// Copyright 2013-2014 The Rust Project Developers. See the COPYRIGHT
// file at the top-level directory of this distribution and at
//
http://rust-lang.org/COPYRIGHT.
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
//
http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or
http://opensource.org/licenses/MIT>, at your
// option. This file may not be copied, modified, or distributed
// except according to those terms.
//! Integer trait and functions.
//!
//! ## Compatibility
//!
//! The `num-integer` crate is tested for rustc 1.8 and greater.
#![doc(html_root_url = "
https://docs.rs/num-integer/0.1")]
#![no_std]
#[cfg(feature = "std")]
extern crate std;
extern crate num_traits as traits;
use core::mem;
use core::ops::Add;
use traits::{Num, Signed, Zero};
mod roots;
pub use roots::Roots;
pub use roots::{cbrt, nth_root, sqrt};
mod average;
pub use average::Average;
pub use average::{average_ceil, average_floor};
pub trait Integer: Sized + Num + PartialOrd + Ord + Eq {
/// Floored integer division.
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert!(( 8).div_floor(& 3) == 2);
/// assert!(( 8).div_floor(&-3) == -3);
/// assert!((-8).div_floor(& 3) == -3);
/// assert!((-8).div_floor(&-3) == 2);
///
/// assert!(( 1).div_floor(& 2) == 0);
/// assert!(( 1).div_floor(&-2) == -1);
/// assert!((-1).div_floor(& 2) == -1);
/// assert!((-1).div_floor(&-2) == 0);
/// ~~~
fn div_floor(&self, other: &Self) -> Self;
/// Floored integer modulo, satisfying:
///
/// ~~~
/// # use num_integer::Integer;
/// # let n = 1; let d = 1;
/// assert!(n.div_floor(&d) * d + n.mod_floor(&d) == n)
/// ~~~
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert!(( 8).mod_floor(& 3) == 2);
/// assert!(( 8).mod_floor(&-3) == -1);
/// assert!((-8).mod_floor(& 3) == 1);
/// assert!((-8).mod_floor(&-3) == -2);
///
/// assert!(( 1).mod_floor(& 2) == 1);
/// assert!(( 1).mod_floor(&-2) == -1);
/// assert!((-1).mod_floor(& 2) == 1);
/// assert!((-1).mod_floor(&-2) == -1);
/// ~~~
fn mod_floor(&self, other: &Self) -> Self;
/// Ceiled integer division.
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert_eq!(( 8).div_ceil( &3), 3);
/// assert_eq!(( 8).div_ceil(&-3), -2);
/// assert_eq!((-8).div_ceil( &3), -2);
/// assert_eq!((-8).div_ceil(&-3), 3);
///
/// assert_eq!(( 1).div_ceil( &2), 1);
/// assert_eq!(( 1).div_ceil(&-2), 0);
/// assert_eq!((-1).div_ceil( &2), 0);
/// assert_eq!((-1).div_ceil(&-2), 1);
/// ~~~
fn div_ceil(&self, other: &Self) -> Self {
let (q, r) = self.div_mod_floor(other);
if r.is_zero() {
q
} else {
q + Self::one()
}
}
/// Greatest Common Divisor (GCD).
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert_eq!(6.gcd(&8), 2);
/// assert_eq!(7.gcd(&3), 1);
/// ~~~
fn gcd(&self, other: &Self) -> Self;
/// Lowest Common Multiple (LCM).
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert_eq!(7.lcm(&3), 21);
/// assert_eq!(2.lcm(&4), 4);
/// assert_eq!(0.lcm(&0), 0);
/// ~~~
fn lcm(&self, other: &Self) -> Self;
/// Greatest Common Divisor (GCD) and
/// Lowest Common Multiple (LCM) together.
///
/// Potentially more efficient than calling `gcd` and `lcm`
/// individually for identical inputs.
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert_eq!(10.gcd_lcm(&4), (2, 20));
/// assert_eq!(8.gcd_lcm(&9), (1, 72));
/// ~~~
#[inline]
fn gcd_lcm(&self, other: &Self) -> (Self, Self) {
(self.gcd(other), self.lcm(other))
}
/// Greatest common divisor and Bézout coefficients.
///
/// # Examples
///
/// ~~~
/// # extern crate num_integer;
/// # extern crate num_traits;
/// # fn main() {
/// # use num_integer::{ExtendedGcd, Integer};
/// # use num_traits::NumAssign;
/// fn check<A: Copy + Integer + NumAssign>(a: A, b: A) -> bool {
/// let ExtendedGcd { gcd, x, y, .. } = a.extended_gcd(&b);
/// gcd == x * a + y * b
/// }
/// assert!(check(10isize, 4isize));
/// assert!(check(8isize, 9isize));
/// # }
/// ~~~
#[inline]
fn extended_gcd(&self, other: &Self) -> ExtendedGcd<Self>
where
Self: Clone,
{
let mut s = (Self::zero(), Self::one());
let mut t = (Self::one(), Self::zero());
let mut r = (other.clone(), self.clone());
while !r.0.is_zero() {
let q = r.1.clone() / r.0.clone();
let f = |mut r: (Self, Self)| {
mem::swap(&mut r.0, &mut r.1);
r.0 = r.0 - q.clone() * r.1.clone();
r
};
r = f(r);
s = f(s);
t = f(t);
}
if r.1 >= Self::zero() {
ExtendedGcd {
gcd: r.1,
x: s.1,
y: t.1,
}
} else {
ExtendedGcd {
gcd: Self::zero() - r.1,
x: Self::zero() - s.1,
y: Self::zero() - t.1,
}
}
}
/// Greatest common divisor, least common multiple, and Bézout coefficients.
#[inline]
fn extended_gcd_lcm(&self, other: &Self) -> (ExtendedGcd<Self>, Self)
where
Self: Clone + Signed,
{
(self.extended_gcd(other), self.lcm(other))
}
/// Deprecated, use `is_multiple_of` instead.
fn divides(&self, other: &Self) -> bool;
/// Returns `true` if `self` is a multiple of `other`.
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert_eq!(9.is_multiple_of(&3), true);
/// assert_eq!(3.is_multiple_of(&9), false);
/// ~~~
fn is_multiple_of(&self, other: &Self) -> bool;
/// Returns `true` if the number is even.
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert_eq!(3.is_even(), false);
/// assert_eq!(4.is_even(), true);
/// ~~~
fn is_even(&self) -> bool;
/// Returns `true` if the number is odd.
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert_eq!(3.is_odd(), true);
/// assert_eq!(4.is_odd(), false);
/// ~~~
fn is_odd(&self) -> bool;
/// Simultaneous truncated integer division and modulus.
/// Returns `(quotient, remainder)`.
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert_eq!(( 8).div_rem( &3), ( 2, 2));
/// assert_eq!(( 8).div_rem(&-3), (-2, 2));
/// assert_eq!((-8).div_rem( &3), (-2, -2));
/// assert_eq!((-8).div_rem(&-3), ( 2, -2));
///
/// assert_eq!(( 1).div_rem( &2), ( 0, 1));
/// assert_eq!(( 1).div_rem(&-2), ( 0, 1));
/// assert_eq!((-1).div_rem( &2), ( 0, -1));
/// assert_eq!((-1).div_rem(&-2), ( 0, -1));
/// ~~~
fn div_rem(&self, other: &Self) -> (Self, Self);
/// Simultaneous floored integer division and modulus.
/// Returns `(quotient, remainder)`.
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert_eq!(( 8).div_mod_floor( &3), ( 2, 2));
/// assert_eq!(( 8).div_mod_floor(&-3), (-3, -1));
/// assert_eq!((-8).div_mod_floor( &3), (-3, 1));
/// assert_eq!((-8).div_mod_floor(&-3), ( 2, -2));
///
/// assert_eq!(( 1).div_mod_floor( &2), ( 0, 1));
/// assert_eq!(( 1).div_mod_floor(&-2), (-1, -1));
/// assert_eq!((-1).div_mod_floor( &2), (-1, 1));
/// assert_eq!((-1).div_mod_floor(&-2), ( 0, -1));
/// ~~~
fn div_mod_floor(&self, other: &Self) -> (Self, Self) {
(self.div_floor(other), self.mod_floor(other))
}
/// Rounds up to nearest multiple of argument.
///
/// # Notes
///
/// For signed types, `a.next_multiple_of(b) = a.prev_multiple_of(b.neg())`.
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert_eq!(( 16).next_multiple_of(& 8), 16);
/// assert_eq!(( 23).next_multiple_of(& 8), 24);
/// assert_eq!(( 16).next_multiple_of(&-8), 16);
/// assert_eq!(( 23).next_multiple_of(&-8), 16);
/// assert_eq!((-16).next_multiple_of(& 8), -16);
/// assert_eq!((-23).next_multiple_of(& 8), -16);
/// assert_eq!((-16).next_multiple_of(&-8), -16);
/// assert_eq!((-23).next_multiple_of(&-8), -24);
/// ~~~
#[inline]
fn next_multiple_of(&self, other: &Self) -> Self
where
Self: Clone,
{
let m = self.mod_floor(other);
self.clone()
+ if m.is_zero() {
Self::zero()
} else {
other.clone() - m
}
}
/// Rounds down to nearest multiple of argument.
///
/// # Notes
///
/// For signed types, `a.prev_multiple_of(b) = a.next_multiple_of(b.neg())`.
///
/// # Examples
///
/// ~~~
/// # use num_integer::Integer;
/// assert_eq!(( 16).prev_multiple_of(& 8), 16);
/// assert_eq!(( 23).prev_multiple_of(& 8), 16);
/// assert_eq!(( 16).prev_multiple_of(&-8), 16);
/// assert_eq!(( 23).prev_multiple_of(&-8), 24);
/// assert_eq!((-16).prev_multiple_of(& 8), -16);
/// assert_eq!((-23).prev_multiple_of(& 8), -24);
/// assert_eq!((-16).prev_multiple_of(&-8), -16);
/// assert_eq!((-23).prev_multiple_of(&-8), -16);
/// ~~~
#[inline]
fn prev_multiple_of(&self, other: &Self) -> Self
where
Self: Clone,
{
self.clone() - self.mod_floor(other)
}
}
/// Greatest common divisor and Bézout coefficients
///
/// ```no_build
/// let e = isize::extended_gcd(a, b);
/// assert_eq!(e.gcd, e.x*a + e.y*b);
/// ```
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub struct ExtendedGcd<A> {
pub gcd: A,
pub x: A,
pub y: A,
}
/// Simultaneous integer division and modulus
#[inline]
pub fn div_rem<T: Integer>(x: T, y: T) -> (T, T) {
x.div_rem(&y)
}
/// Floored integer division
#[inline]
pub fn div_floor<T: Integer>(x: T, y: T) -> T {
x.div_floor(&y)
}
/// Floored integer modulus
#[inline]
pub fn mod_floor<T: Integer>(x: T, y: T) -> T {
x.mod_floor(&y)
}
/// Simultaneous floored integer division and modulus
#[inline]
pub fn div_mod_floor<T: Integer>(x: T, y: T) -> (T, T) {
x.div_mod_floor(&y)
}
/// Ceiled integer division
#[inline]
pub fn div_ceil<T: Integer>(x: T, y: T) -> T {
x.div_ceil(&y)
}
/// Calculates the Greatest Common Divisor (GCD) of the number and `other`. The
/// result is always non-negative.
#[inline(always)]
pub fn gcd<T: Integer>(x: T, y: T) -> T {
x.gcd(&y)
}
/// Calculates the Lowest Common Multiple (LCM) of the number and `other`.
#[inline(always)]
pub fn lcm<T: Integer>(x: T, y: T) -> T {
x.lcm(&y)
}
/// Calculates the Greatest Common Divisor (GCD) and
/// Lowest Common Multiple (LCM) of the number and `other`.
#[inline(always)]
pub fn gcd_lcm<T: Integer>(x: T, y: T) -> (T, T) {
x.gcd_lcm(&y)
}
macro_rules! impl_integer_for_isize {
($T:ty, $test_mod:ident) => {
impl Integer for $T {
/// Floored integer division
#[inline]
fn div_floor(&self, other: &Self) -> Self {
// Algorithm from [Daan Leijen. _Division and Modulus for Computer Scientists_,
// December 2001](
http://research.microsoft.com/pubs/151917/divmodnote-letter.pdf)
let (d, r) = self.div_rem(other);
if (r > 0 && *other < 0) || (r < 0 && *other > 0) {
d - 1
} else {
d
}
}
/// Floored integer modulo
#[inline]
fn mod_floor(&self, other: &Self) -> Self {
// Algorithm from [Daan Leijen. _Division and Modulus for Computer Scientists_,
// December 2001](
http://research.microsoft.com/pubs/151917/divmodnote-letter.pdf)
let r = *self % *other;
if (r > 0 && *other < 0) || (r < 0 && *other > 0) {
r + *other
} else {
r
}
}
/// Calculates `div_floor` and `mod_floor` simultaneously
#[inline]
fn div_mod_floor(&self, other: &Self) -> (Self, Self) {
// Algorithm from [Daan Leijen. _Division and Modulus for Computer Scientists_,
// December 2001](
http://research.microsoft.com/pubs/151917/divmodnote-letter.pdf)
let (d, r) = self.div_rem(other);
if (r > 0 && *other < 0) || (r < 0 && *other > 0) {
(d - 1, r + *other)
} else {
(d, r)
}
}
#[inline]
fn div_ceil(&self, other: &Self) -> Self {
let (d, r) = self.div_rem(other);
if (r > 0 && *other > 0) || (r < 0 && *other < 0) {
d + 1
} else {
d
}
}
/// Calculates the Greatest Common Divisor (GCD) of the number and
/// `other`. The result is always non-negative.
#[inline]
fn gcd(&self, other: &Self) -> Self {
// Use Stein's algorithm
let mut m = *self;
let mut n = *other;
if m == 0 || n == 0 {
return (m | n).abs();
}
// find common factors of 2
let shift = (m | n).trailing_zeros();
// The algorithm needs positive numbers, but the minimum value
// can't be represented as a positive one.
// It's also a power of two, so the gcd can be
// calculated by bitshifting in that case
// Assuming two's complement, the number created by the shift
// is positive for all numbers except gcd = abs(min value)
// The call to .abs() causes a panic in debug mode
if m == Self::min_value() || n == Self::min_value() {
return (1 << shift).abs();
}
// guaranteed to be positive now, rest like unsigned algorithm
m = m.abs();
n = n.abs();
// divide n and m by 2 until odd
m >>= m.trailing_zeros();
n >>= n.trailing_zeros();
while m != n {
if m > n {
m -= n;
m >>= m.trailing_zeros();
} else {
n -= m;
n >>= n.trailing_zeros();
}
}
m << shift
}
#[inline]
fn extended_gcd_lcm(&self, other: &Self) -> (ExtendedGcd<Self>, Self) {
let egcd = self.extended_gcd(other);
// should not have to recalculate abs
let lcm = if egcd.gcd.is_zero() {
Self::zero()
} else {
(*self * (*other / egcd.gcd)).abs()
};
(egcd, lcm)
}
/// Calculates the Lowest Common Multiple (LCM) of the number and
/// `other`.
#[inline]
fn lcm(&self, other: &Self) -> Self {
self.gcd_lcm(other).1
}
/// Calculates the Greatest Common Divisor (GCD) and
/// Lowest Common Multiple (LCM) of the number and `other`.
#[inline]
fn gcd_lcm(&self, other: &Self) -> (Self, Self) {
if self.is_zero() && other.is_zero() {
return (Self::zero(), Self::zero());
}
let gcd = self.gcd(other);
// should not have to recalculate abs
let lcm = (*self * (*other / gcd)).abs();
(gcd, lcm)
}
/// Deprecated, use `is_multiple_of` instead.
#[inline]
fn divides(&self, other: &Self) -> bool {
self.is_multiple_of(other)
}
/// Returns `true` if the number is a multiple of `other`.
#[inline]
fn is_multiple_of(&self, other: &Self) -> bool {
if other.is_zero() {
return self.is_zero();
}
*self % *other == 0
}
/// Returns `true` if the number is divisible by `2`
#[inline]
fn is_even(&self) -> bool {
(*self) & 1 == 0
}
/// Returns `true` if the number is not divisible by `2`
#[inline]
fn is_odd(&self) -> bool {
!self.is_even()
}
/// Simultaneous truncated integer division and modulus.
#[inline]
fn div_rem(&self, other: &Self) -> (Self, Self) {
(*self / *other, *self % *other)
}
/// Rounds up to nearest multiple of argument.
#[inline]
fn next_multiple_of(&self, other: &Self) -> Self {
// Avoid the overflow of `MIN % -1`
if *other == -1 {
return *self;
}
let m = Integer::mod_floor(self, other);
*self + if m == 0 { 0 } else { other - m }
}
/// Rounds down to nearest multiple of argument.
#[inline]
fn prev_multiple_of(&self, other: &Self) -> Self {
// Avoid the overflow of `MIN % -1`
if *other == -1 {
return *self;
}
*self - Integer::mod_floor(self, other)
}
}
#[cfg(test)]
mod $test_mod {
use core::mem;
use Integer;
/// Checks that the division rule holds for:
///
/// - `n`: numerator (dividend)
/// - `d`: denominator (divisor)
/// - `qr`: quotient and remainder
#[cfg(test)]
fn test_division_rule((n, d): ($T, $T), (q, r): ($T, $T)) {
assert_eq!(d * q + r, n);
}
#[test]
fn test_div_rem() {
fn test_nd_dr(nd: ($T, $T), qr: ($T, $T)) {
let (n, d) = nd;
let separate_div_rem = (n / d, n % d);
let combined_div_rem = n.div_rem(&d);
assert_eq!(separate_div_rem, qr);
assert_eq!(combined_div_rem, qr);
test_division_rule(nd, separate_div_rem);
test_division_rule(nd, combined_div_rem);
}
test_nd_dr((8, 3), (2, 2));
test_nd_dr((8, -3), (-2, 2));
test_nd_dr((-8, 3), (-2, -2));
test_nd_dr((-8, -3), (2, -2));
test_nd_dr((1, 2), (0, 1));
test_nd_dr((1, -2), (0, 1));
test_nd_dr((-1, 2), (0, -1));
test_nd_dr((-1, -2), (0, -1));
}
#[test]
fn test_div_mod_floor() {
fn test_nd_dm(nd: ($T, $T), dm: ($T, $T)) {
let (n, d) = nd;
let separate_div_mod_floor =
(Integer::div_floor(&n, &d), Integer::mod_floor(&n, &d));
let combined_div_mod_floor = Integer::div_mod_floor(&n, &d);
assert_eq!(separate_div_mod_floor, dm);
assert_eq!(combined_div_mod_floor, dm);
test_division_rule(nd, separate_div_mod_floor);
test_division_rule(nd, combined_div_mod_floor);
}
test_nd_dm((8, 3), (2, 2));
test_nd_dm((8, -3), (-3, -1));
test_nd_dm((-8, 3), (-3, 1));
test_nd_dm((-8, -3), (2, -2));
test_nd_dm((1, 2), (0, 1));
test_nd_dm((1, -2), (-1, -1));
test_nd_dm((-1, 2), (-1, 1));
test_nd_dm((-1, -2), (0, -1));
}
#[test]
fn test_gcd() {
assert_eq!((10 as $T).gcd(&2), 2 as $T);
assert_eq!((10 as $T).gcd(&3), 1 as $T);
assert_eq!((0 as $T).gcd(&3), 3 as $T);
assert_eq!((3 as $T).gcd(&3), 3 as $T);
assert_eq!((56 as $T).gcd(&42), 14 as $T);
assert_eq!((3 as $T).gcd(&-3), 3 as $T);
assert_eq!((-6 as $T).gcd(&3), 3 as $T);
assert_eq!((-4 as $T).gcd(&-2), 2 as $T);
}
#[test]
fn test_gcd_cmp_with_euclidean() {
fn euclidean_gcd(mut m: $T, mut n: $T) -> $T {
while m != 0 {
mem::swap(&mut m, &mut n);
m %= n;
}
n.abs()
}
// gcd(-128, b) = 128 is not representable as positive value
// for i8
for i in -127..127 {
for j in -127..127 {
assert_eq!(euclidean_gcd(i, j), i.gcd(&j));
}
}
// last value
// FIXME: Use inclusive ranges for above loop when implemented
let i = 127;
for j in -127..127 {
assert_eq!(euclidean_gcd(i, j), i.gcd(&j));
}
assert_eq!(127.gcd(&127), 127);
}
#[test]
fn test_gcd_min_val() {
let min = <$T>::min_value();
let max = <$T>::max_value();
let max_pow2 = max / 2 + 1;
assert_eq!(min.gcd(&max), 1 as $T);
assert_eq!(max.gcd(&min), 1 as $T);
assert_eq!(min.gcd(&max_pow2), max_pow2);
assert_eq!(max_pow2.gcd(&min), max_pow2);
assert_eq!(min.gcd(&42), 2 as $T);
assert_eq!((42 as $T).gcd(&min), 2 as $T);
}
#[test]
#[should_panic]
fn test_gcd_min_val_min_val() {
let min = <$T>::min_value();
assert!(min.gcd(&min) >= 0);
}
#[test]
#[should_panic]
fn test_gcd_min_val_0() {
let min = <$T>::min_value();
assert!(min.gcd(&0) >= 0);
}
#[test]
#[should_panic]
fn test_gcd_0_min_val() {
let min = <$T>::min_value();
assert!((0 as $T).gcd(&min) >= 0);
}
#[test]
fn test_lcm() {
assert_eq!((1 as $T).lcm(&0), 0 as $T);
assert_eq!((0 as $T).lcm(&1), 0 as $T);
assert_eq!((1 as $T).lcm(&1), 1 as $T);
assert_eq!((-1 as $T).lcm(&1), 1 as $T);
assert_eq!((1 as $T).lcm(&-1), 1 as $T);
assert_eq!((-1 as $T).lcm(&-1), 1 as $T);
assert_eq!((8 as $T).lcm(&9), 72 as $T);
assert_eq!((11 as $T).lcm(&5), 55 as $T);
}
#[test]
fn test_gcd_lcm() {
use core::iter::once;
for i in once(0)
.chain((1..).take(127).flat_map(|a| once(a).chain(once(-a))))
.chain(once(-128))
{
for j in once(0)
.chain((1..).take(127).flat_map(|a| once(a).chain(once(-a))))
.chain(once(-128))
{
assert_eq!(i.gcd_lcm(&j), (i.gcd(&j), i.lcm(&j)));
}
}
}
#[test]
fn test_extended_gcd_lcm() {
use core::fmt::Debug;
use traits::NumAssign;
use ExtendedGcd;
fn check<A: Copy + Debug + Integer + NumAssign>(a: A, b: A) {
let ExtendedGcd { gcd, x, y, .. } = a.extended_gcd(&b);
assert_eq!(gcd, x * a + y * b);
}
use core::iter::once;
for i in once(0)
.chain((1..).take(127).flat_map(|a| once(a).chain(once(-a))))
.chain(once(-128))
{
for j in once(0)
.chain((1..).take(127).flat_map(|a| once(a).chain(once(-a))))
.chain(once(-128))
{
check(i, j);
let (ExtendedGcd { gcd, .. }, lcm) = i.extended_gcd_lcm(&j);
assert_eq!((gcd, lcm), (i.gcd(&j), i.lcm(&j)));
}
}
}
#[test]
fn test_even() {
assert_eq!((-4 as $T).is_even(), true);
assert_eq!((-3 as $T).is_even(), false);
assert_eq!((-2 as $T).is_even(), true);
assert_eq!((-1 as $T).is_even(), false);
assert_eq!((0 as $T).is_even(), true);
assert_eq!((1 as $T).is_even(), false);
assert_eq!((2 as $T).is_even(), true);
assert_eq!((3 as $T).is_even(), false);
assert_eq!((4 as $T).is_even(), true);
}
#[test]
fn test_odd() {
assert_eq!((-4 as $T).is_odd(), false);
assert_eq!((-3 as $T).is_odd(), true);
assert_eq!((-2 as $T).is_odd(), false);
assert_eq!((-1 as $T).is_odd(), true);
assert_eq!((0 as $T).is_odd(), false);
assert_eq!((1 as $T).is_odd(), true);
assert_eq!((2 as $T).is_odd(), false);
assert_eq!((3 as $T).is_odd(), true);
assert_eq!((4 as $T).is_odd(), false);
}
#[test]
fn test_multiple_of_one_limits() {
for x in &[<$T>::min_value(), <$T>::max_value()] {
for one in &[1, -1] {
assert_eq!(Integer::next_multiple_of(x, one), *x);
assert_eq!(Integer::prev_multiple_of(x, one), *x);
}
}
}
}
};
}
impl_integer_for_isize!(i8, test_integer_i8);
impl_integer_for_isize!(i16, test_integer_i16);
impl_integer_for_isize!(i32, test_integer_i32);
impl_integer_for_isize!(i64, test_integer_i64);
impl_integer_for_isize!(isize, test_integer_isize);
#[cfg(has_i128)]
impl_integer_for_isize!(i128, test_integer_i128);
macro_rules! impl_integer_for_usize {
($T:ty, $test_mod:ident) => {
impl Integer for $T {
/// Unsigned integer division. Returns the same result as `div` (`/`).
#[inline]
fn div_floor(&self, other: &Self) -> Self {
*self / *other
}
/// Unsigned integer modulo operation. Returns the same result as `rem` (`%`).
#[inline]
fn mod_floor(&self, other: &Self) -> Self {
*self % *other
}
#[inline]
fn div_ceil(&self, other: &Self) -> Self {
*self / *other + (0 != *self % *other) as Self
}
/// Calculates the Greatest Common Divisor (GCD) of the number and `other`
#[inline]
fn gcd(&self, other: &Self) -> Self {
// Use Stein's algorithm
let mut m = *self;
let mut n = *other;
if m == 0 || n == 0 {
return m | n;
}
// find common factors of 2
let shift = (m | n).trailing_zeros();
// divide n and m by 2 until odd
m >>= m.trailing_zeros();
n >>= n.trailing_zeros();
while m != n {
if m > n {
m -= n;
m >>= m.trailing_zeros();
} else {
n -= m;
n >>= n.trailing_zeros();
}
}
m << shift
}
#[inline]
fn extended_gcd_lcm(&self, other: &Self) -> (ExtendedGcd<Self>, Self) {
let egcd = self.extended_gcd(other);
// should not have to recalculate abs
let lcm = if egcd.gcd.is_zero() {
Self::zero()
} else {
*self * (*other / egcd.gcd)
};
(egcd, lcm)
}
/// Calculates the Lowest Common Multiple (LCM) of the number and `other`.
#[inline]
fn lcm(&self, other: &Self) -> Self {
self.gcd_lcm(other).1
}
/// Calculates the Greatest Common Divisor (GCD) and
/// Lowest Common Multiple (LCM) of the number and `other`.
#[inline]
fn gcd_lcm(&self, other: &Self) -> (Self, Self) {
if self.is_zero() && other.is_zero() {
return (Self::zero(), Self::zero());
}
let gcd = self.gcd(other);
let lcm = *self * (*other / gcd);
(gcd, lcm)
}
/// Deprecated, use `is_multiple_of` instead.
#[inline]
fn divides(&self, other: &Self) -> bool {
self.is_multiple_of(other)
}
/// Returns `true` if the number is a multiple of `other`.
#[inline]
fn is_multiple_of(&self, other: &Self) -> bool {
if other.is_zero() {
return self.is_zero();
}
*self % *other == 0
}
/// Returns `true` if the number is divisible by `2`.
#[inline]
fn is_even(&self) -> bool {
*self % 2 == 0
}
/// Returns `true` if the number is not divisible by `2`.
#[inline]
fn is_odd(&self) -> bool {
!self.is_even()
}
/// Simultaneous truncated integer division and modulus.
#[inline]
fn div_rem(&self, other: &Self) -> (Self, Self) {
(*self / *other, *self % *other)
}
}
#[cfg(test)]
mod $test_mod {
use core::mem;
use Integer;
#[test]
fn test_div_mod_floor() {
assert_eq!(<$T as Integer>::div_floor(&10, &3), 3 as $T);
assert_eq!(<$T as Integer>::mod_floor(&10, &3), 1 as $T);
assert_eq!(<$T as Integer>::div_mod_floor(&10, &3), (3 as $T, 1 as $T));
assert_eq!(<$T as Integer>::div_floor(&5, &5), 1 as $T);
assert_eq!(<$T as Integer>::mod_floor(&5, &5), 0 as $T);
assert_eq!(<$T as Integer>::div_mod_floor(&5, &5), (1 as $T, 0 as $T));
assert_eq!(<$T as Integer>::div_floor(&3, &7), 0 as $T);
assert_eq!(<$T as Integer>::div_floor(&3, &7), 0 as $T);
assert_eq!(<$T as Integer>::mod_floor(&3, &7), 3 as $T);
assert_eq!(<$T as Integer>::div_mod_floor(&3, &7), (0 as $T, 3 as $T));
}
#[test]
fn test_gcd() {
assert_eq!((10 as $T).gcd(&2), 2 as $T);
assert_eq!((10 as $T).gcd(&3), 1 as $T);
assert_eq!((0 as $T).gcd(&3), 3 as $T);
assert_eq!((3 as $T).gcd(&3), 3 as $T);
assert_eq!((56 as $T).gcd(&42), 14 as $T);
}
#[test]
fn test_gcd_cmp_with_euclidean() {
fn euclidean_gcd(mut m: $T, mut n: $T) -> $T {
while m != 0 {
mem::swap(&mut m, &mut n);
m %= n;
}
n
}
for i in 0..255 {
for j in 0..255 {
assert_eq!(euclidean_gcd(i, j), i.gcd(&j));
}
}
// last value
// FIXME: Use inclusive ranges for above loop when implemented
let i = 255;
for j in 0..255 {
assert_eq!(euclidean_gcd(i, j), i.gcd(&j));
}
assert_eq!(255.gcd(&255), 255);
}
#[test]
fn test_lcm() {
assert_eq!((1 as $T).lcm(&0), 0 as $T);
assert_eq!((0 as $T).lcm(&1), 0 as $T);
assert_eq!((1 as $T).lcm(&1), 1 as $T);
assert_eq!((8 as $T).lcm(&9), 72 as $T);
assert_eq!((11 as $T).lcm(&5), 55 as $T);
assert_eq!((15 as $T).lcm(&17), 255 as $T);
}
#[test]
fn test_gcd_lcm() {
for i in (0..).take(256) {
for j in (0..).take(256) {
assert_eq!(i.gcd_lcm(&j), (i.gcd(&j), i.lcm(&j)));
}
}
}
#[test]
fn test_is_multiple_of() {
assert!((0 as $T).is_multiple_of(&(0 as $T)));
assert!((6 as $T).is_multiple_of(&(6 as $T)));
assert!((6 as $T).is_multiple_of(&(3 as $T)));
assert!((6 as $T).is_multiple_of(&(1 as $T)));
assert!(!(42 as $T).is_multiple_of(&(5 as $T)));
assert!(!(5 as $T).is_multiple_of(&(3 as $T)));
assert!(!(42 as $T).is_multiple_of(&(0 as $T)));
}
#[test]
fn test_even() {
assert_eq!((0 as $T).is_even(), true);
assert_eq!((1 as $T).is_even(), false);
assert_eq!((2 as $T).is_even(), true);
assert_eq!((3 as $T).is_even(), false);
assert_eq!((4 as $T).is_even(), true);
}
#[test]
fn test_odd() {
assert_eq!((0 as $T).is_odd(), false);
assert_eq!((1 as $T).is_odd(), true);
assert_eq!((2 as $T).is_odd(), false);
assert_eq!((3 as $T).is_odd(), true);
assert_eq!((4 as $T).is_odd(), false);
}
}
};
}
impl_integer_for_usize!(u8, test_integer_u8);
impl_integer_for_usize!(u16, test_integer_u16);
impl_integer_for_usize!(u32, test_integer_u32);
impl_integer_for_usize!(u64, test_integer_u64);
impl_integer_for_usize!(usize, test_integer_usize);
#[cfg(has_i128)]
impl_integer_for_usize!(u128, test_integer_u128);
/// An iterator over binomial coefficients.
pub struct IterBinomial<T> {
a: T,
n: T,
k: T,
}
impl<T> IterBinomial<T>
where
T: Integer,
{
/// For a given n, iterate over all binomial coefficients binomial(n, k), for k=0...n.
///
/// Note that this might overflow, depending on `T`. For the primitive
/// integer types, the following n are the largest ones for which there will
/// be no overflow:
///
/// type | n
/// -----|---
/// u8 | 10
/// i8 | 9
/// u16 | 18
/// i16 | 17
/// u32 | 34
/// i32 | 33
/// u64 | 67
/// i64 | 66
///
/// For larger n, `T` should be a bigint type.
pub fn new(n: T) -> IterBinomial<T> {
IterBinomial {
k: T::zero(),
a: T::one(),
n: n,
}
}
}
impl<T> Iterator for IterBinomial<T>
where
T: Integer + Clone,
{
type Item = T;
fn next(&mut self) -> Option<T> {
if self.k > self.n {
return None;
}
self.a = if !self.k.is_zero() {
multiply_and_divide(
self.a.clone(),
self.n.clone() - self.k.clone() + T::one(),
self.k.clone(),
)
} else {
T::one()
};
self.k = self.k.clone() + T::one();
Some(self.a.clone())
}
}
/// Calculate r * a / b, avoiding overflows and fractions.
///
/// Assumes that b divides r * a evenly.
fn multiply_and_divide<T: Integer + Clone>(r: T, a: T, b: T) -> T {
// See
http://blog.plover.com/math/choose-2.html for the idea.
let g = gcd(r.clone(), b.clone());
r / g.clone() * (a / (b / g))
}
/// Calculate the binomial coefficient.
///
/// Note that this might overflow, depending on `T`. For the primitive integer
/// types, the following n are the largest ones possible such that there will
/// be no overflow for any k:
///
/// type | n
/// -----|---
/// u8 | 10
/// i8 | 9
/// u16 | 18
/// i16 | 17
/// u32 | 34
/// i32 | 33
/// u64 | 67
/// i64 | 66
///
/// For larger n, consider using a bigint type for `T`.
pub fn binomial<T: Integer + Clone>(mut n: T, k: T) -> T {
// See
http://blog.plover.com/math/choose.html for the idea.
if k > n {
return T::zero();
}
if k > n.clone() - k.clone() {
return binomial(n.clone(), n - k);
}
let mut r = T::one();
let mut d = T::one();
loop {
if d > k {
break;
}
r = multiply_and_divide(r, n.clone(), d.clone());
n = n - T::one();
d = d + T::one();
}
r
}
/// Calculate the multinomial coefficient.
pub fn multinomial<T: Integer + Clone>(k: &[T]) -> T
where
for<'a> T: Add<&'a T, Output = T>,
{
let mut r = T::one();
let mut p = T::zero();
for i in k {
p = p + i;
r = r * binomial(p.clone(), i.clone());
}
r
}
#[test]
fn test_lcm_overflow() {
macro_rules! check {
($t:ty, $x:expr, $y:expr, $r:expr) => {{
let x: $t = $x;
let y: $t = $y;
let o = x.checked_mul(y);
assert!(
o.is_none(),
"sanity checking that {} input {} * {} overflows",
stringify!($t),
x,
y
);
assert_eq!(x.lcm(&y), $r);
assert_eq!(y.lcm(&x), $r);
}};
}
// Original bug (Issue #166)
check!(i64, 46656000000000000, 600, 46656000000000000);
check!(i8, 0x40, 0x04, 0x40);
check!(u8, 0x80, 0x02, 0x80);
check!(i16, 0x40_00, 0x04, 0x40_00);
check!(u16, 0x80_00, 0x02, 0x80_00);
check!(i32, 0x4000_0000, 0x04, 0x4000_0000);
check!(u32, 0x8000_0000, 0x02, 0x8000_0000);
check!(i64, 0x4000_0000_0000_0000, 0x04, 0x4000_0000_0000_0000);
check!(u64, 0x8000_0000_0000_0000, 0x02, 0x8000_0000_0000_0000);
}
#[test]
fn test_iter_binomial() {
macro_rules! check_simple {
($t:ty) => {{
let n: $t = 3;
let expected = [1, 3, 3, 1];
for (b, &e) in IterBinomial::new(n).zip(&expected) {
assert_eq!(b, e);
}
}};
}
check_simple!(u8);
check_simple!(i8);
check_simple!(u16);
check_simple!(i16);
check_simple!(u32);
check_simple!(i32);
check_simple!(u64);
check_simple!(i64);
macro_rules! check_binomial {
($t:ty, $n:expr) => {{
let n: $t = $n;
let mut k: $t = 0;
for b in IterBinomial::new(n) {
assert_eq!(b, binomial(n, k));
k += 1;
}
}};
}
// Check the largest n for which there is no overflow.
check_binomial!(u8, 10);
check_binomial!(i8, 9);
check_binomial!(u16, 18);
check_binomial!(i16, 17);
check_binomial!(u32, 34);
check_binomial!(i32, 33);
check_binomial!(u64, 67);
check_binomial!(i64, 66);
}
#[test]
fn test_binomial() {
macro_rules! check {
($t:ty, $x:expr, $y:expr, $r:expr) => {{
let x: $t = $x;
let y: $t = $y;
let expected: $t = $r;
assert_eq!(binomial(x, y), expected);
if y <= x {
assert_eq!(binomial(x, x - y), expected);
}
}};
}
check!(u8, 9, 4, 126);
check!(u8, 0, 0, 1);
check!(u8, 2, 3, 0);
check!(i8, 9, 4, 126);
check!(i8, 0, 0, 1);
check!(i8, 2, 3, 0);
check!(u16, 100, 2, 4950);
check!(u16, 14, 4, 1001);
check!(u16, 0, 0, 1);
check!(u16, 2, 3, 0);
check!(i16, 100, 2, 4950);
check!(i16, 14, 4, 1001);
check!(i16, 0, 0, 1);
check!(i16, 2, 3, 0);
check!(u32, 100, 2, 4950);
check!(u32, 35, 11, 417225900);
check!(u32, 14, 4, 1001);
check!(u32, 0, 0, 1);
check!(u32, 2, 3, 0);
check!(i32, 100, 2, 4950);
check!(i32, 35, 11, 417225900);
check!(i32, 14, 4, 1001);
check!(i32, 0, 0, 1);
check!(i32, 2, 3, 0);
check!(u64, 100, 2, 4950);
check!(u64, 35, 11, 417225900);
check!(u64, 14, 4, 1001);
check!(u64, 0, 0, 1);
check!(u64, 2, 3, 0);
check!(i64, 100, 2, 4950);
check!(i64, 35, 11, 417225900);
check!(i64, 14, 4, 1001);
check!(i64, 0, 0, 1);
check!(i64, 2, 3, 0);
}
#[test]
fn test_multinomial() {
macro_rules! check_binomial {
($t:ty, $k:expr) => {{
let n: $t = $k.iter().fold(0, |acc, &x| acc + x);
let k: &[$t] = $k;
assert_eq!(k.len(), 2);
assert_eq!(multinomial(k), binomial(n, k[0]));
}};
}
check_binomial!(u8, &[4, 5]);
check_binomial!(i8, &[4, 5]);
check_binomial!(u16, &[2, 98]);
check_binomial!(u16, &[4, 10]);
check_binomial!(i16, &[2, 98]);
check_binomial!(i16, &[4, 10]);
check_binomial!(u32, &[2, 98]);
check_binomial!(u32, &[11, 24]);
check_binomial!(u32, &[4, 10]);
check_binomial!(i32, &[2, 98]);
check_binomial!(i32, &[11, 24]);
check_binomial!(i32, &[4, 10]);
check_binomial!(u64, &[2, 98]);
check_binomial!(u64, &[11, 24]);
check_binomial!(u64, &[4, 10]);
check_binomial!(i64, &[2, 98]);
check_binomial!(i64, &[11, 24]);
check_binomial!(i64, &[4, 10]);
macro_rules! check_multinomial {
($t:ty, $k:expr, $r:expr) => {{
let k: &[$t] = $k;
let expected: $t = $r;
assert_eq!(multinomial(k), expected);
}};
}
check_multinomial!(u8, &[2, 1, 2], 30);
check_multinomial!(u8, &[2, 3, 0], 10);
check_multinomial!(i8, &[2, 1, 2], 30);
check_multinomial!(i8, &[2, 3, 0], 10);
check_multinomial!(u16, &[2, 1, 2], 30);
check_multinomial!(u16, &[2, 3, 0], 10);
check_multinomial!(i16, &[2, 1, 2], 30);
check_multinomial!(i16, &[2, 3, 0], 10);
check_multinomial!(u32, &[2, 1, 2], 30);
check_multinomial!(u32, &[2, 3, 0], 10);
check_multinomial!(i32, &[2, 1, 2], 30);
check_multinomial!(i32, &[2, 3, 0], 10);
check_multinomial!(u64, &[2, 1, 2], 30);
check_multinomial!(u64, &[2, 3, 0], 10);
check_multinomial!(i64, &[2, 1, 2], 30);
check_multinomial!(i64, &[2, 3, 0], 10);
check_multinomial!(u64, &[], 1);
check_multinomial!(u64, &[0], 1);
check_multinomial!(u64, &[12345], 1);
}
