src/lib.rs - platform/external/rust/crates/num-traits - Git at Google

 // 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.

 //! Numeric traits for generic mathematics
 //!
 //! ## Compatibility
 //!
 //! The `num-traits` crate is tested for rustc 1.8 and greater.

 #![doc(html_root_url = "https://docs.rs/num-traits/0.2")]
 #![deny(unconditional_recursion)]
 #![no_std]
 #[cfg(feature = "std")]
 extern crate std;

 // Only `no_std` builds actually use `libm`.
 #[cfg(all(not(feature = "std"), feature = "libm"))]
 extern crate libm;

 use core::fmt;
 use core::num::Wrapping;
 use core::ops::{Add, Div, Mul, Rem, Sub};
 use core::ops::{AddAssign, DivAssign, MulAssign, RemAssign, SubAssign};

 pub use bounds::Bounded;
 #[cfg(any(feature = "std", feature = "libm"))]
 pub use float::Float;
 pub use float::FloatConst;
 // pub use real::{FloatCore, Real}; // NOTE: Don't do this, it breaks `use num_traits::*;`.
 pub use cast::{cast, AsPrimitive, FromPrimitive, NumCast, ToPrimitive};
 pub use identities::{one, zero, One, Zero};
 pub use int::PrimInt;
 pub use ops::checked::{
     CheckedAdd, CheckedDiv, CheckedMul, CheckedNeg, CheckedRem, CheckedShl, CheckedShr, CheckedSub,
 };
 pub use ops::inv::Inv;
 pub use ops::mul_add::{MulAdd, MulAddAssign};
 pub use ops::saturating::{Saturating, SaturatingAdd, SaturatingMul, SaturatingSub};
 pub use ops::wrapping::{
     WrappingAdd, WrappingMul, WrappingNeg, WrappingShl, WrappingShr, WrappingSub,
 };
 pub use pow::{checked_pow, pow, Pow};
 pub use sign::{abs, abs_sub, signum, Signed, Unsigned};

 #[macro_use]
 mod macros;

 pub mod bounds;
 pub mod cast;
 pub mod float;
 pub mod identities;
 pub mod int;
 pub mod ops;
 pub mod pow;
 pub mod real;
 pub mod sign;

 /// The base trait for numeric types, covering `0` and `1` values,
 /// comparisons, basic numeric operations, and string conversion.
 pub trait Num: PartialEq + Zero + One + NumOps {
     type FromStrRadixErr;

     /// Convert from a string and radix <= 36.
     ///
     /// # Examples
     ///
     /// ```rust
     /// use num_traits::Num;
     ///
     /// let result = <i32 as Num>::from_str_radix("27", 10);
     /// assert_eq!(result, Ok(27));
     ///
     /// let result = <i32 as Num>::from_str_radix("foo", 10);
     /// assert!(result.is_err());
     /// ```
     fn from_str_radix(str: &str, radix: u32) -> Result<Self, Self::FromStrRadixErr>;
 }

 /// The trait for types implementing basic numeric operations
 ///
 /// This is automatically implemented for types which implement the operators.
 pub trait NumOps<Rhs = Self, Output = Self>:
     Add<Rhs, Output = Output>
     + Sub<Rhs, Output = Output>
     + Mul<Rhs, Output = Output>
     + Div<Rhs, Output = Output>
     + Rem<Rhs, Output = Output>
 {
 }

 impl<T, Rhs, Output> NumOps<Rhs, Output> for T where
     T: Add<Rhs, Output = Output>
         + Sub<Rhs, Output = Output>
         + Mul<Rhs, Output = Output>
         + Div<Rhs, Output = Output>
         + Rem<Rhs, Output = Output>
 {
 }

 /// The trait for `Num` types which also implement numeric operations taking
 /// the second operand by reference.
 ///
 /// This is automatically implemented for types which implement the operators.
 pub trait NumRef: Num + for<'r> NumOps<&'r Self> {}
 impl<T> NumRef for T where T: Num + for<'r> NumOps<&'r T> {}

 /// The trait for references which implement numeric operations, taking the
 /// second operand either by value or by reference.
 ///
 /// This is automatically implemented for types which implement the operators.
 pub trait RefNum<Base>: NumOps<Base, Base> + for<'r> NumOps<&'r Base, Base> {}
 impl<T, Base> RefNum<Base> for T where T: NumOps<Base, Base> + for<'r> NumOps<&'r Base, Base> {}

 /// The trait for types implementing numeric assignment operators (like `+=`).
 ///
 /// This is automatically implemented for types which implement the operators.
 pub trait NumAssignOps<Rhs = Self>:
     AddAssign<Rhs> + SubAssign<Rhs> + MulAssign<Rhs> + DivAssign<Rhs> + RemAssign<Rhs>
 {
 }

 impl<T, Rhs> NumAssignOps<Rhs> for T where
     T: AddAssign<Rhs> + SubAssign<Rhs> + MulAssign<Rhs> + DivAssign<Rhs> + RemAssign<Rhs>
 {
 }

 /// The trait for `Num` types which also implement assignment operators.
 ///
 /// This is automatically implemented for types which implement the operators.
 pub trait NumAssign: Num + NumAssignOps {}
 impl<T> NumAssign for T where T: Num + NumAssignOps {}

 /// The trait for `NumAssign` types which also implement assignment operations
 /// taking the second operand by reference.
 ///
 /// This is automatically implemented for types which implement the operators.
 pub trait NumAssignRef: NumAssign + for<'r> NumAssignOps<&'r Self> {}
 impl<T> NumAssignRef for T where T: NumAssign + for<'r> NumAssignOps<&'r T> {}

 macro_rules! int_trait_impl {
     ($name:ident for $($t:ty)*) => ($(
         impl $name for $t {
             type FromStrRadixErr = ::core::num::ParseIntError;
             #[inline]
             fn from_str_radix(s: &str, radix: u32)
                               -> Result<Self, ::core::num::ParseIntError>
             {
                 <$t>::from_str_radix(s, radix)
             }
         }
     )*)
 }
 int_trait_impl!(Num for usize u8 u16 u32 u64 isize i8 i16 i32 i64);
 #[cfg(has_i128)]
 int_trait_impl!(Num for u128 i128);

 impl<T: Num> Num for Wrapping<T>
 where
     Wrapping<T>: Add<Output = Wrapping<T>>
         + Sub<Output = Wrapping<T>>
         + Mul<Output = Wrapping<T>>
         + Div<Output = Wrapping<T>>
         + Rem<Output = Wrapping<T>>,
 {
     type FromStrRadixErr = T::FromStrRadixErr;
     fn from_str_radix(str: &str, radix: u32) -> Result<Self, Self::FromStrRadixErr> {
         T::from_str_radix(str, radix).map(Wrapping)
     }
 }

 #[derive(Debug)]
 pub enum FloatErrorKind {
     Empty,
     Invalid,
 }
 // FIXME: core::num::ParseFloatError is stable in 1.0, but opaque to us,
 // so there's not really any way for us to reuse it.
 #[derive(Debug)]
 pub struct ParseFloatError {
     pub kind: FloatErrorKind,
 }

 impl fmt::Display for ParseFloatError {
     fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
         let description = match self.kind {
             FloatErrorKind::Empty => "cannot parse float from empty string",
             FloatErrorKind::Invalid => "invalid float literal",
         };

         description.fmt(f)
     }
 }

 // FIXME: The standard library from_str_radix on floats was deprecated, so we're stuck
 // with this implementation ourselves until we want to make a breaking change.
 // (would have to drop it from `Num` though)
 macro_rules! float_trait_impl {
     ($name:ident for $($t:ident)*) => ($(
         impl $name for $t {
             type FromStrRadixErr = ParseFloatError;

             fn from_str_radix(src: &str, radix: u32)
                               -> Result<Self, Self::FromStrRadixErr>
             {
                 use self::FloatErrorKind::*;
                 use self::ParseFloatError as PFE;

                 // Special values
                 match src {
                     "inf"   => return Ok(core::$t::INFINITY),
                     "-inf"  => return Ok(core::$t::NEG_INFINITY),
                     "NaN"   => return Ok(core::$t::NAN),
                     _       => {},
                 }

                 fn slice_shift_char(src: &str) -> Option<(char, &str)> {
                     let mut chars = src.chars();
                     if let Some(ch) = chars.next() {
                         Some((ch, chars.as_str()))
                     } else {
                         None
                     }
                 }

                 let (is_positive, src) =  match slice_shift_char(src) {
                     None             => return Err(PFE { kind: Empty }),
                     Some(('-', ""))  => return Err(PFE { kind: Empty }),
                     Some(('-', src)) => (false, src),
                     Some((_, _))     => (true,  src),
                 };

                 // The significand to accumulate
                 let mut sig = if is_positive { 0.0 } else { -0.0 };
                 // Necessary to detect overflow
                 let mut prev_sig = sig;
                 let mut cs = src.chars().enumerate();
                 // Exponent prefix and exponent index offset
                 let mut exp_info = None::<(char, usize)>;

                 // Parse the integer part of the significand
                 for (i, c) in cs.by_ref() {
                     match c.to_digit(radix) {
                         Some(digit) => {
                             // shift significand one digit left
                             sig = sig * (radix as $t);

                             // add/subtract current digit depending on sign
                             if is_positive {
                                 sig = sig + ((digit as isize) as $t);
                             } else {
                                 sig = sig - ((digit as isize) as $t);
                             }

                             // Detect overflow by comparing to last value, except
                             // if we've not seen any non-zero digits.
                             if prev_sig != 0.0 {
                                 if is_positive && sig <= prev_sig
                                     { return Ok(core::$t::INFINITY); }
                                 if !is_positive && sig >= prev_sig
                                     { return Ok(core::$t::NEG_INFINITY); }

                                 // Detect overflow by reversing the shift-and-add process
                                 if is_positive && (prev_sig != (sig - digit as $t) / radix as $t)
                                     { return Ok(core::$t::INFINITY); }
                                 if !is_positive && (prev_sig != (sig + digit as $t) / radix as $t)
                                     { return Ok(core::$t::NEG_INFINITY); }
                             }
                             prev_sig = sig;
                         },
                         None => match c {
                             'e' | 'E' | 'p' | 'P' => {
                                 exp_info = Some((c, i + 1));
                                 break;  // start of exponent
                             },
                             '.' => {
                                 break;  // start of fractional part
                             },
                             _ => {
                                 return Err(PFE { kind: Invalid });
                             },
                         },
                     }
                 }

                 // If we are not yet at the exponent parse the fractional
                 // part of the significand
                 if exp_info.is_none() {
                     let mut power = 1.0;
                     for (i, c) in cs.by_ref() {
                         match c.to_digit(radix) {
                             Some(digit) => {
                                 // Decrease power one order of magnitude
                                 power = power / (radix as $t);
                                 // add/subtract current digit depending on sign
                                 sig = if is_positive {
                                     sig + (digit as $t) * power
                                 } else {
                                     sig - (digit as $t) * power
                                 };
                                 // Detect overflow by comparing to last value
                                 if is_positive && sig < prev_sig
                                     { return Ok(core::$t::INFINITY); }
                                 if !is_positive && sig > prev_sig
                                     { return Ok(core::$t::NEG_INFINITY); }
                                 prev_sig = sig;
                             },
                             None => match c {
                                 'e' | 'E' | 'p' | 'P' => {
                                     exp_info = Some((c, i + 1));
                                     break; // start of exponent
                                 },
                                 _ => {
                                     return Err(PFE { kind: Invalid });
                                 },
                             },
                         }
                     }
                 }

                 // Parse and calculate the exponent
                 let exp = match exp_info {
                     Some((c, offset)) => {
                         let base = match c {
                             'E' | 'e' if radix == 10 => 10.0,
                             'P' | 'p' if radix == 16 => 2.0,
                             _ => return Err(PFE { kind: Invalid }),
                         };

                         // Parse the exponent as decimal integer
                         let src = &src[offset..];
                         let (is_positive, exp) = match slice_shift_char(src) {
                             Some(('-', src)) => (false, src.parse::<usize>()),
                             Some(('+', src)) => (true,  src.parse::<usize>()),
                             Some((_, _))     => (true,  src.parse::<usize>()),
                             None             => return Err(PFE { kind: Invalid }),
                         };

                         #[cfg(feature = "std")]
                         fn pow(base: $t, exp: usize) -> $t {
                             Float::powi(base, exp as i32)
                         }
                         // otherwise uses the generic `pow` from the root

                         match (is_positive, exp) {
                             (true,  Ok(exp)) => pow(base, exp),
                             (false, Ok(exp)) => 1.0 / pow(base, exp),
                             (_, Err(_))      => return Err(PFE { kind: Invalid }),
                         }
                     },
                     None => 1.0, // no exponent
                 };

                 Ok(sig * exp)
             }
         }
     )*)
 }
 float_trait_impl!(Num for f32 f64);

 /// A value bounded by a minimum and a maximum
 ///
 ///  If input is less than min then this returns min.
 ///  If input is greater than max then this returns max.
 ///  Otherwise this returns input.
 ///
 /// **Panics** in debug mode if `!(min <= max)`.
 #[inline]
 pub fn clamp<T: PartialOrd>(input: T, min: T, max: T) -> T {
     debug_assert!(min <= max, "min must be less than or equal to max");
     if input < min {
         min
     } else if input > max {
         max
     } else {
         input
     }
 }

 /// A value bounded by a minimum value
 ///
 ///  If input is less than min then this returns min.
 ///  Otherwise this returns input.
 ///  `clamp_min(std::f32::NAN, 1.0)` preserves `NAN` different from `f32::min(std::f32::NAN, 1.0)`.
 ///
 /// **Panics** in debug mode if `!(min == min)`. (This occurs if `min` is `NAN`.)
 #[inline]
 pub fn clamp_min<T: PartialOrd>(input: T, min: T) -> T {
     debug_assert!(min == min, "min must not be NAN");
     if input < min {
         min
     } else {
         input
     }
 }

 /// A value bounded by a maximum value
 ///
 ///  If input is greater than max then this returns max.
 ///  Otherwise this returns input.
 ///  `clamp_max(std::f32::NAN, 1.0)` preserves `NAN` different from `f32::max(std::f32::NAN, 1.0)`.
 ///
 /// **Panics** in debug mode if `!(max == max)`. (This occurs if `max` is `NAN`.)
 #[inline]
 pub fn clamp_max<T: PartialOrd>(input: T, max: T) -> T {
     debug_assert!(max == max, "max must not be NAN");
     if input > max {
         max
     } else {
         input
     }
 }

 #[test]
 fn clamp_test() {
     // Int test
     assert_eq!(1, clamp(1, -1, 2));
     assert_eq!(-1, clamp(-2, -1, 2));
     assert_eq!(2, clamp(3, -1, 2));
     assert_eq!(1, clamp_min(1, -1));
     assert_eq!(-1, clamp_min(-2, -1));
     assert_eq!(-1, clamp_max(1, -1));
     assert_eq!(-2, clamp_max(-2, -1));

     // Float test
     assert_eq!(1.0, clamp(1.0, -1.0, 2.0));
     assert_eq!(-1.0, clamp(-2.0, -1.0, 2.0));
     assert_eq!(2.0, clamp(3.0, -1.0, 2.0));
     assert_eq!(1.0, clamp_min(1.0, -1.0));
     assert_eq!(-1.0, clamp_min(-2.0, -1.0));
     assert_eq!(-1.0, clamp_max(1.0, -1.0));
     assert_eq!(-2.0, clamp_max(-2.0, -1.0));
     assert!(clamp(::core::f32::NAN, -1.0, 1.0).is_nan());
     assert!(clamp_min(::core::f32::NAN, 1.0).is_nan());
     assert!(clamp_max(::core::f32::NAN, 1.0).is_nan());
 }

 #[test]
 #[should_panic]
 #[cfg(debug_assertions)]
 fn clamp_nan_min() {
     clamp(0., ::core::f32::NAN, 1.);
 }

 #[test]
 #[should_panic]
 #[cfg(debug_assertions)]
 fn clamp_nan_max() {
     clamp(0., -1., ::core::f32::NAN);
 }

 #[test]
 #[should_panic]
 #[cfg(debug_assertions)]
 fn clamp_nan_min_max() {
     clamp(0., ::core::f32::NAN, ::core::f32::NAN);
 }

 #[test]
 #[should_panic]
 #[cfg(debug_assertions)]
 fn clamp_min_nan_min() {
     clamp_min(0., ::core::f32::NAN);
 }

 #[test]
 #[should_panic]
 #[cfg(debug_assertions)]
 fn clamp_max_nan_max() {
     clamp_max(0., ::core::f32::NAN);
 }

 #[test]
 fn from_str_radix_unwrap() {
     // The Result error must impl Debug to allow unwrap()

     let i: i32 = Num::from_str_radix("0", 10).unwrap();
     assert_eq!(i, 0);

     let f: f32 = Num::from_str_radix("0.0", 10).unwrap();
     assert_eq!(f, 0.0);
 }

 #[test]
 fn from_str_radix_multi_byte_fail() {
     // Ensure parsing doesn't panic, even on invalid sign characters
     assert!(f32::from_str_radix("™0.2", 10).is_err());

     // Even when parsing the exponent sign
     assert!(f32::from_str_radix("0.2E™1", 10).is_err());
 }

 #[test]
 fn wrapping_is_num() {
     fn require_num<T: Num>(_: &T) {}
     require_num(&Wrapping(42_u32));
     require_num(&Wrapping(-42));
 }

 #[test]
 fn wrapping_from_str_radix() {
     macro_rules! test_wrapping_from_str_radix {
         ($($t:ty)+) => {
             $(
                 for &(s, r) in &[("42", 10), ("42", 2), ("-13.0", 10), ("foo", 10)] {
                     let w = Wrapping::<$t>::from_str_radix(s, r).map(|w| w.0);
                     assert_eq!(w, <$t as Num>::from_str_radix(s, r));
                 }
             )+
         };
     }

     test_wrapping_from_str_radix!(usize u8 u16 u32 u64 isize i8 i16 i32 i64);
 }

 #[test]
 fn check_num_ops() {
     fn compute<T: Num + Copy>(x: T, y: T) -> T {
         x * y / y % y + y - y
     }
     assert_eq!(compute(1, 2), 1)
 }

 #[test]
 fn check_numref_ops() {
     fn compute<T: NumRef>(x: T, y: &T) -> T {
         x * y / y % y + y - y
     }
     assert_eq!(compute(1, &2), 1)
 }

 #[test]
 fn check_refnum_ops() {
     fn compute<T: Copy>(x: &T, y: T) -> T
     where
         for<'a> &'a T: RefNum<T>,
     {
         &(&(&(&(x * y) / y) % y) + y) - y
     }
     assert_eq!(compute(&1, 2), 1)
 }

 #[test]
 fn check_refref_ops() {
     fn compute<T>(x: &T, y: &T) -> T
     where
         for<'a> &'a T: RefNum<T>,
     {
         &(&(&(&(x * y) / y) % y) + y) - y
     }
     assert_eq!(compute(&1, &2), 1)
 }

 #[test]
 fn check_numassign_ops() {
     fn compute<T: NumAssign + Copy>(mut x: T, y: T) -> T {
         x *= y;
         x /= y;
         x %= y;
         x += y;
         x -= y;
         x
     }
     assert_eq!(compute(1, 2), 1)
 }

 // TODO test `NumAssignRef`, but even the standard numeric types don't
 // implement this yet. (see rust pr41336)
	// 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.

	//! Numeric traits for generic mathematics
	//!
	//! ## Compatibility
	//!
	//! The `num-traits` crate is tested for rustc 1.8 and greater.

	#![doc(html_root_url = "https://docs.rs/num-traits/0.2")]
	#![deny(unconditional_recursion)]
	#![no_std]
	#[cfg(feature = "std")]
	extern crate std;

	// Only `no_std` builds actually use `libm`.
	#[cfg(all(not(feature = "std"), feature = "libm"))]
	extern crate libm;

	use core::fmt;
	use core::num::Wrapping;
	use core::ops::{Add, Div, Mul, Rem, Sub};
	use core::ops::{AddAssign, DivAssign, MulAssign, RemAssign, SubAssign};

	pub use bounds::Bounded;
	#[cfg(any(feature = "std", feature = "libm"))]
	pub use float::Float;
	pub use float::FloatConst;
	// pub use real::{FloatCore, Real}; // NOTE: Don't do this, it breaks `use num_traits::*;`.
	pub use cast::{cast, AsPrimitive, FromPrimitive, NumCast, ToPrimitive};
	pub use identities::{one, zero, One, Zero};
	pub use int::PrimInt;
	pub use ops::checked::{
	CheckedAdd, CheckedDiv, CheckedMul, CheckedNeg, CheckedRem, CheckedShl, CheckedShr, CheckedSub,
	};
	pub use ops::inv::Inv;
	pub use ops::mul_add::{MulAdd, MulAddAssign};
	pub use ops::saturating::{Saturating, SaturatingAdd, SaturatingMul, SaturatingSub};
	pub use ops::wrapping::{
	WrappingAdd, WrappingMul, WrappingNeg, WrappingShl, WrappingShr, WrappingSub,
	};
	pub use pow::{checked_pow, pow, Pow};
	pub use sign::{abs, abs_sub, signum, Signed, Unsigned};

	#[macro_use]
	mod macros;

	pub mod bounds;
	pub mod cast;
	pub mod float;
	pub mod identities;
	pub mod int;
	pub mod ops;
	pub mod pow;
	pub mod real;
	pub mod sign;

	/// The base trait for numeric types, covering `0` and `1` values,
	/// comparisons, basic numeric operations, and string conversion.
	pub trait Num: PartialEq + Zero + One + NumOps {
	type FromStrRadixErr;

	/// Convert from a string and radix <= 36.
	///
	/// # Examples
	///
	/// ```rust
	/// use num_traits::Num;
	///
	/// let result = <i32 as Num>::from_str_radix("27", 10);
	/// assert_eq!(result, Ok(27));
	///
	/// let result = <i32 as Num>::from_str_radix("foo", 10);
	/// assert!(result.is_err());
	/// ```
	fn from_str_radix(str: &str, radix: u32) -> Result<Self, Self::FromStrRadixErr>;
	}

	/// The trait for types implementing basic numeric operations
	///
	/// This is automatically implemented for types which implement the operators.
	pub trait NumOps<Rhs = Self, Output = Self>:
	Add<Rhs, Output = Output>
	+ Sub<Rhs, Output = Output>
	+ Mul<Rhs, Output = Output>
	+ Div<Rhs, Output = Output>
	+ Rem<Rhs, Output = Output>
	{
	}

	impl<T, Rhs, Output> NumOps<Rhs, Output> for T where
	T: Add<Rhs, Output = Output>
	+ Sub<Rhs, Output = Output>
	+ Mul<Rhs, Output = Output>
	+ Div<Rhs, Output = Output>
	+ Rem<Rhs, Output = Output>
	{
	}

	/// The trait for `Num` types which also implement numeric operations taking
	/// the second operand by reference.
	///
	/// This is automatically implemented for types which implement the operators.
	pub trait NumRef: Num + for<'r> NumOps<&'r Self> {}
	impl<T> NumRef for T where T: Num + for<'r> NumOps<&'r T> {}

	/// The trait for references which implement numeric operations, taking the
	/// second operand either by value or by reference.
	///
	/// This is automatically implemented for types which implement the operators.
	pub trait RefNum<Base>: NumOps<Base, Base> + for<'r> NumOps<&'r Base, Base> {}
	impl<T, Base> RefNum<Base> for T where T: NumOps<Base, Base> + for<'r> NumOps<&'r Base, Base> {}

	/// The trait for types implementing numeric assignment operators (like `+=`).
	///
	/// This is automatically implemented for types which implement the operators.
	pub trait NumAssignOps<Rhs = Self>:
	AddAssign<Rhs> + SubAssign<Rhs> + MulAssign<Rhs> + DivAssign<Rhs> + RemAssign<Rhs>
	{
	}

	impl<T, Rhs> NumAssignOps<Rhs> for T where
	T: AddAssign<Rhs> + SubAssign<Rhs> + MulAssign<Rhs> + DivAssign<Rhs> + RemAssign<Rhs>
	{
	}

	/// The trait for `Num` types which also implement assignment operators.
	///
	/// This is automatically implemented for types which implement the operators.
	pub trait NumAssign: Num + NumAssignOps {}
	impl<T> NumAssign for T where T: Num + NumAssignOps {}

	/// The trait for `NumAssign` types which also implement assignment operations
	/// taking the second operand by reference.
	///
	/// This is automatically implemented for types which implement the operators.
	pub trait NumAssignRef: NumAssign + for<'r> NumAssignOps<&'r Self> {}
	impl<T> NumAssignRef for T where T: NumAssign + for<'r> NumAssignOps<&'r T> {}

	macro_rules! int_trait_impl {
	($name:ident for $($t:ty)*) => ($(
	impl $name for $t {
	type FromStrRadixErr = ::core::num::ParseIntError;
	#[inline]
	fn from_str_radix(s: &str, radix: u32)
	-> Result<Self, ::core::num::ParseIntError>
	{
	<$t>::from_str_radix(s, radix)
	}
	}
	)*)
	}
	int_trait_impl!(Num for usize u8 u16 u32 u64 isize i8 i16 i32 i64);
	#[cfg(has_i128)]
	int_trait_impl!(Num for u128 i128);

	impl<T: Num> Num for Wrapping<T>
	where
	Wrapping<T>: Add<Output = Wrapping<T>>
	+ Sub<Output = Wrapping<T>>
	+ Mul<Output = Wrapping<T>>
	+ Div<Output = Wrapping<T>>
	+ Rem<Output = Wrapping<T>>,
	{
	type FromStrRadixErr = T::FromStrRadixErr;
	fn from_str_radix(str: &str, radix: u32) -> Result<Self, Self::FromStrRadixErr> {
	T::from_str_radix(str, radix).map(Wrapping)
	}
	}

	#[derive(Debug)]
	pub enum FloatErrorKind {
	Empty,
	Invalid,
	}
	// FIXME: core::num::ParseFloatError is stable in 1.0, but opaque to us,
	// so there's not really any way for us to reuse it.
	#[derive(Debug)]
	pub struct ParseFloatError {
	pub kind: FloatErrorKind,
	}

	impl fmt::Display for ParseFloatError {
	fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
	let description = match self.kind {
	FloatErrorKind::Empty => "cannot parse float from empty string",
	FloatErrorKind::Invalid => "invalid float literal",
	};

	description.fmt(f)
	}
	}

	// FIXME: The standard library from_str_radix on floats was deprecated, so we're stuck
	// with this implementation ourselves until we want to make a breaking change.
	// (would have to drop it from `Num` though)
	macro_rules! float_trait_impl {
	($name:ident for $($t:ident)*) => ($(
	impl $name for $t {
	type FromStrRadixErr = ParseFloatError;

	fn from_str_radix(src: &str, radix: u32)
	-> Result<Self, Self::FromStrRadixErr>
	{
	use self::FloatErrorKind::*;
	use self::ParseFloatError as PFE;

	// Special values
	match src {
	"inf" => return Ok(core::$t::INFINITY),
	"-inf" => return Ok(core::$t::NEG_INFINITY),
	"NaN" => return Ok(core::$t::NAN),
	_ => {},
	}

	fn slice_shift_char(src: &str) -> Option<(char, &str)> {
	let mut chars = src.chars();
	if let Some(ch) = chars.next() {
	Some((ch, chars.as_str()))
	} else {
	None
	}
	}

	let (is_positive, src) = match slice_shift_char(src) {
	None => return Err(PFE { kind: Empty }),
	Some(('-', "")) => return Err(PFE { kind: Empty }),
	Some(('-', src)) => (false, src),
	Some((_, _)) => (true, src),
	};

	// The significand to accumulate
	let mut sig = if is_positive { 0.0 } else { -0.0 };
	// Necessary to detect overflow
	let mut prev_sig = sig;
	let mut cs = src.chars().enumerate();
	// Exponent prefix and exponent index offset
	let mut exp_info = None::<(char, usize)>;

	// Parse the integer part of the significand
	for (i, c) in cs.by_ref() {
	match c.to_digit(radix) {
	Some(digit) => {
	// shift significand one digit left
	sig = sig * (radix as $t);

	// add/subtract current digit depending on sign
	if is_positive {
	sig = sig + ((digit as isize) as $t);
	} else {
	sig = sig - ((digit as isize) as $t);
	}

	// Detect overflow by comparing to last value, except
	// if we've not seen any non-zero digits.
	if prev_sig != 0.0 {
	if is_positive && sig <= prev_sig
	{ return Ok(core::$t::INFINITY); }
	if !is_positive && sig >= prev_sig
	{ return Ok(core::$t::NEG_INFINITY); }

	// Detect overflow by reversing the shift-and-add process
	if is_positive && (prev_sig != (sig - digit as $t) / radix as $t)
	{ return Ok(core::$t::INFINITY); }
	if !is_positive && (prev_sig != (sig + digit as $t) / radix as $t)
	{ return Ok(core::$t::NEG_INFINITY); }
	}
	prev_sig = sig;
	},
	None => match c {
	'e' \| 'E' \| 'p' \| 'P' => {
	exp_info = Some((c, i + 1));
	break; // start of exponent
	},
	'.' => {
	break; // start of fractional part
	},
	_ => {
	return Err(PFE { kind: Invalid });
	},
	},
	}
	}

	// If we are not yet at the exponent parse the fractional
	// part of the significand
	if exp_info.is_none() {
	let mut power = 1.0;
	for (i, c) in cs.by_ref() {
	match c.to_digit(radix) {
	Some(digit) => {
	// Decrease power one order of magnitude
	power = power / (radix as $t);
	// add/subtract current digit depending on sign
	sig = if is_positive {
	sig + (digit as $t) * power
	} else {
	sig - (digit as $t) * power
	};
	// Detect overflow by comparing to last value
	if is_positive && sig < prev_sig
	{ return Ok(core::$t::INFINITY); }
	if !is_positive && sig > prev_sig
	{ return Ok(core::$t::NEG_INFINITY); }
	prev_sig = sig;
	},
	None => match c {
	'e' \| 'E' \| 'p' \| 'P' => {
	exp_info = Some((c, i + 1));
	break; // start of exponent
	},
	_ => {
	return Err(PFE { kind: Invalid });
	},
	},
	}
	}
	}

	// Parse and calculate the exponent
	let exp = match exp_info {
	Some((c, offset)) => {
	let base = match c {
	'E' \| 'e' if radix == 10 => 10.0,
	'P' \| 'p' if radix == 16 => 2.0,
	_ => return Err(PFE { kind: Invalid }),
	};

	// Parse the exponent as decimal integer
	let src = &src[offset..];
	let (is_positive, exp) = match slice_shift_char(src) {
	Some(('-', src)) => (false, src.parse::<usize>()),
	Some(('+', src)) => (true, src.parse::<usize>()),
	Some((_, _)) => (true, src.parse::<usize>()),
	None => return Err(PFE { kind: Invalid }),
	};

	#[cfg(feature = "std")]
	fn pow(base: $t, exp: usize) -> $t {
	Float::powi(base, exp as i32)
	}
	// otherwise uses the generic `pow` from the root

	match (is_positive, exp) {
	(true, Ok(exp)) => pow(base, exp),
	(false, Ok(exp)) => 1.0 / pow(base, exp),
	(_, Err(_)) => return Err(PFE { kind: Invalid }),
	}
	},
	None => 1.0, // no exponent
	};

	Ok(sig * exp)
	}
	}
	)*)
	}
	float_trait_impl!(Num for f32 f64);

	/// A value bounded by a minimum and a maximum
	///
	/// If input is less than min then this returns min.
	/// If input is greater than max then this returns max.
	/// Otherwise this returns input.
	///
	/// Panics in debug mode if `!(min <= max)`.
	#[inline]
	pub fn clamp<T: PartialOrd>(input: T, min: T, max: T) -> T {
	debug_assert!(min <= max, "min must be less than or equal to max");
	if input < min {
	min
	} else if input > max {
	max
	} else {
	input
	}
	}

	/// A value bounded by a minimum value
	///
	/// If input is less than min then this returns min.
	/// Otherwise this returns input.
	/// `clamp_min(std::f32::NAN, 1.0)` preserves `NAN` different from `f32::min(std::f32::NAN, 1.0)`.
	///
	/// Panics in debug mode if `!(min == min)`. (This occurs if `min` is `NAN`.)
	#[inline]
	pub fn clamp_min<T: PartialOrd>(input: T, min: T) -> T {
	debug_assert!(min == min, "min must not be NAN");
	if input < min {
	min
	} else {
	input
	}
	}

	/// A value bounded by a maximum value
	///
	/// If input is greater than max then this returns max.
	/// Otherwise this returns input.
	/// `clamp_max(std::f32::NAN, 1.0)` preserves `NAN` different from `f32::max(std::f32::NAN, 1.0)`.
	///
	/// Panics in debug mode if `!(max == max)`. (This occurs if `max` is `NAN`.)
	#[inline]
	pub fn clamp_max<T: PartialOrd>(input: T, max: T) -> T {
	debug_assert!(max == max, "max must not be NAN");
	if input > max {
	max
	} else {
	input
	}
	}

	#[test]
	fn clamp_test() {
	// Int test
	assert_eq!(1, clamp(1, -1, 2));
	assert_eq!(-1, clamp(-2, -1, 2));
	assert_eq!(2, clamp(3, -1, 2));
	assert_eq!(1, clamp_min(1, -1));
	assert_eq!(-1, clamp_min(-2, -1));
	assert_eq!(-1, clamp_max(1, -1));
	assert_eq!(-2, clamp_max(-2, -1));

	// Float test
	assert_eq!(1.0, clamp(1.0, -1.0, 2.0));
	assert_eq!(-1.0, clamp(-2.0, -1.0, 2.0));
	assert_eq!(2.0, clamp(3.0, -1.0, 2.0));
	assert_eq!(1.0, clamp_min(1.0, -1.0));
	assert_eq!(-1.0, clamp_min(-2.0, -1.0));
	assert_eq!(-1.0, clamp_max(1.0, -1.0));
	assert_eq!(-2.0, clamp_max(-2.0, -1.0));
	assert!(clamp(::core::f32::NAN, -1.0, 1.0).is_nan());
	assert!(clamp_min(::core::f32::NAN, 1.0).is_nan());
	assert!(clamp_max(::core::f32::NAN, 1.0).is_nan());
	}

	#[test]
	#[should_panic]
	#[cfg(debug_assertions)]
	fn clamp_nan_min() {
	clamp(0., ::core::f32::NAN, 1.);
	}

	#[test]
	#[should_panic]
	#[cfg(debug_assertions)]
	fn clamp_nan_max() {
	clamp(0., -1., ::core::f32::NAN);
	}

	#[test]
	#[should_panic]
	#[cfg(debug_assertions)]
	fn clamp_nan_min_max() {
	clamp(0., ::core::f32::NAN, ::core::f32::NAN);
	}

	#[test]
	#[should_panic]
	#[cfg(debug_assertions)]
	fn clamp_min_nan_min() {
	clamp_min(0., ::core::f32::NAN);
	}

	#[test]
	#[should_panic]
	#[cfg(debug_assertions)]
	fn clamp_max_nan_max() {
	clamp_max(0., ::core::f32::NAN);
	}

	#[test]
	fn from_str_radix_unwrap() {
	// The Result error must impl Debug to allow unwrap()

	let i: i32 = Num::from_str_radix("0", 10).unwrap();
	assert_eq!(i, 0);

	let f: f32 = Num::from_str_radix("0.0", 10).unwrap();
	assert_eq!(f, 0.0);
	}

	#[test]
	fn from_str_radix_multi_byte_fail() {
	// Ensure parsing doesn't panic, even on invalid sign characters
	assert!(f32::from_str_radix("™0.2", 10).is_err());

	// Even when parsing the exponent sign
	assert!(f32::from_str_radix("0.2E™1", 10).is_err());
	}

	#[test]
	fn wrapping_is_num() {
	fn require_num<T: Num>(_: &T) {}
	require_num(&Wrapping(42_u32));
	require_num(&Wrapping(-42));
	}

	#[test]
	fn wrapping_from_str_radix() {
	macro_rules! test_wrapping_from_str_radix {
	($($t:ty)+) => {
	$(
	for &(s, r) in &[("42", 10), ("42", 2), ("-13.0", 10), ("foo", 10)] {
	let w = Wrapping::<$t>::from_str_radix(s, r).map(\|w\| w.0);
	assert_eq!(w, <$t as Num>::from_str_radix(s, r));
	}
	)+
	};
	}

	test_wrapping_from_str_radix!(usize u8 u16 u32 u64 isize i8 i16 i32 i64);
	}

	#[test]
	fn check_num_ops() {
	fn compute<T: Num + Copy>(x: T, y: T) -> T {
	x * y / y % y + y - y
	}
	assert_eq!(compute(1, 2), 1)
	}

	#[test]
	fn check_numref_ops() {
	fn compute<T: NumRef>(x: T, y: &T) -> T {
	x * y / y % y + y - y
	}
	assert_eq!(compute(1, &2), 1)
	}

	#[test]
	fn check_refnum_ops() {
	fn compute<T: Copy>(x: &T, y: T) -> T
	where
	for<'a> &'a T: RefNum<T>,
	{
	&(&(&(&(x * y) / y) % y) + y) - y
	}
	assert_eq!(compute(&1, 2), 1)
	}

	#[test]
	fn check_refref_ops() {
	fn compute<T>(x: &T, y: &T) -> T
	where
	for<'a> &'a T: RefNum<T>,
	{
	&(&(&(&(x * y) / y) % y) + y) - y
	}
	assert_eq!(compute(&1, &2), 1)
	}

	#[test]
	fn check_numassign_ops() {
	fn compute<T: NumAssign + Copy>(mut x: T, y: T) -> T {
	x *= y;
	x /= y;
	x %= y;
	x += y;
	x -= y;
	x
	}
	assert_eq!(compute(1, 2), 1)
	}

	// TODO test `NumAssignRef`, but even the standard numeric types don't
	// implement this yet. (see rust pr41336)