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//! Building-blocks for arbitrary-precision math.
//!
//! These algorithms assume little-endian order for the large integer
//! buffers, so for a `vec![0, 1, 2, 3]`, `3` is the most significant limb,
//! and `0` is the least significant limb.
use super::large_powers;
use super::num::*;
use super::small_powers::*;
use alloc::vec::Vec;
use core::{cmp, iter, mem};
// ALIASES
// -------
// Type for a single limb of the big integer.
//
// A limb is analogous to a digit in base10, except, it stores 32-bit
// or 64-bit numbers instead.
//
// This should be all-known 64-bit platforms supported by Rust.
//
// Platforms where native 128-bit multiplication is explicitly supported:
// - x86_64 (Supported via `MUL`).
// - mips64 (Supported via `DMULTU`, which `HI` and `LO` can be read-from).
//
// Platforms where native 64-bit multiplication is supported and
// you can extract hi-lo for 64-bit multiplications.
// aarch64 (Requires `UMULH` and `MUL` to capture high and low bits).
// powerpc64 (Requires `MULHDU` and `MULLD` to capture high and low bits).
//
// Platforms where native 128-bit multiplication is not supported,
// requiring software emulation.
// sparc64 (`UMUL` only supported double-word arguments).
// 32-BIT LIMB
#[cfg(limb_width_32)]
pub type Limb = u32;
#[cfg(limb_width_32)]
pub const POW5_LIMB: &[Limb] = &POW5_32;
#[cfg(limb_width_32)]
pub const POW10_LIMB: &[Limb] = &POW10_32;
#[cfg(limb_width_32)]
type Wide = u64;
// 64-BIT LIMB
#[cfg(limb_width_64)]
pub type Limb = u64;
#[cfg(limb_width_64)]
pub const POW5_LIMB: &[Limb] = &POW5_64;
#[cfg(limb_width_64)]
pub const POW10_LIMB: &[Limb] = &POW10_64;
#[cfg(limb_width_64)]
type Wide = u128;
/// Cast to limb type.
#[inline]
pub(crate) fn as_limb<T: Integer>(t: T) -> Limb {
Limb::as_cast(t)
}
/// Cast to wide type.
#[inline]
fn as_wide<T: Integer>(t: T) -> Wide {
Wide::as_cast(t)
}
// SPLIT
// -----
/// Split u64 into limbs, in little-endian order.
#[inline]
#[cfg(limb_width_32)]
fn split_u64(x: u64) -> [Limb; 2] {
[as_limb(x), as_limb(x >> 32)]
}
/// Split u64 into limbs, in little-endian order.
#[inline]
#[cfg(limb_width_64)]
fn split_u64(x: u64) -> [Limb; 1] {
[as_limb(x)]
}
// HI64
// ----
// NONZERO
/// Check if any of the remaining bits are non-zero.
#[inline]
pub fn nonzero<T: Integer>(x: &[T], rindex: usize) -> bool {
let len = x.len();
let slc = &x[..len - rindex];
slc.iter().rev().any(|&x| x != T::ZERO)
}
/// Shift 64-bit integer to high 64-bits.
#[inline]
fn u64_to_hi64_1(r0: u64) -> (u64, bool) {
debug_assert!(r0 != 0);
let ls = r0.leading_zeros();
(r0 << ls, false)
}
/// Shift 2 64-bit integers to high 64-bits.
#[inline]
fn u64_to_hi64_2(r0: u64, r1: u64) -> (u64, bool) {
debug_assert!(r0 != 0);
let ls = r0.leading_zeros();
let rs = 64 - ls;
let v = match ls {
0 => r0,
_ => (r0 << ls) | (r1 >> rs),
};
let n = r1 << ls != 0;
(v, n)
}
/// Trait to export the high 64-bits from a little-endian slice.
trait Hi64<T>: AsRef<[T]> {
/// Get the hi64 bits from a 1-limb slice.
fn hi64_1(&self) -> (u64, bool);
/// Get the hi64 bits from a 2-limb slice.
fn hi64_2(&self) -> (u64, bool);
/// Get the hi64 bits from a 3-limb slice.
fn hi64_3(&self) -> (u64, bool);
/// High-level exporter to extract the high 64 bits from a little-endian slice.
#[inline]
fn hi64(&self) -> (u64, bool) {
match self.as_ref().len() {
0 => (0, false),
1 => self.hi64_1(),
2 => self.hi64_2(),
_ => self.hi64_3(),
}
}
}
impl Hi64<u32> for [u32] {
#[inline]
fn hi64_1(&self) -> (u64, bool) {
debug_assert!(self.len() == 1);
let r0 = self[0] as u64;
u64_to_hi64_1(r0)
}
#[inline]
fn hi64_2(&self) -> (u64, bool) {
debug_assert!(self.len() == 2);
let r0 = (self[1] as u64) << 32;
let r1 = self[0] as u64;
u64_to_hi64_1(r0 | r1)
}
#[inline]
fn hi64_3(&self) -> (u64, bool) {
debug_assert!(self.len() >= 3);
let r0 = self[self.len() - 1] as u64;
let r1 = (self[self.len() - 2] as u64) << 32;
let r2 = self[self.len() - 3] as u64;
let (v, n) = u64_to_hi64_2(r0, r1 | r2);
(v, n || nonzero(self, 3))
}
}
impl Hi64<u64> for [u64] {
#[inline]
fn hi64_1(&self) -> (u64, bool) {
debug_assert!(self.len() == 1);
let r0 = self[0];
u64_to_hi64_1(r0)
}
#[inline]
fn hi64_2(&self) -> (u64, bool) {
debug_assert!(self.len() >= 2);
let r0 = self[self.len() - 1];
let r1 = self[self.len() - 2];
let (v, n) = u64_to_hi64_2(r0, r1);
(v, n || nonzero(self, 2))
}
#[inline]
fn hi64_3(&self) -> (u64, bool) {
self.hi64_2()
}
}
// SCALAR
// ------
// Scalar-to-scalar operations, for building-blocks for arbitrary-precision
// operations.
mod scalar {
use super::*;
// ADDITION
/// Add two small integers and return the resulting value and if overflow happens.
#[inline]
pub fn add(x: Limb, y: Limb) -> (Limb, bool) {
x.overflowing_add(y)
}
/// AddAssign two small integers and return if overflow happens.
#[inline]
pub fn iadd(x: &mut Limb, y: Limb) -> bool {
let t = add(*x, y);
*x = t.0;
t.1
}
// SUBTRACTION
/// Subtract two small integers and return the resulting value and if overflow happens.
#[inline]
pub fn sub(x: Limb, y: Limb) -> (Limb, bool) {
x.overflowing_sub(y)
}
/// SubAssign two small integers and return if overflow happens.
#[inline]
pub fn isub(x: &mut Limb, y: Limb) -> bool {
let t = sub(*x, y);
*x = t.0;
t.1
}
// MULTIPLICATION
/// Multiply two small integers (with carry) (and return the overflow contribution).
///
/// Returns the (low, high) components.
#[inline]
pub fn mul(x: Limb, y: Limb, carry: Limb) -> (Limb, Limb) {
// Cannot overflow, as long as wide is 2x as wide. This is because
// the following is always true:
// `Wide::max_value() - (Narrow::max_value() * Narrow::max_value()) >= Narrow::max_value()`
let z: Wide = as_wide(x) * as_wide(y) + as_wide(carry);
let bits = mem::size_of::<Limb>() * 8;
(as_limb(z), as_limb(z >> bits))
}
/// Multiply two small integers (with carry) (and return if overflow happens).
#[inline]
pub fn imul(x: &mut Limb, y: Limb, carry: Limb) -> Limb {
let t = mul(*x, y, carry);
*x = t.0;
t.1
}
} // scalar
// SMALL
// -----
// Large-to-small operations, to modify a big integer from a native scalar.
mod small {
use super::*;
// MULTIPLICATIION
/// ADDITION
/// Implied AddAssign implementation for adding a small integer to bigint.
///
/// Allows us to choose a start-index in x to store, to allow incrementing
/// from a non-zero start.
#[inline]
pub fn iadd_impl(x: &mut Vec<Limb>, y: Limb, xstart: usize) {
if x.len() <= xstart {
x.push(y);
} else {
// Initial add
let mut carry = scalar::iadd(&mut x[xstart], y);
// Increment until overflow stops occurring.
let mut size = xstart + 1;
while carry && size < x.len() {
carry = scalar::iadd(&mut x[size], 1);
size += 1;
}
// If we overflowed the buffer entirely, need to add 1 to the end
// of the buffer.
if carry {
x.push(1);
}
}
}
/// AddAssign small integer to bigint.
#[inline]
pub fn iadd(x: &mut Vec<Limb>, y: Limb) {
iadd_impl(x, y, 0);
}
// SUBTRACTION
/// SubAssign small integer to bigint.
/// Does not do overflowing subtraction.
#[inline]
pub fn isub_impl(x: &mut Vec<Limb>, y: Limb, xstart: usize) {
debug_assert!(x.len() > xstart && (x[xstart] >= y || x.len() > xstart + 1));
// Initial subtraction
let mut carry = scalar::isub(&mut x[xstart], y);
// Increment until overflow stops occurring.
let mut size = xstart + 1;
while carry && size < x.len() {
carry = scalar::isub(&mut x[size], 1);
size += 1;
}
normalize(x);
}
// MULTIPLICATION
/// MulAssign small integer to bigint.
#[inline]
pub fn imul(x: &mut Vec<Limb>, y: Limb) {
// Multiply iteratively over all elements, adding the carry each time.
let mut carry: Limb = 0;
for xi in &mut *x {
carry = scalar::imul(xi, y, carry);
}
// Overflow of value, add to end.
if carry != 0 {
x.push(carry);
}
}
/// Mul small integer to bigint.
#[inline]
pub fn mul(x: &[Limb], y: Limb) -> Vec<Limb> {
let mut z = Vec::<Limb>::default();
z.extend_from_slice(x);
imul(&mut z, y);
z
}
/// MulAssign by a power.
///
/// Theoretically...
///
/// Use an exponentiation by squaring method, since it reduces the time
/// complexity of the multiplication to ~`O(log(n))` for the squaring,
/// and `O(n*m)` for the result. Since `m` is typically a lower-order
/// factor, this significantly reduces the number of multiplications
/// we need to do. Iteratively multiplying by small powers follows
/// the nth triangular number series, which scales as `O(p^2)`, but
/// where `p` is `n+m`. In short, it scales very poorly.
///
/// Practically....
///
/// Exponentiation by Squaring:
/// running 2 tests
/// test bigcomp_f32_lexical ... bench: 1,018 ns/iter (+/- 78)
/// test bigcomp_f64_lexical ... bench: 3,639 ns/iter (+/- 1,007)
///
/// Exponentiation by Iterative Small Powers:
/// running 2 tests
/// test bigcomp_f32_lexical ... bench: 518 ns/iter (+/- 31)
/// test bigcomp_f64_lexical ... bench: 583 ns/iter (+/- 47)
///
/// Exponentiation by Iterative Large Powers (of 2):
/// running 2 tests
/// test bigcomp_f32_lexical ... bench: 671 ns/iter (+/- 31)
/// test bigcomp_f64_lexical ... bench: 1,394 ns/iter (+/- 47)
///
/// Even using worst-case scenarios, exponentiation by squaring is
/// significantly slower for our workloads. Just multiply by small powers,
/// in simple cases, and use precalculated large powers in other cases.
pub fn imul_pow5(x: &mut Vec<Limb>, n: u32) {
use super::large::KARATSUBA_CUTOFF;
let small_powers = POW5_LIMB;
let large_powers = large_powers::POW5;
if n == 0 {
// No exponent, just return.
// The 0-index of the large powers is `2^0`, which is 1, so we want
// to make sure we don't take that path with a literal 0.
return;
}
// We want to use the asymptotically faster algorithm if we're going
// to be using Karabatsu multiplication sometime during the result,
// otherwise, just use exponentiation by squaring.
let bit_length = 32 - n.leading_zeros() as usize;
debug_assert!(bit_length != 0 && bit_length <= large_powers.len());
if x.len() + large_powers[bit_length - 1].len() < 2 * KARATSUBA_CUTOFF {
// We can use iterative small powers to make this faster for the
// easy cases.
// Multiply by the largest small power until n < step.
let step = small_powers.len() - 1;
let power = small_powers[step];
let mut n = n as usize;
while n >= step {
imul(x, power);
n -= step;
}
// Multiply by the remainder.
imul(x, small_powers[n]);
} else {
// In theory, this code should be asymptotically a lot faster,
// in practice, our small::imul seems to be the limiting step,
// and large imul is slow as well.
// Multiply by higher order powers.
let mut idx: usize = 0;
let mut bit: usize = 1;
let mut n = n as usize;
while n != 0 {
if n & bit != 0 {
debug_assert!(idx < large_powers.len());
large::imul(x, large_powers[idx]);
n ^= bit;
}
idx += 1;
bit <<= 1;
}
}
}
// BIT LENGTH
/// Get number of leading zero bits in the storage.
#[inline]
pub fn leading_zeros(x: &[Limb]) -> usize {
x.last().map_or(0, |x| x.leading_zeros() as usize)
}
/// Calculate the bit-length of the big-integer.
#[inline]
pub fn bit_length(x: &[Limb]) -> usize {
let bits = mem::size_of::<Limb>() * 8;
// Avoid overflowing, calculate via total number of bits
// minus leading zero bits.
let nlz = leading_zeros(x);
bits.checked_mul(x.len())
.map_or_else(usize::max_value, |v| v - nlz)
}
// SHL
/// Shift-left bits inside a buffer.
///
/// Assumes `n < Limb::BITS`, IE, internally shifting bits.
#[inline]
pub fn ishl_bits(x: &mut Vec<Limb>, n: usize) {
// Need to shift by the number of `bits % Limb::BITS)`.
let bits = mem::size_of::<Limb>() * 8;
debug_assert!(n < bits);
if n == 0 {
return;
}
// Internally, for each item, we shift left by n, and add the previous
// right shifted limb-bits.
// For example, we transform (for u8) shifted left 2, to:
// b10100100 b01000010
// b10 b10010001 b00001000
let rshift = bits - n;
let lshift = n;
let mut prev: Limb = 0;
for xi in &mut *x {
let tmp = *xi;
*xi <<= lshift;
*xi |= prev >> rshift;
prev = tmp;
}
// Always push the carry, even if it creates a non-normal result.
let carry = prev >> rshift;
if carry != 0 {
x.push(carry);
}
}
/// Shift-left `n` digits inside a buffer.
///
/// Assumes `n` is not 0.
#[inline]
pub fn ishl_limbs(x: &mut Vec<Limb>, n: usize) {
debug_assert!(n != 0);
if !x.is_empty() {
x.reserve(n);
x.splice(..0, iter::repeat(0).take(n));
}
}
/// Shift-left buffer by n bits.
#[inline]
pub fn ishl(x: &mut Vec<Limb>, n: usize) {
let bits = mem::size_of::<Limb>() * 8;
// Need to pad with zeros for the number of `bits / Limb::BITS`,
// and shift-left with carry for `bits % Limb::BITS`.
let rem = n % bits;
let div = n / bits;
ishl_bits(x, rem);
if div != 0 {
ishl_limbs(x, div);
}
}
// NORMALIZE
/// Normalize the container by popping any leading zeros.
#[inline]
pub fn normalize(x: &mut Vec<Limb>) {
// Remove leading zero if we cause underflow. Since we're dividing
// by a small power, we have at max 1 int removed.
while x.last() == Some(&0) {
x.pop();
}
}
} // small
// LARGE
// -----
// Large-to-large operations, to modify a big integer from a native scalar.
mod large {
use super::*;
// RELATIVE OPERATORS
/// Compare `x` to `y`, in little-endian order.
#[inline]
pub fn compare(x: &[Limb], y: &[Limb]) -> cmp::Ordering {
if x.len() > y.len() {
cmp::Ordering::Greater
} else if x.len() < y.len() {
cmp::Ordering::Less
} else {
let iter = x.iter().rev().zip(y.iter().rev());
for (&xi, &yi) in iter {
if xi > yi {
return cmp::Ordering::Greater;
} else if xi < yi {
return cmp::Ordering::Less;
}
}
// Equal case.
cmp::Ordering::Equal
}
}
/// Check if x is less than y.
#[inline]
pub fn less(x: &[Limb], y: &[Limb]) -> bool {
compare(x, y) == cmp::Ordering::Less
}
/// Check if x is greater than or equal to y.
#[inline]
pub fn greater_equal(x: &[Limb], y: &[Limb]) -> bool {
!less(x, y)
}
// ADDITION
/// Implied AddAssign implementation for bigints.
///
/// Allows us to choose a start-index in x to store, so we can avoid
/// padding the buffer with zeros when not needed, optimized for vectors.
pub fn iadd_impl(x: &mut Vec<Limb>, y: &[Limb], xstart: usize) {
// The effective x buffer is from `xstart..x.len()`, so we need to treat
// that as the current range. If the effective y buffer is longer, need
// to resize to that, + the start index.
if y.len() > x.len() - xstart {
x.resize(y.len() + xstart, 0);
}
// Iteratively add elements from y to x.
let mut carry = false;
for (xi, yi) in x[xstart..].iter_mut().zip(y.iter()) {
// Only one op of the two can overflow, since we added at max
// Limb::max_value() + Limb::max_value(). Add the previous carry,
// and store the current carry for the next.
let mut tmp = scalar::iadd(xi, *yi);
if carry {
tmp |= scalar::iadd(xi, 1);
}
carry = tmp;
}
// Overflow from the previous bit.
if carry {
small::iadd_impl(x, 1, y.len() + xstart);
}
}
/// AddAssign bigint to bigint.
#[inline]
pub fn iadd(x: &mut Vec<Limb>, y: &[Limb]) {
iadd_impl(x, y, 0);
}
/// Add bigint to bigint.
#[inline]
pub fn add(x: &[Limb], y: &[Limb]) -> Vec<Limb> {
let mut z = Vec::<Limb>::default();
z.extend_from_slice(x);
iadd(&mut z, y);
z
}
// SUBTRACTION
/// SubAssign bigint to bigint.
pub fn isub(x: &mut Vec<Limb>, y: &[Limb]) {
// Basic underflow checks.
debug_assert!(greater_equal(x, y));
// Iteratively add elements from y to x.
let mut carry = false;
for (xi, yi) in x.iter_mut().zip(y.iter()) {
// Only one op of the two can overflow, since we added at max
// Limb::max_value() + Limb::max_value(). Add the previous carry,
// and store the current carry for the next.
let mut tmp = scalar::isub(xi, *yi);
if carry {
tmp |= scalar::isub(xi, 1);
}
carry = tmp;
}
if carry {
small::isub_impl(x, 1, y.len());
} else {
small::normalize(x);
}
}
// MULTIPLICATION
/// Number of digits to bottom-out to asymptotically slow algorithms.
///
/// Karatsuba tends to out-perform long-multiplication at ~320-640 bits,
/// so we go halfway, while Newton division tends to out-perform
/// Algorithm D at ~1024 bits. We can toggle this for optimal performance.
pub const KARATSUBA_CUTOFF: usize = 32;
/// Grade-school multiplication algorithm.
///
/// Slow, naive algorithm, using limb-bit bases and just shifting left for
/// each iteration. This could be optimized with numerous other algorithms,
/// but it's extremely simple, and works in O(n*m) time, which is fine
/// by me. Each iteration, of which there are `m` iterations, requires
/// `n` multiplications, and `n` additions, or grade-school multiplication.
fn long_mul(x: &[Limb], y: &[Limb]) -> Vec<Limb> {
// Using the immutable value, multiply by all the scalars in y, using
// the algorithm defined above. Use a single buffer to avoid
// frequent reallocations. Handle the first case to avoid a redundant
// addition, since we know y.len() >= 1.
let mut z: Vec<Limb> = small::mul(x, y[0]);
z.resize(x.len() + y.len(), 0);
// Handle the iterative cases.
for (i, &yi) in y[1..].iter().enumerate() {
let zi: Vec<Limb> = small::mul(x, yi);
iadd_impl(&mut z, &zi, i + 1);
}
small::normalize(&mut z);
z
}
/// Split two buffers into halfway, into (lo, hi).
#[inline]
pub fn karatsuba_split(z: &[Limb], m: usize) -> (&[Limb], &[Limb]) {
(&z[..m], &z[m..])
}
/// Karatsuba multiplication algorithm with roughly equal input sizes.
///
/// Assumes `y.len() >= x.len()`.
fn karatsuba_mul(x: &[Limb], y: &[Limb]) -> Vec<Limb> {
if y.len() <= KARATSUBA_CUTOFF {
// Bottom-out to long division for small cases.
long_mul(x, y)
} else if x.len() < y.len() / 2 {
karatsuba_uneven_mul(x, y)
} else {
// Do our 3 multiplications.
let m = y.len() / 2;
let (xl, xh) = karatsuba_split(x, m);
let (yl, yh) = karatsuba_split(y, m);
let sumx = add(xl, xh);
let sumy = add(yl, yh);
let z0 = karatsuba_mul(xl, yl);
let mut z1 = karatsuba_mul(&sumx, &sumy);
let z2 = karatsuba_mul(xh, yh);
// Properly scale z1, which is `z1 - z2 - zo`.
isub(&mut z1, &z2);
isub(&mut z1, &z0);
// Create our result, which is equal to, in little-endian order:
// [z0, z1 - z2 - z0, z2]
// z1 must be shifted m digits (2^(32m)) over.
// z2 must be shifted 2*m digits (2^(64m)) over.
let len = z0.len().max(m + z1.len()).max(2 * m + z2.len());
let mut result = z0;
result.reserve_exact(len - result.len());
iadd_impl(&mut result, &z1, m);
iadd_impl(&mut result, &z2, 2 * m);
result
}
}
/// Karatsuba multiplication algorithm where y is substantially larger than x.
///
/// Assumes `y.len() >= x.len()`.
fn karatsuba_uneven_mul(x: &[Limb], mut y: &[Limb]) -> Vec<Limb> {
let mut result = Vec::<Limb>::default();
result.resize(x.len() + y.len(), 0);
// This effectively is like grade-school multiplication between
// two numbers, except we're using splits on `y`, and the intermediate
// step is a Karatsuba multiplication.
let mut start = 0;
while !y.is_empty() {
let m = x.len().min(y.len());
let (yl, yh) = karatsuba_split(y, m);
let prod = karatsuba_mul(x, yl);
iadd_impl(&mut result, &prod, start);
y = yh;
start += m;
}
small::normalize(&mut result);
result
}
/// Forwarder to the proper Karatsuba algorithm.
#[inline]
fn karatsuba_mul_fwd(x: &[Limb], y: &[Limb]) -> Vec<Limb> {
if x.len() < y.len() {
karatsuba_mul(x, y)
} else {
karatsuba_mul(y, x)
}
}
/// MulAssign bigint to bigint.
#[inline]
pub fn imul(x: &mut Vec<Limb>, y: &[Limb]) {
if y.len() == 1 {
small::imul(x, y[0]);
} else {
// We're not really in a condition where using Karatsuba
// multiplication makes sense, so we're just going to use long
// division. ~20% speedup compared to:
// *x = karatsuba_mul_fwd(x, y);
*x = karatsuba_mul_fwd(x, y);
}
}
} // large
// TRAITS
// ------
/// Traits for shared operations for big integers.
///
/// None of these are implemented using normal traits, since these
/// are very expensive operations, and we want to deliberately
/// and explicitly use these functions.
pub(crate) trait Math: Clone + Sized + Default {
// DATA
/// Get access to the underlying data
fn data(&self) -> &Vec<Limb>;
/// Get access to the underlying data
fn data_mut(&mut self) -> &mut Vec<Limb>;
// RELATIVE OPERATIONS
/// Compare self to y.
#[inline]
fn compare(&self, y: &Self) -> cmp::Ordering {
large::compare(self.data(), y.data())
}
// PROPERTIES
/// Get the high 64-bits from the bigint and if there are remaining bits.
#[inline]
fn hi64(&self) -> (u64, bool) {
self.data().as_slice().hi64()
}
/// Calculate the bit-length of the big-integer.
/// Returns usize::max_value() if the value overflows,
/// IE, if `self.data().len() > usize::max_value() / 8`.
#[inline]
fn bit_length(&self) -> usize {
small::bit_length(self.data())
}
// INTEGER CONVERSIONS
/// Create new big integer from u64.
#[inline]
fn from_u64(x: u64) -> Self {
let mut v = Self::default();
let slc = split_u64(x);
v.data_mut().extend_from_slice(&slc);
v.normalize();
v
}
// NORMALIZE
/// Normalize the integer, so any leading zero values are removed.
#[inline]
fn normalize(&mut self) {
small::normalize(self.data_mut());
}
// ADDITION
/// AddAssign small integer.
#[inline]
fn iadd_small(&mut self, y: Limb) {
small::iadd(self.data_mut(), y);
}
// MULTIPLICATION
/// MulAssign small integer.
#[inline]
fn imul_small(&mut self, y: Limb) {
small::imul(self.data_mut(), y);
}
/// Multiply by a power of 2.
#[inline]
fn imul_pow2(&mut self, n: u32) {
self.ishl(n as usize);
}
/// Multiply by a power of 5.
#[inline]
fn imul_pow5(&mut self, n: u32) {
small::imul_pow5(self.data_mut(), n);
}
/// MulAssign by a power of 10.
#[inline]
fn imul_pow10(&mut self, n: u32) {
self.imul_pow5(n);
self.imul_pow2(n);
}
// SHIFTS
/// Shift-left the entire buffer n bits.
#[inline]
fn ishl(&mut self, n: usize) {
small::ishl(self.data_mut(), n);
}
}