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::flt2dec::{Decoded, MAX_SIG_DIGITS, round_up};
23 /// A custom 64-bit floating point type, representing `f * 2^e`.
24 #[derive(Copy, Clone, Debug)]
27 /// The integer mantissa.
29 /// The exponent in base 2.
34 /// Returns a correctly rounded product of itself and `other`.
35 pub fn mul(&self, other: &Fp) -> Fp {
36 const MASK: u64 = 0xffffffff;
38 let b = self.f & MASK;
39 let c = other.f >> 32;
40 let d = other.f & MASK;
45 let tmp = (bd >> 32) + (ad & MASK) + (bc & MASK) + (1 << 31) /* round */;
46 let f = ac + (ad >> 32) + (bc >> 32) + (tmp >> 32);
47 let e = self.e + other.e + 64;
51 /// Normalizes itself so that the resulting mantissa is at least `2^63`.
52 pub fn normalize(&self) -> Fp {
55 if f >> (64 - 32) == 0 { f <<= 32; e -= 32; }
56 if f >> (64 - 16) == 0 { f <<= 16; e -= 16; }
57 if f >> (64 - 8) == 0 { f <<= 8; e -= 8; }
58 if f >> (64 - 4) == 0 { f <<= 4; e -= 4; }
59 if f >> (64 - 2) == 0 { f <<= 2; e -= 2; }
60 if f >> (64 - 1) == 0 { f <<= 1; e -= 1; }
61 debug_assert!(f >= (1 >> 63));
65 /// Normalizes itself to have the shared exponent.
66 /// It can only decrease the exponent (and thus increase the mantissa).
67 pub fn normalize_to(&self, e: i16) -> Fp {
68 let edelta = self.e - e;
70 let edelta = edelta as usize;
71 assert_eq!(self.f << edelta >> edelta, self.f);
72 Fp { f: self.f << edelta, e: e }
76 // see the comments in `format_shortest_opt` for the rationale.
77 #[doc(hidden)] pub const ALPHA: i16 = -60;
78 #[doc(hidden)] pub const GAMMA: i16 = -32;
81 # the following Python code generates this table:
82 for i in xrange(-308, 333, 8):
83 if i >= 0: f = 10**i; e = 0
84 else: f = 2**(80-4*i) // 10**-i; e = 4 * i - 80
86 f = ((f << 64 >> (l-1)) + 1) >> 1; e += l - 64
87 print ' (%#018x, %5d, %4d),' % (f, e, i)
91 pub static CACHED_POW10: [(u64, i16, i16); 81] = [ // (f, e, k)
92 (0xe61acf033d1a45df, -1087, -308),
93 (0xab70fe17c79ac6ca, -1060, -300),
94 (0xff77b1fcbebcdc4f, -1034, -292),
95 (0xbe5691ef416bd60c, -1007, -284),
96 (0x8dd01fad907ffc3c, -980, -276),
97 (0xd3515c2831559a83, -954, -268),
98 (0x9d71ac8fada6c9b5, -927, -260),
99 (0xea9c227723ee8bcb, -901, -252),
100 (0xaecc49914078536d, -874, -244),
101 (0x823c12795db6ce57, -847, -236),
102 (0xc21094364dfb5637, -821, -228),
103 (0x9096ea6f3848984f, -794, -220),
104 (0xd77485cb25823ac7, -768, -212),
105 (0xa086cfcd97bf97f4, -741, -204),
106 (0xef340a98172aace5, -715, -196),
107 (0xb23867fb2a35b28e, -688, -188),
108 (0x84c8d4dfd2c63f3b, -661, -180),
109 (0xc5dd44271ad3cdba, -635, -172),
110 (0x936b9fcebb25c996, -608, -164),
111 (0xdbac6c247d62a584, -582, -156),
112 (0xa3ab66580d5fdaf6, -555, -148),
113 (0xf3e2f893dec3f126, -529, -140),
114 (0xb5b5ada8aaff80b8, -502, -132),
115 (0x87625f056c7c4a8b, -475, -124),
116 (0xc9bcff6034c13053, -449, -116),
117 (0x964e858c91ba2655, -422, -108),
118 (0xdff9772470297ebd, -396, -100),
119 (0xa6dfbd9fb8e5b88f, -369, -92),
120 (0xf8a95fcf88747d94, -343, -84),
121 (0xb94470938fa89bcf, -316, -76),
122 (0x8a08f0f8bf0f156b, -289, -68),
123 (0xcdb02555653131b6, -263, -60),
124 (0x993fe2c6d07b7fac, -236, -52),
125 (0xe45c10c42a2b3b06, -210, -44),
126 (0xaa242499697392d3, -183, -36),
127 (0xfd87b5f28300ca0e, -157, -28),
128 (0xbce5086492111aeb, -130, -20),
129 (0x8cbccc096f5088cc, -103, -12),
130 (0xd1b71758e219652c, -77, -4),
131 (0x9c40000000000000, -50, 4),
132 (0xe8d4a51000000000, -24, 12),
133 (0xad78ebc5ac620000, 3, 20),
134 (0x813f3978f8940984, 30, 28),
135 (0xc097ce7bc90715b3, 56, 36),
136 (0x8f7e32ce7bea5c70, 83, 44),
137 (0xd5d238a4abe98068, 109, 52),
138 (0x9f4f2726179a2245, 136, 60),
139 (0xed63a231d4c4fb27, 162, 68),
140 (0xb0de65388cc8ada8, 189, 76),
141 (0x83c7088e1aab65db, 216, 84),
142 (0xc45d1df942711d9a, 242, 92),
143 (0x924d692ca61be758, 269, 100),
144 (0xda01ee641a708dea, 295, 108),
145 (0xa26da3999aef774a, 322, 116),
146 (0xf209787bb47d6b85, 348, 124),
147 (0xb454e4a179dd1877, 375, 132),
148 (0x865b86925b9bc5c2, 402, 140),
149 (0xc83553c5c8965d3d, 428, 148),
150 (0x952ab45cfa97a0b3, 455, 156),
151 (0xde469fbd99a05fe3, 481, 164),
152 (0xa59bc234db398c25, 508, 172),
153 (0xf6c69a72a3989f5c, 534, 180),
154 (0xb7dcbf5354e9bece, 561, 188),
155 (0x88fcf317f22241e2, 588, 196),
156 (0xcc20ce9bd35c78a5, 614, 204),
157 (0x98165af37b2153df, 641, 212),
158 (0xe2a0b5dc971f303a, 667, 220),
159 (0xa8d9d1535ce3b396, 694, 228),
160 (0xfb9b7cd9a4a7443c, 720, 236),
161 (0xbb764c4ca7a44410, 747, 244),
162 (0x8bab8eefb6409c1a, 774, 252),
163 (0xd01fef10a657842c, 800, 260),
164 (0x9b10a4e5e9913129, 827, 268),
165 (0xe7109bfba19c0c9d, 853, 276),
166 (0xac2820d9623bf429, 880, 284),
167 (0x80444b5e7aa7cf85, 907, 292),
168 (0xbf21e44003acdd2d, 933, 300),
169 (0x8e679c2f5e44ff8f, 960, 308),
170 (0xd433179d9c8cb841, 986, 316),
171 (0x9e19db92b4e31ba9, 1013, 324),
172 (0xeb96bf6ebadf77d9, 1039, 332),
175 #[doc(hidden)] pub const CACHED_POW10_FIRST_E: i16 = -1087;
176 #[doc(hidden)] pub const CACHED_POW10_LAST_E: i16 = 1039;
179 pub fn cached_power(alpha: i16, gamma: i16) -> (i16, Fp) {
180 let offset = CACHED_POW10_FIRST_E as i32;
181 let range = (CACHED_POW10.len() as i32) - 1;
182 let domain = (CACHED_POW10_LAST_E - CACHED_POW10_FIRST_E) as i32;
183 let idx = ((gamma as i32) - offset) * range / domain;
184 let (f, e, k) = CACHED_POW10[idx as usize];
185 debug_assert!(alpha <= e && e <= gamma);
186 (k, Fp { f: f, e: e })
189 /// Given `x > 0`, returns `(k, 10^k)` such that `10^k <= x < 10^(k+1)`.
191 pub fn max_pow10_no_more_than(x: u32) -> (u8, u32) {
192 debug_assert!(x > 0);
194 const X9: u32 = 10_0000_0000;
195 const X8: u32 = 1_0000_0000;
196 const X7: u32 = 1000_0000;
197 const X6: u32 = 100_0000;
198 const X5: u32 = 10_0000;
199 const X4: u32 = 1_0000;
200 const X3: u32 = 1000;
205 if x < X2 { if x < X1 {(0, 1)} else {(1, X1)} }
206 else { if x < X3 {(2, X2)} else {(3, X3)} }
208 if x < X6 { if x < X5 {(4, X4)} else {(5, X5)} }
209 else if x < X8 { if x < X7 {(6, X6)} else {(7, X7)} }
210 else { if x < X9 {(8, X8)} else {(9, X9)} }
214 /// The shortest mode implementation for Grisu.
216 /// It returns `None` when it would return an inexact representation otherwise.
217 pub fn format_shortest_opt(d: &Decoded,
218 buf: &mut [u8]) -> Option<(/*#digits*/ usize, /*exp*/ i16)> {
220 assert!(d.minus > 0);
222 assert!(d.mant.checked_add(d.plus).is_some());
223 assert!(d.mant.checked_sub(d.minus).is_some());
224 assert!(buf.len() >= MAX_SIG_DIGITS);
225 assert!(d.mant + d.plus < (1 << 61)); // we need at least three bits of additional precision
227 // start with the normalized values with the shared exponent
228 let plus = Fp { f: d.mant + d.plus, e: d.exp }.normalize();
229 let minus = Fp { f: d.mant - d.minus, e: d.exp }.normalize_to(plus.e);
230 let v = Fp { f: d.mant, e: d.exp }.normalize_to(plus.e);
232 // find any `cached = 10^minusk` such that `ALPHA <= minusk + plus.e + 64 <= GAMMA`.
233 // since `plus` is normalized, this means `2^(62 + ALPHA) <= plus * cached < 2^(64 + GAMMA)`;
234 // given our choices of `ALPHA` and `GAMMA`, this puts `plus * cached` into `[4, 2^32)`.
236 // it is obviously desirable to maximize `GAMMA - ALPHA`,
237 // so that we don't need many cached powers of 10, but there are some considerations:
239 // 1. we want to keep `floor(plus * cached)` within `u32` since it needs a costly division.
240 // (this is not really avoidable, remainder is required for accuracy estimation.)
241 // 2. the remainder of `floor(plus * cached)` repeatedly gets multiplied by 10,
242 // and it should not overflow.
244 // the first gives `64 + GAMMA <= 32`, while the second gives `10 * 2^-ALPHA <= 2^64`;
245 // -60 and -32 is the maximal range with this constraint, and V8 also uses them.
246 let (minusk, cached) = cached_power(ALPHA - plus.e - 64, GAMMA - plus.e - 64);
248 // scale fps. this gives the maximal error of 1 ulp (proved from Theorem 5.1).
249 let plus = plus.mul(&cached);
250 let minus = minus.mul(&cached);
251 let v = v.mul(&cached);
252 debug_assert_eq!(plus.e, minus.e);
253 debug_assert_eq!(plus.e, v.e);
255 // +- actual range of minus
256 // | <---|---------------------- unsafe region --------------------------> |
258 // | |<--->| | <--------------- safe region ---------------> | |
260 // |1 ulp|1 ulp| |1 ulp|1 ulp| |1 ulp|1 ulp|
261 // |<--->|<--->| |<--->|<--->| |<--->|<--->|
262 // |-----|-----|-------...-------|-----|-----|-------...-------|-----|-----|
263 // | minus | | v | | plus |
264 // minus1 minus0 v - 1 ulp v + 1 ulp plus0 plus1
266 // above `minus`, `v` and `plus` are *quantized* approximations (error < 1 ulp).
267 // as we don't know the error is positive or negative, we use two approximations spaced equally
268 // and have the maximal error of 2 ulps.
270 // the "unsafe region" is a liberal interval which we initially generate.
271 // the "safe region" is a conservative interval which we only accept.
272 // we start with the correct repr within the unsafe region, and try to find the closest repr
273 // to `v` which is also within the safe region. if we can't, we give up.
274 let plus1 = plus.f + 1;
275 // let plus0 = plus.f - 1; // only for explanation
276 // let minus0 = minus.f + 1; // only for explanation
277 let minus1 = minus.f - 1;
278 let e = -plus.e as usize; // shared exponent
280 // divide `plus1` into integral and fractional parts.
281 // integral parts are guaranteed to fit in u32, since cached power guarantees `plus < 2^32`
282 // and normalized `plus.f` is always less than `2^64 - 2^4` due to the precision requirement.
283 let plus1int = (plus1 >> e) as u32;
284 let plus1frac = plus1 & ((1 << e) - 1);
286 // calculate the largest `10^max_kappa` no more than `plus1` (thus `plus1 < 10^(max_kappa+1)`).
287 // this is an upper bound of `kappa` below.
288 let (max_kappa, max_ten_kappa) = max_pow10_no_more_than(plus1int);
291 let exp = max_kappa as i16 - minusk + 1;
293 // Theorem 6.2: if `k` is the greatest integer s.t. `0 <= y mod 10^k <= y - x`,
294 // then `V = floor(y / 10^k) * 10^k` is in `[x, y]` and one of the shortest
295 // representations (with the minimal number of significant digits) in that range.
297 // find the digit length `kappa` between `(minus1, plus1)` as per Theorem 6.2.
298 // Theorem 6.2 can be adopted to exclude `x` by requiring `y mod 10^k < y - x` instead.
299 // (e.g. `x` = 32000, `y` = 32777; `kappa` = 2 since `y mod 10^3 = 777 < y - x = 777`.)
300 // the algorithm relies on the later verification phase to exclude `y`.
301 let delta1 = plus1 - minus1;
302 // let delta1int = (delta1 >> e) as usize; // only for explanation
303 let delta1frac = delta1 & ((1 << e) - 1);
305 // render integral parts, while checking for the accuracy at each step.
306 let mut kappa = max_kappa as i16;
307 let mut ten_kappa = max_ten_kappa; // 10^kappa
308 let mut remainder = plus1int; // digits yet to be rendered
309 loop { // we always have at least one digit to render, as `plus1 >= 10^kappa`
311 // - `delta1int <= remainder < 10^(kappa+1)`
312 // - `plus1int = d[0..n-1] * 10^(kappa+1) + remainder`
313 // (it follows that `remainder = plus1int % 10^(kappa+1)`)
315 // divide `remainder` by `10^kappa`. both are scaled by `2^-e`.
316 let q = remainder / ten_kappa;
317 let r = remainder % ten_kappa;
318 debug_assert!(q < 10);
319 buf[i] = b'0' + q as u8;
322 let plus1rem = ((r as u64) << e) + plus1frac; // == (plus1 % 10^kappa) * 2^e
323 if plus1rem < delta1 {
324 // `plus1 % 10^kappa < delta1 = plus1 - minus1`; we've found the correct `kappa`.
325 let ten_kappa = (ten_kappa as u64) << e; // scale 10^kappa back to the shared exponent
326 return round_and_weed(&mut buf[..i], exp, plus1rem, delta1, plus1 - v.f, ten_kappa, 1);
329 // break the loop when we have rendered all integral digits.
330 // the exact number of digits is `max_kappa + 1` as `plus1 < 10^(max_kappa+1)`.
331 if i > max_kappa as usize {
332 debug_assert_eq!(ten_kappa, 1);
333 debug_assert_eq!(kappa, 0);
337 // restore invariants
343 // render fractional parts, while checking for the accuracy at each step.
344 // this time we rely on repeated multiplications, as division will lose the precision.
345 let mut remainder = plus1frac;
346 let mut threshold = delta1frac;
348 loop { // the next digit should be significant as we've tested that before breaking out
349 // invariants, where `m = max_kappa + 1` (# of digits in the integral part):
350 // - `remainder < 2^e`
351 // - `plus1frac * 10^(n-m) = d[m..n-1] * 2^e + remainder`
353 remainder *= 10; // won't overflow, `2^e * 10 < 2^64`
357 // divide `remainder` by `10^kappa`.
358 // both are scaled by `2^e / 10^kappa`, so the latter is implicit here.
359 let q = remainder >> e;
360 let r = remainder & ((1 << e) - 1);
361 debug_assert!(q < 10);
362 buf[i] = b'0' + q as u8;
366 let ten_kappa = 1 << e; // implicit divisor
367 return round_and_weed(&mut buf[..i], exp, r, threshold,
368 (plus1 - v.f) * ulp, ten_kappa, ulp);
371 // restore invariants
376 // we've generated all significant digits of `plus1`, but not sure if it's the optimal one.
377 // for example, if `minus1` is 3.14153... and `plus1` is 3.14158..., there are 5 different
378 // shortest representation from 3.14154 to 3.14158 but we only have the greatest one.
379 // we have to successively decrease the last digit and check if this is the optimal repr.
380 // there are at most 9 candidates (..1 to ..9), so this is fairly quick. ("rounding" phase)
382 // the function checks if this "optimal" repr is actually within the ulp ranges,
383 // and also, it is possible that the "second-to-optimal" repr can actually be optimal
384 // due to the rounding error. in either cases this returns `None`. ("weeding" phase)
386 // all arguments here are scaled by the common (but implicit) value `k`, so that:
387 // - `remainder = (plus1 % 10^kappa) * k`
388 // - `threshold = (plus1 - minus1) * k` (and also, `remainder < threshold`)
389 // - `plus1v = (plus1 - v) * k` (and also, `threshold > plus1v` from prior invariants)
390 // - `ten_kappa = 10^kappa * k`
391 // - `ulp = 2^-e * k`
392 fn round_and_weed(buf: &mut [u8], exp: i16, remainder: u64, threshold: u64, plus1v: u64,
393 ten_kappa: u64, ulp: u64) -> Option<(usize, i16)> {
394 assert!(!buf.is_empty());
396 // produce two approximations to `v` (actually `plus1 - v`) within 1.5 ulps.
397 // the resulting representation should be the closest representation to both.
399 // here `plus1 - v` is used since calculations are done with respect to `plus1`
400 // in order to avoid overflow/underflow (hence the seemingly swapped names).
401 let plus1v_down = plus1v + ulp; // plus1 - (v - 1 ulp)
402 let plus1v_up = plus1v - ulp; // plus1 - (v + 1 ulp)
404 // decrease the last digit and stop at the closest representation to `v + 1 ulp`.
405 let mut plus1w = remainder; // plus1w(n) = plus1 - w(n)
407 let last = buf.last_mut().unwrap();
409 // we work with the approximated digits `w(n)`, which is initially equal to `plus1 -
410 // plus1 % 10^kappa`. after running the loop body `n` times, `w(n) = plus1 -
411 // plus1 % 10^kappa - n * 10^kappa`. we set `plus1w(n) = plus1 - w(n) =
412 // plus1 % 10^kappa + n * 10^kappa` (thus `remainder = plus1w(0)`) to simplify checks.
413 // note that `plus1w(n)` is always increasing.
415 // we have three conditions to terminate. any of them will make the loop unable to
416 // proceed, but we then have at least one valid representation known to be closest to
417 // `v + 1 ulp` anyway. we will denote them as TC1 through TC3 for brevity.
419 // TC1: `w(n) <= v + 1 ulp`, i.e. this is the last repr that can be the closest one.
420 // this is equivalent to `plus1 - w(n) = plus1w(n) >= plus1 - (v + 1 ulp) = plus1v_up`.
421 // combined with TC2 (which checks if `w(n+1)` is valid), this prevents the possible
422 // overflow on the calculation of `plus1w(n)`.
424 // TC2: `w(n+1) < minus1`, i.e. the next repr definitely does not round to `v`.
425 // this is equivalent to `plus1 - w(n) + 10^kappa = plus1w(n) + 10^kappa >
426 // plus1 - minus1 = threshold`. the left hand side can overflow, but we know
427 // `threshold > plus1v`, so if TC1 is false, `threshold - plus1w(n) >
428 // threshold - (plus1v - 1 ulp) > 1 ulp` and we can safely test if
429 // `threshold - plus1w(n) < 10^kappa` instead.
431 // TC3: `abs(w(n) - (v + 1 ulp)) <= abs(w(n+1) - (v + 1 ulp))`, i.e. the next repr is
432 // no closer to `v + 1 ulp` than the current repr. given `z(n) = plus1v_up - plus1w(n)`,
433 // this becomes `abs(z(n)) <= abs(z(n+1))`. again assuming that TC1 is false, we have
434 // `z(n) > 0`. we have two cases to consider:
436 // - when `z(n+1) >= 0`: TC3 becomes `z(n) <= z(n+1)`. as `plus1w(n)` is increasing,
437 // `z(n)` should be decreasing and this is clearly false.
438 // - when `z(n+1) < 0`:
439 // - TC3a: the precondition is `plus1v_up < plus1w(n) + 10^kappa`. assuming TC2 is
440 // false, `threshold >= plus1w(n) + 10^kappa` so it cannot overflow.
441 // - TC3b: TC3 becomes `z(n) <= -z(n+1)`, i.e. `plus1v_up - plus1w(n) >=
442 // plus1w(n+1) - plus1v_up = plus1w(n) + 10^kappa - plus1v_up`. the negated TC1
443 // gives `plus1v_up > plus1w(n)`, so it cannot overflow or underflow when
444 // combined with TC3a.
446 // consequently, we should stop when `TC1 || TC2 || (TC3a && TC3b)`. the following is
447 // equal to its inverse, `!TC1 && !TC2 && (!TC3a || !TC3b)`.
448 while plus1w < plus1v_up &&
449 threshold - plus1w >= ten_kappa &&
450 (plus1w + ten_kappa < plus1v_up ||
451 plus1v_up - plus1w >= plus1w + ten_kappa - plus1v_up) {
453 debug_assert!(*last > b'0'); // the shortest repr cannot end with `0`
458 // check if this representation is also the closest representation to `v - 1 ulp`.
460 // this is simply same to the terminating conditions for `v + 1 ulp`, with all `plus1v_up`
461 // replaced by `plus1v_down` instead. overflow analysis equally holds.
462 if plus1w < plus1v_down &&
463 threshold - plus1w >= ten_kappa &&
464 (plus1w + ten_kappa < plus1v_down ||
465 plus1v_down - plus1w >= plus1w + ten_kappa - plus1v_down) {
469 // now we have the closest representation to `v` between `plus1` and `minus1`.
470 // this is too liberal, though, so we reject any `w(n)` not between `plus0` and `minus0`,
471 // i.e. `plus1 - plus1w(n) <= minus0` or `plus1 - plus1w(n) >= plus0`. we utilize the facts
472 // that `threshold = plus1 - minus1` and `plus1 - plus0 = minus0 - minus1 = 2 ulp`.
473 if 2 * ulp <= plus1w && plus1w <= threshold - 4 * ulp {
474 Some((buf.len(), exp))
481 /// The shortest mode implementation for Grisu with Dragon fallback.
483 /// This should be used for most cases.
484 pub fn format_shortest(d: &Decoded, buf: &mut [u8]) -> (/*#digits*/ usize, /*exp*/ i16) {
485 use num::flt2dec::strategy::dragon::format_shortest as fallback;
486 match format_shortest_opt(d, buf) {
488 None => fallback(d, buf),
492 /// The exact and fixed mode implementation for Grisu.
494 /// It returns `None` when it would return an inexact representation otherwise.
495 pub fn format_exact_opt(d: &Decoded, buf: &mut [u8], limit: i16)
496 -> Option<(/*#digits*/ usize, /*exp*/ i16)> {
498 assert!(d.mant < (1 << 61)); // we need at least three bits of additional precision
499 assert!(!buf.is_empty());
501 // normalize and scale `v`.
502 let v = Fp { f: d.mant, e: d.exp }.normalize();
503 let (minusk, cached) = cached_power(ALPHA - v.e - 64, GAMMA - v.e - 64);
504 let v = v.mul(&cached);
506 // divide `v` into integral and fractional parts.
507 let e = -v.e as usize;
508 let vint = (v.f >> e) as u32;
509 let vfrac = v.f & ((1 << e) - 1);
511 // both old `v` and new `v` (scaled by `10^-k`) has an error of < 1 ulp (Theorem 5.1).
512 // as we don't know the error is positive or negative, we use two approximations
513 // spaced equally and have the maximal error of 2 ulps (same to the shortest case).
515 // the goal is to find the exactly rounded series of digits that are common to
516 // both `v - 1 ulp` and `v + 1 ulp`, so that we are maximally confident.
517 // if this is not possible, we don't know which one is the correct output for `v`,
518 // so we give up and fall back.
520 // `err` is defined as `1 ulp * 2^e` here (same to the ulp in `vfrac`),
521 // and we will scale it whenever `v` gets scaled.
524 // calculate the largest `10^max_kappa` no more than `v` (thus `v < 10^(max_kappa+1)`).
525 // this is an upper bound of `kappa` below.
526 let (max_kappa, max_ten_kappa) = max_pow10_no_more_than(vint);
529 let exp = max_kappa as i16 - minusk + 1;
531 // if we are working with the last-digit limitation, we need to shorten the buffer
532 // before the actual rendering in order to avoid double rounding.
533 // note that we have to enlarge the buffer again when rounding up happens!
534 let len = if exp <= limit {
535 // oops, we cannot even produce *one* digit.
536 // this is possible when, say, we've got something like 9.5 and it's being rounded to 10.
538 // in principle we can immediately call `possibly_round` with an empty buffer,
539 // but scaling `max_ten_kappa << e` by 10 can result in overflow.
540 // thus we are being sloppy here and widen the error range by a factor of 10.
541 // this will increase the false negative rate, but only very, *very* slightly;
542 // it can only matter noticably when the mantissa is bigger than 60 bits.
543 return possibly_round(buf, 0, exp, limit, v.f / 10, (max_ten_kappa as u64) << e, err << e);
544 } else if ((exp as i32 - limit as i32) as usize) < buf.len() {
545 (exp - limit) as usize
549 debug_assert!(len > 0);
551 // render integral parts.
552 // the error is entirely fractional, so we don't need to check it in this part.
553 let mut kappa = max_kappa as i16;
554 let mut ten_kappa = max_ten_kappa; // 10^kappa
555 let mut remainder = vint; // digits yet to be rendered
556 loop { // we always have at least one digit to render
558 // - `remainder < 10^(kappa+1)`
559 // - `vint = d[0..n-1] * 10^(kappa+1) + remainder`
560 // (it follows that `remainder = vint % 10^(kappa+1)`)
562 // divide `remainder` by `10^kappa`. both are scaled by `2^-e`.
563 let q = remainder / ten_kappa;
564 let r = remainder % ten_kappa;
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 let vrem = ((r as u64) << e) + vfrac; // == (v % 10^kappa) * 2^e
572 return possibly_round(buf, len, exp, limit, vrem, (ten_kappa as u64) << e, err << e);
575 // break the loop when we have rendered all integral digits.
576 // the exact number of digits is `max_kappa + 1` as `plus1 < 10^(max_kappa+1)`.
577 if i > max_kappa as usize {
578 debug_assert_eq!(ten_kappa, 1);
579 debug_assert_eq!(kappa, 0);
583 // restore invariants
589 // render fractional parts.
591 // in principle we can continue to the last available digit and check for the accuracy.
592 // unfortunately we are working with the finite-sized integers, so we need some criterion
593 // to detect the overflow. V8 uses `remainder > err`, which becomes false when
594 // the first `i` significant digits of `v - 1 ulp` and `v` differ. however this rejects
595 // too many otherwise valid input.
597 // since the later phase has a correct overflow detection, we instead use tighter criterion:
598 // we continue til `err` exceeds `10^kappa / 2`, so that the range between `v - 1 ulp` and
599 // `v + 1 ulp` definitely contains two or more rounded representations. this is same to
600 // the first two comparisons from `possibly_round`, for the reference.
601 let mut remainder = vfrac;
602 let maxerr = 1 << (e - 1);
604 // invariants, where `m = max_kappa + 1` (# of digits in the integral part):
605 // - `remainder < 2^e`
606 // - `vfrac * 10^(n-m) = d[m..n-1] * 2^e + remainder`
607 // - `err = 10^(n-m)`
609 remainder *= 10; // won't overflow, `2^e * 10 < 2^64`
610 err *= 10; // won't overflow, `err * 10 < 2^e * 5 < 2^64`
612 // divide `remainder` by `10^kappa`.
613 // both are scaled by `2^e / 10^kappa`, so the latter is implicit here.
614 let q = remainder >> e;
615 let r = remainder & ((1 << e) - 1);
616 debug_assert!(q < 10);
617 buf[i] = b'0' + q as u8;
620 // is the buffer full? run the rounding pass with the remainder.
622 return possibly_round(buf, len, exp, limit, r, 1 << e, err);
625 // restore invariants
629 // further calculation is useless (`possibly_round` definitely fails), so we give up.
632 // we've generated all requested digits of `v`, which should be also same to corresponding
633 // digits of `v - 1 ulp`. now we check if there is a unique representation shared by
634 // both `v - 1 ulp` and `v + 1 ulp`; this can be either same to generated digits, or
635 // to the rounded-up version of those digits. if the range contains multiple representations
636 // of the same length, we cannot be sure and should return `None` instead.
638 // all arguments here are scaled by the common (but implicit) value `k`, so that:
639 // - `remainder = (v % 10^kappa) * k`
640 // - `ten_kappa = 10^kappa * k`
641 // - `ulp = 2^-e * k`
642 fn possibly_round(buf: &mut [u8], mut len: usize, mut exp: i16, limit: i16,
643 remainder: u64, ten_kappa: u64, ulp: u64) -> Option<(usize, i16)> {
644 debug_assert!(remainder < ten_kappa);
651 // ----|-----|-----|----
653 // v - 1 ulp v + 1 ulp
655 // (for the reference, the dotted line indicates the exact value for
656 // possible representations in given number of digits.)
658 // error is too large that there are at least three possible representations
659 // between `v - 1 ulp` and `v + 1 ulp`. we cannot determine which one is correct.
660 if ulp >= ten_kappa { return None; }
667 // ----|-----|-----|----
669 // v - 1 ulp v + 1 ulp
671 // in fact, 1/2 ulp is enough to introduce two possible representations.
672 // (remember that we need a unique representation for both `v - 1 ulp` and `v + 1 ulp`.)
673 // this won't overflow, as `ulp < ten_kappa` from the first check.
674 if ten_kappa - ulp <= ulp { return None; }
679 // :<--------- 10^kappa ---------->:
683 // ----|-----|-----|------------------------
685 // v - 1 ulp v + 1 ulp
687 // if `v + 1 ulp` is closer to the rounded-down representation (which is already in `buf`),
688 // then we can safely return. note that `v - 1 ulp` *can* be less than the current
689 // representation, but as `1 ulp < 10^kappa / 2`, this condition is enough:
690 // the distance between `v - 1 ulp` and the current representation
691 // cannot exceed `10^kappa / 2`.
693 // the condition equals to `remainder + ulp < 10^kappa / 2`.
694 // since this can easily overflow, first check if `remainder < 10^kappa / 2`.
695 // we've already verified that `ulp < 10^kappa / 2`, so as long as
696 // `10^kappa` did not overflow after all, the second check is fine.
697 if ten_kappa - remainder > remainder && ten_kappa - 2 * remainder >= 2 * ulp {
698 return Some((len, exp));
701 // :<------- remainder ------>| :
703 // :<--------- 10^kappa --------->:
707 // -----------------------|-----|-----|-----
709 // v - 1 ulp v + 1 ulp
711 // on the other hands, if `v - 1 ulp` is closer to the rounded-up representation,
712 // we should round up and return. for the same reason we don't need to check `v + 1 ulp`.
714 // the condition equals to `remainder - ulp >= 10^kappa / 2`.
715 // again we first check if `remainder > ulp` (note that this is not `remainder >= ulp`,
716 // as `10^kappa` is never zero). also note that `remainder - ulp <= 10^kappa`,
717 // so the second check does not overflow.
718 if remainder > ulp && ten_kappa - (remainder - ulp) <= remainder - ulp {
719 if let Some(c) = round_up(buf, len) {
720 // only add an additional digit when we've been requested the fixed precision.
721 // we also need to check that, if the original buffer was empty,
722 // the additional digit can only be added when `exp == limit` (edge case).
724 if exp > limit && len < buf.len() {
729 return Some((len, exp));
732 // otherwise we are doomed (i.e. some values between `v - 1 ulp` and `v + 1 ulp` are
733 // rounding down and others are rounding up) and give up.
738 /// The exact and fixed mode implementation for Grisu with Dragon fallback.
740 /// This should be used for most cases.
741 pub fn format_exact(d: &Decoded, buf: &mut [u8], limit: i16) -> (/*#digits*/ usize, /*exp*/ i16) {
742 use num::flt2dec::strategy::dragon::format_exact as fallback;
743 match format_exact_opt(d, buf, limit) {
745 None => fallback(d, buf, limit),