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//! `strength_reduce` implements integer division and modulo via "arithmetic strength reduction"
//!
//! Modern processors can do multiplication and shifts much faster than division, and "arithmetic strength reduction" is an algorithm to transform divisions into multiplications and shifts.
//! Compilers already perform this optimization for divisors that are known at compile time; this library enables this optimization for divisors that are only known at runtime.
//!
//! Benchmarking shows a 5-10x speedup or integer division and modulo operations.
//!
//! # Example:
//! ```
//! use strength_reduce::StrengthReducedU64;
//!
//! let mut my_array: Vec<u64> = (0..500).collect();
//! let divisor = 3;
//! let modulo = 14;
//!
//! // slow naive division and modulo
//! for element in &mut my_array {
//! *element = (*element / divisor) % modulo;
//! }
//!
//! // fast strength-reduced division and modulo
//! let reduced_divisor = StrengthReducedU64::new(divisor);
//! let reduced_modulo = StrengthReducedU64::new(modulo);
//! for element in &mut my_array {
//! *element = (*element / reduced_divisor) % reduced_modulo;
//! }
//! ```
//!
//! This library is intended for hot loops like the example above, where a division is repeated many times in a loop with the divisor remaining unchanged.
//! There is a setup cost associated with creating stength-reduced division instances, so using strength-reduced division for 1-2 divisions is not worth the setup cost.
//! The break-even point differs by use-case, but is typically low: Benchmarking has shown that takes 3 to 4 repeated divisions with the same StengthReduced## instance to be worth it.
//!
//! `strength_reduce` is `#![no_std]`
//!
//! The optimizations that this library provides are inherently dependent on architecture, compiler, and platform,
//! so test before you use.
#![no_std]
#[cfg(test)]
extern crate num_bigint;
#[cfg(test)]
extern crate rand;
use core::ops::{Div, Rem};
mod long_division;
mod long_multiplication;
/// Implements unsigned division and modulo via mutiplication and shifts.
///
/// Creating a an instance of this struct is more expensive than a single division, but if the division is repeated,
/// this version will be several times faster than naive division.
#[derive(Clone, Copy, Debug)]
pub struct StrengthReducedU8 {
multiplier: u16,
divisor: u8,
}
impl StrengthReducedU8 {
/// Creates a new divisor instance.
///
/// If possible, avoid calling new() from an inner loop: The intended usage is to create an instance of this struct outside the loop, and use it for divison and remainders inside the loop.
///
/// # Panics:
///
/// Panics if `divisor` is 0
#[inline]
pub fn new(divisor: u8) -> Self {
assert!(divisor > 0);
if divisor.is_power_of_two() {
Self{ multiplier: 0, divisor }
} else {
let divided = core::u16::MAX / (divisor as u16);
Self{ multiplier: divided + 1, divisor }
}
}
/// Simultaneous truncated integer division and modulus.
/// Returns `(quotient, remainder)`.
#[inline]
pub fn div_rem(numerator: u8, denom: Self) -> (u8, u8) {
let quotient = numerator / denom;
let remainder = numerator % denom;
(quotient, remainder)
}
/// Retrieve the value used to create this struct
#[inline]
pub fn get(&self) -> u8 {
self.divisor
}
}
impl Div<StrengthReducedU8> for u8 {
type Output = u8;
#[inline]
fn div(self, rhs: StrengthReducedU8) -> Self::Output {
if rhs.multiplier == 0 {
(self as u16 >> rhs.divisor.trailing_zeros()) as u8
} else {
let numerator = self as u16;
let multiplied_hi = numerator * (rhs.multiplier >> 8);
let multiplied_lo = (numerator * rhs.multiplier as u8 as u16) >> 8;
((multiplied_hi + multiplied_lo) >> 8) as u8
}
}
}
impl Rem<StrengthReducedU8> for u8 {
type Output = u8;
#[inline]
fn rem(self, rhs: StrengthReducedU8) -> Self::Output {
if rhs.multiplier == 0 {
self & (rhs.divisor - 1)
} else {
let product = rhs.multiplier.wrapping_mul(self as u16) as u32;
let divisor = rhs.divisor as u32;
let shifted = (product * divisor) >> 16;
shifted as u8
}
}
}
// small types prefer to do work in the intermediate type
macro_rules! strength_reduced_u16 {
($struct_name:ident, $primitive_type:ident) => (
/// Implements unsigned division and modulo via mutiplication and shifts.
///
/// Creating a an instance of this struct is more expensive than a single division, but if the division is repeated,
/// this version will be several times faster than naive division.
#[derive(Clone, Copy, Debug)]
pub struct $struct_name {
multiplier: u32,
divisor: $primitive_type,
}
impl $struct_name {
/// Creates a new divisor instance.
///
/// If possible, avoid calling new() from an inner loop: The intended usage is to create an instance of this struct outside the loop, and use it for divison and remainders inside the loop.
///
/// # Panics:
///
/// Panics if `divisor` is 0
#[inline]
pub fn new(divisor: $primitive_type) -> Self {
assert!(divisor > 0);
if divisor.is_power_of_two() {
Self{ multiplier: 0, divisor }
} else {
let divided = core::u32::MAX / (divisor as u32);
Self{ multiplier: divided + 1, divisor }
}
}
/// Simultaneous truncated integer division and modulus.
/// Returns `(quotient, remainder)`.
#[inline]
pub fn div_rem(numerator: $primitive_type, denom: Self) -> ($primitive_type, $primitive_type) {
let quotient = numerator / denom;
let remainder = numerator - quotient * denom.divisor;
(quotient, remainder)
}
/// Retrieve the value used to create this struct
#[inline]
pub fn get(&self) -> $primitive_type {
self.divisor
}
}
impl Div<$struct_name> for $primitive_type {
type Output = $primitive_type;
#[inline]
fn div(self, rhs: $struct_name) -> Self::Output {
if rhs.multiplier == 0 {
self >> rhs.divisor.trailing_zeros()
} else {
let numerator = self as u32;
let multiplied_hi = numerator * (rhs.multiplier >> 16);
let multiplied_lo = (numerator * rhs.multiplier as u16 as u32) >> 16;
((multiplied_hi + multiplied_lo) >> 16) as $primitive_type
}
}
}
impl Rem<$struct_name> for $primitive_type {
type Output = $primitive_type;
#[inline]
fn rem(self, rhs: $struct_name) -> Self::Output {
if rhs.multiplier == 0 {
self & (rhs.divisor - 1)
} else {
let quotient = self / rhs;
self - quotient * rhs.divisor
}
}
}
)
}
// small types prefer to do work in the intermediate type
macro_rules! strength_reduced_u32 {
($struct_name:ident, $primitive_type:ident) => (
/// Implements unsigned division and modulo via mutiplication and shifts.
///
/// Creating a an instance of this struct is more expensive than a single division, but if the division is repeated,
/// this version will be several times faster than naive division.
#[derive(Clone, Copy, Debug)]
pub struct $struct_name {
multiplier: u64,
divisor: $primitive_type,
}
impl $struct_name {
/// Creates a new divisor instance.
///
/// If possible, avoid calling new() from an inner loop: The intended usage is to create an instance of this struct outside the loop, and use it for divison and remainders inside the loop.
///
/// # Panics:
///
/// Panics if `divisor` is 0
#[inline]
pub fn new(divisor: $primitive_type) -> Self {
assert!(divisor > 0);
if divisor.is_power_of_two() {
Self{ multiplier: 0, divisor }
} else {
let divided = core::u64::MAX / (divisor as u64);
Self{ multiplier: divided + 1, divisor }
}
}
/// Simultaneous truncated integer division and modulus.
/// Returns `(quotient, remainder)`.
#[inline]
pub fn div_rem(numerator: $primitive_type, denom: Self) -> ($primitive_type, $primitive_type) {
if denom.multiplier == 0 {
(numerator >> denom.divisor.trailing_zeros(), numerator & (denom.divisor - 1))
}
else {
let numerator64 = numerator as u64;
let multiplied_hi = numerator64 * (denom.multiplier >> 32);
let multiplied_lo = numerator64 * (denom.multiplier as u32 as u64) >> 32;
let quotient = ((multiplied_hi + multiplied_lo) >> 32) as $primitive_type;
let remainder = numerator - quotient * denom.divisor;
(quotient, remainder)
}
}
/// Retrieve the value used to create this struct
#[inline]
pub fn get(&self) -> $primitive_type {
self.divisor
}
}
impl Div<$struct_name> for $primitive_type {
type Output = $primitive_type;
#[inline]
fn div(self, rhs: $struct_name) -> Self::Output {
if rhs.multiplier == 0 {
self >> rhs.divisor.trailing_zeros()
} else {
let numerator = self as u64;
let multiplied_hi = numerator * (rhs.multiplier >> 32);
let multiplied_lo = numerator * (rhs.multiplier as u32 as u64) >> 32;
((multiplied_hi + multiplied_lo) >> 32) as $primitive_type
}
}
}
impl Rem<$struct_name> for $primitive_type {
type Output = $primitive_type;
#[inline]
fn rem(self, rhs: $struct_name) -> Self::Output {
if rhs.multiplier == 0 {
self & (rhs.divisor - 1)
} else {
let product = rhs.multiplier.wrapping_mul(self as u64) as u128;
let divisor = rhs.divisor as u128;
let shifted = (product * divisor) >> 64;
shifted as $primitive_type
}
}
}
)
}
macro_rules! strength_reduced_u64 {
($struct_name:ident, $primitive_type:ident) => (
/// Implements unsigned division and modulo via mutiplication and shifts.
///
/// Creating a an instance of this struct is more expensive than a single division, but if the division is repeated,
/// this version will be several times faster than naive division.
#[derive(Clone, Copy, Debug)]
pub struct $struct_name {
multiplier: u128,
divisor: $primitive_type,
}
impl $struct_name {
/// Creates a new divisor instance.
///
/// If possible, avoid calling new() from an inner loop: The intended usage is to create an instance of this struct outside the loop, and use it for divison and remainders inside the loop.
///
/// # Panics:
///
/// Panics if `divisor` is 0
#[inline]
pub fn new(divisor: $primitive_type) -> Self {
assert!(divisor > 0);
if divisor.is_power_of_two() {
Self{ multiplier: 0, divisor }
} else {
let quotient = long_division::divide_128_max_by_64(divisor as u64);
Self{ multiplier: quotient + 1, divisor }
}
}
/// Simultaneous truncated integer division and modulus.
/// Returns `(quotient, remainder)`.
#[inline]
pub fn div_rem(numerator: $primitive_type, denom: Self) -> ($primitive_type, $primitive_type) {
if denom.multiplier == 0 {
(numerator >> denom.divisor.trailing_zeros(), numerator & (denom.divisor - 1))
}
else {
let numerator128 = numerator as u128;
let multiplied_hi = numerator128 * (denom.multiplier >> 64);
let multiplied_lo = numerator128 * (denom.multiplier as u64 as u128) >> 64;
let quotient = ((multiplied_hi + multiplied_lo) >> 64) as $primitive_type;
let remainder = numerator - quotient * denom.divisor;
(quotient, remainder)
}
}
/// Retrieve the value used to create this struct
#[inline]
pub fn get(&self) -> $primitive_type {
self.divisor
}
}
impl Div<$struct_name> for $primitive_type {
type Output = $primitive_type;
#[inline]
fn div(self, rhs: $struct_name) -> Self::Output {
if rhs.multiplier == 0 {
self >> rhs.divisor.trailing_zeros()
} else {
let numerator = self as u128;
let multiplied_hi = numerator * (rhs.multiplier >> 64);
let multiplied_lo = numerator * (rhs.multiplier as u64 as u128) >> 64;
((multiplied_hi + multiplied_lo) >> 64) as $primitive_type
}
}
}
impl Rem<$struct_name> for $primitive_type {
type Output = $primitive_type;
#[inline]
fn rem(self, rhs: $struct_name) -> Self::Output {
if rhs.multiplier == 0 {
self & (rhs.divisor - 1)
} else {
let quotient = self / rhs;
self - quotient * rhs.divisor
}
}
}
)
}
/// Implements unsigned division and modulo via mutiplication and shifts.
///
/// Creating a an instance of this struct is more expensive than a single division, but if the division is repeated,
/// this version will be several times faster than naive division.
#[derive(Clone, Copy, Debug)]
pub struct StrengthReducedU128 {
multiplier_hi: u128,
multiplier_lo: u128,
divisor: u128,
}
impl StrengthReducedU128 {
/// Creates a new divisor instance.
///
/// If possible, avoid calling new() from an inner loop: The intended usage is to create an instance of this struct outside the loop, and use it for divison and remainders inside the loop.
///
/// # Panics:
///
/// Panics if `divisor` is 0
#[inline]
pub fn new(divisor: u128) -> Self {
assert!(divisor > 0);
if divisor.is_power_of_two() {
Self{ multiplier_hi: 0, multiplier_lo: 0, divisor }
} else {
let (quotient_hi, quotient_lo) = long_division::divide_256_max_by_128(divisor);
let multiplier_lo = quotient_lo.wrapping_add(1);
let multiplier_hi = if multiplier_lo == 0 { quotient_hi + 1 } else { quotient_hi };
Self{ multiplier_hi, multiplier_lo, divisor }
}
}
/// Simultaneous truncated integer division and modulus.
/// Returns `(quotient, remainder)`.
#[inline]
pub fn div_rem(numerator: u128, denom: Self) -> (u128, u128) {
let quotient = numerator / denom;
let remainder = numerator - quotient * denom.divisor;
(quotient, remainder)
}
/// Retrieve the value used to create this struct
#[inline]
pub fn get(&self) -> u128 {
self.divisor
}
}
impl Div<StrengthReducedU128> for u128 {
type Output = u128;
#[inline]
fn div(self, rhs: StrengthReducedU128) -> Self::Output {
if rhs.multiplier_hi == 0 {
self >> rhs.divisor.trailing_zeros()
} else {
long_multiplication::multiply_256_by_128_upperbits(rhs.multiplier_hi, rhs.multiplier_lo, self)
}
}
}
impl Rem<StrengthReducedU128> for u128 {
type Output = u128;
#[inline]
fn rem(self, rhs: StrengthReducedU128) -> Self::Output {
if rhs.multiplier_hi == 0 {
self & (rhs.divisor - 1)
} else {
let quotient = long_multiplication::multiply_256_by_128_upperbits(rhs.multiplier_hi, rhs.multiplier_lo, self);
self - quotient * rhs.divisor
}
}
}
// We just hardcoded u8 and u128 since they will never be a usize. for the rest, we have macros, so we can reuse the same code for usize
strength_reduced_u16!(StrengthReducedU16, u16);
strength_reduced_u32!(StrengthReducedU32, u32);
strength_reduced_u64!(StrengthReducedU64, u64);
// Our definition for usize will depend on how big usize is
#[cfg(target_pointer_width = "16")]
strength_reduced_u16!(StrengthReducedUsize, usize);
#[cfg(target_pointer_width = "32")]
strength_reduced_u32!(StrengthReducedUsize, usize);
#[cfg(target_pointer_width = "64")]
strength_reduced_u64!(StrengthReducedUsize, usize);
#[cfg(test)]
mod unit_tests {
use super::*;
macro_rules! reduction_test {
($test_name:ident, $struct_name:ident, $primitive_type:ident) => (
#[test]
fn $test_name() {
let max = core::$primitive_type::MAX;
let divisors = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,max-1,max];
let numerators = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20];
for &divisor in &divisors {
let reduced_divisor = $struct_name::new(divisor);
for &numerator in &numerators {
let expected_div = numerator / divisor;
let expected_rem = numerator % divisor;
let reduced_div = numerator / reduced_divisor;
assert_eq!(expected_div, reduced_div, "Divide failed with numerator: {}, divisor: {}", numerator, divisor);
let reduced_rem = numerator % reduced_divisor;
let (reduced_combined_div, reduced_combined_rem) = $struct_name::div_rem(numerator, reduced_divisor);
assert_eq!(expected_rem, reduced_rem, "Modulo failed with numerator: {}, divisor: {}", numerator, divisor);
assert_eq!(expected_div, reduced_combined_div, "div_rem divide failed with numerator: {}, divisor: {}", numerator, divisor);
assert_eq!(expected_rem, reduced_combined_rem, "div_rem modulo failed with numerator: {}, divisor: {}", numerator, divisor);
}
}
}
)
}
reduction_test!(test_strength_reduced_u8, StrengthReducedU8, u8);
reduction_test!(test_strength_reduced_u16, StrengthReducedU16, u16);
reduction_test!(test_strength_reduced_u32, StrengthReducedU32, u32);
reduction_test!(test_strength_reduced_u64, StrengthReducedU64, u64);
reduction_test!(test_strength_reduced_usize, StrengthReducedUsize, usize);
reduction_test!(test_strength_reduced_u128, StrengthReducedU128, u128);
}
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