mirror of https://github.com/rust-lang/rust
435 lines
17 KiB
Rust
435 lines
17 KiB
Rust
//! Custom arbitrary-precision number (bignum) implementation.
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//!
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//! This is designed to avoid the heap allocation at expense of stack memory.
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//! The most used bignum type, `Big32x40`, is limited by 32 × 40 = 1,280 bits
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//! and will take at most 160 bytes of stack memory. This is more than enough
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//! for round-tripping all possible finite `f64` values.
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//!
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//! In principle it is possible to have multiple bignum types for different
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//! inputs, but we don't do so to avoid the code bloat. Each bignum is still
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//! tracked for the actual usages, so it normally doesn't matter.
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// This module is only for dec2flt and flt2dec, and only public because of coretests.
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// It is not intended to ever be stabilized.
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#![doc(hidden)]
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#![unstable(
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feature = "core_private_bignum",
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reason = "internal routines only exposed for testing",
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issue = "none"
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)]
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#![macro_use]
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/// Arithmetic operations required by bignums.
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pub trait FullOps: Sized {
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/// Returns `(carry', v')` such that `carry' * 2^W + v' = self * other + other2 + carry`,
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/// where `W` is the number of bits in `Self`.
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fn full_mul_add(self, other: Self, other2: Self, carry: Self) -> (Self /* carry */, Self);
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/// Returns `(quo, rem)` such that `borrow * 2^W + self = quo * other + rem`
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/// and `0 <= rem < other`, where `W` is the number of bits in `Self`.
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fn full_div_rem(self, other: Self, borrow: Self)
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-> (Self /* quotient */, Self /* remainder */);
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}
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macro_rules! impl_full_ops {
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($($ty:ty: add($addfn:path), mul/div($bigty:ident);)*) => (
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$(
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impl FullOps for $ty {
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fn full_mul_add(self, other: $ty, other2: $ty, carry: $ty) -> ($ty, $ty) {
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// This cannot overflow;
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// the output is between `0` and `2^nbits * (2^nbits - 1)`.
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let v = (self as $bigty) * (other as $bigty) + (other2 as $bigty) +
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(carry as $bigty);
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((v >> <$ty>::BITS) as $ty, v as $ty)
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}
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fn full_div_rem(self, other: $ty, borrow: $ty) -> ($ty, $ty) {
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debug_assert!(borrow < other);
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// This cannot overflow; the output is between `0` and `other * (2^nbits - 1)`.
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let lhs = ((borrow as $bigty) << <$ty>::BITS) | (self as $bigty);
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let rhs = other as $bigty;
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((lhs / rhs) as $ty, (lhs % rhs) as $ty)
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}
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}
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)*
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)
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}
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impl_full_ops! {
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u8: add(intrinsics::u8_add_with_overflow), mul/div(u16);
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u16: add(intrinsics::u16_add_with_overflow), mul/div(u32);
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u32: add(intrinsics::u32_add_with_overflow), mul/div(u64);
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// See RFC #521 for enabling this.
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// u64: add(intrinsics::u64_add_with_overflow), mul/div(u128);
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}
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/// Table of powers of 5 representable in digits. Specifically, the largest {u8, u16, u32} value
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/// that's a power of five, plus the corresponding exponent. Used in `mul_pow5`.
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const SMALL_POW5: [(u64, usize); 3] = [(125, 3), (15625, 6), (1_220_703_125, 13)];
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macro_rules! define_bignum {
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($name:ident: type=$ty:ty, n=$n:expr) => {
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/// Stack-allocated arbitrary-precision (up to certain limit) integer.
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///
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/// This is backed by a fixed-size array of given type ("digit").
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/// While the array is not very large (normally some hundred bytes),
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/// copying it recklessly may result in the performance hit.
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/// Thus this is intentionally not `Copy`.
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///
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/// All operations available to bignums panic in the case of overflows.
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/// The caller is responsible to use large enough bignum types.
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pub struct $name {
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/// One plus the offset to the maximum "digit" in use.
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/// This does not decrease, so be aware of the computation order.
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/// `base[size..]` should be zero.
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size: usize,
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/// Digits. `[a, b, c, ...]` represents `a + b*2^W + c*2^(2W) + ...`
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/// where `W` is the number of bits in the digit type.
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base: [$ty; $n],
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}
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impl $name {
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/// Makes a bignum from one digit.
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pub fn from_small(v: $ty) -> $name {
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let mut base = [0; $n];
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base[0] = v;
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$name { size: 1, base }
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}
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/// Makes a bignum from `u64` value.
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pub fn from_u64(mut v: u64) -> $name {
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let mut base = [0; $n];
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let mut sz = 0;
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while v > 0 {
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base[sz] = v as $ty;
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v >>= <$ty>::BITS;
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sz += 1;
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}
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$name { size: sz, base }
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}
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/// Returns the internal digits as a slice `[a, b, c, ...]` such that the numeric
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/// value is `a + b * 2^W + c * 2^(2W) + ...` where `W` is the number of bits in
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/// the digit type.
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pub fn digits(&self) -> &[$ty] {
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&self.base[..self.size]
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}
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/// Returns the `i`-th bit where bit 0 is the least significant one.
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/// In other words, the bit with weight `2^i`.
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pub fn get_bit(&self, i: usize) -> u8 {
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let digitbits = <$ty>::BITS as usize;
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let d = i / digitbits;
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let b = i % digitbits;
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((self.base[d] >> b) & 1) as u8
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}
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/// Returns `true` if the bignum is zero.
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pub fn is_zero(&self) -> bool {
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self.digits().iter().all(|&v| v == 0)
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}
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/// Returns the number of bits necessary to represent this value. Note that zero
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/// is considered to need 0 bits.
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pub fn bit_length(&self) -> usize {
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let digitbits = <$ty>::BITS as usize;
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let digits = self.digits();
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// Find the most significant non-zero digit.
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let msd = digits.iter().rposition(|&x| x != 0);
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match msd {
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Some(msd) => msd * digitbits + digits[msd].ilog2() as usize + 1,
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// There are no non-zero digits, i.e., the number is zero.
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_ => 0,
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}
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}
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/// Adds `other` to itself and returns its own mutable reference.
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pub fn add<'a>(&'a mut self, other: &$name) -> &'a mut $name {
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use crate::cmp;
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use crate::iter;
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let mut sz = cmp::max(self.size, other.size);
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let mut carry = false;
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for (a, b) in iter::zip(&mut self.base[..sz], &other.base[..sz]) {
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let (v, c) = (*a).carrying_add(*b, carry);
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*a = v;
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carry = c;
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}
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if carry {
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self.base[sz] = 1;
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sz += 1;
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}
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self.size = sz;
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self
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}
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pub fn add_small(&mut self, other: $ty) -> &mut $name {
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let (v, mut carry) = self.base[0].carrying_add(other, false);
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self.base[0] = v;
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let mut i = 1;
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while carry {
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let (v, c) = self.base[i].carrying_add(0, carry);
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self.base[i] = v;
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carry = c;
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i += 1;
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}
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if i > self.size {
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self.size = i;
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}
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self
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}
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/// Subtracts `other` from itself and returns its own mutable reference.
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pub fn sub<'a>(&'a mut self, other: &$name) -> &'a mut $name {
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use crate::cmp;
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use crate::iter;
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let sz = cmp::max(self.size, other.size);
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let mut noborrow = true;
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for (a, b) in iter::zip(&mut self.base[..sz], &other.base[..sz]) {
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let (v, c) = (*a).carrying_add(!*b, noborrow);
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*a = v;
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noborrow = c;
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}
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assert!(noborrow);
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self.size = sz;
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self
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}
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/// Multiplies itself by a digit-sized `other` and returns its own
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/// mutable reference.
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pub fn mul_small(&mut self, other: $ty) -> &mut $name {
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let mut sz = self.size;
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let mut carry = 0;
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for a in &mut self.base[..sz] {
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let (v, c) = (*a).carrying_mul(other, carry);
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*a = v;
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carry = c;
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}
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if carry > 0 {
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self.base[sz] = carry;
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sz += 1;
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}
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self.size = sz;
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self
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}
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/// Multiplies itself by `2^bits` and returns its own mutable reference.
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pub fn mul_pow2(&mut self, bits: usize) -> &mut $name {
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let digitbits = <$ty>::BITS as usize;
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let digits = bits / digitbits;
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let bits = bits % digitbits;
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assert!(digits < $n);
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debug_assert!(self.base[$n - digits..].iter().all(|&v| v == 0));
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debug_assert!(bits == 0 || (self.base[$n - digits - 1] >> (digitbits - bits)) == 0);
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// shift by `digits * digitbits` bits
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for i in (0..self.size).rev() {
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self.base[i + digits] = self.base[i];
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}
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for i in 0..digits {
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self.base[i] = 0;
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}
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// shift by `bits` bits
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let mut sz = self.size + digits;
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if bits > 0 {
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let last = sz;
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let overflow = self.base[last - 1] >> (digitbits - bits);
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if overflow > 0 {
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self.base[last] = overflow;
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sz += 1;
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}
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for i in (digits + 1..last).rev() {
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self.base[i] =
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(self.base[i] << bits) | (self.base[i - 1] >> (digitbits - bits));
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}
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self.base[digits] <<= bits;
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// self.base[..digits] is zero, no need to shift
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}
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self.size = sz;
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self
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}
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/// Multiplies itself by `5^e` and returns its own mutable reference.
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pub fn mul_pow5(&mut self, mut e: usize) -> &mut $name {
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use crate::mem;
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use crate::num::bignum::SMALL_POW5;
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// There are exactly n trailing zeros on 2^n, and the only relevant digit sizes
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// are consecutive powers of two, so this is well suited index for the table.
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let table_index = mem::size_of::<$ty>().trailing_zeros() as usize;
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let (small_power, small_e) = SMALL_POW5[table_index];
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let small_power = small_power as $ty;
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// Multiply with the largest single-digit power as long as possible ...
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while e >= small_e {
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self.mul_small(small_power);
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e -= small_e;
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}
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// ... then finish off the remainder.
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let mut rest_power = 1;
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for _ in 0..e {
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rest_power *= 5;
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}
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self.mul_small(rest_power);
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self
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}
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/// Multiplies itself by a number described by `other[0] + other[1] * 2^W +
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/// other[2] * 2^(2W) + ...` (where `W` is the number of bits in the digit type)
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/// and returns its own mutable reference.
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pub fn mul_digits<'a>(&'a mut self, other: &[$ty]) -> &'a mut $name {
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// the internal routine. works best when aa.len() <= bb.len().
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fn mul_inner(ret: &mut [$ty; $n], aa: &[$ty], bb: &[$ty]) -> usize {
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use crate::num::bignum::FullOps;
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let mut retsz = 0;
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for (i, &a) in aa.iter().enumerate() {
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if a == 0 {
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continue;
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}
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let mut sz = bb.len();
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let mut carry = 0;
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for (j, &b) in bb.iter().enumerate() {
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let (c, v) = a.full_mul_add(b, ret[i + j], carry);
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ret[i + j] = v;
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carry = c;
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}
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if carry > 0 {
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ret[i + sz] = carry;
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sz += 1;
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}
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if retsz < i + sz {
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retsz = i + sz;
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}
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}
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retsz
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}
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let mut ret = [0; $n];
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let retsz = if self.size < other.len() {
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mul_inner(&mut ret, &self.digits(), other)
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} else {
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mul_inner(&mut ret, other, &self.digits())
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};
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self.base = ret;
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self.size = retsz;
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self
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}
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/// Divides itself by a digit-sized `other` and returns its own
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/// mutable reference *and* the remainder.
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pub fn div_rem_small(&mut self, other: $ty) -> (&mut $name, $ty) {
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use crate::num::bignum::FullOps;
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assert!(other > 0);
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let sz = self.size;
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let mut borrow = 0;
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for a in self.base[..sz].iter_mut().rev() {
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let (q, r) = (*a).full_div_rem(other, borrow);
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*a = q;
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borrow = r;
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}
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(self, borrow)
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}
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/// Divide self by another bignum, overwriting `q` with the quotient and `r` with the
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/// remainder.
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pub fn div_rem(&self, d: &$name, q: &mut $name, r: &mut $name) {
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// Stupid slow base-2 long division taken from
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// https://en.wikipedia.org/wiki/Division_algorithm
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// FIXME use a greater base ($ty) for the long division.
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assert!(!d.is_zero());
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let digitbits = <$ty>::BITS as usize;
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for digit in &mut q.base[..] {
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*digit = 0;
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}
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for digit in &mut r.base[..] {
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*digit = 0;
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}
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r.size = d.size;
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q.size = 1;
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let mut q_is_zero = true;
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let end = self.bit_length();
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for i in (0..end).rev() {
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r.mul_pow2(1);
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r.base[0] |= self.get_bit(i) as $ty;
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if &*r >= d {
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r.sub(d);
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// Set bit `i` of q to 1.
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let digit_idx = i / digitbits;
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let bit_idx = i % digitbits;
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if q_is_zero {
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q.size = digit_idx + 1;
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q_is_zero = false;
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}
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q.base[digit_idx] |= 1 << bit_idx;
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}
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}
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debug_assert!(q.base[q.size..].iter().all(|&d| d == 0));
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debug_assert!(r.base[r.size..].iter().all(|&d| d == 0));
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}
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}
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impl crate::cmp::PartialEq for $name {
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fn eq(&self, other: &$name) -> bool {
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self.base[..] == other.base[..]
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}
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}
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impl crate::cmp::Eq for $name {}
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impl crate::cmp::PartialOrd for $name {
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fn partial_cmp(&self, other: &$name) -> crate::option::Option<crate::cmp::Ordering> {
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crate::option::Option::Some(self.cmp(other))
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}
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}
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impl crate::cmp::Ord for $name {
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fn cmp(&self, other: &$name) -> crate::cmp::Ordering {
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use crate::cmp::max;
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let sz = max(self.size, other.size);
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let lhs = self.base[..sz].iter().cloned().rev();
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let rhs = other.base[..sz].iter().cloned().rev();
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lhs.cmp(rhs)
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}
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}
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impl crate::clone::Clone for $name {
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fn clone(&self) -> Self {
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Self { size: self.size, base: self.base }
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}
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}
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impl crate::fmt::Debug for $name {
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fn fmt(&self, f: &mut crate::fmt::Formatter<'_>) -> crate::fmt::Result {
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let sz = if self.size < 1 { 1 } else { self.size };
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let digitlen = <$ty>::BITS as usize / 4;
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write!(f, "{:#x}", self.base[sz - 1])?;
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for &v in self.base[..sz - 1].iter().rev() {
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write!(f, "_{:01$x}", v, digitlen)?;
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}
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crate::result::Result::Ok(())
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}
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}
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};
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}
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/// The digit type for `Big32x40`.
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pub type Digit32 = u32;
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define_bignum!(Big32x40: type=Digit32, n=40);
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// this one is used for testing only.
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#[doc(hidden)]
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pub mod tests {
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define_bignum!(Big8x3: type=u8, n=3);
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}
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