num-integer-0.1.41/.gitignore010066400247370024737000000000221344401361400142600ustar0000000000000000Cargo.lock target num-integer-0.1.41/Cargo.toml.orig010066400247370024737000000013371347111627300151760ustar0000000000000000[package] authors = ["The Rust Project Developers"] description = "Integer traits and functions" documentation = "https://docs.rs/num-integer" homepage = "https://github.com/rust-num/num-integer" keywords = ["mathematics", "numerics"] categories = ["algorithms", "science", "no-std"] license = "MIT/Apache-2.0" repository = "https://github.com/rust-num/num-integer" name = "num-integer" version = "0.1.41" readme = "README.md" build = "build.rs" exclude = ["/ci/*", "/.travis.yml", "/bors.toml"] [package.metadata.docs.rs] features = ["std"] [dependencies.num-traits] version = "0.2.4" default-features = false [features] default = ["std"] i128 = ["num-traits/i128"] std = ["num-traits/std"] [build-dependencies] autocfg = "0.1.3" num-integer-0.1.41/Cargo.toml0000644000000023610000000000000114100ustar00# THIS FILE IS AUTOMATICALLY GENERATED BY CARGO # # When uploading crates to the registry Cargo will automatically # "normalize" Cargo.toml files for maximal compatibility # with all versions of Cargo and also rewrite `path` dependencies # to registry (e.g., crates.io) dependencies # # If you believe there's an error in this file please file an # issue against the rust-lang/cargo repository. 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See the License for the specific language governing permissions and limitations under the License. num-integer-0.1.41/LICENSE-MIT010066400247370024737000000020571344401361400137360ustar0000000000000000Copyright (c) 2014 The Rust Project Developers Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. 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IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. num-integer-0.1.41/README.md010066400247370024737000000024031344401361400135540ustar0000000000000000# num-integer [![crate](https://img.shields.io/crates/v/num-integer.svg)](https://crates.io/crates/num-integer) [![documentation](https://docs.rs/num-integer/badge.svg)](https://docs.rs/num-integer) ![minimum rustc 1.8](https://img.shields.io/badge/rustc-1.8+-red.svg) [![Travis status](https://travis-ci.org/rust-num/num-integer.svg?branch=master)](https://travis-ci.org/rust-num/num-integer) `Integer` trait and functions for Rust. ## Usage Add this to your `Cargo.toml`: ```toml [dependencies] num-integer = "0.1" ``` and this to your crate root: ```rust extern crate num_integer; ``` ## Features This crate can be used without the standard library (`#![no_std]`) by disabling the default `std` feature. Use this in `Cargo.toml`: ```toml [dependencies.num-integer] version = "0.1.36" default-features = false ``` There is no functional difference with and without `std` at this time, but there may be in the future. Implementations for `i128` and `u128` are only available with Rust 1.26 and later. The build script automatically detects this, but you can make it mandatory by enabling the `i128` crate feature. ## Releases Release notes are available in [RELEASES.md](RELEASES.md). ## Compatibility The `num-integer` crate is tested for rustc 1.8 and greater. num-integer-0.1.41/RELEASES.md010066400247370024737000000044521347111641000140260ustar0000000000000000# Release 0.1.41 (2019-05-21) - [Fixed feature detection on `no_std` targets][25]. **Contributors**: @cuviper [25]: https://github.com/rust-num/num-integer/pull/25 # Release 0.1.40 (2019-05-20) - [Optimized primitive `gcd` by avoiding memory swaps][11]. - [Fixed `lcm(0, 0)` to return `0`, rather than panicking][18]. - [Added `Integer::div_ceil`, `next_multiple_of`, and `prev_multiple_of`][16]. - [Added `Integer::gcd_lcm`, `extended_gcd`, and `extended_gcd_lcm`][19]. **Contributors**: @cuviper, @ignatenkobrain, @smarnach, @strake [11]: https://github.com/rust-num/num-integer/pull/11 [16]: https://github.com/rust-num/num-integer/pull/16 [18]: https://github.com/rust-num/num-integer/pull/18 [19]: https://github.com/rust-num/num-integer/pull/19 # Release 0.1.39 (2018-06-20) - [The new `Roots` trait provides `sqrt`, `cbrt`, and `nth_root` methods][9], calculating an `Integer`'s principal roots rounded toward zero. **Contributors**: @cuviper [9]: https://github.com/rust-num/num-integer/pull/9 # Release 0.1.38 (2018-05-11) - [Support for 128-bit integers is now automatically detected and enabled.][8] Setting the `i128` crate feature now causes the build script to panic if such support is not detected. **Contributors**: @cuviper [8]: https://github.com/rust-num/num-integer/pull/8 # Release 0.1.37 (2018-05-10) - [`Integer` is now implemented for `i128` and `u128`][7] starting with Rust 1.26, enabled by the new `i128` crate feature. **Contributors**: @cuviper [7]: https://github.com/rust-num/num-integer/pull/7 # Release 0.1.36 (2018-02-06) - [num-integer now has its own source repository][num-356] at [rust-num/num-integer][home]. - [Corrected the argument order documented in `Integer::is_multiple_of`][1] - [There is now a `std` feature][5], enabled by default, along with the implication that building *without* this feature makes this a `#[no_std]` crate. - There is no difference in the API at this time. **Contributors**: @cuviper, @jaystrictor [home]: https://github.com/rust-num/num-integer [num-356]: https://github.com/rust-num/num/pull/356 [1]: https://github.com/rust-num/num-integer/pull/1 [5]: https://github.com/rust-num/num-integer/pull/5 # Prior releases No prior release notes were kept. Thanks all the same to the many contributors that have made this crate what it is! num-integer-0.1.41/benches/gcd.rs010066400247370024737000000115001344700235200150050ustar0000000000000000//! Benchmark comparing the current GCD implemtation against an older one. #![feature(test)] extern crate num_integer; extern crate num_traits; extern crate test; use num_integer::Integer; use num_traits::{AsPrimitive, Bounded, Signed}; use test::{black_box, Bencher}; trait GcdOld: Integer { fn gcd_old(&self, other: &Self) -> Self; } macro_rules! impl_gcd_old_for_isize { ($T:ty) => { impl GcdOld for $T { /// Calculates the Greatest Common Divisor (GCD) of the number and /// `other`. The result is always positive. #[inline] fn gcd_old(&self, other: &Self) -> Self { // Use Stein's algorithm let mut m = *self; let mut n = *other; if m == 0 || n == 0 { return (m | n).abs(); } // find common factors of 2 let shift = (m | n).trailing_zeros(); // The algorithm needs positive numbers, but the minimum value // can't be represented as a positive one. // It's also a power of two, so the gcd can be // calculated by bitshifting in that case // Assuming two's complement, the number created by the shift // is positive for all numbers except gcd = abs(min value) // The call to .abs() causes a panic in debug mode if m == Self::min_value() || n == Self::min_value() { return (1 << shift).abs(); } // guaranteed to be positive now, rest like unsigned algorithm m = m.abs(); n = n.abs(); // divide n and m by 2 until odd // m inside loop n >>= n.trailing_zeros(); while m != 0 { m >>= m.trailing_zeros(); if n > m { std::mem::swap(&mut n, &mut m) } m -= n; } n << shift } } }; } impl_gcd_old_for_isize!(i8); impl_gcd_old_for_isize!(i16); impl_gcd_old_for_isize!(i32); impl_gcd_old_for_isize!(i64); impl_gcd_old_for_isize!(isize); impl_gcd_old_for_isize!(i128); macro_rules! impl_gcd_old_for_usize { ($T:ty) => { impl GcdOld for $T { /// Calculates the Greatest Common Divisor (GCD) of the number and /// `other`. The result is always positive. #[inline] fn gcd_old(&self, other: &Self) -> Self { // Use Stein's algorithm let mut m = *self; let mut n = *other; if m == 0 || n == 0 { return m | n; } // find common factors of 2 let shift = (m | n).trailing_zeros(); // divide n and m by 2 until odd // m inside loop n >>= n.trailing_zeros(); while m != 0 { m >>= m.trailing_zeros(); if n > m { std::mem::swap(&mut n, &mut m) } m -= n; } n << shift } } }; } impl_gcd_old_for_usize!(u8); impl_gcd_old_for_usize!(u16); impl_gcd_old_for_usize!(u32); impl_gcd_old_for_usize!(u64); impl_gcd_old_for_usize!(usize); impl_gcd_old_for_usize!(u128); /// Return an iterator that yields all Fibonacci numbers fitting into a u128. fn fibonacci() -> impl Iterator { (0..185).scan((0, 1), |&mut (ref mut a, ref mut b), _| { let tmp = *a; *a = *b; *b += tmp; Some(*b) }) } fn run_bench(b: &mut Bencher, gcd: fn(&T, &T) -> T) where T: AsPrimitive, u128: AsPrimitive, { let max_value: u128 = T::max_value().as_(); let pairs: Vec<(T, T)> = fibonacci() .collect::>() .windows(2) .filter(|&pair| pair[0] <= max_value && pair[1] <= max_value) .map(|pair| (pair[0].as_(), pair[1].as_())) .collect(); b.iter(|| { for &(ref m, ref n) in &pairs { black_box(gcd(m, n)); } }); } macro_rules! bench_gcd { ($T:ident) => { mod $T { use crate::{run_bench, GcdOld}; use num_integer::Integer; use test::Bencher; #[bench] fn bench_gcd(b: &mut Bencher) { run_bench(b, $T::gcd); } #[bench] fn bench_gcd_old(b: &mut Bencher) { run_bench(b, $T::gcd_old); } } }; } bench_gcd!(u8); bench_gcd!(u16); bench_gcd!(u32); bench_gcd!(u64); bench_gcd!(u128); bench_gcd!(i8); bench_gcd!(i16); bench_gcd!(i32); bench_gcd!(i64); bench_gcd!(i128); num-integer-0.1.41/benches/roots.rs010066400247370024737000000077711344700235200154350ustar0000000000000000//! Benchmark sqrt and cbrt #![feature(test)] extern crate num_integer; extern crate num_traits; extern crate test; use num_integer::Integer; use num_traits::checked_pow; use num_traits::{AsPrimitive, PrimInt, WrappingAdd, WrappingMul}; use test::{black_box, Bencher}; trait BenchInteger: Integer + PrimInt + WrappingAdd + WrappingMul + 'static {} impl BenchInteger for T where T: Integer + PrimInt + WrappingAdd + WrappingMul + 'static {} fn bench(b: &mut Bencher, v: &[T], f: F, n: u32) where T: BenchInteger, F: Fn(&T) -> T, { // Pre-validate the results... for i in v { let rt = f(i); if *i >= T::zero() { let rt1 = rt + T::one(); assert!(rt.pow(n) <= *i); if let Some(x) = checked_pow(rt1, n as usize) { assert!(*i < x); } } else { let rt1 = rt - T::one(); assert!(rt < T::zero()); assert!(*i <= rt.pow(n)); if let Some(x) = checked_pow(rt1, n as usize) { assert!(x < *i); } }; } // Now just run as fast as we can! b.iter(|| { for i in v { black_box(f(i)); } }); } // Simple PRNG so we don't have to worry about rand compatibility fn lcg(x: T) -> T where u32: AsPrimitive, T: BenchInteger, { // LCG parameters from Numerical Recipes // (but we're applying it to arbitrary sizes) const LCG_A: u32 = 1664525; const LCG_C: u32 = 1013904223; x.wrapping_mul(&LCG_A.as_()).wrapping_add(&LCG_C.as_()) } fn bench_rand(b: &mut Bencher, f: F, n: u32) where u32: AsPrimitive, T: BenchInteger, F: Fn(&T) -> T, { let mut x: T = 3u32.as_(); let v: Vec = (0..1000) .map(|_| { x = lcg(x); x }) .collect(); bench(b, &v, f, n); } fn bench_rand_pos(b: &mut Bencher, f: F, n: u32) where u32: AsPrimitive, T: BenchInteger, F: Fn(&T) -> T, { let mut x: T = 3u32.as_(); let v: Vec = (0..1000) .map(|_| { x = lcg(x); while x < T::zero() { x = lcg(x); } x }) .collect(); bench(b, &v, f, n); } fn bench_small(b: &mut Bencher, f: F, n: u32) where u32: AsPrimitive, T: BenchInteger, F: Fn(&T) -> T, { let v: Vec = (0..1000).map(|i| i.as_()).collect(); bench(b, &v, f, n); } fn bench_small_pos(b: &mut Bencher, f: F, n: u32) where u32: AsPrimitive, T: BenchInteger, F: Fn(&T) -> T, { let v: Vec = (0..1000) .map(|i| i.as_().mod_floor(&T::max_value())) .collect(); bench(b, &v, f, n); } macro_rules! bench_roots { ($($T:ident),*) => {$( mod $T { use test::Bencher; use num_integer::Roots; #[bench] fn sqrt_rand(b: &mut Bencher) { ::bench_rand_pos(b, $T::sqrt, 2); } #[bench] fn sqrt_small(b: &mut Bencher) { ::bench_small_pos(b, $T::sqrt, 2); } #[bench] fn cbrt_rand(b: &mut Bencher) { ::bench_rand(b, $T::cbrt, 3); } #[bench] fn cbrt_small(b: &mut Bencher) { ::bench_small(b, $T::cbrt, 3); } #[bench] fn fourth_root_rand(b: &mut Bencher) { ::bench_rand_pos(b, |x: &$T| x.nth_root(4), 4); } #[bench] fn fourth_root_small(b: &mut Bencher) { ::bench_small_pos(b, |x: &$T| x.nth_root(4), 4); } #[bench] fn fifth_root_rand(b: &mut Bencher) { ::bench_rand(b, |x: &$T| x.nth_root(5), 5); } #[bench] fn fifth_root_small(b: &mut Bencher) { ::bench_small(b, |x: &$T| x.nth_root(5), 5); } } )*} } bench_roots!(i8, i16, i32, i64, i128); bench_roots!(u8, u16, u32, u64, u128); num-integer-0.1.41/build.rs010066400247370024737000000004651347062134500137550ustar0000000000000000extern crate autocfg; use std::env; fn main() { let ac = autocfg::new(); if ac.probe_type("i128") { println!("cargo:rustc-cfg=has_i128"); } else if env::var_os("CARGO_FEATURE_I128").is_some() { panic!("i128 support was not detected!"); } autocfg::rerun_path(file!()); } num-integer-0.1.41/src/lib.rs010066400247370024737000001215171347062134500142150ustar0000000000000000// 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 or the MIT license // , 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}; 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: 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 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, _hidden: (), } } else { ExtendedGcd { gcd: Self::zero() - r.1, x: Self::zero() - s.1, y: Self::zero() - t.1, _hidden: (), } } } /// Greatest common divisor, least common multiple, and Bézout coefficients. #[inline] fn extended_gcd_lcm(&self, other: &Self) -> (ExtendedGcd, 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)); /// ~~~ #[inline] 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 { pub gcd: A, pub x: A, pub y: A, _hidden: (), } /// Simultaneous integer division and modulus #[inline] pub fn div_rem(x: T, y: T) -> (T, T) { x.div_rem(&y) } /// Floored integer division #[inline] pub fn div_floor(x: T, y: T) -> T { x.div_floor(&y) } /// Floored integer modulus #[inline] pub fn mod_floor(x: T, y: T) -> T { x.mod_floor(&y) } /// Simultaneous floored integer division and modulus #[inline] pub fn div_mod_floor(x: T, y: T) -> (T, T) { x.div_mod_floor(&y) } /// Ceiled integer division #[inline] pub fn div_ceil(x: T, y: T) -> T { x.div_ceil(&y) } /// Calculates the Greatest Common Divisor (GCD) of the number and `other`. The /// result is always positive. #[inline(always)] pub fn gcd(x: T, y: T) -> T { x.gcd(&y) } /// Calculates the Lowest Common Multiple (LCM) of the number and `other`. #[inline(always)] pub fn lcm(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(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 positive. #[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) { 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 { *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) } } #[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 = (n.div_floor(&d), n.mod_floor(&d)); let combined_div_mod_floor = n.div_mod_floor(&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: 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 $T).is_odd(), false); } } }; } impl_integer_for_isize!(i8, test_integer_i8); impl_integer_for_isize!(i16, test_integer_i16); impl_integer_for_isize!(i32, test_integer_i32); impl_integer_for_isize!(i64, test_integer_i64); impl_integer_for_isize!(isize, test_integer_isize); #[cfg(has_i128)] impl_integer_for_isize!(i128, test_integer_i128); macro_rules! impl_integer_for_usize { ($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) { 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 { *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!((10 as $T).div_floor(&(3 as $T)), 3 as $T); assert_eq!((10 as $T).mod_floor(&(3 as $T)), 1 as $T); assert_eq!((10 as $T).div_mod_floor(&(3 as $T)), (3 as $T, 1 as $T)); assert_eq!((5 as $T).div_floor(&(5 as $T)), 1 as $T); assert_eq!((5 as $T).mod_floor(&(5 as $T)), 0 as $T); assert_eq!((5 as $T).div_mod_floor(&(5 as $T)), (1 as $T, 0 as $T)); assert_eq!((3 as $T).div_floor(&(7 as $T)), 0 as $T); assert_eq!((3 as $T).mod_floor(&(7 as $T)), 3 as $T); assert_eq!((3 as $T).div_mod_floor(&(7 as $T)), (0 as $T, 3 as $T)); } #[test] fn test_gcd() { assert_eq!((10 as $T).gcd(&2), 2 as $T); assert_eq!((10 as $T).gcd(&3), 1 as $T); assert_eq!((0 as $T).gcd(&3), 3 as $T); assert_eq!((3 as $T).gcd(&3), 3 as $T); assert_eq!((56 as $T).gcd(&42), 14 as $T); } #[test] fn test_gcd_cmp_with_euclidean() { fn euclidean_gcd(mut m: $T, mut n: $T) -> $T { while m != 0 { 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!((6 as $T).is_multiple_of(&(6 as $T))); assert!((6 as $T).is_multiple_of(&(3 as $T))); assert!((6 as $T).is_multiple_of(&(1 as $T))); } #[test] fn test_even() { assert_eq!((0 as $T).is_even(), true); assert_eq!((1 as $T).is_even(), false); assert_eq!((2 as $T).is_even(), true); assert_eq!((3 as $T).is_even(), false); assert_eq!((4 as $T).is_even(), true); } #[test] fn test_odd() { assert_eq!((0 as $T).is_odd(), false); assert_eq!((1 as $T).is_odd(), true); assert_eq!((2 as $T).is_odd(), false); assert_eq!((3 as $T).is_odd(), true); assert_eq!((4 as $T).is_odd(), false); } } }; } impl_integer_for_usize!(u8, test_integer_u8); impl_integer_for_usize!(u16, test_integer_u16); impl_integer_for_usize!(u32, test_integer_u32); impl_integer_for_usize!(u64, test_integer_u64); impl_integer_for_usize!(usize, test_integer_usize); #[cfg(has_i128)] impl_integer_for_usize!(u128, test_integer_u128); /// An iterator over binomial coefficients. pub struct IterBinomial { a: T, n: T, k: T, } impl IterBinomial 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 { IterBinomial { k: T::zero(), a: T::one(), n: n, } } } impl Iterator for IterBinomial where T: Integer + Clone, { type Item = T; fn next(&mut self) -> Option { 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(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(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(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); } num-integer-0.1.41/src/roots.rs010066400247370024737000000306441347062134500146150ustar0000000000000000use 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(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(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(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 $U).cbrt() as Self) } } } }; } signed_roots!(i8, u8); signed_roots!(i16, u16); signed_roots!(i32, u32); signed_roots!(i64, u64); #[cfg(has_i128)] signed_roots!(i128, u128); signed_roots!(isize, usize); #[inline] fn fixpoint(mut x: T, f: F) -> T where T: Integer + Copy, F: Fn(T) -> T, { let mut xn = f(x); while x < xn { x = xn; xn = f(x); } while x > xn { x = xn; xn = f(x); } x } #[inline] fn bits() -> u32 { 8 * mem::size_of::() as u32 } #[inline] fn log2(x: T) -> u32 { debug_assert!(x > T::zero()); bits::() - 1 - x.leading_zeros() } macro_rules! unsigned_roots { ($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); num-integer-0.1.41/tests/roots.rs010066400247370024737000000151441344700235200151610ustar0000000000000000extern crate num_integer; extern crate num_traits; use num_integer::Roots; use num_traits::checked_pow; use num_traits::{AsPrimitive, PrimInt, Signed}; use std::f64::MANTISSA_DIGITS; use std::fmt::Debug; use std::mem; trait TestInteger: Roots + PrimInt + Debug + AsPrimitive + 'static {} impl TestInteger for T where T: Roots + PrimInt + Debug + AsPrimitive + 'static {} /// Check that each root is correct /// /// If `x` is positive, check `rⁿ ≤ x < (r+1)ⁿ`. /// If `x` is negative, check `(r-1)ⁿ < x ≤ rⁿ`. fn check(v: &[T], n: u32) where T: TestInteger, { for i in v { let rt = i.nth_root(n); // println!("nth_root({:?}, {}) = {:?}", i, n, rt); if n == 2 { assert_eq!(rt, i.sqrt()); } else if n == 3 { assert_eq!(rt, i.cbrt()); } if *i >= T::zero() { let rt1 = rt + T::one(); assert!(rt.pow(n) <= *i); if let Some(x) = checked_pow(rt1, n as usize) { assert!(*i < x); } } else { let rt1 = rt - T::one(); assert!(rt < T::zero()); assert!(*i <= rt.pow(n)); if let Some(x) = checked_pow(rt1, n as usize) { assert!(x < *i); } }; } } /// Get the maximum value that will round down as `f64` (if any), /// and its successor that will round up. /// /// Important because the `std` implementations cast to `f64` to /// get a close approximation of the roots. fn mantissa_max() -> Option<(T, T)> where T: TestInteger, { let bits = if T::min_value().is_zero() { 8 * mem::size_of::() } else { 8 * mem::size_of::() - 1 }; if bits > MANTISSA_DIGITS as usize { let rounding_bit = T::one() << (bits - MANTISSA_DIGITS as usize - 1); let x = T::max_value() - rounding_bit; let x1 = x + T::one(); let x2 = x1 + T::one(); assert!(x.as_() < x1.as_()); assert_eq!(x1.as_(), x2.as_()); Some((x, x1)) } else { None } } fn extend(v: &mut Vec, start: T, end: T) where T: TestInteger, { let mut i = start; while i < end { v.push(i); i = i + T::one(); } v.push(i); } fn extend_shl(v: &mut Vec, start: T, end: T, mask: T) where T: TestInteger, { let mut i = start; while i != end { v.push(i); i = (i << 1) & mask; } } fn extend_shr(v: &mut Vec, start: T, end: T) where T: TestInteger, { let mut i = start; while i != end { v.push(i); i = i >> 1; } } fn pos() -> Vec where T: TestInteger, i8: AsPrimitive, { let mut v: Vec = vec![]; if mem::size_of::() == 1 { extend(&mut v, T::zero(), T::max_value()); } else { extend(&mut v, T::zero(), i8::max_value().as_()); extend( &mut v, T::max_value() - i8::max_value().as_(), T::max_value(), ); if let Some((i, j)) = mantissa_max::() { v.push(i); v.push(j); } extend_shl(&mut v, T::max_value(), T::zero(), !T::min_value()); extend_shr(&mut v, T::max_value(), T::zero()); } v } fn neg() -> Vec where T: TestInteger + Signed, i8: AsPrimitive, { let mut v: Vec = vec![]; if mem::size_of::() <= 1 { extend(&mut v, T::min_value(), T::zero()); } else { extend(&mut v, i8::min_value().as_(), T::zero()); extend( &mut v, T::min_value(), T::min_value() - i8::min_value().as_(), ); if let Some((i, j)) = mantissa_max::() { v.push(-i); v.push(-j); } extend_shl(&mut v, -T::one(), T::min_value(), !T::zero()); extend_shr(&mut v, T::min_value(), -T::one()); } v } macro_rules! test_roots { ($I:ident, $U:ident) => { mod $I { use check; use neg; use num_integer::Roots; use pos; use std::mem; #[test] #[should_panic] fn zeroth_root() { (123 as $I).nth_root(0); } #[test] fn sqrt() { check(&pos::<$I>(), 2); } #[test] #[should_panic] fn sqrt_neg() { (-123 as $I).sqrt(); } #[test] fn cbrt() { check(&pos::<$I>(), 3); } #[test] fn cbrt_neg() { check(&neg::<$I>(), 3); } #[test] fn nth_root() { let bits = 8 * mem::size_of::<$I>() as u32 - 1; let pos = pos::<$I>(); for n in 4..bits { check(&pos, n); } } #[test] fn nth_root_neg() { let bits = 8 * mem::size_of::<$I>() as u32 - 1; let neg = neg::<$I>(); for n in 2..bits / 2 { check(&neg, 2 * n + 1); } } #[test] fn bit_size() { let bits = 8 * mem::size_of::<$I>() as u32 - 1; assert_eq!($I::max_value().nth_root(bits - 1), 2); assert_eq!($I::max_value().nth_root(bits), 1); assert_eq!($I::min_value().nth_root(bits), -2); assert_eq!(($I::min_value() + 1).nth_root(bits), -1); } } mod $U { use check; use num_integer::Roots; use pos; use std::mem; #[test] #[should_panic] fn zeroth_root() { (123 as $U).nth_root(0); } #[test] fn sqrt() { check(&pos::<$U>(), 2); } #[test] fn cbrt() { check(&pos::<$U>(), 3); } #[test] fn nth_root() { let bits = 8 * mem::size_of::<$I>() as u32 - 1; let pos = pos::<$I>(); for n in 4..bits { check(&pos, n); } } #[test] fn bit_size() { let bits = 8 * mem::size_of::<$U>() as u32; assert_eq!($U::max_value().nth_root(bits - 1), 2); assert_eq!($U::max_value().nth_root(bits), 1); } } }; } test_roots!(i8, u8); test_roots!(i16, u16); test_roots!(i32, u32); test_roots!(i64, u64); #[cfg(has_i128)] test_roots!(i128, u128); test_roots!(isize, usize); num-integer-0.1.41/.cargo_vcs_info.json0000644000000001120000000000000134020ustar00{ "git": { "sha1": "bcce4ccbb293e3c6eee0098a1f74a0479739a48c" } }