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.
19 use num
::diy_float
::Fp
;
20 use num
::flt2dec
::{Decoded, MAX_SIG_DIGITS, round_up}
;
23 // see the comments in `format_shortest_opt` for the rationale.
24 #[doc(hidden)] pub const ALPHA: i16 = -60;
25 #[doc(hidden)] pub const GAMMA: i16 = -32;
28 # the following Python code generates this table:
29 for i in xrange(-308, 333, 8):
30 if i >= 0: f = 10**i; e = 0
31 else: f = 2**(80-4*i) // 10**-i; e = 4 * i - 80
33 f = ((f << 64 >> (l-1)) + 1) >> 1; e += l - 64
34 print ' (%#018x, %5d, %4d),' % (f, e, i)
38 pub static CACHED_POW10
: [(u64, i16, i16); 81] = [ // (f, e, k)
39 (0xe61acf033d1a45df, -1087, -308),
40 (0xab70fe17c79ac6ca, -1060, -300),
41 (0xff77b1fcbebcdc4f, -1034, -292),
42 (0xbe5691ef416bd60c, -1007, -284),
43 (0x8dd01fad907ffc3c, -980, -276),
44 (0xd3515c2831559a83, -954, -268),
45 (0x9d71ac8fada6c9b5, -927, -260),
46 (0xea9c227723ee8bcb, -901, -252),
47 (0xaecc49914078536d, -874, -244),
48 (0x823c12795db6ce57, -847, -236),
49 (0xc21094364dfb5637, -821, -228),
50 (0x9096ea6f3848984f, -794, -220),
51 (0xd77485cb25823ac7, -768, -212),
52 (0xa086cfcd97bf97f4, -741, -204),
53 (0xef340a98172aace5, -715, -196),
54 (0xb23867fb2a35b28e, -688, -188),
55 (0x84c8d4dfd2c63f3b, -661, -180),
56 (0xc5dd44271ad3cdba, -635, -172),
57 (0x936b9fcebb25c996, -608, -164),
58 (0xdbac6c247d62a584, -582, -156),
59 (0xa3ab66580d5fdaf6, -555, -148),
60 (0xf3e2f893dec3f126, -529, -140),
61 (0xb5b5ada8aaff80b8, -502, -132),
62 (0x87625f056c7c4a8b, -475, -124),
63 (0xc9bcff6034c13053, -449, -116),
64 (0x964e858c91ba2655, -422, -108),
65 (0xdff9772470297ebd, -396, -100),
66 (0xa6dfbd9fb8e5b88f, -369, -92),
67 (0xf8a95fcf88747d94, -343, -84),
68 (0xb94470938fa89bcf, -316, -76),
69 (0x8a08f0f8bf0f156b, -289, -68),
70 (0xcdb02555653131b6, -263, -60),
71 (0x993fe2c6d07b7fac, -236, -52),
72 (0xe45c10c42a2b3b06, -210, -44),
73 (0xaa242499697392d3, -183, -36),
74 (0xfd87b5f28300ca0e, -157, -28),
75 (0xbce5086492111aeb, -130, -20),
76 (0x8cbccc096f5088cc, -103, -12),
77 (0xd1b71758e219652c, -77, -4),
78 (0x9c40000000000000, -50, 4),
79 (0xe8d4a51000000000, -24, 12),
80 (0xad78ebc5ac620000, 3, 20),
81 (0x813f3978f8940984, 30, 28),
82 (0xc097ce7bc90715b3, 56, 36),
83 (0x8f7e32ce7bea5c70, 83, 44),
84 (0xd5d238a4abe98068, 109, 52),
85 (0x9f4f2726179a2245, 136, 60),
86 (0xed63a231d4c4fb27, 162, 68),
87 (0xb0de65388cc8ada8, 189, 76),
88 (0x83c7088e1aab65db, 216, 84),
89 (0xc45d1df942711d9a, 242, 92),
90 (0x924d692ca61be758, 269, 100),
91 (0xda01ee641a708dea, 295, 108),
92 (0xa26da3999aef774a, 322, 116),
93 (0xf209787bb47d6b85, 348, 124),
94 (0xb454e4a179dd1877, 375, 132),
95 (0x865b86925b9bc5c2, 402, 140),
96 (0xc83553c5c8965d3d, 428, 148),
97 (0x952ab45cfa97a0b3, 455, 156),
98 (0xde469fbd99a05fe3, 481, 164),
99 (0xa59bc234db398c25, 508, 172),
100 (0xf6c69a72a3989f5c, 534, 180),
101 (0xb7dcbf5354e9bece, 561, 188),
102 (0x88fcf317f22241e2, 588, 196),
103 (0xcc20ce9bd35c78a5, 614, 204),
104 (0x98165af37b2153df, 641, 212),
105 (0xe2a0b5dc971f303a, 667, 220),
106 (0xa8d9d1535ce3b396, 694, 228),
107 (0xfb9b7cd9a4a7443c, 720, 236),
108 (0xbb764c4ca7a44410, 747, 244),
109 (0x8bab8eefb6409c1a, 774, 252),
110 (0xd01fef10a657842c, 800, 260),
111 (0x9b10a4e5e9913129, 827, 268),
112 (0xe7109bfba19c0c9d, 853, 276),
113 (0xac2820d9623bf429, 880, 284),
114 (0x80444b5e7aa7cf85, 907, 292),
115 (0xbf21e44003acdd2d, 933, 300),
116 (0x8e679c2f5e44ff8f, 960, 308),
117 (0xd433179d9c8cb841, 986, 316),
118 (0x9e19db92b4e31ba9, 1013, 324),
119 (0xeb96bf6ebadf77d9, 1039, 332),
122 #[doc(hidden)] pub const CACHED_POW10_FIRST_E: i16 = -1087;
123 #[doc(hidden)] pub const CACHED_POW10_LAST_E: i16 = 1039;
126 pub fn cached_power(alpha
: i16, gamma
: i16) -> (i16, Fp
) {
127 let offset
= CACHED_POW10_FIRST_E
as i32;
128 let range
= (CACHED_POW10
.len() as i32) - 1;
129 let domain
= (CACHED_POW10_LAST_E
- CACHED_POW10_FIRST_E
) as i32;
130 let idx
= ((gamma
as i32) - offset
) * range
/ domain
;
131 let (f
, e
, k
) = CACHED_POW10
[idx
as usize];
132 debug_assert
!(alpha
<= e
&& e
<= gamma
);
133 (k
, Fp { f: f, e: e }
)
136 /// Given `x > 0`, returns `(k, 10^k)` such that `10^k <= x < 10^(k+1)`.
138 pub fn max_pow10_no_more_than(x
: u32) -> (u8, u32) {
139 debug_assert
!(x
> 0);
141 const X9
: u32 = 10_0000_0000;
142 const X8
: u32 = 1_0000_0000;
143 const X7
: u32 = 1000_0000;
144 const X6
: u32 = 100_0000;
145 const X5
: u32 = 10_0000;
146 const X4
: u32 = 1_0000;
147 const X3
: u32 = 1000;
152 if x
< X2 { if x < X1 {(0, 1)}
else {(1, X1)}
}
153 else { if x < X3 {(2, X2)}
else {(3, X3)}
}
155 if x
< X6 { if x < X5 {(4, X4)}
else {(5, X5)}
}
156 else if x
< X8 { if x < X7 {(6, X6)}
else {(7, X7)}
}
157 else { if x < X9 {(8, X8)}
else {(9, X9)}
}
161 /// The shortest mode implementation for Grisu.
163 /// It returns `None` when it would return an inexact representation otherwise.
164 pub fn format_shortest_opt(d
: &Decoded
,
165 buf
: &mut [u8]) -> Option
<(/*#digits*/ usize, /*exp*/ i16)> {
167 assert
!(d
.minus
> 0);
169 assert
!(d
.mant
.checked_add(d
.plus
).is_some());
170 assert
!(d
.mant
.checked_sub(d
.minus
).is_some());
171 assert
!(buf
.len() >= MAX_SIG_DIGITS
);
172 assert
!(d
.mant
+ d
.plus
< (1 << 61)); // we need at least three bits of additional precision
174 // start with the normalized values with the shared exponent
175 let plus
= Fp { f: d.mant + d.plus, e: d.exp }
.normalize();
176 let minus
= Fp { f: d.mant - d.minus, e: d.exp }
.normalize_to(plus
.e
);
177 let v
= Fp { f: d.mant, e: d.exp }
.normalize_to(plus
.e
);
179 // find any `cached = 10^minusk` such that `ALPHA <= minusk + plus.e + 64 <= GAMMA`.
180 // since `plus` is normalized, this means `2^(62 + ALPHA) <= plus * cached < 2^(64 + GAMMA)`;
181 // given our choices of `ALPHA` and `GAMMA`, this puts `plus * cached` into `[4, 2^32)`.
183 // it is obviously desirable to maximize `GAMMA - ALPHA`,
184 // so that we don't need many cached powers of 10, but there are some considerations:
186 // 1. we want to keep `floor(plus * cached)` within `u32` since it needs a costly division.
187 // (this is not really avoidable, remainder is required for accuracy estimation.)
188 // 2. the remainder of `floor(plus * cached)` repeatedly gets multiplied by 10,
189 // and it should not overflow.
191 // the first gives `64 + GAMMA <= 32`, while the second gives `10 * 2^-ALPHA <= 2^64`;
192 // -60 and -32 is the maximal range with this constraint, and V8 also uses them.
193 let (minusk
, cached
) = cached_power(ALPHA
- plus
.e
- 64, GAMMA
- plus
.e
- 64);
195 // scale fps. this gives the maximal error of 1 ulp (proved from Theorem 5.1).
196 let plus
= plus
.mul(&cached
);
197 let minus
= minus
.mul(&cached
);
198 let v
= v
.mul(&cached
);
199 debug_assert_eq
!(plus
.e
, minus
.e
);
200 debug_assert_eq
!(plus
.e
, v
.e
);
202 // +- actual range of minus
203 // | <---|---------------------- unsafe region --------------------------> |
205 // | |<--->| | <--------------- safe region ---------------> | |
207 // |1 ulp|1 ulp| |1 ulp|1 ulp| |1 ulp|1 ulp|
208 // |<--->|<--->| |<--->|<--->| |<--->|<--->|
209 // |-----|-----|-------...-------|-----|-----|-------...-------|-----|-----|
210 // | minus | | v | | plus |
211 // minus1 minus0 v - 1 ulp v + 1 ulp plus0 plus1
213 // above `minus`, `v` and `plus` are *quantized* approximations (error < 1 ulp).
214 // as we don't know the error is positive or negative, we use two approximations spaced equally
215 // and have the maximal error of 2 ulps.
217 // the "unsafe region" is a liberal interval which we initially generate.
218 // the "safe region" is a conservative interval which we only accept.
219 // we start with the correct repr within the unsafe region, and try to find the closest repr
220 // to `v` which is also within the safe region. if we can't, we give up.
221 let plus1
= plus
.f
+ 1;
222 // let plus0 = plus.f - 1; // only for explanation
223 // let minus0 = minus.f + 1; // only for explanation
224 let minus1
= minus
.f
- 1;
225 let e
= -plus
.e
as usize; // shared exponent
227 // divide `plus1` into integral and fractional parts.
228 // integral parts are guaranteed to fit in u32, since cached power guarantees `plus < 2^32`
229 // and normalized `plus.f` is always less than `2^64 - 2^4` due to the precision requirement.
230 let plus1int
= (plus1
>> e
) as u32;
231 let plus1frac
= plus1
& ((1 << e
) - 1);
233 // calculate the largest `10^max_kappa` no more than `plus1` (thus `plus1 < 10^(max_kappa+1)`).
234 // this is an upper bound of `kappa` below.
235 let (max_kappa
, max_ten_kappa
) = max_pow10_no_more_than(plus1int
);
238 let exp
= max_kappa
as i16 - minusk
+ 1;
240 // Theorem 6.2: if `k` is the greatest integer s.t. `0 <= y mod 10^k <= y - x`,
241 // then `V = floor(y / 10^k) * 10^k` is in `[x, y]` and one of the shortest
242 // representations (with the minimal number of significant digits) in that range.
244 // find the digit length `kappa` between `(minus1, plus1)` as per Theorem 6.2.
245 // Theorem 6.2 can be adopted to exclude `x` by requiring `y mod 10^k < y - x` instead.
246 // (e.g. `x` = 32000, `y` = 32777; `kappa` = 2 since `y mod 10^3 = 777 < y - x = 777`.)
247 // the algorithm relies on the later verification phase to exclude `y`.
248 let delta1
= plus1
- minus1
;
249 // let delta1int = (delta1 >> e) as usize; // only for explanation
250 let delta1frac
= delta1
& ((1 << e
) - 1);
252 // render integral parts, while checking for the accuracy at each step.
253 let mut kappa
= max_kappa
as i16;
254 let mut ten_kappa
= max_ten_kappa
; // 10^kappa
255 let mut remainder
= plus1int
; // digits yet to be rendered
256 loop { // we always have at least one digit to render, as `plus1 >= 10^kappa`
258 // - `delta1int <= remainder < 10^(kappa+1)`
259 // - `plus1int = d[0..n-1] * 10^(kappa+1) + remainder`
260 // (it follows that `remainder = plus1int % 10^(kappa+1)`)
262 // divide `remainder` by `10^kappa`. both are scaled by `2^-e`.
263 let q
= remainder
/ ten_kappa
;
264 let r
= remainder
% ten_kappa
;
265 debug_assert
!(q
< 10);
266 buf
[i
] = b'
0'
+ q
as u8;
269 let plus1rem
= ((r
as u64) << e
) + plus1frac
; // == (plus1 % 10^kappa) * 2^e
270 if plus1rem
< delta1
{
271 // `plus1 % 10^kappa < delta1 = plus1 - minus1`; we've found the correct `kappa`.
272 let ten_kappa
= (ten_kappa
as u64) << e
; // scale 10^kappa back to the shared exponent
273 return round_and_weed(&mut buf
[..i
], exp
, plus1rem
, delta1
, plus1
- v
.f
, ten_kappa
, 1);
276 // break the loop when we have rendered all integral digits.
277 // the exact number of digits is `max_kappa + 1` as `plus1 < 10^(max_kappa+1)`.
278 if i
> max_kappa
as usize {
279 debug_assert_eq
!(ten_kappa
, 1);
280 debug_assert_eq
!(kappa
, 0);
284 // restore invariants
290 // render fractional parts, while checking for the accuracy at each step.
291 // this time we rely on repeated multiplications, as division will lose the precision.
292 let mut remainder
= plus1frac
;
293 let mut threshold
= delta1frac
;
295 loop { // the next digit should be significant as we've tested that before breaking out
296 // invariants, where `m = max_kappa + 1` (# of digits in the integral part):
297 // - `remainder < 2^e`
298 // - `plus1frac * 10^(n-m) = d[m..n-1] * 2^e + remainder`
300 remainder
*= 10; // won't overflow, `2^e * 10 < 2^64`
304 // divide `remainder` by `10^kappa`.
305 // both are scaled by `2^e / 10^kappa`, so the latter is implicit here.
306 let q
= remainder
>> e
;
307 let r
= remainder
& ((1 << e
) - 1);
308 debug_assert
!(q
< 10);
309 buf
[i
] = b'
0'
+ q
as u8;
313 let ten_kappa
= 1 << e
; // implicit divisor
314 return round_and_weed(&mut buf
[..i
], exp
, r
, threshold
,
315 (plus1
- v
.f
) * ulp
, ten_kappa
, ulp
);
318 // restore invariants
323 // we've generated all significant digits of `plus1`, but not sure if it's the optimal one.
324 // for example, if `minus1` is 3.14153... and `plus1` is 3.14158..., there are 5 different
325 // shortest representation from 3.14154 to 3.14158 but we only have the greatest one.
326 // we have to successively decrease the last digit and check if this is the optimal repr.
327 // there are at most 9 candidates (..1 to ..9), so this is fairly quick. ("rounding" phase)
329 // the function checks if this "optimal" repr is actually within the ulp ranges,
330 // and also, it is possible that the "second-to-optimal" repr can actually be optimal
331 // due to the rounding error. in either cases this returns `None`. ("weeding" phase)
333 // all arguments here are scaled by the common (but implicit) value `k`, so that:
334 // - `remainder = (plus1 % 10^kappa) * k`
335 // - `threshold = (plus1 - minus1) * k` (and also, `remainder < threshold`)
336 // - `plus1v = (plus1 - v) * k` (and also, `threshold > plus1v` from prior invariants)
337 // - `ten_kappa = 10^kappa * k`
338 // - `ulp = 2^-e * k`
339 fn round_and_weed(buf
: &mut [u8], exp
: i16, remainder
: u64, threshold
: u64, plus1v
: u64,
340 ten_kappa
: u64, ulp
: u64) -> Option
<(usize, i16)> {
341 assert
!(!buf
.is_empty());
343 // produce two approximations to `v` (actually `plus1 - v`) within 1.5 ulps.
344 // the resulting representation should be the closest representation to both.
346 // here `plus1 - v` is used since calculations are done with respect to `plus1`
347 // in order to avoid overflow/underflow (hence the seemingly swapped names).
348 let plus1v_down
= plus1v
+ ulp
; // plus1 - (v - 1 ulp)
349 let plus1v_up
= plus1v
- ulp
; // plus1 - (v + 1 ulp)
351 // decrease the last digit and stop at the closest representation to `v + 1 ulp`.
352 let mut plus1w
= remainder
; // plus1w(n) = plus1 - w(n)
354 let last
= buf
.last_mut().unwrap();
356 // we work with the approximated digits `w(n)`, which is initially equal to `plus1 -
357 // plus1 % 10^kappa`. after running the loop body `n` times, `w(n) = plus1 -
358 // plus1 % 10^kappa - n * 10^kappa`. we set `plus1w(n) = plus1 - w(n) =
359 // plus1 % 10^kappa + n * 10^kappa` (thus `remainder = plus1w(0)`) to simplify checks.
360 // note that `plus1w(n)` is always increasing.
362 // we have three conditions to terminate. any of them will make the loop unable to
363 // proceed, but we then have at least one valid representation known to be closest to
364 // `v + 1 ulp` anyway. we will denote them as TC1 through TC3 for brevity.
366 // TC1: `w(n) <= v + 1 ulp`, i.e. this is the last repr that can be the closest one.
367 // this is equivalent to `plus1 - w(n) = plus1w(n) >= plus1 - (v + 1 ulp) = plus1v_up`.
368 // combined with TC2 (which checks if `w(n+1)` is valid), this prevents the possible
369 // overflow on the calculation of `plus1w(n)`.
371 // TC2: `w(n+1) < minus1`, i.e. the next repr definitely does not round to `v`.
372 // this is equivalent to `plus1 - w(n) + 10^kappa = plus1w(n) + 10^kappa >
373 // plus1 - minus1 = threshold`. the left hand side can overflow, but we know
374 // `threshold > plus1v`, so if TC1 is false, `threshold - plus1w(n) >
375 // threshold - (plus1v - 1 ulp) > 1 ulp` and we can safely test if
376 // `threshold - plus1w(n) < 10^kappa` instead.
378 // TC3: `abs(w(n) - (v + 1 ulp)) <= abs(w(n+1) - (v + 1 ulp))`, i.e. the next repr is
379 // no closer to `v + 1 ulp` than the current repr. given `z(n) = plus1v_up - plus1w(n)`,
380 // this becomes `abs(z(n)) <= abs(z(n+1))`. again assuming that TC1 is false, we have
381 // `z(n) > 0`. we have two cases to consider:
383 // - when `z(n+1) >= 0`: TC3 becomes `z(n) <= z(n+1)`. as `plus1w(n)` is increasing,
384 // `z(n)` should be decreasing and this is clearly false.
385 // - when `z(n+1) < 0`:
386 // - TC3a: the precondition is `plus1v_up < plus1w(n) + 10^kappa`. assuming TC2 is
387 // false, `threshold >= plus1w(n) + 10^kappa` so it cannot overflow.
388 // - TC3b: TC3 becomes `z(n) <= -z(n+1)`, i.e. `plus1v_up - plus1w(n) >=
389 // plus1w(n+1) - plus1v_up = plus1w(n) + 10^kappa - plus1v_up`. the negated TC1
390 // gives `plus1v_up > plus1w(n)`, so it cannot overflow or underflow when
391 // combined with TC3a.
393 // consequently, we should stop when `TC1 || TC2 || (TC3a && TC3b)`. the following is
394 // equal to its inverse, `!TC1 && !TC2 && (!TC3a || !TC3b)`.
395 while plus1w
< plus1v_up
&&
396 threshold
- plus1w
>= ten_kappa
&&
397 (plus1w
+ ten_kappa
< plus1v_up
||
398 plus1v_up
- plus1w
>= plus1w
+ ten_kappa
- plus1v_up
) {
400 debug_assert
!(*last
> b'
0'
); // the shortest repr cannot end with `0`
405 // check if this representation is also the closest representation to `v - 1 ulp`.
407 // this is simply same to the terminating conditions for `v + 1 ulp`, with all `plus1v_up`
408 // replaced by `plus1v_down` instead. overflow analysis equally holds.
409 if plus1w
< plus1v_down
&&
410 threshold
- plus1w
>= ten_kappa
&&
411 (plus1w
+ ten_kappa
< plus1v_down
||
412 plus1v_down
- plus1w
>= plus1w
+ ten_kappa
- plus1v_down
) {
416 // now we have the closest representation to `v` between `plus1` and `minus1`.
417 // this is too liberal, though, so we reject any `w(n)` not between `plus0` and `minus0`,
418 // i.e. `plus1 - plus1w(n) <= minus0` or `plus1 - plus1w(n) >= plus0`. we utilize the facts
419 // that `threshold = plus1 - minus1` and `plus1 - plus0 = minus0 - minus1 = 2 ulp`.
420 if 2 * ulp
<= plus1w
&& plus1w
<= threshold
- 4 * ulp
{
421 Some((buf
.len(), exp
))
428 /// The shortest mode implementation for Grisu with Dragon fallback.
430 /// This should be used for most cases.
431 pub fn format_shortest(d
: &Decoded
, buf
: &mut [u8]) -> (/*#digits*/ usize, /*exp*/ i16) {
432 use num
::flt2dec
::strategy
::dragon
::format_shortest
as fallback
;
433 match format_shortest_opt(d
, buf
) {
435 None
=> fallback(d
, buf
),
439 /// The exact and fixed mode implementation for Grisu.
441 /// It returns `None` when it would return an inexact representation otherwise.
442 pub fn format_exact_opt(d
: &Decoded
, buf
: &mut [u8], limit
: i16)
443 -> Option
<(/*#digits*/ usize, /*exp*/ i16)> {
445 assert
!(d
.mant
< (1 << 61)); // we need at least three bits of additional precision
446 assert
!(!buf
.is_empty());
448 // normalize and scale `v`.
449 let v
= Fp { f: d.mant, e: d.exp }
.normalize();
450 let (minusk
, cached
) = cached_power(ALPHA
- v
.e
- 64, GAMMA
- v
.e
- 64);
451 let v
= v
.mul(&cached
);
453 // divide `v` into integral and fractional parts.
454 let e
= -v
.e
as usize;
455 let vint
= (v
.f
>> e
) as u32;
456 let vfrac
= v
.f
& ((1 << e
) - 1);
458 // both old `v` and new `v` (scaled by `10^-k`) has an error of < 1 ulp (Theorem 5.1).
459 // as we don't know the error is positive or negative, we use two approximations
460 // spaced equally and have the maximal error of 2 ulps (same to the shortest case).
462 // the goal is to find the exactly rounded series of digits that are common to
463 // both `v - 1 ulp` and `v + 1 ulp`, so that we are maximally confident.
464 // if this is not possible, we don't know which one is the correct output for `v`,
465 // so we give up and fall back.
467 // `err` is defined as `1 ulp * 2^e` here (same to the ulp in `vfrac`),
468 // and we will scale it whenever `v` gets scaled.
471 // calculate the largest `10^max_kappa` no more than `v` (thus `v < 10^(max_kappa+1)`).
472 // this is an upper bound of `kappa` below.
473 let (max_kappa
, max_ten_kappa
) = max_pow10_no_more_than(vint
);
476 let exp
= max_kappa
as i16 - minusk
+ 1;
478 // if we are working with the last-digit limitation, we need to shorten the buffer
479 // before the actual rendering in order to avoid double rounding.
480 // note that we have to enlarge the buffer again when rounding up happens!
481 let len
= if exp
<= limit
{
482 // oops, we cannot even produce *one* digit.
483 // this is possible when, say, we've got something like 9.5 and it's being rounded to 10.
485 // in principle we can immediately call `possibly_round` with an empty buffer,
486 // but scaling `max_ten_kappa << e` by 10 can result in overflow.
487 // thus we are being sloppy here and widen the error range by a factor of 10.
488 // this will increase the false negative rate, but only very, *very* slightly;
489 // it can only matter noticeably when the mantissa is bigger than 60 bits.
490 return possibly_round(buf
, 0, exp
, limit
, v
.f
/ 10, (max_ten_kappa
as u64) << e
, err
<< e
);
491 } else if ((exp
as i32 - limit
as i32) as usize) < buf
.len() {
492 (exp
- limit
) as usize
496 debug_assert
!(len
> 0);
498 // render integral parts.
499 // the error is entirely fractional, so we don't need to check it in this part.
500 let mut kappa
= max_kappa
as i16;
501 let mut ten_kappa
= max_ten_kappa
; // 10^kappa
502 let mut remainder
= vint
; // digits yet to be rendered
503 loop { // we always have at least one digit to render
505 // - `remainder < 10^(kappa+1)`
506 // - `vint = d[0..n-1] * 10^(kappa+1) + remainder`
507 // (it follows that `remainder = vint % 10^(kappa+1)`)
509 // divide `remainder` by `10^kappa`. both are scaled by `2^-e`.
510 let q
= remainder
/ ten_kappa
;
511 let r
= remainder
% ten_kappa
;
512 debug_assert
!(q
< 10);
513 buf
[i
] = b'
0'
+ q
as u8;
516 // is the buffer full? run the rounding pass with the remainder.
518 let vrem
= ((r
as u64) << e
) + vfrac
; // == (v % 10^kappa) * 2^e
519 return possibly_round(buf
, len
, exp
, limit
, vrem
, (ten_kappa
as u64) << e
, err
<< e
);
522 // break the loop when we have rendered all integral digits.
523 // the exact number of digits is `max_kappa + 1` as `plus1 < 10^(max_kappa+1)`.
524 if i
> max_kappa
as usize {
525 debug_assert_eq
!(ten_kappa
, 1);
526 debug_assert_eq
!(kappa
, 0);
530 // restore invariants
536 // render fractional parts.
538 // in principle we can continue to the last available digit and check for the accuracy.
539 // unfortunately we are working with the finite-sized integers, so we need some criterion
540 // to detect the overflow. V8 uses `remainder > err`, which becomes false when
541 // the first `i` significant digits of `v - 1 ulp` and `v` differ. however this rejects
542 // too many otherwise valid input.
544 // since the later phase has a correct overflow detection, we instead use tighter criterion:
545 // we continue til `err` exceeds `10^kappa / 2`, so that the range between `v - 1 ulp` and
546 // `v + 1 ulp` definitely contains two or more rounded representations. this is same to
547 // the first two comparisons from `possibly_round`, for the reference.
548 let mut remainder
= vfrac
;
549 let maxerr
= 1 << (e
- 1);
551 // invariants, where `m = max_kappa + 1` (# of digits in the integral part):
552 // - `remainder < 2^e`
553 // - `vfrac * 10^(n-m) = d[m..n-1] * 2^e + remainder`
554 // - `err = 10^(n-m)`
556 remainder
*= 10; // won't overflow, `2^e * 10 < 2^64`
557 err
*= 10; // won't overflow, `err * 10 < 2^e * 5 < 2^64`
559 // divide `remainder` by `10^kappa`.
560 // both are scaled by `2^e / 10^kappa`, so the latter is implicit here.
561 let q
= remainder
>> e
;
562 let r
= remainder
& ((1 << e
) - 1);
563 debug_assert
!(q
< 10);
564 buf
[i
] = b'
0'
+ q
as u8;
567 // is the buffer full? run the rounding pass with the remainder.
569 return possibly_round(buf
, len
, exp
, limit
, r
, 1 << e
, err
);
572 // restore invariants
576 // further calculation is useless (`possibly_round` definitely fails), so we give up.
579 // we've generated all requested digits of `v`, which should be also same to corresponding
580 // digits of `v - 1 ulp`. now we check if there is a unique representation shared by
581 // both `v - 1 ulp` and `v + 1 ulp`; this can be either same to generated digits, or
582 // to the rounded-up version of those digits. if the range contains multiple representations
583 // of the same length, we cannot be sure and should return `None` instead.
585 // all arguments here are scaled by the common (but implicit) value `k`, so that:
586 // - `remainder = (v % 10^kappa) * k`
587 // - `ten_kappa = 10^kappa * k`
588 // - `ulp = 2^-e * k`
589 fn possibly_round(buf
: &mut [u8], mut len
: usize, mut exp
: i16, limit
: i16,
590 remainder
: u64, ten_kappa
: u64, ulp
: u64) -> Option
<(usize, i16)> {
591 debug_assert
!(remainder
< ten_kappa
);
598 // ----|-----|-----|----
600 // v - 1 ulp v + 1 ulp
602 // (for the reference, the dotted line indicates the exact value for
603 // possible representations in given number of digits.)
605 // error is too large that there are at least three possible representations
606 // between `v - 1 ulp` and `v + 1 ulp`. we cannot determine which one is correct.
607 if ulp
>= ten_kappa { return None; }
614 // ----|-----|-----|----
616 // v - 1 ulp v + 1 ulp
618 // in fact, 1/2 ulp is enough to introduce two possible representations.
619 // (remember that we need a unique representation for both `v - 1 ulp` and `v + 1 ulp`.)
620 // this won't overflow, as `ulp < ten_kappa` from the first check.
621 if ten_kappa
- ulp
<= ulp { return None; }
626 // :<--------- 10^kappa ---------->:
630 // ----|-----|-----|------------------------
632 // v - 1 ulp v + 1 ulp
634 // if `v + 1 ulp` is closer to the rounded-down representation (which is already in `buf`),
635 // then we can safely return. note that `v - 1 ulp` *can* be less than the current
636 // representation, but as `1 ulp < 10^kappa / 2`, this condition is enough:
637 // the distance between `v - 1 ulp` and the current representation
638 // cannot exceed `10^kappa / 2`.
640 // the condition equals to `remainder + ulp < 10^kappa / 2`.
641 // since this can easily overflow, first check if `remainder < 10^kappa / 2`.
642 // we've already verified that `ulp < 10^kappa / 2`, so as long as
643 // `10^kappa` did not overflow after all, the second check is fine.
644 if ten_kappa
- remainder
> remainder
&& ten_kappa
- 2 * remainder
>= 2 * ulp
{
645 return Some((len
, exp
));
648 // :<------- remainder ------>| :
650 // :<--------- 10^kappa --------->:
654 // -----------------------|-----|-----|-----
656 // v - 1 ulp v + 1 ulp
658 // on the other hands, if `v - 1 ulp` is closer to the rounded-up representation,
659 // we should round up and return. for the same reason we don't need to check `v + 1 ulp`.
661 // the condition equals to `remainder - ulp >= 10^kappa / 2`.
662 // again we first check if `remainder > ulp` (note that this is not `remainder >= ulp`,
663 // as `10^kappa` is never zero). also note that `remainder - ulp <= 10^kappa`,
664 // so the second check does not overflow.
665 if remainder
> ulp
&& ten_kappa
- (remainder
- ulp
) <= remainder
- ulp
{
666 if let Some(c
) = round_up(buf
, len
) {
667 // only add an additional digit when we've been requested the fixed precision.
668 // we also need to check that, if the original buffer was empty,
669 // the additional digit can only be added when `exp == limit` (edge case).
671 if exp
> limit
&& len
< buf
.len() {
676 return Some((len
, exp
));
679 // otherwise we are doomed (i.e. some values between `v - 1 ulp` and `v + 1 ulp` are
680 // rounding down and others are rounding up) and give up.
685 /// The exact and fixed mode implementation for Grisu with Dragon fallback.
687 /// This should be used for most cases.
688 pub fn format_exact(d
: &Decoded
, buf
: &mut [u8], limit
: i16) -> (/*#digits*/ usize, /*exp*/ i16) {
689 use num
::flt2dec
::strategy
::dragon
::format_exact
as fallback
;
690 match format_exact_opt(d
, buf
, limit
) {
692 None
=> fallback(d
, buf
, limit
),