c4ca7a44410, 747, 244),
108 (0x8bab8eefb6409c1a, 774, 252),
109 (0xd01fef10a657842c, 800, 260),
110 (0x9b10a4e5e9913129, 827, 268),
111 (0xe7109bfba19c0c9d, 853, 276),
112 (0xac2820d9623bf429, 880, 284),
113 (0x80444b5e7aa7cf85, 907, 292),
114 (0xbf21e44003acdd2d, 933, 300),
115 (0x8e679c2f5e44ff8f, 960, 308),
116 (0xd433179d9c8cb841, 986, 316),
117 (0x9e19db92b4e31ba9, 1013, 324),
118 (0xeb96bf6ebadf77d9, 1039, 332),
121 #[doc(hidden)] pub const CACHED_POW10_FIRST_E: i16 = -1087;
122 #[doc(hidden)] pub const CACHED_POW10_LAST_E: i16 = 1039;
125 pub fn cached_power(alpha: i16, gamma: i16) -> (i16, Fp) {
126 let offset = CACHED_POW10_FIRST_E as i32;
127 let range = (CACHED_POW10.len() as i32) - 1;
128 let domain = (CACHED_POW10_LAST_E - CACHED_POW10_FIRST_E) as i32;
129 let idx = ((gamma as i32) - offset) * range / domain;
130 let (f, e, k) = CACHED_POW10[idx as usize];
131 debug_assert!(alpha <= e && e <= gamma);
135 /// Given `x > 0`, returns `(k, 10^k)` such that `10^k <= x < 10^(k+1)`.
137 pub fn max_pow10_no_more_than(x: u32) -> (u8, u32) {
138 debug_assert!(x > 0);
140 const X9: u32 = 10_0000_0000;
141 const X8: u32 = 1_0000_0000;
142 const X7: u32 = 1000_0000;
143 const X6: u32 = 100_0000;
144 const X5: u32 = 10_0000;
145 const X4: u32 = 1_0000;
146 const X3: u32 = 1000;
151 if x < X2 { if x < X1 {(0, 1)} else {(1, X1)} }
152 else { if x < X3 {(2, X2)} else {(3, X3)} }
154 if x < X6 { if x < X5 {(4, X4)} else {(5, X5)} }
155 else if x < X8 { if x < X7 {(6, X6)} else {(7, X7)} }
156 else { if x < X9 {(8, X8)} else {(9, X9)} }
160 /// The shortest mode implementation for Grisu.
162 /// It returns `None` when it would return an inexact representation otherwise.
163 pub fn format_shortest_opt(d: &Decoded,
164 buf: &mut [u8]) -> Option<(/*#digits*/ usize, /*exp*/ i16)> {
166 assert!(d.minus > 0);
168 assert!(d.mant.checked_add(d.plus).is_some());
169 assert!(d.mant.checked_sub(d.minus).is_some());
170 assert!(buf.len() >= MAX_SIG_DIGITS);
171 assert!(d.mant + d.plus < (1 << 61)); // we need at least three bits of additional precision
173 // start with the normalized values with the shared exponent
174 let plus = Fp { f: d.mant + d.plus, e: d.exp }.normalize();
175 let minus = Fp { f: d.mant - d.minus, e: d.exp }.normalize_to(plus.e);
176 let v = Fp { f: d.mant, e: d.exp }.normalize_to(plus.e);
178 // find any `cached = 10^minusk` such that `ALPHA <= minusk + plus.e + 64 <= GAMMA`.
179 // since `plus` is normalized, this means `2^(62 + ALPHA) <= plus * cached < 2^(64 + GAMMA)`;
180 // given our choices of `ALPHA` and `GAMMA`, this puts `plus * cached` into `[4, 2^32)`.
182 // it is obviously desirable to maximize `GAMMA - ALPHA`,
183 // so that we don't need many cached powers of 10, but there are some considerations:
185 // 1. we want to keep `floor(plus * cached)` within `u32` since it needs a costly division.
186 // (this is not really avoidable, remainder is required for accuracy estimation.)
187 // 2. the remainder of `floor(plus * cached)` repeatedly gets multiplied by 10,
188 // and it should not overflow.
190 // the first gives `64 + GAMMA <= 32`, while the second gives `10 * 2^-ALPHA <= 2^64`;
191 // -60 and -32 is the maximal range with this constraint, and V8 also uses them.
192 let (minusk, cached) = cached_power(ALPHA - plus.e - 64, GAMMA - plus.e - 64);
194 // scale fps. this gives the maximal error of 1 ulp (proved from Theorem 5.1).
195 let plus = plus.mul(&cached);
196 let minus = minus.mul(&cached);
197 let v = v.mul(&cached);
198 debug_assert_eq!(plus.e, minus.e);
199 debug_assert_eq!(plus.e, v.e);
201 // +- actual range of minus
202 // | <---|---------------------- unsafe region --------------------------> |
204 // | |<--->| | <--------------- safe region ---------------> | |
206 // |1 ulp|1 ulp| |1 ulp|1 ulp| |1 ulp|1 ulp|
207 // |<--->|<--->| |<--->|<--->| |<--->|<--->|
208 // |-----|-----|-------...-------|-----|-----|-------...-------|-----|-----|
209 // | minus | | v | | plus |
210 // minus1 minus0 v - 1 ulp v + 1 ulp plus0 plus1
212 // above `minus`, `v` and `plus` are *quantized* approximations (error < 1 ulp).
213 // as we don't know the error is positive or negative, we use two approximations spaced equally
214 // and have the maximal error of 2 ulps.
216 // the "unsafe region" is a liberal interval which we initially generate.
217 // the "safe region" is a conservative interval which we only accept.
218 // we start with the correct repr within the unsafe region, and try to find the closest repr
219 // to `v` which is also within the safe region. if we can't, we give up.
220 let plus1 = plus.f + 1;
221 // let plus0 = plus.f - 1; // only for explanation
222 // let minus0 = minus.f + 1; // only for explanation
223 let minus1 = minus.f - 1;
224 let e = -plus.e as usize; // shared exponent
226 // divide `plus1` into integral and fractional parts.
227 // integral parts are guaranteed to fit in u32, since cached power guarantees `plus < 2^32`
228 // and normalized `plus.f` is always less than `2^64 - 2^4` due to the precision requirement.
229 let plus1int = (plus1 >> e) as u32;
230 let plus1frac = plus1 & ((1 << e) - 1);
232 // calculate the largest `10^max_kappa` no more than `plus1` (thus `plus1 < 10^(max_kappa+1)`).
233 // this is an upper bound of `kappa` below.
234 let (max_kappa, max_ten_kappa) = max_pow10_no_more_than(plus1int);
237 let exp = max_kappa as i16 - minusk + 1;
239 // Theorem 6.2: if `k` is the greatest integer s.t. `0 <= y mod 10^k <= y - x`,
240 // then `V = floor(y / 10^k) * 10^k` is in `[x, y]` and one of the shortest
241 // representations (with the minimal number of significant digits) in that range.
243 // find the digit length `kappa` between `(minus1, plus1)` as per Theorem 6.2.
244 // Theorem 6.2 can be adopted to exclude `x` by requiring `y mod 10^k < y - x` instead.
245 // (e.g., `x` = 32000, `y` = 32777; `kappa` = 2 since `y mod 10^3 = 777 < y - x = 777`.)
246 // the algorithm relies on the later verification phase to exclude `y`.
247 let delta1 = plus1 - minus1;
248 // let delta1int = (delta1 >> e) as usize; // only for explanation
249 let delta1frac = delta1 & ((1 << e) - 1);
251 // render integral parts, while checking for the accuracy at each step.
252 let mut kappa = max_kappa as i16;
253 let mut ten_kappa = max_ten_kappa; // 10^kappa
254 let mut remainder = plus1int; // digits yet to be rendered
255 loop { // we always have at least one digit to render, as `plus1 >= 10^kappa`
257 // - `delta1int <= remainder < 10^(kappa+1)`
258 // - `plus1int = d[0..n-1] * 10^(kappa+1) + remainder`
259 // (it follows that `remainder = plus1int % 10^(kappa+1)`)
261 // divide `remainder` by `10^kappa`. both are scaled by `2^-e`.
262 let q = remainder / ten_kappa;
263 let r = remainder % ten_kappa;
264 debug_assert!(q < 10);
265 buf[i] = b'0' + q as u8;
268 let plus1rem = ((r as u64) << e) + plus1frac; // == (plus1 % 10^kappa) * 2^e
269 if plus1rem < delta1 {
270 // `plus1 % 10^kappa < delta1 = plus1 - minus1`; we've found the correct `kappa`.
271 let ten_kappa = (ten_kappa as u64) << e; // scale 10^kappa back to the shared exponent
272 return round_and_weed(&mut buf[..i], exp, plus1rem, delta1, plus1 - v.f, ten_kappa, 1);
275 // break the loop when we have rendered all integral digits.
276 // the exact number of digits is `max_kappa + 1` as `plus1 < 10^(max_kappa+1)`.
277 if i > max_kappa as usize {
278 debug_assert_eq!(ten_kappa, 1);
279 debug_assert_eq!(kappa, 0);
283 // restore invariants
289 // render fractional parts, while checking for the accuracy at each step.
290 // this time we rely on repeated multiplications, as division will lose the precision.
291 let mut remainder = plus1frac;
292 let mut threshold = delta1frac;
294 loop { // the next digit should be significant as we've tested that before breaking out
295 // invariants, where `m = max_kappa + 1` (# of digits in the integral part):
296 // - `remainder < 2^e`
297 // - `plus1frac * 10^(n-m) = d[m..n-1] * 2^e + remainder`
299 remainder *= 10; // won't overflow, `2^e * 10 < 2^64`
303 // divide `remainder` by `10^kappa`.
304 // both are scaled by `2^e / 10^kappa`, so the latter is implicit here.
305 let q = remainder >> e;
306 let r = remainder & ((1 << e) - 1);
307 debug_assert!(q < 10);
308 buf[i] = b'0' + q as u8;
312 let ten_kappa = 1 << e; // implicit divisor
313 return round_and_weed(&mut buf[..i], exp, r, threshold,
314 (plus1 - v.f) * ulp, ten_kappa, ulp);
317 // restore invariants
322 // we've generated all significant digits of `plus1`, but not sure if it's the optimal one.
323 // for example, if `minus1` is 3.14153... and `plus1` is 3.14158..., there are 5 different
324 // shortest representation from 3.14154 to 3.14158 but we only have the greatest one.
325 // we have to successively decrease the last digit and check if this is the optimal repr.
326 // there are at most 9 candidates (..1 to ..9), so this is fairly quick. ("rounding" phase)
328 // the function checks if this "optimal" repr is actually within the ulp ranges,
329 // and also, it is possible that the "second-to-optimal" repr can actually be optimal
330 // due to the rounding error. in either cases this returns `None`. ("weeding" phase)
332 // all arguments here are scaled by the common (but implicit) value `k`, so that:
333 // - `remainder = (plus1 % 10^kappa) * k`
334 // - `threshold = (plus1 - minus1) * k` (and also, `remainder < threshold`)
335 // - `plus1v = (plus1 - v) * k` (and also, `threshold > plus1v` from prior invariants)
336 // - `ten_kappa = 10^kappa * k`
337 // - `ulp = 2^-e * k`
338 fn round_and_weed(buf: &mut [u8], exp: i16, remainder: u64, threshold: u64, plus1v: u64,
339 ten_kappa: u64, ulp: u64) -> Option<(usize, i16)> {
340 assert!(!buf.is_empty());
342 // produce two approximations to `v` (actually `plus1 - v`) within 1.5 ulps.
343 // the resulting representation should be the closest representation to both.
345 // here `plus1 - v` is used since calculations are done with respect to `plus1`
346 // in order to avoid overflow/underflow (hence the seemingly swapped names).
347 let plus1v_down = plus1v + ulp; // plus1 - (v - 1 ulp)
348 let plus1v_up = plus1v - ulp; // plus1 - (v + 1 ulp)
350 // decrease the last digit and stop at the closest representation to `v + 1 ulp`.
351 let mut plus1w = remainder; // plus1w(n) = plus1 - w(n)
353 let last = buf.last_mut().unwrap();
355 // we work with the approximated digits `w(n)`, which is initially equal to `plus1 -
356 // plus1 % 10^kappa`. after running the loop body `n` times, `w(n) = plus1 -
357 // plus1 % 10^kappa - n * 10^kappa`. we set `plus1w(n) = plus1 - w(n) =
358 // plus1 % 10^kappa + n * 10^kappa` (thus `remainder = plus1w(0)`) to simplify checks.
359 // note that `plus1w(n)` is always increasing.
361 // we have three conditions to terminate. any of them will make the loop unable to
362 // proceed, but we then have at least one valid representation known to be closest to
363 // `v + 1 ulp` anyway. we will denote them as TC1 through TC3 for brevity.
365 // TC1: `w(n) <= v + 1 ulp`, i.e., this is the last repr that can be the closest one.
366 // this is equivalent to `plus1 - w(n) = plus1w(n) >= plus1 - (v + 1 ulp) = plus1v_up`.
367 // combined with TC2 (which checks if `w(n+1)` is valid), this prevents the possible
368 // overflow on the calculation of `plus1w(n)`.
370 // TC2: `w(n+1) < minus1`, i.e., the next repr definitely does not round to `v`.
371 // this is equivalent to `plus1 - w(n) + 10^kappa = plus1w(n) + 10^kappa >
372 // plus1 - minus1 = threshold`. the left hand side can overflow, but we know
373 // `threshold > plus1v`, so if TC1 is false, `threshold - plus1w(n) >
374 // threshold - (plus1v - 1 ulp) > 1 ulp` and we can safely test if
375 // `threshold - plus1w(n) < 10^kappa` instead.
377 // TC3: `abs(w(n) - (v + 1 ulp)) <= abs(w(n+1) - (v + 1 ulp))`, i.e., the next repr is
378 // no closer to `v + 1 ulp` than the current repr. given `z(n) = plus1v_up - plus1w(n)`,
379 // this becomes `abs(z(n)) <= abs(z(n+1))`. again assuming that TC1 is false, we have
380 // `z(n) > 0`. we have two cases to consider:
382 // - when `z(n+1) >= 0`: TC3 becomes `z(n) <= z(n+1)`. as `plus1w(n)` is increasing,
383 // `z(n)` should be decreasing and this is clearly false.
384 // - when `z(n+1) < 0`:
385 // - TC3a: the precondition is `plus1v_up < plus1w(n) + 10^kappa`. assuming TC2 is
386 // false, `threshold >= plus1w(n) + 10^kappa` so it cannot overflow.
387 // - TC3b: TC3 becomes `z(n) <= -z(n+1)`, i.e., `plus1v_up - plus1w(n) >=
388 // plus1w(n+1) - plus1v_up = plus1w(n) + 10^kappa - plus1v_up`. the negated TC1
389 // gives `plus1v_up > plus1w(n)`, so it cannot overflow or underflow when
390 // combined with TC3a.
392 // consequently, we should stop when `TC1 || TC2 || (TC3a && TC3b)`. the following is
393 // equal to its inverse, `!TC1 && !TC2 && (!TC3a || !TC3b)`.
394 while plus1w < plus1v_up &&
395 threshold - plus1w >= ten_kappa &&
396 (plus1w + ten_kappa < plus1v_up ||
397 plus1v_up - plus1w >= plus1w + ten_kappa - plus1v_up) {
399 debug_assert!(*last > b'0'); // the shortest repr cannot end with `0`
404 // check if this representation is also the closest representation to `v - 1 ulp`.
406 // this is simply same to the terminating conditions for `v + 1 ulp`, with all `plus1v_up`
407 // replaced by `plus1v_down` instead. overflow analysis equally holds.
408 if plus1w < plus1v_down &&
409 threshold - plus1w >= ten_kappa &&
410 (plus1w + ten_kappa < plus1v_down ||
411 plus1v_down - plus1w >= plus1w + ten_kappa - plus1v_down) {
415 // now we have the closest representation to `v` between `plus1` and `minus1`.
416 // this is too liberal, though, so we reject any `w(n)` not between `plus0` and `minus0`,
417 // i.e., `plus1 - plus1w(n) <= minus0` or `plus1 - plus1w(n) >= plus0`. we utilize the facts
418 // that `threshold = plus1 - minus1` and `plus1 - plus0 = minus0 - minus1 = 2 ulp`.
419 if 2 * ulp <= plus1w && plus1w <= threshold - 4 * ulp {
420 Some((buf.len(), exp))
427 /// The shortest mode implementation for Grisu with Dragon fallback.
429 /// This should be used for most cases.
430 pub fn format_shortest(d: &Decoded, buf: &mut [u8]) -> (/*#digits*/ usize, /*exp*/ i16) {
431 use num::flt2dec::strategy::dragon::format_shortest as fallback;
432 match format_shortest_opt(d, buf) {
434 None => fallback(d, buf),
438 /// The exact and fixed mode implementation for Grisu.
440 /// It returns `None` when it would return an inexact representation otherwise.
441 pub fn format_exact_opt(d: &Decoded, buf: &mut [u8], limit: i16)
442 -> Option<(/*#digits*/ usize, /*exp*/ i16)> {
444 assert!(d.mant < (1 << 61)); // we need at least three bits of additional precision
445 assert!(!buf.is_empty());
447 // normalize and scale `v`.
448 let v = Fp { f: d.mant, e: d.exp }.normalize();
449 let (minusk, cached) = cached_power(ALPHA - v.e - 64, GAMMA - v.e - 64);
450 let v = v.mul(&cached);
452 // divide `v` into integral and fractional parts.
453 let e = -v.e as usize;
454 let vint = (v.f >> e) as u32;
455 let vfrac = v.f & ((1 << e) - 1);
457 // both old `v` and new `v` (scaled by `10^-k`) has an error of < 1 ulp (Theorem 5.1).
458 // as we don't know the error is positive or negative, we use two approximations
459 // spaced equally and have the maximal error of 2 ulps (same to the shortest case).
461 // the goal is to find the exactly rounded series of digits that are common to
462 // both `v - 1 ulp` and `v + 1 ulp`, so that we are maximally confident.
463 // if this is not possible, we don't know which one is the correct output for `v`,
464 // so we give up and fall back.
466 // `err` is defined as `1 ulp * 2^e` here (same to the ulp in `vfrac`),
467 // and we will scale it whenever `v` gets scaled.
470 // calculate the largest `10^max_kappa` no more than `v` (thus `v < 10^(max_kappa+1)`).
471 // this is an upper bound of `kappa` below.
472 let (max_kappa, max_ten_kappa) = max_pow10_no_more_than(vint);
475 let exp = max_kappa as i16 - minusk + 1;
477 // if we are working with the last-digit limitation, we need to shorten the buffer
478 // before the actual rendering in order to avoid double rounding.
479 // note that we have to enlarge the buffer again when rounding up happens!
480 let len = if exp <= limit {
481 // oops, we cannot even produce *one* digit.
482 // this is possible when, say, we've got something like 9.5 and it's being rounded to 10.
484 // in principle we can immediately call `possibly_round` with an empty buffer,
485 // but scaling `max_ten_kappa << e` by 10 can result in overflow.
486 // thus we are being sloppy here and widen the error range by a factor of 10.
487 // this will increase the false negative rate, but only very, *very* slightly;
488 // it can only matter noticeably when the mantissa is bigger than 60 bits.
489 return possibly_round(buf, 0, exp, limit, v.f / 10, (max_ten_kappa as u64) << e, err << e);
490 } else if ((exp as i32 - limit as i32) as usize) < buf.len() {
491 (exp - limit) as usize
495 debug_assert!(len > 0);
497 // render integral parts.
498 // the error is entirely fractional, so we don't need to check it in this part.
499 let mut kappa = max_kappa as i16;
500 let mut ten_kappa = max_ten_kappa; // 10^kappa
501 let mut remainder = vint; // digits yet to be rendered
502 loop { // we always have at least one digit to render
504 // - `remainder < 10^(kappa+1)`
505 // - `vint = d[0..n-1] * 10^(kappa+1) + remainder`
506 // (it follows that `remainder = vint % 10^(kappa+1)`)
508 // divide `remainder` by `10^kappa`. both are scaled by `2^-e`.
509 let q = remainder / ten_kappa;
510 let r = remainder % ten_kappa;
511 debug_assert!(q < 10);
512 buf[i] = b'0' + q as u8;
515 // is the buffer full? run the rounding pass with the remainder.
517 let vrem = ((r as u64) << e) + vfrac; // == (v % 10^kappa) * 2^e
518 return possibly_round(buf, len, exp, limit, vrem, (ten_kappa as u64) << e, err << e);
521 // break the loop when we have rendered all integral digits.
522 // the exact number of digits is `max_kappa + 1` as `plus1 < 10^(max_kappa+1)`.
523 if i > max_kappa as usize {
524 debug_assert_eq!(ten_kappa, 1);
525 debug_assert_eq!(kappa, 0);
529 // restore invariants
535 // render fractional parts.
537 // in principle we can continue to the last available digit and check for the accuracy.
538 // unfortunately we are working with the finite-sized integers, so we need some criterion
539 // to detect the overflow. V8 uses `remainder > err`, which becomes false when
540 // the first `i` significant digits of `v - 1 ulp` and `v` differ. however this rejects
541 // too many otherwise valid input.
543 // since the later phase has a correct overflow detection, we instead use tighter criterion:
544 // we continue til `err` exceeds `10^kappa / 2`, so that the range between `v - 1 ulp` and
545 // `v + 1 ulp` definitely contains two or more rounded representations. this is same to
546 // the first two comparisons from `possibly_round`, for the reference.
547 let mut remainder = vfrac;
548 let maxerr = 1 << (e - 1);
550 // invariants, where `m = max_kappa + 1` (# of digits in the integral part):
551 // - `remainder < 2^e`
552 // - `vfrac * 10^(n-m) = d[m..n-1] * 2^e + remainder`
553 // - `err = 10^(n-m)`
555 remainder *= 10; // won't overflow, `2^e * 10 < 2^64`
556 err *= 10; // won't overflow, `err * 10 < 2^e * 5 < 2^64`
558 // divide `remainder` by `10^kappa`.
559 // both are scaled by `2^e / 10^kappa`, so the latter is implicit here.
560 let q = remainder >> e;
561 let r = remainder & ((1 << e) - 1);
562 debug_assert!(q < 10);
563 buf[i] = b'0' + q as u8;
566 // is the buffer full? run the rounding pass with the remainder.
568 return possibly_round(buf, len, exp, limit, r, 1 << e, err);
571 // restore invariants
575 // further calculation is useless (`possibly_round` definitely fails), so we give up.
578 // we've generated all requested digits of `v`, which should be also same to corresponding
579 // digits of `v - 1 ulp`. now we check if there is a unique representation shared by
580 // both `v - 1 ulp` and `v + 1 ulp`; this can be either same to generated digits, or
581 // to the rounded-up version of those digits. if the range contains multiple representations
582 // of the same length, we cannot be sure and should return `None` instead.
584 // all arguments here are scaled by the common (but implicit) value `k`, so that:
585 // - `remainder = (v % 10^kappa) * k`
586 // - `ten_kappa = 10^kappa * k`
587 // - `ulp = 2^-e * k`
588 fn possibly_round(buf: &mut [u8], mut len: usize, mut exp: i16, limit: i16,
589 remainder: u64, ten_kappa: u64, ulp: u64) -> Option<(usize, i16)> {
590 debug_assert!(remainder < ten_kappa);
597 // ----|-----|-----|----
599 // v - 1 ulp v + 1 ulp
601 // (for the reference, the dotted line indicates the exact value for
602 // possible representations in given number of digits.)
604 // error is too large that there are at least three possible representations
605 // between `v - 1 ulp` and `v + 1 ulp`. we cannot determine which one is correct.
606 if ulp >= ten_kappa { return None; }
613 // ----|-----|-----|----
615 // v - 1 ulp v + 1 ulp
617 // in fact, 1/2 ulp is enough to introduce two possible representations.
618 // (remember that we need a unique representation for both `v - 1 ulp` and `v + 1 ulp`.)
619 // this won't overflow, as `ulp < ten_kappa` from the first check.
620 if ten_kappa - ulp <= ulp { return None; }
625 // :<--------- 10^kappa ---------->:
629 // ----|-----|-----|------------------------
631 // v - 1 ulp v + 1 ulp
633 // if `v + 1 ulp` is closer to the rounded-down representation (which is already in `buf`),
634 // then we can safely return. note that `v - 1 ulp` *can* be less than the current
635 // representation, but as `1 ulp < 10^kappa / 2`, this condition is enough:
636 // the distance between `v - 1 ulp` and the current representation
637 // cannot exceed `10^kappa / 2`.
639 // the condition equals to `remainder + ulp < 10^kappa / 2`.
640 // since this can easily overflow, first check if `remainder < 10^kappa / 2`.
641 // we've already verified that `ulp < 10^kappa / 2`, so as long as
642 // `10^kappa` did not overflow after all, the second check is fine.
643 if ten_kappa - remainder > remainder && ten_kappa - 2 * remainder >= 2 * ulp {
644 return Some((len, exp));
647 // :<------- remainder ------>| :
649 // :<--------- 10^kappa --------->:
653 // -----------------------|-----|-----|-----
655 // v - 1 ulp v + 1 ulp
657 // on the other hands, if `v - 1 ulp` is closer to the rounded-up representation,
658 // we should round up and return. for the same reason we don't need to check `v + 1 ulp`.
660 // the condition equals to `remainder - ulp >= 10^kappa / 2`.
661 // again we first check if `remainder > ulp` (note that this is not `remainder >= ulp`,
662 // as `10^kappa` is never zero). also note that `remainder - ulp <= 10^kappa`,
663 // so the second check does not overflow.
664 if remainder > ulp && ten_kappa - (remainder - ulp) <= remainder - ulp {
665 if let Some(c) = round_up(buf, len) {
666 // only add an additional digit when we've been requested the fixed precision.
667 // we also need to check that, if the original buffer was empty,
668 // the additional digit can only be added when `exp == limit` (edge case).
670 if exp > limit && len < buf.len() {
675 return Some((len, exp));
678 // otherwise we are doomed (i.e., some values between `v - 1 ulp` and `v + 1 ulp` are
679 // rounding down and others are rounding up) and give up.
684 /// The exact and fixed mode implementation for Grisu with Dragon fallback.
686 /// This should be used for most cases.
687 pub fn format_exact(d: &Decoded, buf: &mut [u8], limit: i16) -> (/*#digits*/ usize, /*exp*/ i16) {
688 use num::flt2dec::strategy::dragon::format_exact as fallback;
689 match format_exact_opt(d, buf, limit) {
691 None => fallback(d, buf, limit),