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-// 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);
-}
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