1 //! Custom arbitrary-precision number (bignum) implementation.
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.
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.
12 // This module is only for dec2flt and flt2dec, and only public because of coretests.
13 // It is not intended to ever be stabilized.
16 feature = "core_private_bignum",
17 reason = "internal routines only exposed for testing",
22 /// Arithmetic operations required by bignums.
23 pub 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);
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 */);
34 macro_rules! impl_full_ops {
35 ($($ty:ty: add($addfn:path), mul/div($bigty:ident);)*) => (
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 $bigty) +
43 ((v >> <$ty>::BITS) as $ty, v as $ty)
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 $ty)
59 u8: add(intrinsics::u8_add_with_overflow), mul/div(u16);
60 u16: add(intrinsics::u16_add_with_overflow), mul/div(u32);
61 u32: add(intrinsics::u32_add_with_overflow), mul/div(u64);
62 // See RFC #521 for enabling this.
63 // u64: add(intrinsics::u64_add_with_overflow), mul/div(u128);
66 /// Table of powers of 5 representable in digits. Specifically, the largest {u8, u16, u32} value
67 /// that's a power of five, plus the corresponding exponent. Used in `mul_pow5`.
68 const SMALL_POW5: [(u64, usize); 3] = [(125, 3), (15625, 6), (1_220_703_125, 13)];
70 macro_rules! define_bignum {
71 ($name:ident: type=$ty:ty, n=$n:expr) => {
72 /// Stack-allocated arbitrary-precision (up to certain limit) integer.
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`.
79 /// All operations available to bignums panic in the case of overflows.
80 /// The caller is responsible to use large enough bignum types.
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.
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.
92 /// Makes a bignum from one digit.
93 pub fn from_small(v: $ty) -> $name {
94 let mut base = [0; $n];
96 $name { size: 1, base }
99 /// Makes a bignum from `u64` value.
100 pub fn from_u64(mut v: u64) -> $name {
101 let mut base = [0; $n];
108 $name { size: sz, base }
111 /// Returns the internal digits as a slice `[a, b, c, ...]` such that the numeric
112 /// value is `a + b * 2^W + c * 2^(2W) + ...` where `W` is the number of bits in
114 pub fn digits(&self) -> &[$ty] {
115 &self.base[..self.size]
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 = <$ty>::BITS as usize;
122 let d = i / digitbits;
123 let b = i % digitbits;
124 ((self.base[d] >> b) & 1) as u8
127 /// Returns `true` if the bignum is zero.
128 pub fn is_zero(&self) -> bool {
129 self.digits().iter().all(|&v| v == 0)
132 /// Returns the number of bits necessary to represent this value. Note that zero
133 /// is considered to need 0 bits.
134 pub fn bit_length(&self) -> usize {
135 let digitbits = <$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);
140 Some(msd) => msd * digitbits + digits[msd].ilog2() as usize + 1,
141 // There are no non-zero digits, i.e., the number is zero.
146 /// Adds `other` to itself and returns its own mutable reference.
147 pub fn add<'a>(&'a mut self, other: &$name) -> &'a mut $name {
151 let mut sz = cmp::max(self.size, other.size);
152 let mut carry = false;
153 for (a, b) in iter::zip(&mut self.base[..sz], &other.base[..sz]) {
154 let (v, c) = (*a).carrying_add(*b, carry);
166 pub fn add_small(&mut self, other: $ty) -> &mut $name {
167 let (v, mut carry) = self.base[0].carrying_add(other, false);
171 let (v, c) = self.base[i].carrying_add(0, carry);
182 /// Subtracts `other` from itself and returns its own mutable reference.
183 pub fn sub<'a>(&'a mut self, other: &$name) -> &'a mut $name {
187 let sz = cmp::max(self.size, other.size);
188 let mut noborrow = true;
189 for (a, b) in iter::zip(&mut self.base[..sz], &other.base[..sz]) {
190 let (v, c) = (*a).carrying_add(!*b, noborrow);
199 /// Multiplies itself by a digit-sized `other` and returns its own
200 /// mutable reference.
201 pub fn mul_small(&mut self, other: $ty) -> &mut $name {
202 let mut sz = self.size;
204 for a in &mut self.base[..sz] {
205 let (v, c) = (*a).carrying_mul(other, carry);
210 self.base[sz] = carry;
217 /// Multiplies itself by `2^bits` and returns its own mutable reference.
218 pub fn mul_pow2(&mut self, bits: usize) -> &mut $name {
219 let digitbits = <$ty>::BITS as usize;
220 let digits = bits / digitbits;
221 let bits = bits % digitbits;
223 assert!(digits < $n);
224 debug_assert!(self.base[$n - digits..].iter().all(|&v| v == 0));
225 debug_assert!(bits == 0 || (self.base[$n - digits - 1] >> (digitbits - bits)) == 0);
227 // shift by `digits * digitbits` bits
228 for i in (0..self.size).rev() {
229 self.base[i + digits] = self.base[i];
235 // shift by `bits` bits
236 let mut sz = self.size + digits;
239 let overflow = self.base[last - 1] >> (digitbits - bits);
241 self.base[last] = overflow;
244 for i in (digits + 1..last).rev() {
246 (self.base[i] << bits) | (self.base[i - 1] >> (digitbits - bits));
248 self.base[digits] <<= bits;
249 // self.base[..digits] is zero, no need to shift
256 /// Multiplies itself by `5^e` and returns its own mutable reference.
257 pub fn mul_pow5(&mut self, mut e: usize) -> &mut $name {
259 use crate::num::bignum::SMALL_POW5;
261 // There are exactly n trailing zeros on 2^n, and the only relevant digit sizes
262 // are consecutive powers of two, so this is well suited index for the table.
263 let table_index = mem::size_of::<$ty>().trailing_zeros() as usize;
264 let (small_power, small_e) = SMALL_POW5[table_index];
265 let small_power = small_power as $ty;
267 // Multiply with the largest single-digit power as long as possible ...
269 self.mul_small(small_power);
273 // ... then finish off the remainder.
274 let mut rest_power = 1;
278 self.mul_small(rest_power);
283 /// Multiplies itself by a number described by `other[0] + other[1] * 2^W +
284 /// other[2] * 2^(2W) + ...` (where `W` is the number of bits in the digit type)
285 /// and returns its own mutable reference.
286 pub fn mul_digits<'a>(&'a mut self, other: &[$ty]) -> &'a mut $name {
287 // the internal routine. works best when aa.len() <= bb.len().
288 fn mul_inner(ret: &mut [$ty; $n], aa: &[$ty], bb: &[$ty]) -> usize {
289 use crate::num::bignum::FullOps;
292 for (i, &a) in aa.iter().enumerate() {
296 let mut sz = bb.len();
298 for (j, &b) in bb.iter().enumerate() {
299 let (c, v) = a.full_mul_add(b, ret[i + j], carry);
314 let mut ret = [0; $n];
315 let retsz = if self.size < other.len() {
316 mul_inner(&mut ret, &self.digits(), other)
318 mul_inner(&mut ret, other, &self.digits())
325 /// Divides itself by a digit-sized `other` and returns its own
326 /// mutable reference *and* the remainder.
327 pub fn div_rem_small(&mut self, other: $ty) -> (&mut $name, $ty) {
328 use crate::num::bignum::FullOps;
334 for a in self.base[..sz].iter_mut().rev() {
335 let (q, r) = (*a).full_div_rem(other, borrow);
342 /// Divide self by another bignum, overwriting `q` with the quotient and `r` with the
344 pub fn div_rem(&self, d: &$name, q: &mut $name, r: &mut $name) {
345 // Stupid slow base-2 long division taken from
346 // https://en.wikipedia.org/wiki/Division_algorithm
347 // FIXME use a greater base ($ty) for the long division.
348 assert!(!d.is_zero());
349 let digitbits = <$ty>::BITS as usize;
350 for digit in &mut q.base[..] {
353 for digit in &mut r.base[..] {
358 let mut q_is_zero = true;
359 let end = self.bit_length();
360 for i in (0..end).rev() {
362 r.base[0] |= self.get_bit(i) as $ty;
365 // Set bit `i` of q to 1.
366 let digit_idx = i / digitbits;
367 let bit_idx = i % digitbits;
369 q.size = digit_idx + 1;
372 q.base[digit_idx] |= 1 << bit_idx;
375 debug_assert!(q.base[q.size..].iter().all(|&d| d == 0));
376 debug_assert!(r.base[r.size..].iter().all(|&d| d == 0));
380 impl crate::cmp::PartialEq for $name {
381 fn eq(&self, other: &$name) -> bool {
382 self.base[..] == other.base[..]
386 impl crate::cmp::Eq for $name {}
388 impl crate::cmp::PartialOrd for $name {
389 fn partial_cmp(&self, other: &$name) -> crate::option::Option<crate::cmp::Ordering> {
390 crate::option::Option::Some(self.cmp(other))
394 impl crate::cmp::Ord for $name {
395 fn cmp(&self, other: &$name) -> crate::cmp::Ordering {
397 let sz = max(self.size, other.size);
398 let lhs = self.base[..sz].iter().cloned().rev();
399 let rhs = other.base[..sz].iter().cloned().rev();
404 impl crate::clone::Clone for $name {
405 fn clone(&self) -> Self {
406 Self { size: self.size, base: self.base }
410 impl crate::fmt::Debug for $name {
411 fn fmt(&self, f: &mut crate::fmt::Formatter<'_>) -> crate::fmt::Result {
412 let sz = if self.size < 1 { 1 } else { self.size };
413 let digitlen = <$ty>::BITS as usize / 4;
415 write!(f, "{:#x}", self.base[sz - 1])?;
416 for &v in self.base[..sz - 1].iter().rev() {
417 write!(f, "_{:01$x}", v, digitlen)?;
419 crate::result::Result::Ok(())
425 /// The digit type for `Big32x40`.
426 pub type Digit32 = u32;
428 define_bignum!(Big32x40: type=Digit32, n=40);
430 // this one is used for testing only.
433 define_bignum!(Big8x3: type=u8, n=3);