num-traits/rational/src/lib.rs

// 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.

//! Rational numbers
#![doc(html_logo_url = "https://rust-num.github.io/num/rust-logo-128x128-blk-v2.png",
       html_favicon_url = "https://rust-num.github.io/num/favicon.ico",
       html_root_url = "https://rust-num.github.io/num/",
       html_playground_url = "http://play.integer32.com/")]

#[cfg(feature = "rustc-serialize")]
extern crate rustc_serialize;
#[cfg(feature = "serde")]
extern crate serde;
#[cfg(feature = "num-bigint")]
extern crate num_bigint as bigint;

extern crate num_traits as traits;
extern crate num_integer as integer;

use std::cmp;
use std::error::Error;
use std::fmt;
#[cfg(test)]
use std::hash;
use std::ops::{Add, Div, Mul, Neg, Rem, Sub};
use std::str::FromStr;

#[cfg(feature = "num-bigint")]
use bigint::{BigInt, BigUint, Sign};

use integer::Integer;
use traits::{FromPrimitive, Float, PrimInt, Num, Signed, Zero, One, Bounded, NumCast};

/// Represents the ratio between 2 numbers.
#[derive(Copy, Clone, Hash, Debug)]
#[cfg_attr(feature = "rustc-serialize", derive(RustcEncodable, RustcDecodable))]
#[allow(missing_docs)]
pub struct Ratio<T> {
    numer: T,
    denom: T,
}

/// Alias for a `Ratio` of machine-sized integers.
pub type Rational = Ratio<isize>;
pub type Rational32 = Ratio<i32>;
pub type Rational64 = Ratio<i64>;

#[cfg(feature = "num-bigint")]
/// Alias for arbitrary precision rationals.
pub type BigRational = Ratio<BigInt>;

impl<T: Clone + Integer> Ratio<T> {
    /// Creates a new `Ratio`. Fails if `denom` is zero.
    #[inline]
    pub fn new(numer: T, denom: T) -> Ratio<T> {
        if denom.is_zero() {
            panic!("denominator == 0");
        }
        let mut ret = Ratio::new_raw(numer, denom);
        ret.reduce();
        ret
    }

    /// Creates a `Ratio` representing the integer `t`.
    #[inline]
    pub fn from_integer(t: T) -> Ratio<T> {
        Ratio::new_raw(t, One::one())
    }

    /// Creates a `Ratio` without checking for `denom == 0` or reducing.
    #[inline]
    pub fn new_raw(numer: T, denom: T) -> Ratio<T> {
        Ratio {
            numer: numer,
            denom: denom,
        }
    }

    /// Converts to an integer, rounding towards zero.
    #[inline]
    pub fn to_integer(&self) -> T {
        self.trunc().numer
    }

    /// Gets an immutable reference to the numerator.
    #[inline]
    pub fn numer<'a>(&'a self) -> &'a T {
        &self.numer
    }

    /// Gets an immutable reference to the denominator.
    #[inline]
    pub fn denom<'a>(&'a self) -> &'a T {
        &self.denom
    }

    /// Returns true if the rational number is an integer (denominator is 1).
    #[inline]
    pub fn is_integer(&self) -> bool {
        self.denom == One::one()
    }

    /// Puts self into lowest terms, with denom > 0.
    fn reduce(&mut self) {
        let g: T = self.numer.gcd(&self.denom);

        // FIXME(#5992): assignment operator overloads
        // self.numer /= g;
        self.numer = self.numer.clone() / g.clone();
        // FIXME(#5992): assignment operator overloads
        // self.denom /= g;
        self.denom = self.denom.clone() / g;

        // keep denom positive!
        if self.denom < T::zero() {
            self.numer = T::zero() - self.numer.clone();
            self.denom = T::zero() - self.denom.clone();
        }
    }

    /// Returns a reduced copy of self.
    ///
    /// In general, it is not necessary to use this method, as the only
    /// method of procuring a non-reduced fraction is through `new_raw`.
    pub fn reduced(&self) -> Ratio<T> {
        let mut ret = self.clone();
        ret.reduce();
        ret
    }

    /// Returns the reciprocal.
    ///
    /// Fails if the `Ratio` is zero.
    #[inline]
    pub fn recip(&self) -> Ratio<T> {
        match self.numer.cmp(&T::zero()) {
            cmp::Ordering::Equal => panic!("numerator == 0"),
            cmp::Ordering::Greater => Ratio::new_raw(self.denom.clone(), self.numer.clone()),
            cmp::Ordering::Less => Ratio::new_raw(T::zero() - self.denom.clone(),
                                                  T::zero() - self.numer.clone())
        }
    }

    /// Rounds towards minus infinity.
    #[inline]
    pub fn floor(&self) -> Ratio<T> {
        if *self < Zero::zero() {
            let one: T = One::one();
            Ratio::from_integer((self.numer.clone() - self.denom.clone() + one) /
                                self.denom.clone())
        } else {
            Ratio::from_integer(self.numer.clone() / self.denom.clone())
        }
    }

    /// Rounds towards plus infinity.
    #[inline]
    pub fn ceil(&self) -> Ratio<T> {
        if *self < Zero::zero() {
            Ratio::from_integer(self.numer.clone() / self.denom.clone())
        } else {
            let one: T = One::one();
            Ratio::from_integer((self.numer.clone() + self.denom.clone() - one) /
                                self.denom.clone())
        }
    }

    /// Rounds to the nearest integer. Rounds half-way cases away from zero.
    #[inline]
    pub fn round(&self) -> Ratio<T> {
        let zero: Ratio<T> = Zero::zero();
        let one: T = One::one();
        let two: T = one.clone() + one.clone();

        // Find unsigned fractional part of rational number
        let mut fractional = self.fract();
        if fractional < zero {
            fractional = zero - fractional
        };

        // The algorithm compares the unsigned fractional part with 1/2, that
        // is, a/b >= 1/2, or a >= b/2. For odd denominators, we use
        // a >= (b/2)+1. This avoids overflow issues.
        let half_or_larger = if fractional.denom().is_even() {
            *fractional.numer() >= fractional.denom().clone() / two.clone()
        } else {
            *fractional.numer() >= (fractional.denom().clone() / two.clone()) + one.clone()
        };

        if half_or_larger {
            let one: Ratio<T> = One::one();
            if *self >= Zero::zero() {
                self.trunc() + one
            } else {
                self.trunc() - one
            }
        } else {
            self.trunc()
        }
    }

    /// Rounds towards zero.
    #[inline]
    pub fn trunc(&self) -> Ratio<T> {
        Ratio::from_integer(self.numer.clone() / self.denom.clone())
    }

    /// Returns the fractional part of a number, with division rounded towards zero.
    ///
    /// Satisfies `self == self.trunc() + self.fract()`.
    #[inline]
    pub fn fract(&self) -> Ratio<T> {
        Ratio::new_raw(self.numer.clone() % self.denom.clone(), self.denom.clone())
    }
}

impl<T: Clone + Integer + PrimInt> Ratio<T> {
    /// Raises the `Ratio` to the power of an exponent.
    #[inline]
    pub fn pow(&self, expon: i32) -> Ratio<T> {
        match expon.cmp(&0) {
            cmp::Ordering::Equal => One::one(),
            cmp::Ordering::Less => self.recip().pow(-expon),
            cmp::Ordering::Greater => {
                Ratio::new_raw(self.numer.pow(expon as u32), self.denom.pow(expon as u32))
            }
        }
    }
}

#[cfg(feature = "num-bigint")]
impl Ratio<BigInt> {
    /// Converts a float into a rational number.
    pub fn from_float<T: Float>(f: T) -> Option<BigRational> {
        if !f.is_finite() {
            return None;
        }
        let (mantissa, exponent, sign) = f.integer_decode();
        let bigint_sign = if sign == 1 {
            Sign::Plus
        } else {
            Sign::Minus
        };
        if exponent < 0 {
            let one: BigInt = One::one();
            let denom: BigInt = one << ((-exponent) as usize);
            let numer: BigUint = FromPrimitive::from_u64(mantissa).unwrap();
            Some(Ratio::new(BigInt::from_biguint(bigint_sign, numer), denom))
        } else {
            let mut numer: BigUint = FromPrimitive::from_u64(mantissa).unwrap();
            numer = numer << (exponent as usize);
            Some(Ratio::from_integer(BigInt::from_biguint(bigint_sign, numer)))
        }
    }
}

// From integer
impl<T> From<T> for Ratio<T> where T: Clone + Integer {
    fn from(x: T) -> Ratio<T> {
        Ratio::from_integer(x)
    }
}


// From pair (through the `new` constructor)
impl<T> From<(T, T)> for Ratio<T> where T: Clone + Integer {
    fn from(pair: (T, T)) -> Ratio<T> {
        Ratio::new(pair.0, pair.1)
    }
}

// Comparisons

// Mathematically, comparing a/b and c/d is the same as comparing a*d and b*c, but it's very easy
// for those multiplications to overflow fixed-size integers, so we need to take care.

impl<T: Clone + Integer> Ord for Ratio<T> {
    #[inline]
    fn cmp(&self, other: &Self) -> cmp::Ordering {
        // With equal denominators, the numerators can be directly compared
        if self.denom == other.denom {
            let ord = self.numer.cmp(&other.numer);
            return if self.denom < T::zero() {
                ord.reverse()
            } else {
                ord
            };
        }

        // With equal numerators, the denominators can be inversely compared
        if self.numer == other.numer {
            let ord = self.denom.cmp(&other.denom);
            return if self.numer < T::zero() {
                ord
            } else {
                ord.reverse()
            };
        }

        // Unfortunately, we don't have CheckedMul to try.  That could sometimes avoid all the
        // division below, or even always avoid it for BigInt and BigUint.
        // FIXME- future breaking change to add Checked* to Integer?

        // Compare as floored integers and remainders
        let (self_int, self_rem) = self.numer.div_mod_floor(&self.denom);
        let (other_int, other_rem) = other.numer.div_mod_floor(&other.denom);
        match self_int.cmp(&other_int) {
            cmp::Ordering::Greater => cmp::Ordering::Greater,
            cmp::Ordering::Less => cmp::Ordering::Less,
            cmp::Ordering::Equal => {
                match (self_rem.is_zero(), other_rem.is_zero()) {
                    (true, true) => cmp::Ordering::Equal,
                    (true, false) => cmp::Ordering::Less,
                    (false, true) => cmp::Ordering::Greater,
                    (false, false) => {
                        // Compare the reciprocals of the remaining fractions in reverse
                        let self_recip = Ratio::new_raw(self.denom.clone(), self_rem);
                        let other_recip = Ratio::new_raw(other.denom.clone(), other_rem);
                        self_recip.cmp(&other_recip).reverse()
                    }
                }
            }
        }
    }
}

impl<T: Clone + Integer> PartialOrd for Ratio<T> {
    #[inline]
    fn partial_cmp(&self, other: &Self) -> Option<cmp::Ordering> {
        Some(self.cmp(other))
    }
}

impl<T: Clone + Integer> PartialEq for Ratio<T> {
    #[inline]
    fn eq(&self, other: &Self) -> bool {
        self.cmp(other) == cmp::Ordering::Equal
    }
}

impl<T: Clone + Integer> Eq for Ratio<T> {}


macro_rules! forward_val_val_binop {
    (impl $imp:ident, $method:ident) => {
        impl<T: Clone + Integer> $imp<Ratio<T>> for Ratio<T> {
            type Output = Ratio<T>;

            #[inline]
            fn $method(self, other: Ratio<T>) -> Ratio<T> {
                (&self).$method(&other)
            }
        }
    }
}

macro_rules! forward_ref_val_binop {
    (impl $imp:ident, $method:ident) => {
        impl<'a, T> $imp<Ratio<T>> for &'a Ratio<T> where
            T: Clone + Integer
        {
            type Output = Ratio<T>;

            #[inline]
            fn $method(self, other: Ratio<T>) -> Ratio<T> {
                self.$method(&other)
            }
        }
    }
}

macro_rules! forward_val_ref_binop {
    (impl $imp:ident, $method:ident) => {
        impl<'a, T> $imp<&'a Ratio<T>> for Ratio<T> where
            T: Clone + Integer
        {
            type Output = Ratio<T>;

            #[inline]
            fn $method(self, other: &Ratio<T>) -> Ratio<T> {
                (&self).$method(other)
            }
        }
    }
}

macro_rules! forward_all_binop {
    (impl $imp:ident, $method:ident) => {
        forward_val_val_binop!(impl $imp, $method);
        forward_ref_val_binop!(impl $imp, $method);
        forward_val_ref_binop!(impl $imp, $method);
    };
}

// Arithmetic
forward_all_binop!(impl Mul, mul);
// a/b * c/d = (a*c)/(b*d)
impl<'a, 'b, T> Mul<&'b Ratio<T>> for &'a Ratio<T>
    where T: Clone + Integer
{
    type Output = Ratio<T>;
    #[inline]
    fn mul(self, rhs: &Ratio<T>) -> Ratio<T> {
        Ratio::new(self.numer.clone() * rhs.numer.clone(),
                   self.denom.clone() * rhs.denom.clone())
    }
}

forward_all_binop!(impl Div, div);
// (a/b) / (c/d) = (a*d)/(b*c)
impl<'a, 'b, T> Div<&'b Ratio<T>> for &'a Ratio<T>
    where T: Clone + Integer
{
    type Output = Ratio<T>;

    #[inline]
    fn div(self, rhs: &Ratio<T>) -> Ratio<T> {
        Ratio::new(self.numer.clone() * rhs.denom.clone(),
                   self.denom.clone() * rhs.numer.clone())
    }
}

// Abstracts the a/b `op` c/d = (a*d `op` b*d) / (b*d) pattern
macro_rules! arith_impl {
    (impl $imp:ident, $method:ident) => {
        forward_all_binop!(impl $imp, $method);
        impl<'a, 'b, T: Clone + Integer>
            $imp<&'b Ratio<T>> for &'a Ratio<T> {
            type Output = Ratio<T>;
            #[inline]
            fn $method(self, rhs: &Ratio<T>) -> Ratio<T> {
                Ratio::new((self.numer.clone() * rhs.denom.clone()).$method(self.denom.clone() * rhs.numer.clone()),
                           self.denom.clone() * rhs.denom.clone())
            }
        }
    }
}

// a/b + c/d = (a*d + b*c)/(b*d)
arith_impl!(impl Add, add);

// a/b - c/d = (a*d - b*c)/(b*d)
arith_impl!(impl Sub, sub);

// a/b % c/d = (a*d % b*c)/(b*d)
arith_impl!(impl Rem, rem);

impl<T> Neg for Ratio<T>
    where T: Clone + Integer + Neg<Output = T>
{
    type Output = Ratio<T>;

    #[inline]
    fn neg(self) -> Ratio<T> {
        Ratio::new_raw(-self.numer, self.denom)
    }
}

impl<'a, T> Neg for &'a Ratio<T>
    where T: Clone + Integer + Neg<Output = T>
{
    type Output = Ratio<T>;

    #[inline]
    fn neg(self) -> Ratio<T> {
        -self.clone()
    }
}

// Constants
impl<T: Clone + Integer> Zero for Ratio<T> {
    #[inline]
    fn zero() -> Ratio<T> {
        Ratio::new_raw(Zero::zero(), One::one())
    }

    #[inline]
    fn is_zero(&self) -> bool {
        self.numer.is_zero()
    }
}

impl<T: Clone + Integer> One for Ratio<T> {
    #[inline]
    fn one() -> Ratio<T> {
        Ratio::new_raw(One::one(), One::one())
    }
}

impl<T: Clone + Integer> Num for Ratio<T> {
    type FromStrRadixErr = ParseRatioError;

    /// Parses `numer/denom` where the numbers are in base `radix`.
    fn from_str_radix(s: &str, radix: u32) -> Result<Ratio<T>, ParseRatioError> {
        let split: Vec<&str> = s.splitn(2, '/').collect();
        if split.len() < 2 {
            Err(ParseRatioError { kind: RatioErrorKind::ParseError })
        } else {
            let a_result: Result<T, _> = T::from_str_radix(split[0], radix).map_err(|_| {
                ParseRatioError { kind: RatioErrorKind::ParseError }
            });
            a_result.and_then(|a| {
                let b_result: Result<T, _> = T::from_str_radix(split[1], radix).map_err(|_| {
                    ParseRatioError { kind: RatioErrorKind::ParseError }
                });
                b_result.and_then(|b| {
                    if b.is_zero() {
                        Err(ParseRatioError { kind: RatioErrorKind::ZeroDenominator })
                    } else {
                        Ok(Ratio::new(a.clone(), b.clone()))
                    }
                })
            })
        }
    }
}

impl<T: Clone + Integer + Signed> Signed for Ratio<T> {
    #[inline]
    fn abs(&self) -> Ratio<T> {
        if self.is_negative() {
            -self.clone()
        } else {
            self.clone()
        }
    }

    #[inline]
    fn abs_sub(&self, other: &Ratio<T>) -> Ratio<T> {
        if *self <= *other {
            Zero::zero()
        } else {
            self - other
        }
    }

    #[inline]
    fn signum(&self) -> Ratio<T> {
        if self.is_positive() {
            Self::one()
        } else if self.is_zero() {
            Self::zero()
        } else {
            -Self::one()
        }
    }

    #[inline]
    fn is_positive(&self) -> bool {
        (self.numer.is_positive() && self.denom.is_positive()) ||
        (self.numer.is_negative() && self.denom.is_negative())
    }

    #[inline]
    fn is_negative(&self) -> bool {
        (self.numer.is_negative() && self.denom.is_positive()) ||
        (self.numer.is_positive() && self.denom.is_negative())
    }
}

// String conversions
impl<T> fmt::Display for Ratio<T>
    where T: fmt::Display + Eq + One
{
    /// Renders as `numer/denom`. If denom=1, renders as numer.
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        if self.denom == One::one() {
            write!(f, "{}", self.numer)
        } else {
            write!(f, "{}/{}", self.numer, self.denom)
        }
    }
}

impl<T: FromStr + Clone + Integer> FromStr for Ratio<T> {
    type Err = ParseRatioError;

    /// Parses `numer/denom` or just `numer`.
    fn from_str(s: &str) -> Result<Ratio<T>, ParseRatioError> {
        let mut split = s.splitn(2, '/');

        let n = try!(split.next().ok_or(ParseRatioError { kind: RatioErrorKind::ParseError }));
        let num = try!(FromStr::from_str(n)
                           .map_err(|_| ParseRatioError { kind: RatioErrorKind::ParseError }));

        let d = split.next().unwrap_or("1");
        let den = try!(FromStr::from_str(d)
                           .map_err(|_| ParseRatioError { kind: RatioErrorKind::ParseError }));

        if Zero::is_zero(&den) {
            Err(ParseRatioError { kind: RatioErrorKind::ZeroDenominator })
        } else {
            Ok(Ratio::new(num, den))
        }
    }
}

impl<T> Into<(T, T)> for Ratio<T> {
    fn into(self) -> (T, T) {
        (self.numer, self.denom)
    }
}

#[cfg(feature = "serde")]
impl<T> serde::Serialize for Ratio<T>
    where T: serde::Serialize + Clone + Integer + PartialOrd
{
    fn serialize<S>(&self, serializer: &mut S) -> Result<(), S::Error>
        where S: serde::Serializer
    {
        (self.numer(), self.denom()).serialize(serializer)
    }
}

#[cfg(feature = "serde")]
impl<T> serde::Deserialize for Ratio<T>
    where T: serde::Deserialize + Clone + Integer + PartialOrd
{
    fn deserialize<D>(deserializer: &mut D) -> Result<Self, D::Error>
        where D: serde::Deserializer
    {
        let (numer, denom): (T,T) = try!(serde::Deserialize::deserialize(deserializer));
        if denom.is_zero() {
            Err(serde::de::Error::invalid_value("denominator is zero"))
        } else {
            Ok(Ratio::new_raw(numer, denom))
        }
    }
}

// FIXME: Bubble up specific errors
#[derive(Copy, Clone, Debug, PartialEq)]
pub struct ParseRatioError {
    kind: RatioErrorKind,
}

#[derive(Copy, Clone, Debug, PartialEq)]
enum RatioErrorKind {
    ParseError,
    ZeroDenominator,
}

impl fmt::Display for ParseRatioError {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        self.description().fmt(f)
    }
}

impl Error for ParseRatioError {
    fn description(&self) -> &str {
        self.kind.description()
    }
}

impl RatioErrorKind {
    fn description(&self) -> &'static str {
        match *self {
            RatioErrorKind::ParseError => "failed to parse integer",
            RatioErrorKind::ZeroDenominator => "zero value denominator",
        }
    }
}

impl FromPrimitive for Ratio<BigInt> {
    fn from_i64(n: i64) -> Option<Self> {
        Some(Ratio::from_integer(n.into()))
    }

    fn from_u64(n: u64) -> Option<Self> {
        Some(Ratio::from_integer(n.into()))
    }

    fn from_f32(n: f32) -> Option<Self> {
        Ratio::from_float(n)
    }

    fn from_f64(n: f64) -> Option<Self> {
        Ratio::from_float(n)
    }
}

macro_rules! from_primitive_integer {
    ($typ:ty, $approx:ident) => {
        impl FromPrimitive for Ratio<$typ> {
            fn from_i64(n: i64) -> Option<Self> {
                <$typ as FromPrimitive>::from_i64(n).map(Ratio::from_integer)
            }

            fn from_u64(n: u64) -> Option<Self> {
                <$typ as FromPrimitive>::from_u64(n).map(Ratio::from_integer)
            }

            fn from_f32(n: f32) -> Option<Self> {
                $approx(n, 10e-20, 30)
            }

            fn from_f64(n: f64) -> Option<Self> {
                $approx(n, 10e-20, 30)
            }
        }
    }
}

from_primitive_integer!(i8, approximate_float);
from_primitive_integer!(i16, approximate_float);
from_primitive_integer!(i32, approximate_float);
from_primitive_integer!(i64, approximate_float);
from_primitive_integer!(isize, approximate_float);

from_primitive_integer!(u8, approximate_float_unsigned);
from_primitive_integer!(u16, approximate_float_unsigned);
from_primitive_integer!(u32, approximate_float_unsigned);
from_primitive_integer!(u64, approximate_float_unsigned);
from_primitive_integer!(usize, approximate_float_unsigned);

impl<T: Integer + Signed + Bounded + NumCast + Clone> Ratio<T> {
    pub fn approximate_float<F: Float + NumCast>(f: F) -> Option<Ratio<T>> {
        // 1/10e-20 < 1/2**32 which seems like a good default, and 30 seems
        // to work well. Might want to choose something based on the types in the future, e.g.
        // T::max().recip() and T::bits() or something similar.
        let epsilon = <F as NumCast>::from(10e-20).expect("Can't convert 10e-20");
        approximate_float(f, epsilon, 30)
    }
}

fn approximate_float<T, F>(val: F, max_error: F, max_iterations: usize) -> Option<Ratio<T>>
    where T: Integer + Signed + Bounded + NumCast + Clone,
          F: Float + NumCast
{
    let negative = val.is_sign_negative();
    let abs_val = val.abs();

    let r = approximate_float_unsigned(abs_val, max_error, max_iterations);

    // Make negative again if needed
    if negative {
        r.map(|r| r.neg())
    } else {
        r
    }
}

// No Unsigned constraint because this also works on positive integers and is called
// like that, see above
fn approximate_float_unsigned<T, F>(val: F, max_error: F, max_iterations: usize) -> Option<Ratio<T>>
    where T: Integer + Bounded + NumCast + Clone,
          F: Float + NumCast
{
    // Continued fractions algorithm
    // http://mathforum.org/dr.math/faq/faq.fractions.html#decfrac

    if val < F::zero() {
        return None;
    }

    let mut q = val;
    let mut n0 = T::zero();
    let mut d0 = T::one();
    let mut n1 = T::one();
    let mut d1 = T::zero();

    let t_max = T::max_value();
    let t_max_f = match <F as NumCast>::from(t_max.clone()) {
        None => return None,
        Some(t_max_f) => t_max_f,
    };

    // 1/epsilon > T::MAX
    let epsilon = t_max_f.recip();

    // Overflow
    if q > t_max_f {
        return None;
    }

    for _ in 0..max_iterations {
        let a = match <T as NumCast>::from(q) {
            None => break,
            Some(a) => a,
        };

        let a_f = match <F as NumCast>::from(a.clone()) {
            None => break,
            Some(a_f) => a_f,
        };
        let f = q - a_f;

        // Prevent overflow
        if !a.is_zero() &&
           (n1 > t_max.clone() / a.clone() ||
            d1 > t_max.clone() / a.clone() ||
            a.clone() * n1.clone() > t_max.clone() - n0.clone() ||
            a.clone() * d1.clone() > t_max.clone() - d0.clone()) {
            break;
        }

        let n = a.clone() * n1.clone() + n0.clone();
        let d = a.clone() * d1.clone() + d0.clone();

        n0 = n1;
        d0 = d1;
        n1 = n.clone();
        d1 = d.clone();

        // Simplify fraction. Doing so here instead of at the end
        // allows us to get closer to the target value without overflows
        let g = Integer::gcd(&n1, &d1);
        if !g.is_zero() {
            n1 = n1 / g.clone();
            d1 = d1 / g.clone();
        }

        // Close enough?
        let (n_f, d_f) = match (<F as NumCast>::from(n), <F as NumCast>::from(d)) {
            (Some(n_f), Some(d_f)) => (n_f, d_f),
            _ => break,
        };
        if (n_f / d_f - val).abs() < max_error {
            break;
        }

        // Prevent division by ~0
        if f < epsilon {
            break;
        }
        q = f.recip();
    }

    // Overflow
    if d1.is_zero() {
        return None;
    }

    Some(Ratio::new(n1, d1))
}

#[cfg(test)]
fn hash<T: hash::Hash>(x: &T) -> u64 {
    use std::hash::Hasher;
    let mut hasher = hash::SipHasher::new();
    x.hash(&mut hasher);
    hasher.finish()
}

#[cfg(test)]
mod test {
    use super::{Ratio, Rational};
    #[cfg(feature = "num-bigint")]
    use super::BigRational;

    use std::str::FromStr;
    use std::i32;
    use std::f64;
    use traits::{Zero, One, Signed, FromPrimitive, Float};

    pub const _0: Rational = Ratio {
        numer: 0,
        denom: 1,
    };
    pub const _1: Rational = Ratio {
        numer: 1,
        denom: 1,
    };
    pub const _2: Rational = Ratio {
        numer: 2,
        denom: 1,
    };
    pub const _NEG2: Rational = Ratio {
        numer: -2,
        denom: 1,
    };
    pub const _1_2: Rational = Ratio {
        numer: 1,
        denom: 2,
    };
    pub const _3_2: Rational = Ratio {
        numer: 3,
        denom: 2,
    };
    pub const _NEG1_2: Rational = Ratio {
        numer: -1,
        denom: 2,
    };
    pub const _1_NEG2: Rational = Ratio {
        numer: 1,
        denom: -2,
    };
    pub const _NEG1_NEG2: Rational = Ratio {
        numer: -1,
        denom: -2,
    };
    pub const _1_3: Rational = Ratio {
        numer: 1,
        denom: 3,
    };
    pub const _NEG1_3: Rational = Ratio {
        numer: -1,
        denom: 3,
    };
    pub const _2_3: Rational = Ratio {
        numer: 2,
        denom: 3,
    };
    pub const _NEG2_3: Rational = Ratio {
        numer: -2,
        denom: 3,
    };

    #[cfg(feature = "num-bigint")]
    pub fn to_big(n: Rational) -> BigRational {
        Ratio::new(FromPrimitive::from_isize(n.numer).unwrap(),
                   FromPrimitive::from_isize(n.denom).unwrap())
    }
    #[cfg(not(feature = "num-bigint"))]
    pub fn to_big(n: Rational) -> Rational {
        Ratio::new(FromPrimitive::from_isize(n.numer).unwrap(),
                   FromPrimitive::from_isize(n.denom).unwrap())
    }

    #[test]
    fn test_test_constants() {
        // check our constants are what Ratio::new etc. would make.
        assert_eq!(_0, Zero::zero());
        assert_eq!(_1, One::one());
        assert_eq!(_2, Ratio::from_integer(2));
        assert_eq!(_1_2, Ratio::new(1, 2));
        assert_eq!(_3_2, Ratio::new(3, 2));
        assert_eq!(_NEG1_2, Ratio::new(-1, 2));
        assert_eq!(_2, From::from(2));
    }

    #[test]
    fn test_new_reduce() {
        let one22 = Ratio::new(2, 2);

        assert_eq!(one22, One::one());
    }
    #[test]
    #[should_panic]
    fn test_new_zero() {
        let _a = Ratio::new(1, 0);
    }

    #[test]
    fn test_approximate_float() {
        assert_eq!(Ratio::from_f32(0.5f32), Some(Ratio::new(1i64, 2)));
        assert_eq!(Ratio::from_f64(0.5f64), Some(Ratio::new(1i32, 2)));
        assert_eq!(Ratio::from_f32(5f32), Some(Ratio::new(5i64, 1)));
        assert_eq!(Ratio::from_f64(5f64), Some(Ratio::new(5i32, 1)));
        assert_eq!(Ratio::from_f32(29.97f32), Some(Ratio::new(2997i64, 100)));
        assert_eq!(Ratio::from_f32(-29.97f32), Some(Ratio::new(-2997i64, 100)));

        assert_eq!(Ratio::<i8>::from_f32(63.5f32), Some(Ratio::new(127i8, 2)));
        assert_eq!(Ratio::<i8>::from_f32(126.5f32), Some(Ratio::new(126i8, 1)));
        assert_eq!(Ratio::<i8>::from_f32(127.0f32), Some(Ratio::new(127i8, 1)));
        assert_eq!(Ratio::<i8>::from_f32(127.5f32), None);
        assert_eq!(Ratio::<i8>::from_f32(-63.5f32), Some(Ratio::new(-127i8, 2)));
        assert_eq!(Ratio::<i8>::from_f32(-126.5f32), Some(Ratio::new(-126i8, 1)));
        assert_eq!(Ratio::<i8>::from_f32(-127.0f32), Some(Ratio::new(-127i8, 1)));
        assert_eq!(Ratio::<i8>::from_f32(-127.5f32), None);

        assert_eq!(Ratio::<u8>::from_f32(-127f32), None);
        assert_eq!(Ratio::<u8>::from_f32(127f32), Some(Ratio::new(127u8, 1)));
        assert_eq!(Ratio::<u8>::from_f32(127.5f32), Some(Ratio::new(255u8, 2)));
        assert_eq!(Ratio::<u8>::from_f32(256f32), None);

        assert_eq!(Ratio::<i64>::from_f64(-10e200), None);
        assert_eq!(Ratio::<i64>::from_f64(10e200), None);
        assert_eq!(Ratio::<i64>::from_f64(f64::INFINITY), None);
        assert_eq!(Ratio::<i64>::from_f64(f64::NEG_INFINITY), None);
        assert_eq!(Ratio::<i64>::from_f64(f64::NAN), None);
        assert_eq!(Ratio::<i64>::from_f64(f64::EPSILON), Some(Ratio::new(1, 4503599627370496)));
        assert_eq!(Ratio::<i64>::from_f64(0.0), Some(Ratio::new(0, 1)));
        assert_eq!(Ratio::<i64>::from_f64(-0.0), Some(Ratio::new(0, 1)));
    }

    #[test]
    fn test_cmp() {
        assert!(_0 == _0 && _1 == _1);
        assert!(_0 != _1 && _1 != _0);
        assert!(_0 < _1 && !(_1 < _0));
        assert!(_1 > _0 && !(_0 > _1));

        assert!(_0 <= _0 && _1 <= _1);
        assert!(_0 <= _1 && !(_1 <= _0));

        assert!(_0 >= _0 && _1 >= _1);
        assert!(_1 >= _0 && !(_0 >= _1));
    }

    #[test]
    fn test_cmp_overflow() {
        use std::cmp::Ordering;

        // issue #7 example:
        let big = Ratio::new(128u8, 1);
        let small = big.recip();
        assert!(big > small);

        // try a few that are closer together
        // (some matching numer, some matching denom, some neither)
        let ratios = vec![
            Ratio::new(125_i8, 127_i8),
            Ratio::new(63_i8, 64_i8),
            Ratio::new(124_i8, 125_i8),
            Ratio::new(125_i8, 126_i8),
            Ratio::new(126_i8, 127_i8),
            Ratio::new(127_i8, 126_i8),
        ];

        fn check_cmp(a: Ratio<i8>, b: Ratio<i8>, ord: Ordering) {
            println!("comparing {} and {}", a, b);
            assert_eq!(a.cmp(&b), ord);
            assert_eq!(b.cmp(&a), ord.reverse());
        }

        for (i, &a) in ratios.iter().enumerate() {
            check_cmp(a, a, Ordering::Equal);
            check_cmp(-a, a, Ordering::Less);
            for &b in &ratios[i + 1..] {
                check_cmp(a, b, Ordering::Less);
                check_cmp(-a, -b, Ordering::Greater);
                check_cmp(a.recip(), b.recip(), Ordering::Greater);
                check_cmp(-a.recip(), -b.recip(), Ordering::Less);
            }
        }
    }

    #[test]
    fn test_to_integer() {
        assert_eq!(_0.to_integer(), 0);
        assert_eq!(_1.to_integer(), 1);
        assert_eq!(_2.to_integer(), 2);
        assert_eq!(_1_2.to_integer(), 0);
        assert_eq!(_3_2.to_integer(), 1);
        assert_eq!(_NEG1_2.to_integer(), 0);
    }


    #[test]
    fn test_numer() {
        assert_eq!(_0.numer(), &0);
        assert_eq!(_1.numer(), &1);
        assert_eq!(_2.numer(), &2);
        assert_eq!(_1_2.numer(), &1);
        assert_eq!(_3_2.numer(), &3);
        assert_eq!(_NEG1_2.numer(), &(-1));
    }
    #[test]
    fn test_denom() {
        assert_eq!(_0.denom(), &1);
        assert_eq!(_1.denom(), &1);
        assert_eq!(_2.denom(), &1);
        assert_eq!(_1_2.denom(), &2);
        assert_eq!(_3_2.denom(), &2);
        assert_eq!(_NEG1_2.denom(), &2);
    }


    #[test]
    fn test_is_integer() {
        assert!(_0.is_integer());
        assert!(_1.is_integer());
        assert!(_2.is_integer());
        assert!(!_1_2.is_integer());
        assert!(!_3_2.is_integer());
        assert!(!_NEG1_2.is_integer());
    }

    #[test]
    fn test_show() {
        assert_eq!(format!("{}", _2), "2".to_string());
        assert_eq!(format!("{}", _1_2), "1/2".to_string());
        assert_eq!(format!("{}", _0), "0".to_string());
        assert_eq!(format!("{}", Ratio::from_integer(-2)), "-2".to_string());
    }

    mod arith {
        use super::{_0, _1, _2, _1_2, _3_2, _NEG1_2, to_big};
        use super::super::{Ratio, Rational};

        #[test]
        fn test_add() {
            fn test(a: Rational, b: Rational, c: Rational) {
                assert_eq!(a + b, c);
                assert_eq!(to_big(a) + to_big(b), to_big(c));
            }

            test(_1, _1_2, _3_2);
            test(_1, _1, _2);
            test(_1_2, _3_2, _2);
            test(_1_2, _NEG1_2, _0);
        }

        #[test]
        fn test_sub() {
            fn test(a: Rational, b: Rational, c: Rational) {
                assert_eq!(a - b, c);
                assert_eq!(to_big(a) - to_big(b), to_big(c))
            }

            test(_1, _1_2, _1_2);
            test(_3_2, _1_2, _1);
            test(_1, _NEG1_2, _3_2);
        }

        #[test]
        fn test_mul() {
            fn test(a: Rational, b: Rational, c: Rational) {
                assert_eq!(a * b, c);
                assert_eq!(to_big(a) * to_big(b), to_big(c))
            }

            test(_1, _1_2, _1_2);
            test(_1_2, _3_2, Ratio::new(3, 4));
            test(_1_2, _NEG1_2, Ratio::new(-1, 4));
        }

        #[test]
        fn test_div() {
            fn test(a: Rational, b: Rational, c: Rational) {
                assert_eq!(a / b, c);
                assert_eq!(to_big(a) / to_big(b), to_big(c))
            }

            test(_1, _1_2, _2);
            test(_3_2, _1_2, _1 + _2);
            test(_1, _NEG1_2, _NEG1_2 + _NEG1_2 + _NEG1_2 + _NEG1_2);
        }

        #[test]
        fn test_rem() {
            fn test(a: Rational, b: Rational, c: Rational) {
                assert_eq!(a % b, c);
                assert_eq!(to_big(a) % to_big(b), to_big(c))
            }

            test(_3_2, _1, _1_2);
            test(_2, _NEG1_2, _0);
            test(_1_2, _2, _1_2);
        }

        #[test]
        fn test_neg() {
            fn test(a: Rational, b: Rational) {
                assert_eq!(-a, b);
                assert_eq!(-to_big(a), to_big(b))
            }

            test(_0, _0);
            test(_1_2, _NEG1_2);
            test(-_1, _1);
        }
        #[test]
        fn test_zero() {
            assert_eq!(_0 + _0, _0);
            assert_eq!(_0 * _0, _0);
            assert_eq!(_0 * _1, _0);
            assert_eq!(_0 / _NEG1_2, _0);
            assert_eq!(_0 - _0, _0);
        }
        #[test]
        #[should_panic]
        fn test_div_0() {
            let _a = _1 / _0;
        }
    }

    #[test]
    fn test_round() {
        assert_eq!(_1_3.ceil(), _1);
        assert_eq!(_1_3.floor(), _0);
        assert_eq!(_1_3.round(), _0);
        assert_eq!(_1_3.trunc(), _0);

        assert_eq!(_NEG1_3.ceil(), _0);
        assert_eq!(_NEG1_3.floor(), -_1);
        assert_eq!(_NEG1_3.round(), _0);
        assert_eq!(_NEG1_3.trunc(), _0);

        assert_eq!(_2_3.ceil(), _1);
        assert_eq!(_2_3.floor(), _0);
        assert_eq!(_2_3.round(), _1);
        assert_eq!(_2_3.trunc(), _0);

        assert_eq!(_NEG2_3.ceil(), _0);
        assert_eq!(_NEG2_3.floor(), -_1);
        assert_eq!(_NEG2_3.round(), -_1);
        assert_eq!(_NEG2_3.trunc(), _0);

        assert_eq!(_1_2.ceil(), _1);
        assert_eq!(_1_2.floor(), _0);
        assert_eq!(_1_2.round(), _1);
        assert_eq!(_1_2.trunc(), _0);

        assert_eq!(_NEG1_2.ceil(), _0);
        assert_eq!(_NEG1_2.floor(), -_1);
        assert_eq!(_NEG1_2.round(), -_1);
        assert_eq!(_NEG1_2.trunc(), _0);

        assert_eq!(_1.ceil(), _1);
        assert_eq!(_1.floor(), _1);
        assert_eq!(_1.round(), _1);
        assert_eq!(_1.trunc(), _1);

        // Overflow checks

        let _neg1 = Ratio::from_integer(-1);
        let _large_rat1 = Ratio::new(i32::MAX, i32::MAX - 1);
        let _large_rat2 = Ratio::new(i32::MAX - 1, i32::MAX);
        let _large_rat3 = Ratio::new(i32::MIN + 2, i32::MIN + 1);
        let _large_rat4 = Ratio::new(i32::MIN + 1, i32::MIN + 2);
        let _large_rat5 = Ratio::new(i32::MIN + 2, i32::MAX);
        let _large_rat6 = Ratio::new(i32::MAX, i32::MIN + 2);
        let _large_rat7 = Ratio::new(1, i32::MIN + 1);
        let _large_rat8 = Ratio::new(1, i32::MAX);

        assert_eq!(_large_rat1.round(), One::one());
        assert_eq!(_large_rat2.round(), One::one());
        assert_eq!(_large_rat3.round(), One::one());
        assert_eq!(_large_rat4.round(), One::one());
        assert_eq!(_large_rat5.round(), _neg1);
        assert_eq!(_large_rat6.round(), _neg1);
        assert_eq!(_large_rat7.round(), Zero::zero());
        assert_eq!(_large_rat8.round(), Zero::zero());
    }

    #[test]
    fn test_fract() {
        assert_eq!(_1.fract(), _0);
        assert_eq!(_NEG1_2.fract(), _NEG1_2);
        assert_eq!(_1_2.fract(), _1_2);
        assert_eq!(_3_2.fract(), _1_2);
    }

    #[test]
    fn test_recip() {
        assert_eq!(_1 * _1.recip(), _1);
        assert_eq!(_2 * _2.recip(), _1);
        assert_eq!(_1_2 * _1_2.recip(), _1);
        assert_eq!(_3_2 * _3_2.recip(), _1);
        assert_eq!(_NEG1_2 * _NEG1_2.recip(), _1);

        assert_eq!(_3_2.recip(), _2_3);
        assert_eq!(_NEG1_2.recip(), _NEG2);
        assert_eq!(_NEG1_2.recip().denom(), &1);
    }

    #[test]
    #[should_panic(expected = "== 0")]
    fn test_recip_fail() {
        let _a = Ratio::new(0, 1).recip();
    }

    #[test]
    fn test_pow() {
        assert_eq!(_1_2.pow(2), Ratio::new(1, 4));
        assert_eq!(_1_2.pow(-2), Ratio::new(4, 1));
        assert_eq!(_1.pow(1), _1);
        assert_eq!(_NEG1_2.pow(2), _1_2.pow(2));
        assert_eq!(_NEG1_2.pow(3), -_1_2.pow(3));
        assert_eq!(_3_2.pow(0), _1);
        assert_eq!(_3_2.pow(-1), _3_2.recip());
        assert_eq!(_3_2.pow(3), Ratio::new(27, 8));
    }

    #[test]
    fn test_to_from_str() {
        fn test(r: Rational, s: String) {
            assert_eq!(FromStr::from_str(&s), Ok(r));
            assert_eq!(r.to_string(), s);
        }
        test(_1, "1".to_string());
        test(_0, "0".to_string());
        test(_1_2, "1/2".to_string());
        test(_3_2, "3/2".to_string());
        test(_2, "2".to_string());
        test(_NEG1_2, "-1/2".to_string());
    }
    #[test]
    fn test_from_str_fail() {
        fn test(s: &str) {
            let rational: Result<Rational, _> = FromStr::from_str(s);
            assert!(rational.is_err());
        }

        let xs = ["0 /1", "abc", "", "1/", "--1/2", "3/2/1", "1/0"];
        for &s in xs.iter() {
            test(s);
        }
    }

    #[cfg(feature = "num-bigint")]
    #[test]
    fn test_from_float() {
        fn test<T: Float>(given: T, (numer, denom): (&str, &str)) {
            let ratio: BigRational = Ratio::from_float(given).unwrap();
            assert_eq!(ratio,
                       Ratio::new(FromStr::from_str(numer).unwrap(),
                                  FromStr::from_str(denom).unwrap()));
        }

        // f32
        test(3.14159265359f32, ("13176795", "4194304"));
        test(2f32.powf(100.), ("1267650600228229401496703205376", "1"));
        test(-2f32.powf(100.), ("-1267650600228229401496703205376", "1"));
        test(1.0 / 2f32.powf(100.),
             ("1", "1267650600228229401496703205376"));
        test(684729.48391f32, ("1369459", "2"));
        test(-8573.5918555f32, ("-4389679", "512"));

        // f64
        test(3.14159265359f64, ("3537118876014453", "1125899906842624"));
        test(2f64.powf(100.), ("1267650600228229401496703205376", "1"));
        test(-2f64.powf(100.), ("-1267650600228229401496703205376", "1"));
        test(684729.48391f64, ("367611342500051", "536870912"));
        test(-8573.5918555f64, ("-4713381968463931", "549755813888"));
        test(1.0 / 2f64.powf(100.),
             ("1", "1267650600228229401496703205376"));
    }

    #[cfg(feature = "num-bigint")]
    #[test]
    fn test_from_float_fail() {
        use std::{f32, f64};

        assert_eq!(Ratio::from_float(f32::NAN), None);
        assert_eq!(Ratio::from_float(f32::INFINITY), None);
        assert_eq!(Ratio::from_float(f32::NEG_INFINITY), None);
        assert_eq!(Ratio::from_float(f64::NAN), None);
        assert_eq!(Ratio::from_float(f64::INFINITY), None);
        assert_eq!(Ratio::from_float(f64::NEG_INFINITY), None);
    }

    #[test]
    fn test_signed() {
        assert_eq!(_NEG1_2.abs(), _1_2);
        assert_eq!(_3_2.abs_sub(&_1_2), _1);
        assert_eq!(_1_2.abs_sub(&_3_2), Zero::zero());
        assert_eq!(_1_2.signum(), One::one());
        assert_eq!(_NEG1_2.signum(), -<Ratio<isize>>::one());
        assert_eq!(_0.signum(), Zero::zero());
        assert!(_NEG1_2.is_negative());
        assert!(_1_NEG2.is_negative());
        assert!(!_NEG1_2.is_positive());
        assert!(!_1_NEG2.is_positive());
        assert!(_1_2.is_positive());
        assert!(_NEG1_NEG2.is_positive());
        assert!(!_1_2.is_negative());
        assert!(!_NEG1_NEG2.is_negative());
        assert!(!_0.is_positive());
        assert!(!_0.is_negative());
    }

    #[test]
    fn test_hash() {
        assert!(::hash(&_0) != ::hash(&_1));
        assert!(::hash(&_0) != ::hash(&_3_2));
    }

    #[test]
    fn test_into_pair() {
        assert_eq! ((0, 1), _0.into());
        assert_eq! ((-2, 1), _NEG2.into());
        assert_eq! ((1, -2), _1_NEG2.into());
    }

    #[test]
    fn test_from_pair() {
        assert_eq! (_0, Ratio::from ((0, 1)));
        assert_eq! (_1, Ratio::from ((1, 1)));
        assert_eq! (_NEG2, Ratio::from ((-2, 1)));
        assert_eq! (_1_NEG2, Ratio::from ((1, -2)));
    }
}