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-rw-r--r-- | library/core/src/num/bignum.rs | 434 |
1 files changed, 434 insertions, 0 deletions
diff --git a/library/core/src/num/bignum.rs b/library/core/src/num/bignum.rs new file mode 100644 index 000000000..de85fdd6e --- /dev/null +++ b/library/core/src/num/bignum.rs @@ -0,0 +1,434 @@ +//! Custom arbitrary-precision number (bignum) implementation. +//! +//! This is designed to avoid the heap allocation at expense of stack memory. +//! The most used bignum type, `Big32x40`, is limited by 32 × 40 = 1,280 bits +//! and will take at most 160 bytes of stack memory. This is more than enough +//! for round-tripping all possible finite `f64` values. +//! +//! In principle it is possible to have multiple bignum types for different +//! inputs, but we don't do so to avoid the code bloat. Each bignum is still +//! tracked for the actual usages, so it normally doesn't matter. + +// This module is only for dec2flt and flt2dec, and only public because of coretests. +// It is not intended to ever be stabilized. +#![doc(hidden)] +#![unstable( + feature = "core_private_bignum", + reason = "internal routines only exposed for testing", + issue = "none" +)] +#![macro_use] + +/// Arithmetic operations required by bignums. +pub trait FullOps: Sized { + /// Returns `(carry', v')` such that `carry' * 2^W + v' = self * other + other2 + carry`, + /// where `W` is the number of bits in `Self`. + fn full_mul_add(self, other: Self, other2: Self, carry: Self) -> (Self /* carry */, Self); + + /// Returns `(quo, rem)` such that `borrow * 2^W + self = quo * other + rem` + /// and `0 <= rem < other`, where `W` is the number of bits in `Self`. + fn full_div_rem(self, other: Self, borrow: Self) + -> (Self /* quotient */, Self /* remainder */); +} + +macro_rules! impl_full_ops { + ($($ty:ty: add($addfn:path), mul/div($bigty:ident);)*) => ( + $( + impl FullOps for $ty { + fn full_mul_add(self, other: $ty, other2: $ty, carry: $ty) -> ($ty, $ty) { + // This cannot overflow; + // the output is between `0` and `2^nbits * (2^nbits - 1)`. + let v = (self as $bigty) * (other as $bigty) + (other2 as $bigty) + + (carry as $bigty); + ((v >> <$ty>::BITS) as $ty, v as $ty) + } + + fn full_div_rem(self, other: $ty, borrow: $ty) -> ($ty, $ty) { + debug_assert!(borrow < other); + // This cannot overflow; the output is between `0` and `other * (2^nbits - 1)`. + let lhs = ((borrow as $bigty) << <$ty>::BITS) | (self as $bigty); + let rhs = other as $bigty; + ((lhs / rhs) as $ty, (lhs % rhs) as $ty) + } + } + )* + ) +} + +impl_full_ops! { + u8: add(intrinsics::u8_add_with_overflow), mul/div(u16); + u16: add(intrinsics::u16_add_with_overflow), mul/div(u32); + u32: add(intrinsics::u32_add_with_overflow), mul/div(u64); + // See RFC #521 for enabling this. + // u64: add(intrinsics::u64_add_with_overflow), mul/div(u128); +} + +/// Table of powers of 5 representable in digits. Specifically, the largest {u8, u16, u32} value +/// that's a power of five, plus the corresponding exponent. Used in `mul_pow5`. +const SMALL_POW5: [(u64, usize); 3] = [(125, 3), (15625, 6), (1_220_703_125, 13)]; + +macro_rules! define_bignum { + ($name:ident: type=$ty:ty, n=$n:expr) => { + /// Stack-allocated arbitrary-precision (up to certain limit) integer. + /// + /// This is backed by a fixed-size array of given type ("digit"). + /// While the array is not very large (normally some hundred bytes), + /// copying it recklessly may result in the performance hit. + /// Thus this is intentionally not `Copy`. + /// + /// All operations available to bignums panic in the case of overflows. + /// The caller is responsible to use large enough bignum types. + pub struct $name { + /// One plus the offset to the maximum "digit" in use. + /// This does not decrease, so be aware of the computation order. + /// `base[size..]` should be zero. + size: usize, + /// Digits. `[a, b, c, ...]` represents `a + b*2^W + c*2^(2W) + ...` + /// where `W` is the number of bits in the digit type. + base: [$ty; $n], + } + + impl $name { + /// Makes a bignum from one digit. + pub fn from_small(v: $ty) -> $name { + let mut base = [0; $n]; + base[0] = v; + $name { size: 1, base } + } + + /// Makes a bignum from `u64` value. + pub fn from_u64(mut v: u64) -> $name { + let mut base = [0; $n]; + let mut sz = 0; + while v > 0 { + base[sz] = v as $ty; + v >>= <$ty>::BITS; + sz += 1; + } + $name { size: sz, base } + } + + /// Returns the internal digits as a slice `[a, b, c, ...]` such that the numeric + /// value is `a + b * 2^W + c * 2^(2W) + ...` where `W` is the number of bits in + /// the digit type. + pub fn digits(&self) -> &[$ty] { + &self.base[..self.size] + } + + /// Returns the `i`-th bit where bit 0 is the least significant one. + /// In other words, the bit with weight `2^i`. + pub fn get_bit(&self, i: usize) -> u8 { + let digitbits = <$ty>::BITS as usize; + let d = i / digitbits; + let b = i % digitbits; + ((self.base[d] >> b) & 1) as u8 + } + + /// Returns `true` if the bignum is zero. + pub fn is_zero(&self) -> bool { + self.digits().iter().all(|&v| v == 0) + } + + /// Returns the number of bits necessary to represent this value. Note that zero + /// is considered to need 0 bits. + pub fn bit_length(&self) -> usize { + let digitbits = <$ty>::BITS as usize; + let digits = self.digits(); + // Find the most significant non-zero digit. + let msd = digits.iter().rposition(|&x| x != 0); + match msd { + Some(msd) => msd * digitbits + digits[msd].log2() as usize + 1, + // There are no non-zero digits, i.e., the number is zero. + _ => 0, + } + } + + /// Adds `other` to itself and returns its own mutable reference. + pub fn add<'a>(&'a mut self, other: &$name) -> &'a mut $name { + use crate::cmp; + use crate::iter; + + let mut sz = cmp::max(self.size, other.size); + let mut carry = false; + for (a, b) in iter::zip(&mut self.base[..sz], &other.base[..sz]) { + let (v, c) = (*a).carrying_add(*b, carry); + *a = v; + carry = c; + } + if carry { + self.base[sz] = 1; + sz += 1; + } + self.size = sz; + self + } + + pub fn add_small(&mut self, other: $ty) -> &mut $name { + let (v, mut carry) = self.base[0].carrying_add(other, false); + self.base[0] = v; + let mut i = 1; + while carry { + let (v, c) = self.base[i].carrying_add(0, carry); + self.base[i] = v; + carry = c; + i += 1; + } + if i > self.size { + self.size = i; + } + self + } + + /// Subtracts `other` from itself and returns its own mutable reference. + pub fn sub<'a>(&'a mut self, other: &$name) -> &'a mut $name { + use crate::cmp; + use crate::iter; + + let sz = cmp::max(self.size, other.size); + let mut noborrow = true; + for (a, b) in iter::zip(&mut self.base[..sz], &other.base[..sz]) { + let (v, c) = (*a).carrying_add(!*b, noborrow); + *a = v; + noborrow = c; + } + assert!(noborrow); + self.size = sz; + self + } + + /// Multiplies itself by a digit-sized `other` and returns its own + /// mutable reference. + pub fn mul_small(&mut self, other: $ty) -> &mut $name { + let mut sz = self.size; + let mut carry = 0; + for a in &mut self.base[..sz] { + let (v, c) = (*a).carrying_mul(other, carry); + *a = v; + carry = c; + } + if carry > 0 { + self.base[sz] = carry; + sz += 1; + } + self.size = sz; + self + } + + /// Multiplies itself by `2^bits` and returns its own mutable reference. + pub fn mul_pow2(&mut self, bits: usize) -> &mut $name { + let digitbits = <$ty>::BITS as usize; + let digits = bits / digitbits; + let bits = bits % digitbits; + + assert!(digits < $n); + debug_assert!(self.base[$n - digits..].iter().all(|&v| v == 0)); + debug_assert!(bits == 0 || (self.base[$n - digits - 1] >> (digitbits - bits)) == 0); + + // shift by `digits * digitbits` bits + for i in (0..self.size).rev() { + self.base[i + digits] = self.base[i]; + } + for i in 0..digits { + self.base[i] = 0; + } + + // shift by `bits` bits + let mut sz = self.size + digits; + if bits > 0 { + let last = sz; + let overflow = self.base[last - 1] >> (digitbits - bits); + if overflow > 0 { + self.base[last] = overflow; + sz += 1; + } + for i in (digits + 1..last).rev() { + self.base[i] = + (self.base[i] << bits) | (self.base[i - 1] >> (digitbits - bits)); + } + self.base[digits] <<= bits; + // self.base[..digits] is zero, no need to shift + } + + self.size = sz; + self + } + + /// Multiplies itself by `5^e` and returns its own mutable reference. + pub fn mul_pow5(&mut self, mut e: usize) -> &mut $name { + use crate::mem; + use crate::num::bignum::SMALL_POW5; + + // There are exactly n trailing zeros on 2^n, and the only relevant digit sizes + // are consecutive powers of two, so this is well suited index for the table. + let table_index = mem::size_of::<$ty>().trailing_zeros() as usize; + let (small_power, small_e) = SMALL_POW5[table_index]; + let small_power = small_power as $ty; + + // Multiply with the largest single-digit power as long as possible ... + while e >= small_e { + self.mul_small(small_power); + e -= small_e; + } + + // ... then finish off the remainder. + let mut rest_power = 1; + for _ in 0..e { + rest_power *= 5; + } + self.mul_small(rest_power); + + self + } + + /// Multiplies itself by a number described by `other[0] + other[1] * 2^W + + /// other[2] * 2^(2W) + ...` (where `W` is the number of bits in the digit type) + /// and returns its own mutable reference. + pub fn mul_digits<'a>(&'a mut self, other: &[$ty]) -> &'a mut $name { + // the internal routine. works best when aa.len() <= bb.len(). + fn mul_inner(ret: &mut [$ty; $n], aa: &[$ty], bb: &[$ty]) -> usize { + use crate::num::bignum::FullOps; + + let mut retsz = 0; + for (i, &a) in aa.iter().enumerate() { + if a == 0 { + continue; + } + let mut sz = bb.len(); + let mut carry = 0; + for (j, &b) in bb.iter().enumerate() { + let (c, v) = a.full_mul_add(b, ret[i + j], carry); + ret[i + j] = v; + carry = c; + } + if carry > 0 { + ret[i + sz] = carry; + sz += 1; + } + if retsz < i + sz { + retsz = i + sz; + } + } + retsz + } + + let mut ret = [0; $n]; + let retsz = if self.size < other.len() { + mul_inner(&mut ret, &self.digits(), other) + } else { + mul_inner(&mut ret, other, &self.digits()) + }; + self.base = ret; + self.size = retsz; + self + } + + /// Divides itself by a digit-sized `other` and returns its own + /// mutable reference *and* the remainder. + pub fn div_rem_small(&mut self, other: $ty) -> (&mut $name, $ty) { + use crate::num::bignum::FullOps; + + assert!(other > 0); + + let sz = self.size; + let mut borrow = 0; + for a in self.base[..sz].iter_mut().rev() { + let (q, r) = (*a).full_div_rem(other, borrow); + *a = q; + borrow = r; + } + (self, borrow) + } + + /// Divide self by another bignum, overwriting `q` with the quotient and `r` with the + /// remainder. + pub fn div_rem(&self, d: &$name, q: &mut $name, r: &mut $name) { + // Stupid slow base-2 long division taken from + // https://en.wikipedia.org/wiki/Division_algorithm + // FIXME use a greater base ($ty) for the long division. + assert!(!d.is_zero()); + let digitbits = <$ty>::BITS as usize; + for digit in &mut q.base[..] { + *digit = 0; + } + for digit in &mut r.base[..] { + *digit = 0; + } + r.size = d.size; + q.size = 1; + let mut q_is_zero = true; + let end = self.bit_length(); + for i in (0..end).rev() { + r.mul_pow2(1); + r.base[0] |= self.get_bit(i) as $ty; + if &*r >= d { + r.sub(d); + // Set bit `i` of q to 1. + let digit_idx = i / digitbits; + let bit_idx = i % digitbits; + if q_is_zero { + q.size = digit_idx + 1; + q_is_zero = false; + } + q.base[digit_idx] |= 1 << bit_idx; + } + } + debug_assert!(q.base[q.size..].iter().all(|&d| d == 0)); + debug_assert!(r.base[r.size..].iter().all(|&d| d == 0)); + } + } + + impl crate::cmp::PartialEq for $name { + fn eq(&self, other: &$name) -> bool { + self.base[..] == other.base[..] + } + } + + impl crate::cmp::Eq for $name {} + + impl crate::cmp::PartialOrd for $name { + fn partial_cmp(&self, other: &$name) -> crate::option::Option<crate::cmp::Ordering> { + crate::option::Option::Some(self.cmp(other)) + } + } + + impl crate::cmp::Ord for $name { + fn cmp(&self, other: &$name) -> crate::cmp::Ordering { + use crate::cmp::max; + let sz = max(self.size, other.size); + let lhs = self.base[..sz].iter().cloned().rev(); + let rhs = other.base[..sz].iter().cloned().rev(); + lhs.cmp(rhs) + } + } + + impl crate::clone::Clone for $name { + fn clone(&self) -> Self { + Self { size: self.size, base: self.base } + } + } + + impl crate::fmt::Debug for $name { + fn fmt(&self, f: &mut crate::fmt::Formatter<'_>) -> crate::fmt::Result { + let sz = if self.size < 1 { 1 } else { self.size }; + let digitlen = <$ty>::BITS as usize / 4; + + write!(f, "{:#x}", self.base[sz - 1])?; + for &v in self.base[..sz - 1].iter().rev() { + write!(f, "_{:01$x}", v, digitlen)?; + } + crate::result::Result::Ok(()) + } + } + }; +} + +/// The digit type for `Big32x40`. +pub type Digit32 = u32; + +define_bignum!(Big32x40: type=Digit32, n=40); + +// this one is used for testing only. +#[doc(hidden)] +pub mod tests { + define_bignum!(Big8x3: type=u8, n=3); +} |