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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);
+}
generated by cgit v1.2.3 (git 2.39.1) at 2024-11-11 23:54:38 +0000