1//! Almost direct (but slightly optimized) Rust translation of Figure 3 of "Printing
2//! Floating-Point Numbers Quickly and Accurately"[^1].
3//!
4//! [^1]: Burger, R. G. and Dybvig, R. K. 1996. Printing floating-point numbers
5//! quickly and accurately. SIGPLAN Not. 31, 5 (May. 1996), 108-116.
6
7use crate::cmp::Ordering;
8use crate::mem::MaybeUninit;
9use crate::num::bignum::{Big32x40 as Big, Digit32 as Digit};
10use crate::num::flt2dec::estimator::estimate_scaling_factor;
11use crate::num::flt2dec::{Decoded, MAX_SIG_DIGITS, round_up};
12
13static POW10: [Digit; 10] =
14 [1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000];
15// precalculated arrays of `Digit`s for 5^(2^n).
16static POW5TO16: [Digit; 2] = [0x86f26fc1, 0x23];
17static POW5TO32: [Digit; 3] = [0x85acef81, 0x2d6d415b, 0x4ee];
18static POW5TO64: [Digit; 5] = [0xbf6a1f01, 0x6e38ed64, 0xdaa797ed, 0xe93ff9f4, 0x184f03];
19static POW5TO128: [Digit; 10] = [
20 0x2e953e01, 0x3df9909, 0xf1538fd, 0x2374e42f, 0xd3cff5ec, 0xc404dc08, 0xbccdb0da, 0xa6337f19,
21 0xe91f2603, 0x24e,
22];
23static POW5TO256: [Digit; 19] = [
24 0x982e7c01, 0xbed3875b, 0xd8d99f72, 0x12152f87, 0x6bde50c6, 0xcf4a6e70, 0xd595d80f, 0x26b2716e,
25 0xadc666b0, 0x1d153624, 0x3c42d35a, 0x63ff540e, 0xcc5573c0, 0x65f9ef17, 0x55bc28f2, 0x80dcc7f7,
26 0xf46eeddc, 0x5fdcefce, 0x553f7,
27];
28
29#[doc(hidden)]
30pub fn mul_pow10(x: &mut Big, n: usize) -> &mut Big {
31 debug_assert!(n < 512);
32 // Save ourself the left shift for the smallest cases.
33 if n < 8 {
34 return x.mul_small(POW10[n & 7]);
35 }
36 // Multiply by the powers of 5 and shift the 2s in at the end.
37 // This keeps the intermediate products smaller and faster.
38 if n & 7 != 0 {
39 x.mul_small(POW10[n & 7] >> (n & 7));
40 }
41 if n & 8 != 0 {
42 x.mul_small(POW10[8] >> 8);
43 }
44 if n & 16 != 0 {
45 x.mul_digits(&POW5TO16);
46 }
47 if n & 32 != 0 {
48 x.mul_digits(&POW5TO32);
49 }
50 if n & 64 != 0 {
51 x.mul_digits(&POW5TO64);
52 }
53 if n & 128 != 0 {
54 x.mul_digits(&POW5TO128);
55 }
56 if n & 256 != 0 {
57 x.mul_digits(&POW5TO256);
58 }
59 x.mul_pow2(n)
60}
61
62fn div_2pow10(x: &mut Big, mut n: usize) -> &mut Big {
63 let largest: usize = POW10.len() - 1;
64 while n > largest {
65 x.div_rem_small(POW10[largest]);
66 n -= largest;
67 }
68 x.div_rem_small(POW10[n] << 1);
69 x
70}
71
72// only usable when `x < 16 * scale`; `scaleN` should be `scale.mul_small(N)`
73fn div_rem_upto_16<'a>(
74 x: &'a mut Big,
75 scale: &Big,
76 scale2: &Big,
77 scale4: &Big,
78 scale8: &Big,
79) -> (u8, &'a mut Big) {
80 let mut d: u8 = 0;
81 if *x >= *scale8 {
82 x.sub(scale8);
83 d += 8;
84 }
85 if *x >= *scale4 {
86 x.sub(scale4);
87 d += 4;
88 }
89 if *x >= *scale2 {
90 x.sub(scale2);
91 d += 2;
92 }
93 if *x >= *scale {
94 x.sub(scale);
95 d += 1;
96 }
97 debug_assert!(*x < *scale);
98 (d, x)
99}
100
101/// The shortest mode implementation for Dragon.
102pub fn format_shortest<'a>(
103 d: &Decoded,
104 buf: &'a mut [MaybeUninit<u8>],
105) -> (/*digits*/ &'a [u8], /*exp*/ i16) {
106 // the number `v` to format is known to be:
107 // - equal to `mant * 2^exp`;
108 // - preceded by `(mant - 2 * minus) * 2^exp` in the original type; and
109 // - followed by `(mant + 2 * plus) * 2^exp` in the original type.
110 //
111 // obviously, `minus` and `plus` cannot be zero. (for infinities, we use out-of-range values.)
112 // also we assume that at least one digit is generated, i.e., `mant` cannot be zero too.
113 //
114 // this also means that any number between `low = (mant - minus) * 2^exp` and
115 // `high = (mant + plus) * 2^exp` will map to this exact floating point number,
116 // with bounds included when the original mantissa was even (i.e., `!mant_was_odd`).
117
118 assert!(d.mant > 0);
119 assert!(d.minus > 0);
120 assert!(d.plus > 0);
121 assert!(d.mant.checked_add(d.plus).is_some());
122 assert!(d.mant.checked_sub(d.minus).is_some());
123 assert!(buf.len() >= MAX_SIG_DIGITS);
124
125 // `a.cmp(&b) < rounding` is `if d.inclusive {a <= b} else {a < b}`
126 let rounding = if d.inclusive { Ordering::Greater } else { Ordering::Equal };
127
128 // estimate `k_0` from original inputs satisfying `10^(k_0-1) < high <= 10^(k_0+1)`.
129 // the tight bound `k` satisfying `10^(k-1) < high <= 10^k` is calculated later.
130 let mut k = estimate_scaling_factor(d.mant + d.plus, d.exp);
131
132 // convert `{mant, plus, minus} * 2^exp` into the fractional form so that:
133 // - `v = mant / scale`
134 // - `low = (mant - minus) / scale`
135 // - `high = (mant + plus) / scale`
136 let mut mant = Big::from_u64(d.mant);
137 let mut minus = Big::from_u64(d.minus);
138 let mut plus = Big::from_u64(d.plus);
139 let mut scale = Big::from_small(1);
140 if d.exp < 0 {
141 scale.mul_pow2(-d.exp as usize);
142 } else {
143 mant.mul_pow2(d.exp as usize);
144 minus.mul_pow2(d.exp as usize);
145 plus.mul_pow2(d.exp as usize);
146 }
147
148 // divide `mant` by `10^k`. now `scale / 10 < mant + plus <= scale * 10`.
149 if k >= 0 {
150 mul_pow10(&mut scale, k as usize);
151 } else {
152 mul_pow10(&mut mant, -k as usize);
153 mul_pow10(&mut minus, -k as usize);
154 mul_pow10(&mut plus, -k as usize);
155 }
156
157 // fixup when `mant + plus > scale` (or `>=`).
158 // we are not actually modifying `scale`, since we can skip the initial multiplication instead.
159 // now `scale < mant + plus <= scale * 10` and we are ready to generate digits.
160 //
161 // note that `d[0]` *can* be zero, when `scale - plus < mant < scale`.
162 // in this case rounding-up condition (`up` below) will be triggered immediately.
163 if scale.cmp(mant.clone().add(&plus)) < rounding {
164 // equivalent to scaling `scale` by 10
165 k += 1;
166 } else {
167 mant.mul_small(10);
168 minus.mul_small(10);
169 plus.mul_small(10);
170 }
171
172 // cache `(2, 4, 8) * scale` for digit generation.
173 let mut scale2 = scale.clone();
174 scale2.mul_pow2(1);
175 let mut scale4 = scale.clone();
176 scale4.mul_pow2(2);
177 let mut scale8 = scale.clone();
178 scale8.mul_pow2(3);
179
180 let mut down;
181 let mut up;
182 let mut i = 0;
183 loop {
184 // invariants, where `d[0..n-1]` are digits generated so far:
185 // - `v = mant / scale * 10^(k-n-1) + d[0..n-1] * 10^(k-n)`
186 // - `v - low = minus / scale * 10^(k-n-1)`
187 // - `high - v = plus / scale * 10^(k-n-1)`
188 // - `(mant + plus) / scale <= 10` (thus `mant / scale < 10`)
189 // where `d[i..j]` is a shorthand for `d[i] * 10^(j-i) + ... + d[j-1] * 10 + d[j]`.
190
191 // generate one digit: `d[n] = floor(mant / scale) < 10`.
192 let (d, _) = div_rem_upto_16(&mut mant, &scale, &scale2, &scale4, &scale8);
193 debug_assert!(d < 10);
194 buf[i] = MaybeUninit::new(b'0' + d);
195 i += 1;
196
197 // this is a simplified description of the modified Dragon algorithm.
198 // many intermediate derivations and completeness arguments are omitted for convenience.
199 //
200 // start with modified invariants, as we've updated `n`:
201 // - `v = mant / scale * 10^(k-n) + d[0..n-1] * 10^(k-n)`
202 // - `v - low = minus / scale * 10^(k-n)`
203 // - `high - v = plus / scale * 10^(k-n)`
204 //
205 // assume that `d[0..n-1]` is the shortest representation between `low` and `high`,
206 // i.e., `d[0..n-1]` satisfies both of the following but `d[0..n-2]` doesn't:
207 // - `low < d[0..n-1] * 10^(k-n) < high` (bijectivity: digits round to `v`); and
208 // - `abs(v / 10^(k-n) - d[0..n-1]) <= 1/2` (the last digit is correct).
209 //
210 // the second condition simplifies to `2 * mant <= scale`.
211 // solving invariants in terms of `mant`, `low` and `high` yields
212 // a simpler version of the first condition: `-plus < mant < minus`.
213 // since `-plus < 0 <= mant`, we have the correct shortest representation
214 // when `mant < minus` and `2 * mant <= scale`.
215 // (the former becomes `mant <= minus` when the original mantissa is even.)
216 //
217 // when the second doesn't hold (`2 * mant > scale`), we need to increase the last digit.
218 // this is enough for restoring that condition: we already know that
219 // the digit generation guarantees `0 <= v / 10^(k-n) - d[0..n-1] < 1`.
220 // in this case, the first condition becomes `-plus < mant - scale < minus`.
221 // since `mant < scale` after the generation, we have `scale < mant + plus`.
222 // (again, this becomes `scale <= mant + plus` when the original mantissa is even.)
223 //
224 // in short:
225 // - stop and round `down` (keep digits as is) when `mant < minus` (or `<=`).
226 // - stop and round `up` (increase the last digit) when `scale < mant + plus` (or `<=`).
227 // - keep generating otherwise.
228 down = mant.cmp(&minus) < rounding;
229 up = scale.cmp(mant.clone().add(&plus)) < rounding;
230 if down || up {
231 break;
232 } // we have the shortest representation, proceed to the rounding
233
234 // restore the invariants.
235 // this makes the algorithm always terminating: `minus` and `plus` always increases,
236 // but `mant` is clipped modulo `scale` and `scale` is fixed.
237 mant.mul_small(10);
238 minus.mul_small(10);
239 plus.mul_small(10);
240 }
241
242 // rounding up happens when
243 // i) only the rounding-up condition was triggered, or
244 // ii) both conditions were triggered and tie breaking prefers rounding up.
245 if up && (!down || *mant.mul_pow2(1) >= scale) {
246 // if rounding up changes the length, the exponent should also change.
247 // it seems that this condition is very hard to satisfy (possibly impossible),
248 // but we are just being safe and consistent here.
249 // SAFETY: we initialized that memory above.
250 if let Some(c) = round_up(unsafe { buf[..i].assume_init_mut() }) {
251 buf[i] = MaybeUninit::new(c);
252 i += 1;
253 k += 1;
254 }
255 }
256
257 // SAFETY: we initialized that memory above.
258 (unsafe { buf[..i].assume_init_ref() }, k)
259}
260
261/// The exact and fixed mode implementation for Dragon.
262pub fn format_exact<'a>(
263 d: &Decoded,
264 buf: &'a mut [MaybeUninit<u8>],
265 limit: i16,
266) -> (/*digits*/ &'a [u8], /*exp*/ i16) {
267 assert!(d.mant > 0);
268 assert!(d.minus > 0);
269 assert!(d.plus > 0);
270 assert!(d.mant.checked_add(d.plus).is_some());
271 assert!(d.mant.checked_sub(d.minus).is_some());
272
273 // estimate `k_0` from original inputs satisfying `10^(k_0-1) < v <= 10^(k_0+1)`.
274 let mut k = estimate_scaling_factor(d.mant, d.exp);
275
276 // `v = mant / scale`.
277 let mut mant = Big::from_u64(d.mant);
278 let mut scale = Big::from_small(1);
279 if d.exp < 0 {
280 scale.mul_pow2(-d.exp as usize);
281 } else {
282 mant.mul_pow2(d.exp as usize);
283 }
284
285 // divide `mant` by `10^k`. now `scale / 10 < mant <= scale * 10`.
286 if k >= 0 {
287 mul_pow10(&mut scale, k as usize);
288 } else {
289 mul_pow10(&mut mant, -k as usize);
290 }
291
292 // fixup when `mant + plus >= scale`, where `plus / scale = 10^-buf.len() / 2`.
293 // in order to keep the fixed-size bignum, we actually use `mant + floor(plus) >= scale`.
294 // we are not actually modifying `scale`, since we can skip the initial multiplication instead.
295 // again with the shortest algorithm, `d[0]` can be zero but will be eventually rounded up.
296 if *div_2pow10(&mut scale.clone(), buf.len()).add(&mant) >= scale {
297 // equivalent to scaling `scale` by 10
298 k += 1;
299 } else {
300 mant.mul_small(10);
301 }
302
303 // if we are working with the last-digit limitation, we need to shorten the buffer
304 // before the actual rendering in order to avoid double rounding.
305 // note that we have to enlarge the buffer again when rounding up happens!
306 let mut len = if k < limit {
307 // oops, we cannot even produce *one* digit.
308 // this is possible when, say, we've got something like 9.5 and it's being rounded to 10.
309 // we return an empty buffer, with an exception of the later rounding-up case
310 // which occurs when `k == limit` and has to produce exactly one digit.
311 0
312 } else if ((k as i32 - limit as i32) as usize) < buf.len() {
313 (k - limit) as usize
314 } else {
315 buf.len()
316 };
317
318 if len > 0 {
319 // cache `(2, 4, 8) * scale` for digit generation.
320 // (this can be expensive, so do not calculate them when the buffer is empty.)
321 let mut scale2 = scale.clone();
322 scale2.mul_pow2(1);
323 let mut scale4 = scale.clone();
324 scale4.mul_pow2(2);
325 let mut scale8 = scale.clone();
326 scale8.mul_pow2(3);
327
328 for i in 0..len {
329 if mant.is_zero() {
330 // following digits are all zeroes, we stop here
331 // do *not* try to perform rounding! rather, fill remaining digits.
332 for c in &mut buf[i..len] {
333 *c = MaybeUninit::new(b'0');
334 }
335 // SAFETY: we initialized that memory above.
336 return (unsafe { buf[..len].assume_init_ref() }, k);
337 }
338
339 let mut d = 0;
340 if mant >= scale8 {
341 mant.sub(&scale8);
342 d += 8;
343 }
344 if mant >= scale4 {
345 mant.sub(&scale4);
346 d += 4;
347 }
348 if mant >= scale2 {
349 mant.sub(&scale2);
350 d += 2;
351 }
352 if mant >= scale {
353 mant.sub(&scale);
354 d += 1;
355 }
356 debug_assert!(mant < scale);
357 debug_assert!(d < 10);
358 buf[i] = MaybeUninit::new(b'0' + d);
359 mant.mul_small(10);
360 }
361 }
362
363 // rounding up if we stop in the middle of digits
364 // if the following digits are exactly 5000..., check the prior digit and try to
365 // round to even (i.e., avoid rounding up when the prior digit is even).
366 let order = mant.cmp(scale.mul_small(5));
367 if order == Ordering::Greater
368 || (order == Ordering::Equal
369 // SAFETY: `buf[len-1]` is initialized.
370 && len > 0 && unsafe { buf[len - 1].assume_init() } & 1 == 1)
371 {
372 // if rounding up changes the length, the exponent should also change.
373 // but we've been requested a fixed number of digits, so do not alter the buffer...
374 // SAFETY: we initialized that memory above.
375 if let Some(c) = round_up(unsafe { buf[..len].assume_init_mut() }) {
376 // ...unless we've been requested the fixed precision instead.
377 // we also need to check that, if the original buffer was empty,
378 // the additional digit can only be added when `k == limit` (edge case).
379 k += 1;
380 if k > limit && len < buf.len() {
381 buf[len] = MaybeUninit::new(c);
382 len += 1;
383 }
384 }
385 }
386
387 // SAFETY: we initialized that memory above.
388 (unsafe { buf[..len].assume_init_ref() }, k)
389}
390