1//! Custom arbitrary-precision number (bignum) implementation.
2//!
3//! This is designed to avoid the heap allocation at expense of stack memory.
4//! The most used bignum type, `Big32x40`, is limited by 32 × 40 = 1,280 bits
5//! and will take at most 160 bytes of stack memory. This is more than enough
6//! for round-tripping all possible finite `f64` values.
7//!
8//! In principle it is possible to have multiple bignum types for different
9//! inputs, but we don't do so to avoid the code bloat. Each bignum is still
10//! tracked for the actual usages, so it normally doesn't matter.
11
12// This module is only for dec2flt and flt2dec, and only public because of coretests.
13// It is not intended to ever be stabilized.
14#![doc(hidden)]
15#![unstable(
16 feature = "core_private_bignum",
17 reason = "internal routines only exposed for testing",
18 issue = "none"
19)]
20#![macro_use]
21
22/// Arithmetic operations required by bignums.
23pub trait FullOps: Sized {
24 /// Returns `(carry', v')` such that `carry' * 2^W + v' = self * other + other2 + carry`,
25 /// where `W` is the number of bits in `Self`.
26 fn full_mul_add(self, other: Self, other2: Self, carry: Self) -> (Self /* carry */, Self);
27
28 /// Returns `(quo, rem)` such that `borrow * 2^W + self = quo * other + rem`
29 /// and `0 <= rem < other`, where `W` is the number of bits in `Self`.
30 fn full_div_rem(self, other: Self, borrow: Self)
31 -> (Self /* quotient */, Self /* remainder */);
32}
33
34macro_rules! impl_full_ops {
35 ($($ty:ty: add($addfn:path), mul/div($bigty:ident);)*) => (
36 $(
37 impl FullOps for $ty {
38 fn full_mul_add(self, other: $ty, other2: $ty, carry: $ty) -> ($ty, $ty) {
39 // This cannot overflow;
40 // the output is between `0` and `2^nbits * (2^nbits - 1)`.
41 let v = (self as $bigty) * (other as $bigty) + (other2 as $bigty) +
42 (carry as $bigty);
43 ((v >> <$ty>::BITS) as $ty, v as $ty)
44 }
45
46 fn full_div_rem(self, other: $ty, borrow: $ty) -> ($ty, $ty) {
47 debug_assert!(borrow < other);
48 // This cannot overflow; the output is between `0` and `other * (2^nbits - 1)`.
49 let lhs = ((borrow as $bigty) << <$ty>::BITS) | (self as $bigty);
50 let rhs = other as $bigty;
51 ((lhs / rhs) as $ty, (lhs % rhs) as $ty)
52 }
53 }
54 )*
55 )
56}
57
58impl_full_ops! {
59 u8: add(intrinsics::u8_add_with_overflow), mul/div(u16);
60 u16: add(intrinsics::u16_add_with_overflow), mul/div(u32);
61 u32: add(intrinsics::u32_add_with_overflow), mul/div(u64);
62 // See RFC #521 for enabling this.
63 // u64: add(intrinsics::u64_add_with_overflow), mul/div(u128);
64}
65
66/// Table of powers of 5 representable in digits. Specifically, the largest {u8, u16, u32} value
67/// that's a power of five, plus the corresponding exponent. Used in `mul_pow5`.
68const SMALL_POW5: [(u64, usize); 3] = [(125, 3), (15625, 6), (1_220_703_125, 13)];
69
70macro_rules! define_bignum {
71 ($name:ident: type=$ty:ty, n=$n:expr) => {
72 /// Stack-allocated arbitrary-precision (up to certain limit) integer.
73 ///
74 /// This is backed by a fixed-size array of given type ("digit").
75 /// While the array is not very large (normally some hundred bytes),
76 /// copying it recklessly may result in the performance hit.
77 /// Thus this is intentionally not `Copy`.
78 ///
79 /// All operations available to bignums panic in the case of overflows.
80 /// The caller is responsible to use large enough bignum types.
81 pub struct $name {
82 /// One plus the offset to the maximum "digit" in use.
83 /// This does not decrease, so be aware of the computation order.
84 /// `base[size..]` should be zero.
85 size: usize,
86 /// Digits. `[a, b, c, ...]` represents `a + b*2^W + c*2^(2W) + ...`
87 /// where `W` is the number of bits in the digit type.
88 base: [$ty; $n],
89 }
90
91 impl $name {
92 /// Makes a bignum from one digit.
93 pub fn from_small(v: $ty) -> $name {
94 let mut base = [0; $n];
95 base[0] = v;
96 $name { size: 1, base }
97 }
98
99 /// Makes a bignum from `u64` value.
100 pub fn from_u64(mut v: u64) -> $name {
101 let mut base = [0; $n];
102 let mut sz = 0;
103 while v > 0 {
104 base[sz] = v as $ty;
105 v >>= <$ty>::BITS;
106 sz += 1;
107 }
108 $name { size: sz, base }
109 }
110
111 /// Returns the internal digits as a slice `[a, b, c, ...]` such that the numeric
112 /// value is `a + b * 2^W + c * 2^(2W) + ...` where `W` is the number of bits in
113 /// the digit type.
114 pub fn digits(&self) -> &[$ty] {
115 &self.base[..self.size]
116 }
117
118 /// Returns the `i`-th bit where bit 0 is the least significant one.
119 /// In other words, the bit with weight `2^i`.
120 pub fn get_bit(&self, i: usize) -> u8 {
121 let digitbits = <$ty>::BITS as usize;
122 let d = i / digitbits;
123 let b = i % digitbits;
124 ((self.base[d] >> b) & 1) as u8
125 }
126
127 /// Returns `true` if the bignum is zero.
128 pub fn is_zero(&self) -> bool {
129 self.digits().iter().all(|&v| v == 0)
130 }
131
132 /// Returns the number of bits necessary to represent this value. Note that zero
133 /// is considered to need 0 bits.
134 pub fn bit_length(&self) -> usize {
135 let digitbits = <$ty>::BITS as usize;
136 let digits = self.digits();
137 // Find the most significant non-zero digit.
138 let msd = digits.iter().rposition(|&x| x != 0);
139 match msd {
140 Some(msd) => msd * digitbits + digits[msd].ilog2() as usize + 1,
141 // There are no non-zero digits, i.e., the number is zero.
142 _ => 0,
143 }
144 }
145
146 /// Adds `other` to itself and returns its own mutable reference.
147 pub fn add<'a>(&'a mut self, other: &$name) -> &'a mut $name {
148 use crate::{cmp, iter};
149
150 let mut sz = cmp::max(self.size, other.size);
151 let mut carry = false;
152 for (a, b) in iter::zip(&mut self.base[..sz], &other.base[..sz]) {
153 let (v, c) = (*a).carrying_add(*b, carry);
154 *a = v;
155 carry = c;
156 }
157 if carry {
158 self.base[sz] = 1;
159 sz += 1;
160 }
161 self.size = sz;
162 self
163 }
164
165 pub fn add_small(&mut self, other: $ty) -> &mut $name {
166 let (v, mut carry) = self.base[0].carrying_add(other, false);
167 self.base[0] = v;
168 let mut i = 1;
169 while carry {
170 let (v, c) = self.base[i].carrying_add(0, carry);
171 self.base[i] = v;
172 carry = c;
173 i += 1;
174 }
175 if i > self.size {
176 self.size = i;
177 }
178 self
179 }
180
181 /// Subtracts `other` from itself and returns its own mutable reference.
182 pub fn sub<'a>(&'a mut self, other: &$name) -> &'a mut $name {
183 use crate::{cmp, iter};
184
185 let sz = cmp::max(self.size, other.size);
186 let mut noborrow = true;
187 for (a, b) in iter::zip(&mut self.base[..sz], &other.base[..sz]) {
188 let (v, c) = (*a).carrying_add(!*b, noborrow);
189 *a = v;
190 noborrow = c;
191 }
192 assert!(noborrow);
193 self.size = sz;
194 self
195 }
196
197 /// Multiplies itself by a digit-sized `other` and returns its own
198 /// mutable reference.
199 pub fn mul_small(&mut self, other: $ty) -> &mut $name {
200 let mut sz = self.size;
201 let mut carry = 0;
202 for a in &mut self.base[..sz] {
203 let (v, c) = (*a).carrying_mul(other, carry);
204 *a = v;
205 carry = c;
206 }
207 if carry > 0 {
208 self.base[sz] = carry;
209 sz += 1;
210 }
211 self.size = sz;
212 self
213 }
214
215 /// Multiplies itself by `2^bits` and returns its own mutable reference.
216 pub fn mul_pow2(&mut self, bits: usize) -> &mut $name {
217 let digitbits = <$ty>::BITS as usize;
218 let digits = bits / digitbits;
219 let bits = bits % digitbits;
220
221 assert!(digits < $n);
222 debug_assert!(self.base[$n - digits..].iter().all(|&v| v == 0));
223 debug_assert!(bits == 0 || (self.base[$n - digits - 1] >> (digitbits - bits)) == 0);
224
225 // shift by `digits * digitbits` bits
226 for i in (0..self.size).rev() {
227 self.base[i + digits] = self.base[i];
228 }
229 for i in 0..digits {
230 self.base[i] = 0;
231 }
232
233 // shift by `bits` bits
234 let mut sz = self.size + digits;
235 if bits > 0 {
236 let last = sz;
237 let overflow = self.base[last - 1] >> (digitbits - bits);
238 if overflow > 0 {
239 self.base[last] = overflow;
240 sz += 1;
241 }
242 for i in (digits + 1..last).rev() {
243 self.base[i] =
244 (self.base[i] << bits) | (self.base[i - 1] >> (digitbits - bits));
245 }
246 self.base[digits] <<= bits;
247 // self.base[..digits] is zero, no need to shift
248 }
249
250 self.size = sz;
251 self
252 }
253
254 /// Multiplies itself by `5^e` and returns its own mutable reference.
255 pub fn mul_pow5(&mut self, mut e: usize) -> &mut $name {
256 use crate::num::bignum::SMALL_POW5;
257
258 // There are exactly n trailing zeros on 2^n, and the only relevant digit sizes
259 // are consecutive powers of two, so this is well suited index for the table.
260 let table_index = size_of::<$ty>().trailing_zeros() as usize;
261 let (small_power, small_e) = SMALL_POW5[table_index];
262 let small_power = small_power as $ty;
263
264 // Multiply with the largest single-digit power as long as possible ...
265 while e >= small_e {
266 self.mul_small(small_power);
267 e -= small_e;
268 }
269
270 // ... then finish off the remainder.
271 let mut rest_power = 1;
272 for _ in 0..e {
273 rest_power *= 5;
274 }
275 self.mul_small(rest_power);
276
277 self
278 }
279
280 /// Multiplies itself by a number described by `other[0] + other[1] * 2^W +
281 /// other[2] * 2^(2W) + ...` (where `W` is the number of bits in the digit type)
282 /// and returns its own mutable reference.
283 pub fn mul_digits<'a>(&'a mut self, other: &[$ty]) -> &'a mut $name {
284 // the internal routine. works best when aa.len() <= bb.len().
285 fn mul_inner(ret: &mut [$ty; $n], aa: &[$ty], bb: &[$ty]) -> usize {
286 use crate::num::bignum::FullOps;
287
288 let mut retsz = 0;
289 for (i, &a) in aa.iter().enumerate() {
290 if a == 0 {
291 continue;
292 }
293 let mut sz = bb.len();
294 let mut carry = 0;
295 for (j, &b) in bb.iter().enumerate() {
296 let (c, v) = a.full_mul_add(b, ret[i + j], carry);
297 ret[i + j] = v;
298 carry = c;
299 }
300 if carry > 0 {
301 ret[i + sz] = carry;
302 sz += 1;
303 }
304 if retsz < i + sz {
305 retsz = i + sz;
306 }
307 }
308 retsz
309 }
310
311 let mut ret = [0; $n];
312 let retsz = if self.size < other.len() {
313 mul_inner(&mut ret, &self.digits(), other)
314 } else {
315 mul_inner(&mut ret, other, &self.digits())
316 };
317 self.base = ret;
318 self.size = retsz;
319 self
320 }
321
322 /// Divides itself by a digit-sized `other` and returns its own
323 /// mutable reference *and* the remainder.
324 pub fn div_rem_small(&mut self, other: $ty) -> (&mut $name, $ty) {
325 use crate::num::bignum::FullOps;
326
327 assert!(other > 0);
328
329 let sz = self.size;
330 let mut borrow = 0;
331 for a in self.base[..sz].iter_mut().rev() {
332 let (q, r) = (*a).full_div_rem(other, borrow);
333 *a = q;
334 borrow = r;
335 }
336 (self, borrow)
337 }
338
339 /// Divide self by another bignum, overwriting `q` with the quotient and `r` with the
340 /// remainder.
341 pub fn div_rem(&self, d: &$name, q: &mut $name, r: &mut $name) {
342 // Stupid slow base-2 long division taken from
343 // https://en.wikipedia.org/wiki/Division_algorithm
344 // FIXME use a greater base ($ty) for the long division.
345 assert!(!d.is_zero());
346 let digitbits = <$ty>::BITS as usize;
347 for digit in &mut q.base[..] {
348 *digit = 0;
349 }
350 for digit in &mut r.base[..] {
351 *digit = 0;
352 }
353 r.size = d.size;
354 q.size = 1;
355 let mut q_is_zero = true;
356 let end = self.bit_length();
357 for i in (0..end).rev() {
358 r.mul_pow2(1);
359 r.base[0] |= self.get_bit(i) as $ty;
360 if &*r >= d {
361 r.sub(d);
362 // Set bit `i` of q to 1.
363 let digit_idx = i / digitbits;
364 let bit_idx = i % digitbits;
365 if q_is_zero {
366 q.size = digit_idx + 1;
367 q_is_zero = false;
368 }
369 q.base[digit_idx] |= 1 << bit_idx;
370 }
371 }
372 debug_assert!(q.base[q.size..].iter().all(|&d| d == 0));
373 debug_assert!(r.base[r.size..].iter().all(|&d| d == 0));
374 }
375 }
376
377 impl crate::cmp::PartialEq for $name {
378 fn eq(&self, other: &$name) -> bool {
379 self.base[..] == other.base[..]
380 }
381 }
382
383 impl crate::cmp::Eq for $name {}
384
385 impl crate::cmp::PartialOrd for $name {
386 fn partial_cmp(&self, other: &$name) -> crate::option::Option<crate::cmp::Ordering> {
387 crate::option::Option::Some(self.cmp(other))
388 }
389 }
390
391 impl crate::cmp::Ord for $name {
392 fn cmp(&self, other: &$name) -> crate::cmp::Ordering {
393 use crate::cmp::max;
394 let sz = max(self.size, other.size);
395 let lhs = self.base[..sz].iter().cloned().rev();
396 let rhs = other.base[..sz].iter().cloned().rev();
397 lhs.cmp(rhs)
398 }
399 }
400
401 impl crate::clone::Clone for $name {
402 fn clone(&self) -> Self {
403 Self { size: self.size, base: self.base }
404 }
405 }
406
407 impl crate::clone::UseCloned for $name {}
408
409 impl crate::fmt::Debug for $name {
410 fn fmt(&self, f: &mut crate::fmt::Formatter<'_>) -> crate::fmt::Result {
411 let sz = if self.size < 1 { 1 } else { self.size };
412 let digitlen = <$ty>::BITS as usize / 4;
413
414 write!(f, "{:#x}", self.base[sz - 1])?;
415 for &v in self.base[..sz - 1].iter().rev() {
416 write!(f, "_{:01$x}", v, digitlen)?;
417 }
418 crate::result::Result::Ok(())
419 }
420 }
421 };
422}
423
424/// The digit type for `Big32x40`.
425pub type Digit32 = u32;
426
427define_bignum!(Big32x40: type=Digit32, n=40);
428
429// this one is used for testing only.
430#[doc(hidden)]
431pub mod tests {
432 define_bignum!(Big8x3: type=u8, n=3);
433}
434