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