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.
12 Rust adaptation of Grisu3 algorithm described in [1]. It uses about
13 1KB of precomputed table, and in turn, it's 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.
21 use num::diy_float::Fp;
22 use num::flt2dec::{Decoded, MAX_SIG_DIGITS, round_up};
25 // see the comments in `format_shortest_opt` for the rationale.
26 #[doc(hidden)] pub const ALPHA: i16 = -60;
27 #[doc(hidden)] pub const GAMMA: i16 = -32;
30 # the following Python code generates this table:
31 for i in xrange(-308, 333, 8):
32 if i >= 0: f = 10**i; e = 0
33 else: f = 2**(80-4*i) // 10**-i; e = 4 * i - 80
35 f = ((f << 64 >> (l-1)) + 1) >> 1; e += l - 64
36 print ' (%#018x, %5d, %4d),' % (f, e, i)
40 pub static CACHED_POW10: [(u64, i16, i16); 81] = [ // (f, e, k)
41 (0xe61acf033d1a45df, -1087, -308),
42 (0xab70fe17c79ac6ca, -1060, -300),
43 (0xff77b1fcbebcdc4f, -1034, -292),
44 (0xbe5691ef416bd60c, -1007, -284),
45 (0x8dd01fad907ffc3c, -980, -276),
46 (0xd3515c2831559a83, -954, -268),
47 (0x9d71ac8fada6c9b5, -927, -260),
48 (0xea9c227723ee8bcb, -901, -252),
49 (0xaecc49914078536d, -874, -244),
50 (0x823c12795db6ce57, -847, -236),
51 (0xc21094364dfb5637, -821, -228),
52 (0x9096ea6f3848984f, -794, -220),
53 (0xd77485cb25823ac7, -768, -212),
54 (0xa086cfcd97bf97f4, -741, -204),
55 (0xef340a98172aace5, -715, -196),
56 (0xb23867fb2a35b28e, -688, -188),
57 (0x84c8d4dfd2c63f3b, -661, -180),
58 (0xc5dd44271ad3cdba, -635, -172),
59 (0x936b9fcebb25c996, -608, -164),
60 (0xdbac6c247d62a584, -582, -156),
61 (0xa3ab66580d5fdaf6, -555, -148),
62 (0xf3e2f893dec3f126, -529, -140),
63 (0xb5b5ada8aaff80b8, -502, -132),
64 (0x87625f056c7c4a8b, -475, -124),
65 (0xc9bcff6034c13053, -449, -116),
66 (0x964e858c91ba2655, -422, -108),
67 (0xdff9772470297ebd, -396, -100),
68 (0xa6dfbd9fb8e5b88f, -369, -92),
69 (0xf8a95fcf88747d94, -343, -84),
70 (0xb94470938fa89bcf, -316, -76),
71 (0x8a08f0f8bf0f156b, -289, -68),
72 (0xcdb02555653131b6, -263, -60),
73 (0x993fe2c6d07b7fac, -236, -52),
74 (0xe45c10c42a2b3b06, -210, -44),
75 (0xaa242499697392d3, -183, -36),
76 (0xfd87b5f28300ca0e, -157, -28),
77 (0xbce5086492111aeb, -130, -20),
78 (0x8cbccc096f5088cc, -103, -12),
79 (0xd1b71758e219652c, -77, -4),
80 (0x9c40000000000000, -50, 4),
81 (0xe8d4a51000000000, -24, 12),
82 (0xad78ebc5ac620000, 3, 20),
83 (0x813f3978f8940984, 30, 28),
84 (0xc097ce7bc90715b3, 56, 36),
85 (0x8f7e32ce7bea5c70, 83, 44),
86 (0xd5d238a4abe98068, 109, 52),
87 (0x9f4f2726179a2245, 136, 60),
88 (0xed63a231d4c4fb27, 162, 68),
89 (0xb0de65388cc8ada8, 189, 76),
90 (0x83c7088e1aab65db, 216, 84),
91 (0xc45d1df942711d9a, 242, 92),
92 (0x924d692ca61be758, 269, 100),
93 (0xda01ee641a708dea, 295, 108),
94 (0xa26da3999aef774a, 322, 116),
95 (0xf209787bb47d6b85, 348, 124),
96 (0xb454e4a179dd1877, 375, 132),
97 (0x865b86925b9bc5c2, 402, 140),
98 (0xc83553c5c8965d3d, 428, 148),
99 (0x952ab45cfa97a0b3, 455, 156),
100 (0xde469fbd99a05fe3, 481, 164),
101 (0xa59bc234db398c25, 508, 172),
102 (0xf6c69a72a3989f5c, 534, 180),
103 (0xb7dcbf5354e9bece, 561, 188),
104 (0x88fcf317f22241e2, 588, 196),
105 (0xcc20ce9bd35c78a5, 614, 204),
106 (0x98165af37b2153df, 641, 212),
107 (0xe2a0b5dc971f303a, 667, 220),
108 (0xa8d9d1535ce3b396, 694, 228),
109 (0xfb9b7cd9a4a7443c, 720, 236),
110 (0xbb764c4ca7a44410, 747, 244),
111 (0x8bab8eefb6409c1a, 774, 252),
112 (0xd01fef10a657842c, 800, 260),
113 (0x9b10a4e5e9913129, 827, 268),
114 (0xe7109bfba19c0c9d, 853, 276),
115 (0xac2820d9623bf429, 880, 284),
116 (0x80444b5e7aa7cf85, 907, 292),
117 (0xbf21e44003acdd2d, 933, 300),
118 (0x8e679c2f5e44ff8f, 960, 308),
119 (0xd433179d9c8cb841, 986, 316),
120 (0x9e19db92b4e31ba9, 1013, 324),
121 (0xeb96bf6ebadf77d9, 1039, 332),
124 #[doc(hidden)] pub const CACHED_POW10_FIRST_E: i16 = -1087;
125 #[doc(hidden)] pub const CACHED_POW10_LAST_E: i16 = 1039;
128 pub fn cached_power(alpha: i16, gamma: i16) -> (i16, Fp) {
129 let offset = CACHED_POW10_FIRST_E as i32;
130 let range = (CACHED_POW10.len() as i32) - 1;
131 let domain = (CACHED_POW10_LAST_E - CACHED_POW10_FIRST_E) as i32;
132 let idx = ((gamma as i32) - offset) * range / domain;
133 let (f, e, k) = CACHED_POW10[idx as usize];
134 debug_assert!(alpha <= e && e <= gamma);
135 (k, Fp { f: f, e: e })
138 /// Given `x > 0`, returns `(k, 10^k)` such that `10^k <= x < 10^(k+1)`.
140 pub fn max_pow10_no_more_than(x: u32) -> (u8, u32) {
141 debug_assert!(x > 0);
143 const X9: u32 = 10_0000_0000;
144 const X8: u32 = 1_0000_0000;
145 const X7: u32 = 1000_0000;
146 const X6: u32 = 100_0000;
147 const X5: u32 = 10_0000;
148 const X4: u32 = 1_0000;
149 const X3: u32 = 1000;
154 if x < X2 { if x < X1 {(0, 1)} else {(1, X1)} }
155 else { if x < X3 {(2, X2)} else {(3, X3)} }
157 if x < X6 { if x < X5 {(4, X4)} else {(5, X5)} }
158 else if x < X8 { if x < X7 {(6, X6)} else {(7, X7)} }
159 else { if x < X9 {(8, X8)} else {(9, X9)} }
163 /// The shortest mode implementation for Grisu.
165 /// It returns `None` when it would return an inexact representation otherwise.
166 pub fn format_shortest_opt(d: &Decoded,
167 buf: &mut [u8]) -> Option<(/*#digits*/ usize, /*exp*/ i16)> {
169 assert!(d.minus > 0);
171 assert!(d.mant.checked_add(d.plus).is_some());
172 assert!(d.mant.checked_sub(d.minus).is_some());
173 assert!(buf.len() >= MAX_SIG_DIGITS);
174 assert!(d.mant + d.plus < (1 << 61)); // we need at least three bits of additional precision
176 // start with the normalized values with the shared exponent
177 let plus = Fp { f: d.mant + d.plus, e: d.exp }.normalize();
178 let minus = Fp { f: d.mant - d.minus, e: d.exp }.normalize_to(plus.e);
179 let v = Fp { f: d.mant, e: d.exp }.normalize_to(plus.e);
181 // find any `cached = 10^minusk` such that `ALPHA <= minusk + plus.e + 64 <= GAMMA`.
182 // since `plus` is normalized, this means `2^(62 + ALPHA) <= plus * cached < 2^(64 + GAMMA)`;
183 // given our choices of `ALPHA` and `GAMMA`, this puts `plus * cached` into `[4, 2^32)`.
185 // it is obviously desirable to maximize `GAMMA - ALPHA`,
186 // so that we don't need many cached powers of 10, but there are some considerations:
188 // 1. we want to keep `floor(plus * cached)` within `u32` since it needs a costly division.
189 // (this is not really avoidable, remainder is required for accuracy estimation.)
190 // 2. the remainder of `floor(plus * cached)` repeatedly gets multiplied by 10,
191 // and it should not overflow.
193 // the first gives `64 + GAMMA <= 32`, while the second gives `10 * 2^-ALPHA <= 2^64`;
194 // -60 and -32 is the maximal range with this constraint, and V8 also uses them.
195 let (minusk, cached) = cached_power(ALPHA - plus.e - 64, GAMMA - plus.e - 64);
197 // scale fps. this gives the maximal error of 1 ulp (proved from Theorem 5.1).
198 let plus = plus.mul(&cached);
199 let minus = minus.mul(&cached);
200 let v = v.mul(&cached);
201 debug_assert_eq!(plus.e, minus.e);
202 debug_assert_eq!(plus.e, v.e);
204 // +- actual range of minus
205 // | <---|---------------------- unsafe region --------------------------> |
207 // | |<--->| | <--------------- safe region ---------------> | |
209 // |1 ulp|1 ulp| |1 ulp|1 ulp| |1 ulp|1 ulp|
210 // |<--->|<--->| |<--->|<--->| |<--->|<--->|
211 // |-----|-----|-------...-------|-----|-----|-------...-------|-----|-----|
212 // | minus | | v | | plus |
213 // minus1 minus0 v - 1 ulp v + 1 ulp plus0 plus1
215 // above `minus`, `v` and `plus` are *quantized* approximations (error < 1 ulp).
216 // as we don't know the error is positive or negative, we use two approximations spaced equally
217 // and have the maximal error of 2 ulps.
219 // the "unsafe region" is a liberal interval which we initially generate.
220 // the "safe region" is a conservative interval which we only accept.
221 // we start with the correct repr within the unsafe region, and try to find the closest repr
222 // to `v` which is also within the safe region. if we can't, we give up.
223 let plus1 = plus.f + 1;
224 // let plus0 = plus.f - 1; // only for explanation
225 // let minus0 = minus.f + 1; // only for explanation
226 let minus1 = minus.f - 1;
227 let e = -plus.e as usize; // shared exponent
229 // divide `plus1` into integral and fractional parts.
230 // integral parts are guaranteed to fit in u32, since cached power guarantees `plus < 2^32`
231 // and normalized `plus.f` is always less than `2^64 - 2^4` due to the precision requirement.
232 let plus1int = (plus1 >> e) as u32;
233 let plus1frac = plus1 & ((1 << e) - 1);
235 // calculate the largest `10^max_kappa` no more than `plus1` (thus `plus1 < 10^(max_kappa+1)`).
236 // this is an upper bound of `kappa` below.
237 let (max_kappa, max_ten_kappa) = max_pow10_no_more_than(plus1int);
240 let exp = max_kappa as i16 - minusk + 1;
242 // Theorem 6.2: if `k` is the greatest integer s.t. `0 <= y mod 10^k <= y - x`,
243 // then `V = floor(y / 10^k) * 10^k` is in `[x, y]` and one of the shortest
244 // representations (with the minimal number of significant digits) in that range.
246 // find the digit length `kappa` between `(minus1, plus1)` as per Theorem 6.2.
247 // Theorem 6.2 can be adopted to exclude `x` by requiring `y mod 10^k < y - x` instead.
248 // (e.g. `x` = 32000, `y` = 32777; `kappa` = 2 since `y mod 10^3 = 777 < y - x = 777`.)
249 // the algorithm relies on the later verification phase to exclude `y`.
250 let delta1 = plus1 - minus1;
251 // let delta1int = (delta1 >> e) as usize; // only for explanation
252 let delta1frac = delta1 & ((1 << e) - 1);
254 // render integral parts, while checking for the accuracy at each step.
255 let mut kappa = max_kappa as i16;
256 let mut ten_kappa = max_ten_kappa; // 10^kappa
257 let mut remainder = plus1int; // digits yet to be rendered
258 loop { // we always have at least one digit to render, as `plus1 >= 10^kappa`
260 // - `delta1int <= remainder < 10^(kappa+1)`
261 // - `plus1int = d[0..n-1] * 10^(kappa+1) + remainder`
262 // (it follows that `remainder = plus1int % 10^(kappa+1)`)
264 // divide `remainder` by `10^kappa`. both are scaled by `2^-e`.
265 let q = remainder / ten_kappa;
266 let r = remainder % ten_kappa;
267 debug_assert!(q < 10);
268 buf[i] = b'0' + q as u8;
271 let plus1rem = ((r as u64) << e) + plus1frac; // == (plus1 % 10^kappa) * 2^e
272 if plus1rem < delta1 {
273 // `plus1 % 10^kappa < delta1 = plus1 - minus1`; we've found the correct `kappa`.
274 let ten_kappa = (ten_kappa as u64) << e; // scale 10^kappa back to the shared exponent
275 return round_and_weed(&mut buf[..i], exp, plus1rem, delta1, plus1 - v.f, ten_kappa, 1);
278 // break the loop when we have rendered all integral digits.
279 // the exact number of digits is `max_kappa + 1` as `plus1 < 10^(max_kappa+1)`.
280 if i > max_kappa as usize {
281 debug_assert_eq!(ten_kappa, 1);
282 debug_assert_eq!(kappa, 0);
286 // restore invariants
292 // render fractional parts, while checking for the accuracy at each step.
293 // this time we rely on repeated multiplications, as division will lose the precision.
294 let mut remainder = plus1frac;
295 let mut threshold = delta1frac;
297 loop { // the next digit should be significant as we've tested that before breaking out
298 // invariants, where `m = max_kappa + 1` (# of digits in the integral part):
299 // - `remainder < 2^e`
300 // - `plus1frac * 10^(n-m) = d[m..n-1] * 2^e + remainder`
302 remainder *= 10; // won't overflow, `2^e * 10 < 2^64`
306 // divide `remainder` by `10^kappa`.
307 // both are scaled by `2^e / 10^kappa`, so the latter is implicit here.
308 let q = remainder >> e;
309 let r = remainder & ((1 << e) - 1);
310 debug_assert!(q < 10);
311 buf[i] = b'0' + q as u8;
315 let ten_kappa = 1 << e; // implicit divisor
316 return round_and_weed(&mut buf[..i], exp, r, threshold,
317 (plus1 - v.f) * ulp, ten_kappa, ulp);
320 // restore invariants
325 // we've generated all significant digits of `plus1`, but not sure if it's the optimal one.
326 // for example, if `minus1` is 3.14153... and `plus1` is 3.14158..., there are 5 different
327 // shortest representation from 3.14154 to 3.14158 but we only have the greatest one.
328 // we have to successively decrease the last digit and check if this is the optimal repr.
329 // there are at most 9 candidates (..1 to ..9), so this is fairly quick. ("rounding" phase)
331 // the function checks if this "optimal" repr is actually within the ulp ranges,
332 // and also, it is possible that the "second-to-optimal" repr can actually be optimal
333 // due to the rounding error. in either cases this returns `None`. ("weeding" phase)
335 // all arguments here are scaled by the common (but implicit) value `k`, so that:
336 // - `remainder = (plus1 % 10^kappa) * k`
337 // - `threshold = (plus1 - minus1) * k` (and also, `remainder < threshold`)
338 // - `plus1v = (plus1 - v) * k` (and also, `threshold > plus1v` from prior invariants)
339 // - `ten_kappa = 10^kappa * k`
340 // - `ulp = 2^-e * k`
341 fn round_and_weed(buf: &mut [u8], exp: i16, remainder: u64, threshold: u64, plus1v: u64,
342 ten_kappa: u64, ulp: u64) -> Option<(usize, i16)> {
343 assert!(!buf.is_empty());
345 // produce two approximations to `v` (actually `plus1 - v`) within 1.5 ulps.
346 // the resulting representation should be the closest representation to both.
348 // here `plus1 - v` is used since calculations are done with respect to `plus1`
349 // in order to avoid overflow/underflow (hence the seemingly swapped names).
350 let plus1v_down = plus1v + ulp; // plus1 - (v - 1 ulp)
351 let plus1v_up = plus1v - ulp; // plus1 - (v + 1 ulp)
353 // decrease the last digit and stop at the closest representation to `v + 1 ulp`.
354 let mut plus1w = remainder; // plus1w(n) = plus1 - w(n)
356 let last = buf.last_mut().unwrap();
358 // we work with the approximated digits `w(n)`, which is initially equal to `plus1 -
359 // plus1 % 10^kappa`. after running the loop body `n` times, `w(n) = plus1 -
360 // plus1 % 10^kappa - n * 10^kappa`. we set `plus1w(n) = plus1 - w(n) =
361 // plus1 % 10^kappa + n * 10^kappa` (thus `remainder = plus1w(0)`) to simplify checks.
362 // note that `plus1w(n)` is always increasing.
364 // we have three conditions to terminate. any of them will make the loop unable to
365 // proceed, but we then have at least one valid representation known to be closest to
366 // `v + 1 ulp` anyway. we will denote them as TC1 through TC3 for brevity.
368 // TC1: `w(n) <= v + 1 ulp`, i.e. this is the last repr that can be the closest one.
369 // this is equivalent to `plus1 - w(n) = plus1w(n) >= plus1 - (v + 1 ulp) = plus1v_up`.
370 // combined with TC2 (which checks if `w(n+1)` is valid), this prevents the possible
371 // overflow on the calculation of `plus1w(n)`.
373 // TC2: `w(n+1) < minus1`, i.e. the next repr definitely does not round to `v`.
374 // this is equivalent to `plus1 - w(n) + 10^kappa = plus1w(n) + 10^kappa >
375 // plus1 - minus1 = threshold`. the left hand side can overflow, but we know
376 // `threshold > plus1v`, so if TC1 is false, `threshold - plus1w(n) >
377 // threshold - (plus1v - 1 ulp) > 1 ulp` and we can safely test if
378 // `threshold - plus1w(n) < 10^kappa` instead.
380 // TC3: `abs(w(n) - (v + 1 ulp)) <= abs(w(n+1) - (v + 1 ulp))`, i.e. the next repr is
381 // no closer to `v + 1 ulp` than the current repr. given `z(n) = plus1v_up - plus1w(n)`,
382 // this becomes `abs(z(n)) <= abs(z(n+1))`. again assuming that TC1 is false, we have
383 // `z(n) > 0`. we have two cases to consider:
385 // - when `z(n+1) >= 0`: TC3 becomes `z(n) <= z(n+1)`. as `plus1w(n)` is increasing,
386 // `z(n)` should be decreasing and this is clearly false.
387 // - when `z(n+1) < 0`:
388 // - TC3a: the precondition is `plus1v_up < plus1w(n) + 10^kappa`. assuming TC2 is
389 // false, `threshold >= plus1w(n) + 10^kappa` so it cannot overflow.
390 // - TC3b: TC3 becomes `z(n) <= -z(n+1)`, i.e. `plus1v_up - plus1w(n) >=
391 // plus1w(n+1) - plus1v_up = plus1w(n) + 10^kappa - plus1v_up`. the negated TC1
392 // gives `plus1v_up > plus1w(n)`, so it cannot overflow or underflow when
393 // combined with TC3a.
395 // consequently, we should stop when `TC1 || TC2 || (TC3a && TC3b)`. the following is
396 // equal to its inverse, `!TC1 && !TC2 && (!TC3a || !TC3b)`.
397 while plus1w < plus1v_up &&
398 threshold - plus1w >= ten_kappa &&
399 (plus1w + ten_kappa < plus1v_up ||
400 plus1v_up - plus1w >= plus1w + ten_kappa - plus1v_up) {
402 debug_assert!(*last > b'0'); // the shortest repr cannot end with `0`
407 // check if this representation is also the closest representation to `v - 1 ulp`.
409 // this is simply same to the terminating conditions for `v + 1 ulp`, with all `plus1v_up`
410 // replaced by `plus1v_down` instead. overflow analysis equally holds.
411 if plus1w < plus1v_down &&
412 threshold - plus1w >= ten_kappa &&
413 (plus1w + ten_kappa < plus1v_down ||
414 plus1v_down - plus1w >= plus1w + ten_kappa - plus1v_down) {
418 // now we have the closest representation to `v` between `plus1` and `minus1`.
419 // this is too liberal, though, so we reject any `w(n)` not between `plus0` and `minus0`,
420 // i.e. `plus1 - plus1w(n) <= minus0` or `plus1 - plus1w(n) >= plus0`. we utilize the facts
421 // that `threshold = plus1 - minus1` and `plus1 - plus0 = minus0 - minus1 = 2 ulp`.
422 if 2 * ulp <= plus1w && plus1w <= threshold - 4 * ulp {
423 Some((buf.len(), exp))
430 /// The shortest mode implementation for Grisu with Dragon fallback.
432 /// This should be used for most cases.
433 pub fn format_shortest(d: &Decoded, buf: &mut [u8]) -> (/*#digits*/ usize, /*exp*/ i16) {
434 use num::flt2dec::strategy::dragon::format_shortest as fallback;
435 match format_shortest_opt(d, buf) {
437 None => fallback(d, buf),
441 /// The exact and fixed mode implementation for Grisu.
443 /// It returns `None` when it would return an inexact representation otherwise.
444 pub fn format_exact_opt(d: &Decoded, buf: &mut [u8], limit: i16)
445 -> Option<(/*#digits*/ usize, /*exp*/ i16)> {
447 assert!(d.mant < (1 << 61)); // we need at least three bits of additional precision
448 assert!(!buf.is_empty());
450 // normalize and scale `v`.
451 let v = Fp { f: d.mant, e: d.exp }.normalize();
452 let (minusk, cached) = cached_power(ALPHA - v.e - 64, GAMMA - v.e - 64);
453 let v = v.mul(&cached);
455 // divide `v` into integral and fractional parts.
456 let e = -v.e as usize;
457 let vint = (v.f >> e) as u32;
458 let vfrac = v.f & ((1 << e) - 1);
460 // both old `v` and new `v` (scaled by `10^-k`) has an error of < 1 ulp (Theorem 5.1).
461 // as we don't know the error is positive or negative, we use two approximations
462 // spaced equally and have the maximal error of 2 ulps (same to the shortest case).
464 // the goal is to find the exactly rounded series of digits that are common to
465 // both `v - 1 ulp` and `v + 1 ulp`, so that we are maximally confident.
466 // if this is not possible, we don't know which one is the correct output for `v`,
467 // so we give up and fall back.
469 // `err` is defined as `1 ulp * 2^e` here (same to the ulp in `vfrac`),
470 // and we will scale it whenever `v` gets scaled.
473 // calculate the largest `10^max_kappa` no more than `v` (thus `v < 10^(max_kappa+1)`).
474 // this is an upper bound of `kappa` below.
475 let (max_kappa, max_ten_kappa) = max_pow10_no_more_than(vint);
478 let exp = max_kappa as i16 - minusk + 1;
480 // if we are working with the last-digit limitation, we need to shorten the buffer
481 // before the actual rendering in order to avoid double rounding.
482 // note that we have to enlarge the buffer again when rounding up happens!
483 let len = if exp <= limit {
484 // oops, we cannot even produce *one* digit.
485 // this is possible when, say, we've got something like 9.5 and it's being rounded to 10.
487 // in principle we can immediately call `possibly_round` with an empty buffer,
488 // but scaling `max_ten_kappa << e` by 10 can result in overflow.
489 // thus we are being sloppy here and widen the error range by a factor of 10.
490 // this will increase the false negative rate, but only very, *very* slightly;
491 // it can only matter noticably when the mantissa is bigger than 60 bits.
492 return possibly_round(buf, 0, exp, limit, v.f / 10, (max_ten_kappa as u64) << e, err << e);
493 } else if ((exp as i32 - limit as i32) as usize) < buf.len() {
494 (exp - limit) as usize
498 debug_assert!(len > 0);
500 // render integral parts.
501 // the error is entirely fractional, so we don't need to check it in this part.
502 let mut kappa = max_kappa as i16;
503 let mut ten_kappa = max_ten_kappa; // 10^kappa
504 let mut remainder = vint; // digits yet to be rendered
505 loop { // we always have at least one digit to render
507 // - `remainder < 10^(kappa+1)`
508 // - `vint = d[0..n-1] * 10^(kappa+1) + remainder`
509 // (it follows that `remainder = vint % 10^(kappa+1)`)
511 // divide `remainder` by `10^kappa`. both are scaled by `2^-e`.
512 let q = remainder / ten_kappa;
513 let r = remainder % ten_kappa;
514 debug_assert!(q < 10);
515 buf[i] = b'0' + q as u8;
518 // is the buffer full? run the rounding pass with the remainder.
520 let vrem = ((r as u64) << e) + vfrac; // == (v % 10^kappa) * 2^e
521 return possibly_round(buf, len, exp, limit, vrem, (ten_kappa as u64) << e, err << e);
524 // break the loop when we have rendered all integral digits.
525 // the exact number of digits is `max_kappa + 1` as `plus1 < 10^(max_kappa+1)`.
526 if i > max_kappa as usize {
527 debug_assert_eq!(ten_kappa, 1);
528 debug_assert_eq!(kappa, 0);
532 // restore invariants
538 // render fractional parts.
540 // in principle we can continue to the last available digit and check for the accuracy.
541 // unfortunately we are working with the finite-sized integers, so we need some criterion
542 // to detect the overflow. V8 uses `remainder > err`, which becomes false when
543 // the first `i` significant digits of `v - 1 ulp` and `v` differ. however this rejects
544 // too many otherwise valid input.
546 // since the later phase has a correct overflow detection, we instead use tighter criterion:
547 // we continue til `err` exceeds `10^kappa / 2`, so that the range between `v - 1 ulp` and
548 // `v + 1 ulp` definitely contains two or more rounded representations. this is same to
549 // the first two comparisons from `possibly_round`, for the reference.
550 let mut remainder = vfrac;
551 let maxerr = 1 << (e - 1);
553 // invariants, where `m = max_kappa + 1` (# of digits in the integral part):
554 // - `remainder < 2^e`
555 // - `vfrac * 10^(n-m) = d[m..n-1] * 2^e + remainder`
556 // - `err = 10^(n-m)`
558 remainder *= 10; // won't overflow, `2^e * 10 < 2^64`
559 err *= 10; // won't overflow, `err * 10 < 2^e * 5 < 2^64`
561 // divide `remainder` by `10^kappa`.
562 // both are scaled by `2^e / 10^kappa`, so the latter is implicit here.
563 let q = remainder >> e;
564 let r = remainder & ((1 << e) - 1);
565 debug_assert!(q < 10);
566 buf[i] = b'0' + q as u8;
569 // is the buffer full? run the rounding pass with the remainder.
571 return possibly_round(buf, len, exp, limit, r, 1 << e, err);
574 // restore invariants
578 // further calculation is useless (`possibly_round` definitely fails), so we give up.
581 // we've generated all requested digits of `v`, which should be also same to corresponding
582 // digits of `v - 1 ulp`. now we check if there is a unique representation shared by
583 // both `v - 1 ulp` and `v + 1 ulp`; this can be either same to generated digits, or
584 // to the rounded-up version of those digits. if the range contains multiple representations
585 // of the same length, we cannot be sure and should return `None` instead.
587 // all arguments here are scaled by the common (but implicit) value `k`, so that:
588 // - `remainder = (v % 10^kappa) * k`
589 // - `ten_kappa = 10^kappa * k`
590 // - `ulp = 2^-e * k`
591 fn possibly_round(buf: &mut [u8], mut len: usize, mut exp: i16, limit: i16,
592 remainder: u64, ten_kappa: u64, ulp: u64) -> Option<(usize, i16)> {
593 debug_assert!(remainder < ten_kappa);
600 // ----|-----|-----|----
602 // v - 1 ulp v + 1 ulp
604 // (for the reference, the dotted line indicates the exact value for
605 // possible representations in given number of digits.)
607 // error is too large that there are at least three possible representations
608 // between `v - 1 ulp` and `v + 1 ulp`. we cannot determine which one is correct.
609 if ulp >= ten_kappa { return None; }
616 // ----|-----|-----|----
618 // v - 1 ulp v + 1 ulp
620 // in fact, 1/2 ulp is enough to introduce two possible representations.
621 // (remember that we need a unique representation for both `v - 1 ulp` and `v + 1 ulp`.)
622 // this won't overflow, as `ulp < ten_kappa` from the first check.
623 if ten_kappa - ulp <= ulp { return None; }
628 // :<--------- 10^kappa ---------->:
632 // ----|-----|-----|------------------------
634 // v - 1 ulp v + 1 ulp
636 // if `v + 1 ulp` is closer to the rounded-down representation (which is already in `buf`),
637 // then we can safely return. note that `v - 1 ulp` *can* be less than the current
638 // representation, but as `1 ulp < 10^kappa / 2`, this condition is enough:
639 // the distance between `v - 1 ulp` and the current representation
640 // cannot exceed `10^kappa / 2`.
642 // the condition equals to `remainder + ulp < 10^kappa / 2`.
643 // since this can easily overflow, first check if `remainder < 10^kappa / 2`.
644 // we've already verified that `ulp < 10^kappa / 2`, so as long as
645 // `10^kappa` did not overflow after all, the second check is fine.
646 if ten_kappa - remainder > remainder && ten_kappa - 2 * remainder >= 2 * ulp {
647 return Some((len, exp));
650 // :<------- remainder ------>| :
652 // :<--------- 10^kappa --------->:
656 // -----------------------|-----|-----|-----
658 // v - 1 ulp v + 1 ulp
660 // on the other hands, if `v - 1 ulp` is closer to the rounded-up representation,
661 // we should round up and return. for the same reason we don't need to check `v + 1 ulp`.
663 // the condition equals to `remainder - ulp >= 10^kappa / 2`.
664 // again we first check if `remainder > ulp` (note that this is not `remainder >= ulp`,
665 // as `10^kappa` is never zero). also note that `remainder - ulp <= 10^kappa`,
666 // so the second check does not overflow.
667 if remainder > ulp && ten_kappa - (remainder - ulp) <= remainder - ulp {
668 if let Some(c) = round_up(buf, len) {
669 // only add an additional digit when we've been requested the fixed precision.
670 // we also need to check that, if the original buffer was empty,
671 // the additional digit can only be added when `exp == limit` (edge case).
673 if exp > limit && len < buf.len() {
678 return Some((len, exp));
681 // otherwise we are doomed (i.e. some values between `v - 1 ulp` and `v + 1 ulp` are
682 // rounding down and others are rounding up) and give up.
687 /// The exact and fixed mode implementation for Grisu with Dragon fallback.
689 /// This should be used for most cases.
690 pub fn format_exact(d: &Decoded, buf: &mut [u8], limit: i16) -> (/*#digits*/ usize, /*exp*/ i16) {
691 use num::flt2dec::strategy::dragon::format_exact as fallback;
692 match format_exact_opt(d, buf, limit) {
694 None => fallback(d, buf, limit),