diff options
Diffstat (limited to 'vendor/num-integer/src')
| -rw-r--r-- | vendor/num-integer/src/average.rs | 78 | ||||
| -rw-r--r-- | vendor/num-integer/src/lib.rs | 1386 | ||||
| -rw-r--r-- | vendor/num-integer/src/roots.rs | 391 |
3 files changed, 1855 insertions, 0 deletions
diff --git a/vendor/num-integer/src/average.rs b/vendor/num-integer/src/average.rs new file mode 100644 index 0000000..29cd11e --- /dev/null +++ b/vendor/num-integer/src/average.rs @@ -0,0 +1,78 @@ +use core::ops::{BitAnd, BitOr, BitXor, Shr}; +use Integer; + +/// Provides methods to compute the average of two integers, without overflows. +pub trait Average: Integer { + /// Returns the ceiling value of the average of `self` and `other`. + /// -- `⌈(self + other)/2⌉` + /// + /// # Examples + /// + /// ``` + /// use num_integer::Average; + /// + /// assert_eq!(( 3).average_ceil(&10), 7); + /// assert_eq!((-2).average_ceil(&-5), -3); + /// assert_eq!(( 4).average_ceil(& 4), 4); + /// + /// assert_eq!(u8::max_value().average_ceil(&2), 129); + /// assert_eq!(i8::min_value().average_ceil(&-1), -64); + /// assert_eq!(i8::min_value().average_ceil(&i8::max_value()), 0); + /// ``` + /// + fn average_ceil(&self, other: &Self) -> Self; + + /// Returns the floor value of the average of `self` and `other`. + /// -- `⌊(self + other)/2⌋` + /// + /// # Examples + /// + /// ``` + /// use num_integer::Average; + /// + /// assert_eq!(( 3).average_floor(&10), 6); + /// assert_eq!((-2).average_floor(&-5), -4); + /// assert_eq!(( 4).average_floor(& 4), 4); + /// + /// assert_eq!(u8::max_value().average_floor(&2), 128); + /// assert_eq!(i8::min_value().average_floor(&-1), -65); + /// assert_eq!(i8::min_value().average_floor(&i8::max_value()), -1); + /// ``` + /// + fn average_floor(&self, other: &Self) -> Self; +} + +impl<I> Average for I +where + I: Integer + Shr<usize, Output = I>, + for<'a, 'b> &'a I: + BitAnd<&'b I, Output = I> + BitOr<&'b I, Output = I> + BitXor<&'b I, Output = I>, +{ + // The Henry Gordon Dietz implementation as shown in the Hacker's Delight, + // see http://aggregate.org/MAGIC/#Average%20of%20Integers + + /// Returns the floor value of the average of `self` and `other`. + #[inline] + fn average_floor(&self, other: &I) -> I { + (self & other) + ((self ^ other) >> 1) + } + + /// Returns the ceil value of the average of `self` and `other`. + #[inline] + fn average_ceil(&self, other: &I) -> I { + (self | other) - ((self ^ other) >> 1) + } +} + +/// Returns the floor value of the average of `x` and `y` -- +/// see [Average::average_floor](trait.Average.html#tymethod.average_floor). +#[inline] +pub fn average_floor<T: Average>(x: T, y: T) -> T { + x.average_floor(&y) +} +/// Returns the ceiling value of the average of `x` and `y` -- +/// see [Average::average_ceil](trait.Average.html#tymethod.average_ceil). +#[inline] +pub fn average_ceil<T: Average>(x: T, y: T) -> T { + x.average_ceil(&y) +} diff --git a/vendor/num-integer/src/lib.rs b/vendor/num-integer/src/lib.rs new file mode 100644 index 0000000..5005801 --- /dev/null +++ b/vendor/num-integer/src/lib.rs @@ -0,0 +1,1386 @@ +// 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); +} diff --git a/vendor/num-integer/src/roots.rs b/vendor/num-integer/src/roots.rs new file mode 100644 index 0000000..a9eec1a --- /dev/null +++ b/vendor/num-integer/src/roots.rs @@ -0,0 +1,391 @@ +use core; +use core::mem; +use traits::checked_pow; +use traits::PrimInt; +use Integer; + +/// Provides methods to compute an integer's square root, cube root, +/// and arbitrary `n`th root. +pub trait Roots: Integer { + /// Returns the truncated principal `n`th root of an integer + /// -- `if x >= 0 { ⌊ⁿ√x⌋ } else { ⌈ⁿ√x⌉ }` + /// + /// This is solving for `r` in `rⁿ = x`, rounding toward zero. + /// If `x` is positive, the result will satisfy `rⁿ ≤ x < (r+1)ⁿ`. + /// If `x` is negative and `n` is odd, then `(r-1)ⁿ < x ≤ rⁿ`. + /// + /// # Panics + /// + /// Panics if `n` is zero: + /// + /// ```should_panic + /// # use num_integer::Roots; + /// println!("can't compute ⁰√x : {}", 123.nth_root(0)); + /// ``` + /// + /// or if `n` is even and `self` is negative: + /// + /// ```should_panic + /// # use num_integer::Roots; + /// println!("no imaginary numbers... {}", (-1).nth_root(10)); + /// ``` + /// + /// # Examples + /// + /// ``` + /// use num_integer::Roots; + /// + /// let x: i32 = 12345; + /// assert_eq!(x.nth_root(1), x); + /// assert_eq!(x.nth_root(2), x.sqrt()); + /// assert_eq!(x.nth_root(3), x.cbrt()); + /// assert_eq!(x.nth_root(4), 10); + /// assert_eq!(x.nth_root(13), 2); + /// assert_eq!(x.nth_root(14), 1); + /// assert_eq!(x.nth_root(std::u32::MAX), 1); + /// + /// assert_eq!(std::i32::MAX.nth_root(30), 2); + /// assert_eq!(std::i32::MAX.nth_root(31), 1); + /// assert_eq!(std::i32::MIN.nth_root(31), -2); + /// assert_eq!((std::i32::MIN + 1).nth_root(31), -1); + /// + /// assert_eq!(std::u32::MAX.nth_root(31), 2); + /// assert_eq!(std::u32::MAX.nth_root(32), 1); + /// ``` + fn nth_root(&self, n: u32) -> Self; + + /// Returns the truncated principal square root of an integer -- `⌊√x⌋` + /// + /// This is solving for `r` in `r² = x`, rounding toward zero. + /// The result will satisfy `r² ≤ x < (r+1)²`. + /// + /// # Panics + /// + /// Panics if `self` is less than zero: + /// + /// ```should_panic + /// # use num_integer::Roots; + /// println!("no imaginary numbers... {}", (-1).sqrt()); + /// ``` + /// + /// # Examples + /// + /// ``` + /// use num_integer::Roots; + /// + /// let x: i32 = 12345; + /// assert_eq!((x * x).sqrt(), x); + /// assert_eq!((x * x + 1).sqrt(), x); + /// assert_eq!((x * x - 1).sqrt(), x - 1); + /// ``` + #[inline] + fn sqrt(&self) -> Self { + self.nth_root(2) + } + + /// Returns the truncated principal cube root of an integer -- + /// `if x >= 0 { ⌊∛x⌋ } else { ⌈∛x⌉ }` + /// + /// This is solving for `r` in `r³ = x`, rounding toward zero. + /// If `x` is positive, the result will satisfy `r³ ≤ x < (r+1)³`. + /// If `x` is negative, then `(r-1)³ < x ≤ r³`. + /// + /// # Examples + /// + /// ``` + /// use num_integer::Roots; + /// + /// let x: i32 = 1234; + /// assert_eq!((x * x * x).cbrt(), x); + /// assert_eq!((x * x * x + 1).cbrt(), x); + /// assert_eq!((x * x * x - 1).cbrt(), x - 1); + /// + /// assert_eq!((-(x * x * x)).cbrt(), -x); + /// assert_eq!((-(x * x * x + 1)).cbrt(), -x); + /// assert_eq!((-(x * x * x - 1)).cbrt(), -(x - 1)); + /// ``` + #[inline] + fn cbrt(&self) -> Self { + self.nth_root(3) + } +} + +/// Returns the truncated principal square root of an integer -- +/// see [Roots::sqrt](trait.Roots.html#method.sqrt). +#[inline] +pub fn sqrt<T: Roots>(x: T) -> T { + x.sqrt() +} + +/// Returns the truncated principal cube root of an integer -- +/// see [Roots::cbrt](trait.Roots.html#method.cbrt). +#[inline] +pub fn cbrt<T: Roots>(x: T) -> T { + x.cbrt() +} + +/// Returns the truncated principal `n`th root of an integer -- +/// see [Roots::nth_root](trait.Roots.html#tymethod.nth_root). +#[inline] +pub fn nth_root<T: Roots>(x: T, n: u32) -> T { + x.nth_root(n) +} + +macro_rules! signed_roots { + ($T:ty, $U:ty) => { + impl Roots for $T { + #[inline] + fn nth_root(&self, n: u32) -> Self { + if *self >= 0 { + (*self as $U).nth_root(n) as Self + } else { + assert!(n.is_odd(), "even roots of a negative are imaginary"); + -((self.wrapping_neg() as $U).nth_root(n) as Self) + } + } + + #[inline] + fn sqrt(&self) -> Self { + assert!(*self >= 0, "the square root of a negative is imaginary"); + (*self as $U).sqrt() as Self + } + + #[inline] + fn cbrt(&self) -> Self { + if *self >= 0 { + (*self as $U).cbrt() as Self + } else { + -((self.wrapping_neg() as $U).cbrt() as Self) + } + } + } + }; +} + +signed_roots!(i8, u8); +signed_roots!(i16, u16); +signed_roots!(i32, u32); +signed_roots!(i64, u64); +#[cfg(has_i128)] +signed_roots!(i128, u128); +signed_roots!(isize, usize); + +#[inline] +fn fixpoint<T, F>(mut x: T, f: F) -> T +where + T: Integer + Copy, + F: Fn(T) -> T, +{ + let mut xn = f(x); + while x < xn { + x = xn; + xn = f(x); + } + while x > xn { + x = xn; + xn = f(x); + } + x +} + +#[inline] +fn bits<T>() -> u32 { + 8 * mem::size_of::<T>() as u32 +} + +#[inline] +fn log2<T: PrimInt>(x: T) -> u32 { + debug_assert!(x > T::zero()); + bits::<T>() - 1 - x.leading_zeros() +} + +macro_rules! unsigned_roots { + ($T:ident) => { + impl Roots for $T { + #[inline] + fn nth_root(&self, n: u32) -> Self { + fn go(a: $T, n: u32) -> $T { + // Specialize small roots + match n { + 0 => panic!("can't find a root of degree 0!"), + 1 => return a, + 2 => return a.sqrt(), + 3 => return a.cbrt(), + _ => (), + } + + // The root of values less than 2ⁿ can only be 0 or 1. + if bits::<$T>() <= n || a < (1 << n) { + return (a > 0) as $T; + } + + if bits::<$T>() > 64 { + // 128-bit division is slow, so do a bitwise `nth_root` until it's small enough. + return if a <= core::u64::MAX as $T { + (a as u64).nth_root(n) as $T + } else { + let lo = (a >> n).nth_root(n) << 1; + let hi = lo + 1; + // 128-bit `checked_mul` also involves division, but we can't always + // compute `hiⁿ` without risking overflow. Try to avoid it though... + if hi.next_power_of_two().trailing_zeros() * n >= bits::<$T>() { + match checked_pow(hi, n as usize) { + Some(x) if x <= a => hi, + _ => lo, + } + } else { + if hi.pow(n) <= a { + hi + } else { + lo + } + } + }; + } + + #[cfg(feature = "std")] + #[inline] + fn guess(x: $T, n: u32) -> $T { + // for smaller inputs, `f64` doesn't justify its cost. + if bits::<$T>() <= 32 || x <= core::u32::MAX as $T { + 1 << ((log2(x) + n - 1) / n) + } else { + ((x as f64).ln() / f64::from(n)).exp() as $T + } + } + + #[cfg(not(feature = "std"))] + #[inline] + fn guess(x: $T, n: u32) -> $T { + 1 << ((log2(x) + n - 1) / n) + } + + // https://en.wikipedia.org/wiki/Nth_root_algorithm + let n1 = n - 1; + let next = |x: $T| { + let y = match checked_pow(x, n1 as usize) { + Some(ax) => a / ax, + None => 0, + }; + (y + x * n1 as $T) / n as $T + }; + fixpoint(guess(a, n), next) + } + go(*self, n) + } + + #[inline] + fn sqrt(&self) -> Self { + fn go(a: $T) -> $T { + if bits::<$T>() > 64 { + // 128-bit division is slow, so do a bitwise `sqrt` until it's small enough. + return if a <= core::u64::MAX as $T { + (a as u64).sqrt() as $T + } else { + let lo = (a >> 2u32).sqrt() << 1; + let hi = lo + 1; + if hi * hi <= a { + hi + } else { + lo + } + }; + } + + if a < 4 { + return (a > 0) as $T; + } + + #[cfg(feature = "std")] + #[inline] + fn guess(x: $T) -> $T { + (x as f64).sqrt() as $T + } + + #[cfg(not(feature = "std"))] + #[inline] + fn guess(x: $T) -> $T { + 1 << ((log2(x) + 1) / 2) + } + + // https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method + let next = |x: $T| (a / x + x) >> 1; + fixpoint(guess(a), next) + } + go(*self) + } + + #[inline] + fn cbrt(&self) -> Self { + fn go(a: $T) -> $T { + if bits::<$T>() > 64 { + // 128-bit division is slow, so do a bitwise `cbrt` until it's small enough. + return if a <= core::u64::MAX as $T { + (a as u64).cbrt() as $T + } else { + let lo = (a >> 3u32).cbrt() << 1; + let hi = lo + 1; + if hi * hi * hi <= a { + hi + } else { + lo + } + }; + } + + if bits::<$T>() <= 32 { + // Implementation based on Hacker's Delight `icbrt2` + let mut x = a; + let mut y2 = 0; + let mut y = 0; + let smax = bits::<$T>() / 3; + for s in (0..smax + 1).rev() { + let s = s * 3; + y2 *= 4; + y *= 2; + let b = 3 * (y2 + y) + 1; + if x >> s >= b { + x -= b << s; + y2 += 2 * y + 1; + y += 1; + } + } + return y; + } + + if a < 8 { + return (a > 0) as $T; + } + if a <= core::u32::MAX as $T { + return (a as u32).cbrt() as $T; + } + + #[cfg(feature = "std")] + #[inline] + fn guess(x: $T) -> $T { + (x as f64).cbrt() as $T + } + + #[cfg(not(feature = "std"))] + #[inline] + fn guess(x: $T) -> $T { + 1 << ((log2(x) + 2) / 3) + } + + // https://en.wikipedia.org/wiki/Cube_root#Numerical_methods + let next = |x: $T| (a / (x * x) + x * 2) / 3; + fixpoint(guess(a), next) + } + go(*self) + } + } + }; +} + +unsigned_roots!(u8); +unsigned_roots!(u16); +unsigned_roots!(u32); +unsigned_roots!(u64); +#[cfg(has_i128)] +unsigned_roots!(u128); +unsigned_roots!(usize); |