989 lines
31 KiB
Rust
989 lines
31 KiB
Rust
// Copyright 2013-2014 The Rust Project Developers. See the COPYRIGHT
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// file at the top-level directory of this distribution and at
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// http://rust-lang.org/COPYRIGHT.
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//
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// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
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// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
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// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
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// option. This file may not be copied, modified, or distributed
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// except according to those terms.
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//! Integer trait and functions.
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#![doc(html_logo_url = "https://rust-num.github.io/num/rust-logo-128x128-blk-v2.png",
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html_favicon_url = "https://rust-num.github.io/num/favicon.ico",
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html_root_url = "https://rust-num.github.io/num/",
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html_playground_url = "http://play.integer32.com/")]
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extern crate num_traits as traits;
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use std::ops::Add;
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use traits::{Num, Signed};
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pub trait Integer: Sized + Num + PartialOrd + Ord + Eq {
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/// Floored integer division.
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///
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/// # Examples
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///
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/// ~~~
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/// # use num_integer::Integer;
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/// assert!(( 8).div_floor(& 3) == 2);
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/// assert!(( 8).div_floor(&-3) == -3);
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/// assert!((-8).div_floor(& 3) == -3);
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/// assert!((-8).div_floor(&-3) == 2);
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///
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/// assert!(( 1).div_floor(& 2) == 0);
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/// assert!(( 1).div_floor(&-2) == -1);
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/// assert!((-1).div_floor(& 2) == -1);
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/// assert!((-1).div_floor(&-2) == 0);
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/// ~~~
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fn div_floor(&self, other: &Self) -> Self;
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/// Floored integer modulo, satisfying:
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///
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/// ~~~
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/// # use num_integer::Integer;
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/// # let n = 1; let d = 1;
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/// assert!(n.div_floor(&d) * d + n.mod_floor(&d) == n)
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/// ~~~
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///
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/// # Examples
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///
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/// ~~~
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/// # use num_integer::Integer;
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/// assert!(( 8).mod_floor(& 3) == 2);
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/// assert!(( 8).mod_floor(&-3) == -1);
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/// assert!((-8).mod_floor(& 3) == 1);
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/// assert!((-8).mod_floor(&-3) == -2);
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///
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/// assert!(( 1).mod_floor(& 2) == 1);
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/// assert!(( 1).mod_floor(&-2) == -1);
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/// assert!((-1).mod_floor(& 2) == 1);
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/// assert!((-1).mod_floor(&-2) == -1);
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/// ~~~
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fn mod_floor(&self, other: &Self) -> Self;
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/// Greatest Common Divisor (GCD).
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///
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/// # Examples
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///
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/// ~~~
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/// # use num_integer::Integer;
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/// assert_eq!(6.gcd(&8), 2);
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/// assert_eq!(7.gcd(&3), 1);
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/// ~~~
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fn gcd(&self, other: &Self) -> Self;
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/// Lowest Common Multiple (LCM).
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///
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/// # Examples
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///
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/// ~~~
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/// # use num_integer::Integer;
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/// assert_eq!(7.lcm(&3), 21);
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/// assert_eq!(2.lcm(&4), 4);
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/// ~~~
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fn lcm(&self, other: &Self) -> Self;
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/// Deprecated, use `is_multiple_of` instead.
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fn divides(&self, other: &Self) -> bool;
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/// Returns `true` if `other` is a multiple of `self`.
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///
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/// # Examples
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///
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/// ~~~
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/// # use num_integer::Integer;
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/// assert_eq!(9.is_multiple_of(&3), true);
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/// assert_eq!(3.is_multiple_of(&9), false);
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/// ~~~
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fn is_multiple_of(&self, other: &Self) -> bool;
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/// Returns `true` if the number is even.
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///
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/// # Examples
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///
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/// ~~~
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/// # use num_integer::Integer;
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/// assert_eq!(3.is_even(), false);
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/// assert_eq!(4.is_even(), true);
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/// ~~~
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fn is_even(&self) -> bool;
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/// Returns `true` if the number is odd.
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///
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/// # Examples
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///
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/// ~~~
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/// # use num_integer::Integer;
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/// assert_eq!(3.is_odd(), true);
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/// assert_eq!(4.is_odd(), false);
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/// ~~~
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fn is_odd(&self) -> bool;
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/// Simultaneous truncated integer division and modulus.
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/// Returns `(quotient, remainder)`.
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///
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/// # Examples
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///
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/// ~~~
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/// # use num_integer::Integer;
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/// assert_eq!(( 8).div_rem( &3), ( 2, 2));
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/// assert_eq!(( 8).div_rem(&-3), (-2, 2));
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/// assert_eq!((-8).div_rem( &3), (-2, -2));
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/// assert_eq!((-8).div_rem(&-3), ( 2, -2));
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///
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/// assert_eq!(( 1).div_rem( &2), ( 0, 1));
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/// assert_eq!(( 1).div_rem(&-2), ( 0, 1));
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/// assert_eq!((-1).div_rem( &2), ( 0, -1));
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/// assert_eq!((-1).div_rem(&-2), ( 0, -1));
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/// ~~~
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#[inline]
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fn div_rem(&self, other: &Self) -> (Self, Self);
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/// Simultaneous floored integer division and modulus.
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/// Returns `(quotient, remainder)`.
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///
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/// # Examples
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///
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/// ~~~
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/// # use num_integer::Integer;
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/// assert_eq!(( 8).div_mod_floor( &3), ( 2, 2));
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/// assert_eq!(( 8).div_mod_floor(&-3), (-3, -1));
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/// assert_eq!((-8).div_mod_floor( &3), (-3, 1));
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/// assert_eq!((-8).div_mod_floor(&-3), ( 2, -2));
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///
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/// assert_eq!(( 1).div_mod_floor( &2), ( 0, 1));
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/// assert_eq!(( 1).div_mod_floor(&-2), (-1, -1));
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/// assert_eq!((-1).div_mod_floor( &2), (-1, 1));
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/// assert_eq!((-1).div_mod_floor(&-2), ( 0, -1));
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/// ~~~
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fn div_mod_floor(&self, other: &Self) -> (Self, Self) {
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(self.div_floor(other), self.mod_floor(other))
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}
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}
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/// Simultaneous integer division and modulus
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#[inline]
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pub fn div_rem<T: Integer>(x: T, y: T) -> (T, T) {
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x.div_rem(&y)
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}
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/// Floored integer division
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#[inline]
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pub fn div_floor<T: Integer>(x: T, y: T) -> T {
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x.div_floor(&y)
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}
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/// Floored integer modulus
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#[inline]
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pub fn mod_floor<T: Integer>(x: T, y: T) -> T {
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x.mod_floor(&y)
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}
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/// Simultaneous floored integer division and modulus
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#[inline]
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pub fn div_mod_floor<T: Integer>(x: T, y: T) -> (T, T) {
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x.div_mod_floor(&y)
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}
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/// Calculates the Greatest Common Divisor (GCD) of the number and `other`. The
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/// result is always positive.
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#[inline(always)]
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pub fn gcd<T: Integer>(x: T, y: T) -> T {
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x.gcd(&y)
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}
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/// Calculates the Lowest Common Multiple (LCM) of the number and `other`.
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#[inline(always)]
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pub fn lcm<T: Integer>(x: T, y: T) -> T {
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x.lcm(&y)
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}
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macro_rules! impl_integer_for_isize {
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($T:ty, $test_mod:ident) => (
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impl Integer for $T {
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/// Floored integer division
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#[inline]
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fn div_floor(&self, other: &Self) -> Self {
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// Algorithm from [Daan Leijen. _Division and Modulus for Computer Scientists_,
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// December 2001](http://research.microsoft.com/pubs/151917/divmodnote-letter.pdf)
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match self.div_rem(other) {
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(d, r) if (r > 0 && *other < 0)
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|| (r < 0 && *other > 0) => d - 1,
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(d, _) => d,
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}
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}
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/// Floored integer modulo
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#[inline]
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fn mod_floor(&self, other: &Self) -> Self {
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// Algorithm from [Daan Leijen. _Division and Modulus for Computer Scientists_,
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// December 2001](http://research.microsoft.com/pubs/151917/divmodnote-letter.pdf)
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match *self % *other {
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r if (r > 0 && *other < 0)
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|| (r < 0 && *other > 0) => r + *other,
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r => r,
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}
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}
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/// Calculates `div_floor` and `mod_floor` simultaneously
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#[inline]
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fn div_mod_floor(&self, other: &Self) -> (Self, Self) {
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// Algorithm from [Daan Leijen. _Division and Modulus for Computer Scientists_,
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// December 2001](http://research.microsoft.com/pubs/151917/divmodnote-letter.pdf)
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match self.div_rem(other) {
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(d, r) if (r > 0 && *other < 0)
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|| (r < 0 && *other > 0) => (d - 1, r + *other),
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(d, r) => (d, r),
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}
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}
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/// Calculates the Greatest Common Divisor (GCD) of the number and
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/// `other`. The result is always positive.
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#[inline]
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fn gcd(&self, other: &Self) -> Self {
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// Use Stein's algorithm
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let mut m = *self;
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let mut n = *other;
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if m == 0 || n == 0 { return (m | n).abs() }
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// find common factors of 2
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let shift = (m | n).trailing_zeros();
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// The algorithm needs positive numbers, but the minimum value
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// can't be represented as a positive one.
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// It's also a power of two, so the gcd can be
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// calculated by bitshifting in that case
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// Assuming two's complement, the number created by the shift
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// is positive for all numbers except gcd = abs(min value)
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// The call to .abs() causes a panic in debug mode
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if m == Self::min_value() || n == Self::min_value() {
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return (1 << shift).abs()
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}
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// guaranteed to be positive now, rest like unsigned algorithm
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m = m.abs();
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n = n.abs();
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// divide n and m by 2 until odd
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// m inside loop
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n >>= n.trailing_zeros();
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while m != 0 {
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m >>= m.trailing_zeros();
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if n > m { ::std::mem::swap(&mut n, &mut m) }
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m -= n;
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}
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n << shift
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}
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/// Calculates the Lowest Common Multiple (LCM) of the number and
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/// `other`.
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#[inline]
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fn lcm(&self, other: &Self) -> Self {
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// should not have to recalculate abs
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(*self * (*other / self.gcd(other))).abs()
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}
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/// Deprecated, use `is_multiple_of` instead.
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#[inline]
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fn divides(&self, other: &Self) -> bool {
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self.is_multiple_of(other)
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}
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/// Returns `true` if the number is a multiple of `other`.
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#[inline]
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fn is_multiple_of(&self, other: &Self) -> bool {
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*self % *other == 0
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}
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/// Returns `true` if the number is divisible by `2`
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#[inline]
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fn is_even(&self) -> bool { (*self) & 1 == 0 }
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/// Returns `true` if the number is not divisible by `2`
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#[inline]
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fn is_odd(&self) -> bool { !self.is_even() }
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/// Simultaneous truncated integer division and modulus.
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#[inline]
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fn div_rem(&self, other: &Self) -> (Self, Self) {
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(*self / *other, *self % *other)
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}
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}
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#[cfg(test)]
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mod $test_mod {
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use Integer;
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/// Checks that the division rule holds for:
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///
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/// - `n`: numerator (dividend)
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/// - `d`: denominator (divisor)
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/// - `qr`: quotient and remainder
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#[cfg(test)]
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fn test_division_rule((n,d): ($T, $T), (q,r): ($T, $T)) {
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assert_eq!(d * q + r, n);
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}
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#[test]
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fn test_div_rem() {
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fn test_nd_dr(nd: ($T,$T), qr: ($T,$T)) {
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let (n,d) = nd;
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let separate_div_rem = (n / d, n % d);
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let combined_div_rem = n.div_rem(&d);
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assert_eq!(separate_div_rem, qr);
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assert_eq!(combined_div_rem, qr);
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test_division_rule(nd, separate_div_rem);
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test_division_rule(nd, combined_div_rem);
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}
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test_nd_dr(( 8, 3), ( 2, 2));
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test_nd_dr(( 8, -3), (-2, 2));
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test_nd_dr((-8, 3), (-2, -2));
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test_nd_dr((-8, -3), ( 2, -2));
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test_nd_dr(( 1, 2), ( 0, 1));
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test_nd_dr(( 1, -2), ( 0, 1));
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test_nd_dr((-1, 2), ( 0, -1));
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test_nd_dr((-1, -2), ( 0, -1));
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}
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#[test]
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fn test_div_mod_floor() {
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fn test_nd_dm(nd: ($T,$T), dm: ($T,$T)) {
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let (n,d) = nd;
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let separate_div_mod_floor = (n.div_floor(&d), n.mod_floor(&d));
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let combined_div_mod_floor = n.div_mod_floor(&d);
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assert_eq!(separate_div_mod_floor, dm);
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assert_eq!(combined_div_mod_floor, dm);
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test_division_rule(nd, separate_div_mod_floor);
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test_division_rule(nd, combined_div_mod_floor);
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}
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test_nd_dm(( 8, 3), ( 2, 2));
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test_nd_dm(( 8, -3), (-3, -1));
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test_nd_dm((-8, 3), (-3, 1));
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test_nd_dm((-8, -3), ( 2, -2));
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test_nd_dm(( 1, 2), ( 0, 1));
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test_nd_dm(( 1, -2), (-1, -1));
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test_nd_dm((-1, 2), (-1, 1));
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test_nd_dm((-1, -2), ( 0, -1));
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}
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#[test]
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fn test_gcd() {
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assert_eq!((10 as $T).gcd(&2), 2 as $T);
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assert_eq!((10 as $T).gcd(&3), 1 as $T);
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assert_eq!((0 as $T).gcd(&3), 3 as $T);
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assert_eq!((3 as $T).gcd(&3), 3 as $T);
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assert_eq!((56 as $T).gcd(&42), 14 as $T);
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assert_eq!((3 as $T).gcd(&-3), 3 as $T);
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assert_eq!((-6 as $T).gcd(&3), 3 as $T);
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assert_eq!((-4 as $T).gcd(&-2), 2 as $T);
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}
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#[test]
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fn test_gcd_cmp_with_euclidean() {
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fn euclidean_gcd(mut m: $T, mut n: $T) -> $T {
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while m != 0 {
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::std::mem::swap(&mut m, &mut n);
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m %= n;
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}
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n.abs()
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}
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// gcd(-128, b) = 128 is not representable as positive value
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// for i8
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for i in -127..127 {
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for j in -127..127 {
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assert_eq!(euclidean_gcd(i,j), i.gcd(&j));
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}
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}
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// last value
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// FIXME: Use inclusive ranges for above loop when implemented
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let i = 127;
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for j in -127..127 {
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assert_eq!(euclidean_gcd(i,j), i.gcd(&j));
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}
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assert_eq!(127.gcd(&127), 127);
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}
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#[test]
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fn test_gcd_min_val() {
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let min = <$T>::min_value();
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let max = <$T>::max_value();
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let max_pow2 = max / 2 + 1;
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assert_eq!(min.gcd(&max), 1 as $T);
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assert_eq!(max.gcd(&min), 1 as $T);
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assert_eq!(min.gcd(&max_pow2), max_pow2);
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assert_eq!(max_pow2.gcd(&min), max_pow2);
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assert_eq!(min.gcd(&42), 2 as $T);
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assert_eq!((42 as $T).gcd(&min), 2 as $T);
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}
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#[test]
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#[should_panic]
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fn test_gcd_min_val_min_val() {
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let min = <$T>::min_value();
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assert!(min.gcd(&min) >= 0);
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}
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#[test]
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#[should_panic]
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fn test_gcd_min_val_0() {
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let min = <$T>::min_value();
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assert!(min.gcd(&0) >= 0);
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}
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#[test]
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#[should_panic]
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fn test_gcd_0_min_val() {
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let min = <$T>::min_value();
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assert!((0 as $T).gcd(&min) >= 0);
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}
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#[test]
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fn test_lcm() {
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assert_eq!((1 as $T).lcm(&0), 0 as $T);
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assert_eq!((0 as $T).lcm(&1), 0 as $T);
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assert_eq!((1 as $T).lcm(&1), 1 as $T);
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assert_eq!((-1 as $T).lcm(&1), 1 as $T);
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assert_eq!((1 as $T).lcm(&-1), 1 as $T);
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assert_eq!((-1 as $T).lcm(&-1), 1 as $T);
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assert_eq!((8 as $T).lcm(&9), 72 as $T);
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assert_eq!((11 as $T).lcm(&5), 55 as $T);
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}
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#[test]
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fn test_even() {
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assert_eq!((-4 as $T).is_even(), true);
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assert_eq!((-3 as $T).is_even(), false);
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assert_eq!((-2 as $T).is_even(), true);
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assert_eq!((-1 as $T).is_even(), false);
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assert_eq!((0 as $T).is_even(), true);
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assert_eq!((1 as $T).is_even(), false);
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assert_eq!((2 as $T).is_even(), true);
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assert_eq!((3 as $T).is_even(), false);
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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 $T).is_odd(), false);
|
|
}
|
|
}
|
|
)
|
|
}
|
|
|
|
impl_integer_for_isize!(i8, test_integer_i8);
|
|
impl_integer_for_isize!(i16, test_integer_i16);
|
|
impl_integer_for_isize!(i32, test_integer_i32);
|
|
impl_integer_for_isize!(i64, test_integer_i64);
|
|
impl_integer_for_isize!(isize, test_integer_isize);
|
|
|
|
macro_rules! impl_integer_for_usize {
|
|
($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
|
|
}
|
|
|
|
/// 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 inside loop
|
|
n >>= n.trailing_zeros();
|
|
|
|
while m != 0 {
|
|
m >>= m.trailing_zeros();
|
|
if n > m { ::std::mem::swap(&mut n, &mut m) }
|
|
m -= n;
|
|
}
|
|
|
|
n << shift
|
|
}
|
|
|
|
/// Calculates the Lowest Common Multiple (LCM) of the number and `other`.
|
|
#[inline]
|
|
fn lcm(&self, other: &Self) -> Self {
|
|
*self * (*other / self.gcd(other))
|
|
}
|
|
|
|
/// 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 {
|
|
*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 Integer;
|
|
|
|
#[test]
|
|
fn test_div_mod_floor() {
|
|
assert_eq!((10 as $T).div_floor(&(3 as $T)), 3 as $T);
|
|
assert_eq!((10 as $T).mod_floor(&(3 as $T)), 1 as $T);
|
|
assert_eq!((10 as $T).div_mod_floor(&(3 as $T)), (3 as $T, 1 as $T));
|
|
assert_eq!((5 as $T).div_floor(&(5 as $T)), 1 as $T);
|
|
assert_eq!((5 as $T).mod_floor(&(5 as $T)), 0 as $T);
|
|
assert_eq!((5 as $T).div_mod_floor(&(5 as $T)), (1 as $T, 0 as $T));
|
|
assert_eq!((3 as $T).div_floor(&(7 as $T)), 0 as $T);
|
|
assert_eq!((3 as $T).mod_floor(&(7 as $T)), 3 as $T);
|
|
assert_eq!((3 as $T).div_mod_floor(&(7 as $T)), (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 {
|
|
::std::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_is_multiple_of() {
|
|
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)));
|
|
}
|
|
|
|
#[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);
|
|
|
|
/// 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: $t = 3;
|
|
let c: Vec<_> = IterBinomial::new(n).collect();
|
|
let expected = vec![1, 3, 3, 1];
|
|
assert_eq!(c, expected);
|
|
} }
|
|
}
|
|
|
|
check_simple!(u8);
|
|
check_simple!(i8);
|
|
check_simple!(u16);
|
|
check_simple!(i16);
|
|
check_simple!(u32);
|
|
check_simple!(i32);
|
|
check_simple!(u64);
|
|
check_simple!(i64);
|
|
|
|
macro_rules! check_binomial {
|
|
($t:ty, $n:expr) => { {
|
|
let n: $t = $n;
|
|
let c: Vec<_> = IterBinomial::new(n).collect();
|
|
let mut k: $t = 0;
|
|
for b in c {
|
|
assert_eq!(b, binomial(n, k));
|
|
k += 1;
|
|
}
|
|
} }
|
|
}
|
|
|
|
// Check the largest n for which there is no overflow.
|
|
check_binomial!(u8, 10);
|
|
check_binomial!(i8, 9);
|
|
check_binomial!(u16, 18);
|
|
check_binomial!(i16, 17);
|
|
check_binomial!(u32, 34);
|
|
check_binomial!(i32, 33);
|
|
check_binomial!(u64, 67);
|
|
check_binomial!(i64, 66);
|
|
}
|
|
|
|
#[test]
|
|
fn test_binomial() {
|
|
macro_rules! check {
|
|
($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);
|
|
}
|