Initial vendor packages

Signed-off-by: Valentin Popov <valentin@popov.link>

1 files changed, 3106 insertions, 0 deletions

diff --git a/vendor/num-rational/src/lib.rs b/vendor/num-rational/src/lib.rs
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+// Copyright 2013-2014 The Rust Project Developers. See the COPYRIGHT
+// file at the top-level directory of this distribution and at
+// http://rust-lang.org/COPYRIGHT.
+//
+// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
+// http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
+// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your
+// option. This file may not be copied, modified, or distributed
+// except according to those terms.
+
+//! Rational numbers
+//!
+//! ## Compatibility
+//!
+//! The `num-rational` crate is tested for rustc 1.31 and greater.
+
+#![doc(html_root_url = "https://docs.rs/num-rational/0.4")]
+#![no_std]
+// Ratio ops often use other "suspicious" ops
+#![allow(clippy::suspicious_arithmetic_impl)]
+#![allow(clippy::suspicious_op_assign_impl)]
+
+#[cfg(feature = "std")]
+#[macro_use]
+extern crate std;
+
+use core::cmp;
+use core::fmt;
+use core::fmt::{Binary, Display, Formatter, LowerExp, LowerHex, Octal, UpperExp, UpperHex};
+use core::hash::{Hash, Hasher};
+use core::ops::{Add, Div, Mul, Neg, Rem, ShlAssign, Sub};
+use core::str::FromStr;
+#[cfg(feature = "std")]
+use std::error::Error;
+
+#[cfg(feature = "num-bigint")]
+use num_bigint::{BigInt, BigUint, Sign, ToBigInt};
+
+use num_integer::Integer;
+use num_traits::float::FloatCore;
+use num_traits::ToPrimitive;
+use num_traits::{
+    Bounded, CheckedAdd, CheckedDiv, CheckedMul, CheckedSub, FromPrimitive, Inv, Num, NumCast, One,
+    Pow, Signed, Zero,
+};
+
+mod pow;
+
+/// Represents the ratio between two numbers.
+#[derive(Copy, Clone, Debug)]
+#[allow(missing_docs)]
+pub struct Ratio<T> {
+    /// Numerator.
+    numer: T,
+    /// Denominator.
+    denom: T,
+}
+
+/// Alias for a `Ratio` of machine-sized integers.
+#[deprecated(
+    since = "0.4.0",
+    note = "it's better to use a specific size, like `Rational32` or `Rational64`"
+)]
+pub type Rational = Ratio<isize>;
+/// Alias for a `Ratio` of 32-bit-sized integers.
+pub type Rational32 = Ratio<i32>;
+/// Alias for a `Ratio` of 64-bit-sized integers.
+pub type Rational64 = Ratio<i64>;
+
+#[cfg(feature = "num-bigint")]
+/// Alias for arbitrary precision rationals.
+pub type BigRational = Ratio<BigInt>;
+
+/// These method are `const` for Rust 1.31 and later.
+impl<T> Ratio<T> {
+    /// Creates a `Ratio` without checking for `denom == 0` or reducing.
+    ///
+    /// **There are several methods that will panic if used on a `Ratio` with
+    /// `denom == 0`.**
+    #[inline]
+    pub const fn new_raw(numer: T, denom: T) -> Ratio<T> {
+        Ratio { numer, denom }
+    }
+
+    /// Gets an immutable reference to the numerator.
+    #[inline]
+    pub const fn numer(&self) -> &T {
+        &self.numer
+    }
+
+    /// Gets an immutable reference to the denominator.
+    #[inline]
+    pub const fn denom(&self) -> &T {
+        &self.denom
+    }
+}
+
+impl<T: Clone + Integer> Ratio<T> {
+    /// Creates a new `Ratio`.
+    ///
+    /// **Panics if `denom` is zero.**
+    #[inline]
+    pub fn new(numer: T, denom: T) -> Ratio<T> {
+        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())
+    }
+
+    /// Converts to an integer, rounding towards zero.
+    #[inline]
+    pub fn to_integer(&self) -> T {
+        self.trunc().numer
+    }
+
+    /// Returns true if the rational number is an integer (denominator is 1).
+    #[inline]
+    pub fn is_integer(&self) -> bool {
+        self.denom.is_one()
+    }
+
+    /// Puts self into lowest terms, with `denom` > 0.
+    ///
+    /// **Panics if `denom` is zero.**
+    fn reduce(&mut self) {
+        if self.denom.is_zero() {
+            panic!("denominator == 0");
+        }
+        if self.numer.is_zero() {
+            self.denom.set_one();
+            return;
+        }
+        if self.numer == self.denom {
+            self.set_one();
+            return;
+        }
+        let g: T = self.numer.gcd(&self.denom);
+
+        // FIXME(#5992): assignment operator overloads
+        // T: Clone + Integer != T: Clone + NumAssign
+
+        #[inline]
+        fn replace_with<T: Zero>(x: &mut T, f: impl FnOnce(T) -> T) {
+            let y = core::mem::replace(x, T::zero());
+            *x = f(y);
+        }
+
+        // self.numer /= g;
+        replace_with(&mut self.numer, |x| x / g.clone());
+
+        // self.denom /= g;
+        replace_with(&mut self.denom, |x| x / g);
+
+        // keep denom positive!
+        if self.denom < T::zero() {
+            replace_with(&mut self.numer, |x| T::zero() - x);
+            replace_with(&mut self.denom, |x| T::zero() - x);
+        }
+    }
+
+    /// 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`.
+    ///
+    /// **Panics if `denom` is zero.**
+    pub fn reduced(&self) -> Ratio<T> {
+        let mut ret = self.clone();
+        ret.reduce();
+        ret
+    }
+
+    /// Returns the reciprocal.
+    ///
+    /// **Panics if the `Ratio` is zero.**
+    #[inline]
+    pub fn recip(&self) -> Ratio<T> {
+        self.clone().into_recip()
+    }
+
+    #[inline]
+    fn into_recip(self) -> Ratio<T> {
+        match self.numer.cmp(&T::zero()) {
+            cmp::Ordering::Equal => panic!("division by zero"),
+            cmp::Ordering::Greater => Ratio::new_raw(self.denom, self.numer),
+            cmp::Ordering::Less => Ratio::new_raw(T::zero() - self.denom, T::zero() - self.numer),
+        }
+    }
+
+    /// 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 / two
+        } else {
+            fractional.numer >= (fractional.denom / two) + one
+        };
+
+        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())
+    }
+
+    /// Raises the `Ratio` to the power of an exponent.
+    #[inline]
+    pub fn pow(&self, expon: i32) -> Ratio<T>
+    where
+        for<'a> &'a T: Pow<u32, Output = T>,
+    {
+        Pow::pow(self, expon)
+    }
+}
+
+#[cfg(feature = "num-bigint")]
+impl Ratio<BigInt> {
+    /// Converts a float into a rational number.
+    pub fn from_float<T: FloatCore>(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 <<= exponent as usize;
+            Some(Ratio::from_integer(BigInt::from_biguint(
+                bigint_sign,
+                numer,
+            )))
+        }
+    }
+}
+
+impl<T: Clone + Integer> Default for Ratio<T> {
+    /// Returns zero
+    fn default() -> Self {
+        Ratio::zero()
+    }
+}
+
+// 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 {
+            if self.numer.is_zero() {
+                return cmp::Ordering::Equal;
+            }
+            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> {}
+
+// NB: We can't just `#[derive(Hash)]`, because it needs to agree
+// with `Eq` even for non-reduced ratios.
+impl<T: Clone + Integer + Hash> Hash for Ratio<T> {
+    fn hash<H: Hasher>(&self, state: &mut H) {
+        recurse(&self.numer, &self.denom, state);
+
+        fn recurse<T: Integer + Hash, H: Hasher>(numer: &T, denom: &T, state: &mut H) {
+            if !denom.is_zero() {
+                let (int, rem) = numer.div_mod_floor(denom);
+                int.hash(state);
+                recurse(denom, &rem, state);
+            } else {
+                denom.hash(state);
+            }
+        }
+    }
+}
+
+mod iter_sum_product {
+    use crate::Ratio;
+    use core::iter::{Product, Sum};
+    use num_integer::Integer;
+    use num_traits::{One, Zero};
+
+    impl<T: Integer + Clone> Sum for Ratio<T> {
+        fn sum<I>(iter: I) -> Self
+        where
+            I: Iterator<Item = Ratio<T>>,
+        {
+            iter.fold(Self::zero(), |sum, num| sum + num)
+        }
+    }
+
+    impl<'a, T: Integer + Clone> Sum<&'a Ratio<T>> for Ratio<T> {
+        fn sum<I>(iter: I) -> Self
+        where
+            I: Iterator<Item = &'a Ratio<T>>,
+        {
+            iter.fold(Self::zero(), |sum, num| sum + num)
+        }
+    }
+
+    impl<T: Integer + Clone> Product for Ratio<T> {
+        fn product<I>(iter: I) -> Self
+        where
+            I: Iterator<Item = Ratio<T>>,
+        {
+            iter.fold(Self::one(), |prod, num| prod * num)
+        }
+    }
+
+    impl<'a, T: Integer + Clone> Product<&'a Ratio<T>> for Ratio<T> {
+        fn product<I>(iter: I) -> Self
+        where
+            I: Iterator<Item = &'a Ratio<T>>,
+        {
+            iter.fold(Self::one(), |prod, num| prod * num)
+        }
+    }
+}
+
+mod opassign {
+    use core::ops::{AddAssign, DivAssign, MulAssign, RemAssign, SubAssign};
+
+    use crate::Ratio;
+    use num_integer::Integer;
+    use num_traits::NumAssign;
+
+    impl<T: Clone + Integer + NumAssign> AddAssign for Ratio<T> {
+        fn add_assign(&mut self, other: Ratio<T>) {
+            if self.denom == other.denom {
+                self.numer += other.numer
+            } else {
+                let lcm = self.denom.lcm(&other.denom);
+                let lhs_numer = self.numer.clone() * (lcm.clone() / self.denom.clone());
+                let rhs_numer = other.numer * (lcm.clone() / other.denom);
+                self.numer = lhs_numer + rhs_numer;
+                self.denom = lcm;
+            }
+            self.reduce();
+        }
+    }
+
+    // (a/b) / (c/d) = (a/gcd_ac)*(d/gcd_bd) / ((c/gcd_ac)*(b/gcd_bd))
+    impl<T: Clone + Integer + NumAssign> DivAssign for Ratio<T> {
+        fn div_assign(&mut self, other: Ratio<T>) {
+            let gcd_ac = self.numer.gcd(&other.numer);
+            let gcd_bd = self.denom.gcd(&other.denom);
+            self.numer /= gcd_ac.clone();
+            self.numer *= other.denom / gcd_bd.clone();
+            self.denom /= gcd_bd;
+            self.denom *= other.numer / gcd_ac;
+            self.reduce(); // TODO: remove this line. see #8.
+        }
+    }
+
+    // a/b * c/d = (a/gcd_ad)*(c/gcd_bc) / ((d/gcd_ad)*(b/gcd_bc))
+    impl<T: Clone + Integer + NumAssign> MulAssign for Ratio<T> {
+        fn mul_assign(&mut self, other: Ratio<T>) {
+            let gcd_ad = self.numer.gcd(&other.denom);
+            let gcd_bc = self.denom.gcd(&other.numer);
+            self.numer /= gcd_ad.clone();
+            self.numer *= other.numer / gcd_bc.clone();
+            self.denom /= gcd_bc;
+            self.denom *= other.denom / gcd_ad;
+            self.reduce(); // TODO: remove this line. see #8.
+        }
+    }
+
+    impl<T: Clone + Integer + NumAssign> RemAssign for Ratio<T> {
+        fn rem_assign(&mut self, other: Ratio<T>) {
+            if self.denom == other.denom {
+                self.numer %= other.numer
+            } else {
+                let lcm = self.denom.lcm(&other.denom);
+                let lhs_numer = self.numer.clone() * (lcm.clone() / self.denom.clone());
+                let rhs_numer = other.numer * (lcm.clone() / other.denom);
+                self.numer = lhs_numer % rhs_numer;
+                self.denom = lcm;
+            }
+            self.reduce();
+        }
+    }
+
+    impl<T: Clone + Integer + NumAssign> SubAssign for Ratio<T> {
+        fn sub_assign(&mut self, other: Ratio<T>) {
+            if self.denom == other.denom {
+                self.numer -= other.numer
+            } else {
+                let lcm = self.denom.lcm(&other.denom);
+                let lhs_numer = self.numer.clone() * (lcm.clone() / self.denom.clone());
+                let rhs_numer = other.numer * (lcm.clone() / other.denom);
+                self.numer = lhs_numer - rhs_numer;
+                self.denom = lcm;
+            }
+            self.reduce();
+        }
+    }
+
+    // a/b + c/1 = (a*1 + b*c) / (b*1) = (a + b*c) / b
+    impl<T: Clone + Integer + NumAssign> AddAssign<T> for Ratio<T> {
+        fn add_assign(&mut self, other: T) {
+            self.numer += self.denom.clone() * other;
+            self.reduce();
+        }
+    }
+
+    impl<T: Clone + Integer + NumAssign> DivAssign<T> for Ratio<T> {
+        fn div_assign(&mut self, other: T) {
+            let gcd = self.numer.gcd(&other);
+            self.numer /= gcd.clone();
+            self.denom *= other / gcd;
+            self.reduce(); // TODO: remove this line. see #8.
+        }
+    }
+
+    impl<T: Clone + Integer + NumAssign> MulAssign<T> for Ratio<T> {
+        fn mul_assign(&mut self, other: T) {
+            let gcd = self.denom.gcd(&other);
+            self.denom /= gcd.clone();
+            self.numer *= other / gcd;
+            self.reduce(); // TODO: remove this line. see #8.
+        }
+    }
+
+    // a/b % c/1 = (a*1 % b*c) / (b*1) = (a % b*c) / b
+    impl<T: Clone + Integer + NumAssign> RemAssign<T> for Ratio<T> {
+        fn rem_assign(&mut self, other: T) {
+            self.numer %= self.denom.clone() * other;
+            self.reduce();
+        }
+    }
+
+    // a/b - c/1 = (a*1 - b*c) / (b*1) = (a - b*c) / b
+    impl<T: Clone + Integer + NumAssign> SubAssign<T> for Ratio<T> {
+        fn sub_assign(&mut self, other: T) {
+            self.numer -= self.denom.clone() * other;
+            self.reduce();
+        }
+    }
+
+    macro_rules! forward_op_assign {
+        (impl $imp:ident, $method:ident) => {
+            impl<'a, T: Clone + Integer + NumAssign> $imp<&'a Ratio<T>> for Ratio<T> {
+                #[inline]
+                fn $method(&mut self, other: &Ratio<T>) {
+                    self.$method(other.clone())
+                }
+            }
+            impl<'a, T: Clone + Integer + NumAssign> $imp<&'a T> for Ratio<T> {
+                #[inline]
+                fn $method(&mut self, other: &T) {
+                    self.$method(other.clone())
+                }
+            }
+        };
+    }
+
+    forward_op_assign!(impl AddAssign, add_assign);
+    forward_op_assign!(impl DivAssign, div_assign);
+    forward_op_assign!(impl MulAssign, mul_assign);
+    forward_op_assign!(impl RemAssign, rem_assign);
+    forward_op_assign!(impl SubAssign, sub_assign);
+}
+
+macro_rules! forward_ref_ref_binop {
+    (impl $imp:ident, $method:ident) => {
+        impl<'a, 'b, T: Clone + Integer> $imp<&'b Ratio<T>> for &'a Ratio<T> {
+            type Output = Ratio<T>;
+
+            #[inline]
+            fn $method(self, other: &'b Ratio<T>) -> Ratio<T> {
+                self.clone().$method(other.clone())
+            }
+        }
+        impl<'a, 'b, T: Clone + Integer> $imp<&'b T> for &'a Ratio<T> {
+            type Output = Ratio<T>;
+
+            #[inline]
+            fn $method(self, other: &'b T) -> Ratio<T> {
+                self.clone().$method(other.clone())
+            }
+        }
+    };
+}
+
+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 $method(self, other: Ratio<T>) -> Ratio<T> {
+                self.clone().$method(other)
+            }
+        }
+        impl<'a, T> $imp<T> for &'a Ratio<T>
+        where
+            T: Clone + Integer,
+        {
+            type Output = Ratio<T>;
+
+            #[inline]
+            fn $method(self, other: T) -> Ratio<T> {
+                self.clone().$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.clone())
+            }
+        }
+        impl<'a, T> $imp<&'a T> for Ratio<T>
+        where
+            T: Clone + Integer,
+        {
+            type Output = Ratio<T>;
+
+            #[inline]
+            fn $method(self, other: &T) -> Ratio<T> {
+                self.$method(other.clone())
+            }
+        }
+    };
+}
+
+macro_rules! forward_all_binop {
+    (impl $imp:ident, $method:ident) => {
+        forward_ref_ref_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/gcd_ad)*(c/gcd_bc) / ((d/gcd_ad)*(b/gcd_bc))
+impl<T> Mul<Ratio<T>> for Ratio<T>
+where
+    T: Clone + Integer,
+{
+    type Output = Ratio<T>;
+    #[inline]
+    fn mul(self, rhs: Ratio<T>) -> Ratio<T> {
+        let gcd_ad = self.numer.gcd(&rhs.denom);
+        let gcd_bc = self.denom.gcd(&rhs.numer);
+        Ratio::new(
+            self.numer / gcd_ad.clone() * (rhs.numer / gcd_bc.clone()),
+            self.denom / gcd_bc * (rhs.denom / gcd_ad),
+        )
+    }
+}
+// a/b * c/1 = (a*c) / (b*1) = (a*c) / b
+impl<T> Mul<T> for Ratio<T>
+where
+    T: Clone + Integer,
+{
+    type Output = Ratio<T>;
+    #[inline]
+    fn mul(self, rhs: T) -> Ratio<T> {
+        let gcd = self.denom.gcd(&rhs);
+        Ratio::new(self.numer * (rhs / gcd.clone()), self.denom / gcd)
+    }
+}
+
+forward_all_binop!(impl Div, div);
+// (a/b) / (c/d) = (a/gcd_ac)*(d/gcd_bd) / ((c/gcd_ac)*(b/gcd_bd))
+impl<T> Div<Ratio<T>> for Ratio<T>
+where
+    T: Clone + Integer,
+{
+    type Output = Ratio<T>;
+
+    #[inline]
+    fn div(self, rhs: Ratio<T>) -> Ratio<T> {
+        let gcd_ac = self.numer.gcd(&rhs.numer);
+        let gcd_bd = self.denom.gcd(&rhs.denom);
+        Ratio::new(
+            self.numer / gcd_ac.clone() * (rhs.denom / gcd_bd.clone()),
+            self.denom / gcd_bd * (rhs.numer / gcd_ac),
+        )
+    }
+}
+// (a/b) / (c/1) = (a*1) / (b*c) = a / (b*c)
+impl<T> Div<T> for Ratio<T>
+where
+    T: Clone + Integer,
+{
+    type Output = Ratio<T>;
+
+    #[inline]
+    fn div(self, rhs: T) -> Ratio<T> {
+        let gcd = self.numer.gcd(&rhs);
+        Ratio::new(self.numer / gcd.clone(), self.denom * (rhs / gcd))
+    }
+}
+
+macro_rules! arith_impl {
+    (impl $imp:ident, $method:ident) => {
+        forward_all_binop!(impl $imp, $method);
+        // Abstracts a/b `op` c/d = (a*lcm/b `op` c*lcm/d)/lcm where lcm = lcm(b,d)
+        impl<T: Clone + Integer> $imp<Ratio<T>> for Ratio<T> {
+            type Output = Ratio<T>;
+            #[inline]
+            fn $method(self, rhs: Ratio<T>) -> Ratio<T> {
+                if self.denom == rhs.denom {
+                    return Ratio::new(self.numer.$method(rhs.numer), rhs.denom);
+                }
+                let lcm = self.denom.lcm(&rhs.denom);
+                let lhs_numer = self.numer * (lcm.clone() / self.denom);
+                let rhs_numer = rhs.numer * (lcm.clone() / rhs.denom);
+                Ratio::new(lhs_numer.$method(rhs_numer), lcm)
+            }
+        }
+        // Abstracts the a/b `op` c/1 = (a*1 `op` b*c) / (b*1) = (a `op` b*c) / b pattern
+        impl<T: Clone + Integer> $imp<T> for Ratio<T> {
+            type Output = Ratio<T>;
+            #[inline]
+            fn $method(self, rhs: T) -> Ratio<T> {
+                Ratio::new(self.numer.$method(self.denom.clone() * rhs), self.denom)
+            }
+        }
+    };
+}
+
+arith_impl!(impl Add, add);
+arith_impl!(impl Sub, sub);
+arith_impl!(impl Rem, rem);
+
+// a/b * c/d = (a*c)/(b*d)
+impl<T> CheckedMul for Ratio<T>
+where
+    T: Clone + Integer + CheckedMul,
+{
+    #[inline]
+    fn checked_mul(&self, rhs: &Ratio<T>) -> Option<Ratio<T>> {
+        let gcd_ad = self.numer.gcd(&rhs.denom);
+        let gcd_bc = self.denom.gcd(&rhs.numer);
+        Some(Ratio::new(
+            (self.numer.clone() / gcd_ad.clone())
+                .checked_mul(&(rhs.numer.clone() / gcd_bc.clone()))?,
+            (self.denom.clone() / gcd_bc).checked_mul(&(rhs.denom.clone() / gcd_ad))?,
+        ))
+    }
+}
+
+// (a/b) / (c/d) = (a*d)/(b*c)
+impl<T> CheckedDiv for Ratio<T>
+where
+    T: Clone + Integer + CheckedMul,
+{
+    #[inline]
+    fn checked_div(&self, rhs: &Ratio<T>) -> Option<Ratio<T>> {
+        if rhs.is_zero() {
+            return None;
+        }
+        let (numer, denom) = if self.denom == rhs.denom {
+            (self.numer.clone(), rhs.numer.clone())
+        } else if self.numer == rhs.numer {
+            (rhs.denom.clone(), self.denom.clone())
+        } else {
+            let gcd_ac = self.numer.gcd(&rhs.numer);
+            let gcd_bd = self.denom.gcd(&rhs.denom);
+            (
+                (self.numer.clone() / gcd_ac.clone())
+                    .checked_mul(&(rhs.denom.clone() / gcd_bd.clone()))?,
+                (self.denom.clone() / gcd_bd).checked_mul(&(rhs.numer.clone() / gcd_ac))?,
+            )
+        };
+        // Manual `reduce()`, avoiding sharp edges
+        if denom.is_zero() {
+            None
+        } else if numer.is_zero() {
+            Some(Self::zero())
+        } else if numer == denom {
+            Some(Self::one())
+        } else {
+            let g = numer.gcd(&denom);
+            let numer = numer / g.clone();
+            let denom = denom / g;
+            let raw = if denom < T::zero() {
+                // We need to keep denom positive, but 2's-complement MIN may
+                // overflow negation -- instead we can check multiplying -1.
+                let n1 = T::zero() - T::one();
+                Ratio::new_raw(numer.checked_mul(&n1)?, denom.checked_mul(&n1)?)
+            } else {
+                Ratio::new_raw(numer, denom)
+            };
+            Some(raw)
+        }
+    }
+}
+
+// As arith_impl! but for Checked{Add,Sub} traits
+macro_rules! checked_arith_impl {
+    (impl $imp:ident, $method:ident) => {
+        impl<T: Clone + Integer + CheckedMul + $imp> $imp for Ratio<T> {
+            #[inline]
+            fn $method(&self, rhs: &Ratio<T>) -> Option<Ratio<T>> {
+                let gcd = self.denom.clone().gcd(&rhs.denom);
+                let lcm = (self.denom.clone() / gcd.clone()).checked_mul(&rhs.denom)?;
+                let lhs_numer = (lcm.clone() / self.denom.clone()).checked_mul(&self.numer)?;
+                let rhs_numer = (lcm.clone() / rhs.denom.clone()).checked_mul(&rhs.numer)?;
+                Some(Ratio::new(lhs_numer.$method(&rhs_numer)?, lcm))
+            }
+        }
+    };
+}
+
+// a/b + c/d = (lcm/b*a + lcm/d*c)/lcm, where lcm = lcm(b,d)
+checked_arith_impl!(impl CheckedAdd, checked_add);
+
+// a/b - c/d = (lcm/b*a - lcm/d*c)/lcm, where lcm = lcm(b,d)
+checked_arith_impl!(impl CheckedSub, checked_sub);
+
+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()
+    }
+}
+
+impl<T> Inv for Ratio<T>
+where
+    T: Clone + Integer,
+{
+    type Output = Ratio<T>;
+
+    #[inline]
+    fn inv(self) -> Ratio<T> {
+        self.recip()
+    }
+}
+
+impl<'a, T> Inv for &'a Ratio<T>
+where
+    T: Clone + Integer,
+{
+    type Output = Ratio<T>;
+
+    #[inline]
+    fn inv(self) -> Ratio<T> {
+        self.recip()
+    }
+}
+
+// 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()
+    }
+
+    #[inline]
+    fn set_zero(&mut self) {
+        self.numer.set_zero();
+        self.denom.set_one();
+    }
+}
+
+impl<T: Clone + Integer> One for Ratio<T> {
+    #[inline]
+    fn one() -> Ratio<T> {
+        Ratio::new_raw(One::one(), One::one())
+    }
+
+    #[inline]
+    fn is_one(&self) -> bool {
+        self.numer == self.denom
+    }
+
+    #[inline]
+    fn set_one(&mut self) {
+        self.numer.set_one();
+        self.denom.set_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> {
+        if s.splitn(2, '/').count() == 2 {
+            let mut parts = s.splitn(2, '/').map(|ss| {
+                T::from_str_radix(ss, radix).map_err(|_| ParseRatioError {
+                    kind: RatioErrorKind::ParseError,
+                })
+            });
+            let numer: T = parts.next().unwrap()?;
+            let denom: T = parts.next().unwrap()?;
+            if denom.is_zero() {
+                Err(ParseRatioError {
+                    kind: RatioErrorKind::ZeroDenominator,
+                })
+            } else {
+                Ok(Ratio::new(numer, denom))
+            }
+        } else {
+            Err(ParseRatioError {
+                kind: RatioErrorKind::ParseError,
+            })
+        }
+    }
+}
+
+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
+macro_rules! impl_formatting {
+    ($fmt_trait:ident, $prefix:expr, $fmt_str:expr, $fmt_alt:expr) => {
+        impl<T: $fmt_trait + Clone + Integer> $fmt_trait for Ratio<T> {
+            #[cfg(feature = "std")]
+            fn fmt(&self, f: &mut Formatter<'_>) -> fmt::Result {
+                let pre_pad = if self.denom.is_one() {
+                    format!($fmt_str, self.numer)
+                } else {
+                    if f.alternate() {
+                        format!(concat!($fmt_str, "/", $fmt_alt), self.numer, self.denom)
+                    } else {
+                        format!(concat!($fmt_str, "/", $fmt_str), self.numer, self.denom)
+                    }
+                };
+                // TODO: replace with strip_prefix, when stabalized
+                let (pre_pad, non_negative) = {
+                    if pre_pad.starts_with("-") {
+                        (&pre_pad[1..], false)
+                    } else {
+                        (&pre_pad[..], true)
+                    }
+                };
+                f.pad_integral(non_negative, $prefix, pre_pad)
+            }
+            #[cfg(not(feature = "std"))]
+            fn fmt(&self, f: &mut Formatter<'_>) -> fmt::Result {
+                let plus = if f.sign_plus() && self.numer >= T::zero() {
+                    "+"
+                } else {
+                    ""
+                };
+                if self.denom.is_one() {
+                    if f.alternate() {
+                        write!(f, concat!("{}", $fmt_alt), plus, self.numer)
+                    } else {
+                        write!(f, concat!("{}", $fmt_str), plus, self.numer)
+                    }
+                } else {
+                    if f.alternate() {
+                        write!(
+                            f,
+                            concat!("{}", $fmt_alt, "/", $fmt_alt),
+                            plus, self.numer, self.denom
+                        )
+                    } else {
+                        write!(
+                            f,
+                            concat!("{}", $fmt_str, "/", $fmt_str),
+                            plus, self.numer, self.denom
+                        )
+                    }
+                }
+            }
+        }
+    };
+}
+
+impl_formatting!(Display, "", "{}", "{:#}");
+impl_formatting!(Octal, "0o", "{:o}", "{:#o}");
+impl_formatting!(Binary, "0b", "{:b}", "{:#b}");
+impl_formatting!(LowerHex, "0x", "{:x}", "{:#x}");
+impl_formatting!(UpperHex, "0x", "{:X}", "{:#X}");
+impl_formatting!(LowerExp, "", "{:e}", "{:#e}");
+impl_formatting!(UpperExp, "", "{:E}", "{:#E}");
+
+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 = split.next().ok_or(ParseRatioError {
+            kind: RatioErrorKind::ParseError,
+        })?;
+        let num = FromStr::from_str(n).map_err(|_| ParseRatioError {
+            kind: RatioErrorKind::ParseError,
+        })?;
+
+        let d = split.next().unwrap_or("1");
+        let den = 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: S) -> Result<S::Ok, S::Error>
+    where
+        S: serde::Serializer,
+    {
+        (self.numer(), self.denom()).serialize(serializer)
+    }
+}
+
+#[cfg(feature = "serde")]
+impl<'de, T> serde::Deserialize<'de> for Ratio<T>
+where
+    T: serde::Deserialize<'de> + Clone + Integer + PartialOrd,
+{
+    fn deserialize<D>(deserializer: D) -> Result<Self, D::Error>
+    where
+        D: serde::Deserializer<'de>,
+    {
+        use serde::de::Error;
+        use serde::de::Unexpected;
+        let (numer, denom): (T, T) = serde::Deserialize::deserialize(deserializer)?;
+        if denom.is_zero() {
+            Err(Error::invalid_value(
+                Unexpected::Signed(0),
+                &"a ratio with non-zero denominator",
+            ))
+        } 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.kind.description().fmt(f)
+    }
+}
+
+#[cfg(feature = "std")]
+impl Error for ParseRatioError {
+    #[allow(deprecated)]
+    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",
+        }
+    }
+}
+
+#[cfg(feature = "num-bigint")]
+impl FromPrimitive for Ratio<BigInt> {
+    fn from_i64(n: i64) -> Option<Self> {
+        Some(Ratio::from_integer(n.into()))
+    }
+
+    fn from_i128(n: i128) -> Option<Self> {
+        Some(Ratio::from_integer(n.into()))
+    }
+
+    fn from_u64(n: u64) -> Option<Self> {
+        Some(Ratio::from_integer(n.into()))
+    }
+
+    fn from_u128(n: u128) -> 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> {
+                <$typ as FromPrimitive>::from_i64(n).map(Ratio::from_integer)
+            }
+
+            fn from_i128(n: i128) -> Option<Self> {
+                <$typ as FromPrimitive>::from_i128(n).map(Ratio::from_integer)
+            }
+
+            fn from_u64(n: u64) -> Option<Self> {
+                <$typ as FromPrimitive>::from_u64(n).map(Ratio::from_integer)
+            }
+
+            fn from_u128(n: u128) -> Option<Self> {
+                <$typ as FromPrimitive>::from_u128(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!(i128, 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!(u128, approximate_float_unsigned);
+from_primitive_integer!(usize, approximate_float_unsigned);
+
+impl<T: Integer + Signed + Bounded + NumCast + Clone> Ratio<T> {
+    pub fn approximate_float<F: FloatCore + 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: FloatCore + 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
+    Some(if negative { 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: FloatCore + NumCast,
+{
+    // Continued fractions algorithm
+    // http://mathforum.org/dr.math/faq/faq.fractions.html#decfrac
+
+    if val < F::zero() || val.is_nan() {
+        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 = <F as NumCast>::from(t_max.clone())?;
+
+    // 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(not(feature = "num-bigint"))]
+macro_rules! to_primitive_small {
+    ($($type_name:ty)*) => ($(
+        impl ToPrimitive for Ratio<$type_name> {
+            fn to_i64(&self) -> Option<i64> {
+                self.to_integer().to_i64()
+            }
+
+            fn to_i128(&self) -> Option<i128> {
+                self.to_integer().to_i128()
+            }
+
+            fn to_u64(&self) -> Option<u64> {
+                self.to_integer().to_u64()
+            }
+
+            fn to_u128(&self) -> Option<u128> {
+                self.to_integer().to_u128()
+            }
+
+            fn to_f64(&self) -> Option<f64> {
+                let float = self.numer.to_f64().unwrap() / self.denom.to_f64().unwrap();
+                if float.is_nan() {
+                    None
+                } else {
+                    Some(float)
+                }
+            }
+        }
+    )*)
+}
+
+#[cfg(not(feature = "num-bigint"))]
+to_primitive_small!(u8 i8 u16 i16 u32 i32);
+
+#[cfg(all(target_pointer_width = "32", not(feature = "num-bigint")))]
+to_primitive_small!(usize isize);
+
+#[cfg(not(feature = "num-bigint"))]
+macro_rules! to_primitive_64 {
+    ($($type_name:ty)*) => ($(
+        impl ToPrimitive for Ratio<$type_name> {
+            fn to_i64(&self) -> Option<i64> {
+                self.to_integer().to_i64()
+            }
+
+            fn to_i128(&self) -> Option<i128> {
+                self.to_integer().to_i128()
+            }
+
+            fn to_u64(&self) -> Option<u64> {
+                self.to_integer().to_u64()
+            }
+
+            fn to_u128(&self) -> Option<u128> {
+                self.to_integer().to_u128()
+            }
+
+            fn to_f64(&self) -> Option<f64> {
+                let float = ratio_to_f64(
+                    self.numer as i128,
+                    self.denom as i128
+                );
+                if float.is_nan() {
+                    None
+                } else {
+                    Some(float)
+                }
+            }
+        }
+    )*)
+}
+
+#[cfg(not(feature = "num-bigint"))]
+to_primitive_64!(u64 i64);
+
+#[cfg(all(target_pointer_width = "64", not(feature = "num-bigint")))]
+to_primitive_64!(usize isize);
+
+#[cfg(feature = "num-bigint")]
+impl<T: Clone + Integer + ToPrimitive + ToBigInt> ToPrimitive for Ratio<T> {
+    fn to_i64(&self) -> Option<i64> {
+        self.to_integer().to_i64()
+    }
+
+    fn to_i128(&self) -> Option<i128> {
+        self.to_integer().to_i128()
+    }
+
+    fn to_u64(&self) -> Option<u64> {
+        self.to_integer().to_u64()
+    }
+
+    fn to_u128(&self) -> Option<u128> {
+        self.to_integer().to_u128()
+    }
+
+    fn to_f64(&self) -> Option<f64> {
+        let float = match (self.numer.to_i64(), self.denom.to_i64()) {
+            (Some(numer), Some(denom)) => ratio_to_f64(
+                <i128 as From<_>>::from(numer),
+                <i128 as From<_>>::from(denom),
+            ),
+            _ => {
+                let numer: BigInt = self.numer.to_bigint()?;
+                let denom: BigInt = self.denom.to_bigint()?;
+                ratio_to_f64(numer, denom)
+            }
+        };
+        if float.is_nan() {
+            None
+        } else {
+            Some(float)
+        }
+    }
+}
+
+trait Bits {
+    fn bits(&self) -> u64;
+}
+
+#[cfg(feature = "num-bigint")]
+impl Bits for BigInt {
+    fn bits(&self) -> u64 {
+        self.bits()
+    }
+}
+
+impl Bits for i128 {
+    fn bits(&self) -> u64 {
+        (128 - self.wrapping_abs().leading_zeros()).into()
+    }
+}
+
+/// Converts a ratio of `T` to an f64.
+///
+/// In addition to stated trait bounds, `T` must be able to hold numbers 56 bits larger than
+/// the largest of `numer` and `denom`. This is automatically true if `T` is `BigInt`.
+fn ratio_to_f64<T: Bits + Clone + Integer + Signed + ShlAssign<usize> + ToPrimitive>(
+    numer: T,
+    denom: T,
+) -> f64 {
+    use core::f64::{INFINITY, MANTISSA_DIGITS, MAX_EXP, MIN_EXP, RADIX};
+
+    assert_eq!(
+        RADIX, 2,
+        "only floating point implementations with radix 2 are supported"
+    );
+
+    // Inclusive upper and lower bounds to the range of exactly-representable ints in an f64.
+    const MAX_EXACT_INT: i64 = 1i64 << MANTISSA_DIGITS;
+    const MIN_EXACT_INT: i64 = -MAX_EXACT_INT;
+
+    let flo_sign = numer.signum().to_f64().unwrap() / denom.signum().to_f64().unwrap();
+    if !flo_sign.is_normal() {
+        return flo_sign;
+    }
+
+    // Fast track: both sides can losslessly be converted to f64s. In this case, letting the
+    // FPU do the job is faster and easier. In any other case, converting to f64s may lead
+    // to an inexact result: https://stackoverflow.com/questions/56641441/.
+    if let (Some(n), Some(d)) = (numer.to_i64(), denom.to_i64()) {
+        if MIN_EXACT_INT <= n && n <= MAX_EXACT_INT && MIN_EXACT_INT <= d && d <= MAX_EXACT_INT {
+            return n.to_f64().unwrap() / d.to_f64().unwrap();
+        }
+    }
+
+    // Otherwise, the goal is to obtain a quotient with at least 55 bits. 53 of these bits will
+    // be used as the mantissa of the resulting float, and the remaining two are for rounding.
+    // There's an error of up to 1 on the number of resulting bits, so we may get either 55 or
+    // 56 bits.
+    let mut numer = numer.abs();
+    let mut denom = denom.abs();
+    let (is_diff_positive, absolute_diff) = match numer.bits().checked_sub(denom.bits()) {
+        Some(diff) => (true, diff),
+        None => (false, denom.bits() - numer.bits()),
+    };
+
+    // Filter out overflows and underflows. After this step, the signed difference fits in an
+    // isize.
+    if is_diff_positive && absolute_diff > MAX_EXP as u64 {
+        return INFINITY * flo_sign;
+    }
+    if !is_diff_positive && absolute_diff > -MIN_EXP as u64 + MANTISSA_DIGITS as u64 + 1 {
+        return 0.0 * flo_sign;
+    }
+    let diff = if is_diff_positive {
+        absolute_diff.to_isize().unwrap()
+    } else {
+        -absolute_diff.to_isize().unwrap()
+    };
+
+    // Shift is chosen so that the quotient will have 55 or 56 bits. The exception is if the
+    // quotient is going to be subnormal, in which case it may have fewer bits.
+    let shift: isize = diff.max(MIN_EXP as isize) - MANTISSA_DIGITS as isize - 2;
+    if shift >= 0 {
+        denom <<= shift as usize
+    } else {
+        numer <<= -shift as usize
+    };
+
+    let (quotient, remainder) = numer.div_rem(&denom);
+
+    // This is guaranteed to fit since we've set up quotient to be at most 56 bits.
+    let mut quotient = quotient.to_u64().unwrap();
+    let n_rounding_bits = {
+        let quotient_bits = 64 - quotient.leading_zeros() as isize;
+        let subnormal_bits = MIN_EXP as isize - shift;
+        quotient_bits.max(subnormal_bits) - MANTISSA_DIGITS as isize
+    } as usize;
+    debug_assert!(n_rounding_bits == 2 || n_rounding_bits == 3);
+    let rounding_bit_mask = (1u64 << n_rounding_bits) - 1;
+
+    // Round to 53 bits with round-to-even. For rounding, we need to take into account both
+    // our rounding bits and the division's remainder.
+    let ls_bit = quotient & (1u64 << n_rounding_bits) != 0;
+    let ms_rounding_bit = quotient & (1u64 << (n_rounding_bits - 1)) != 0;
+    let ls_rounding_bits = quotient & (rounding_bit_mask >> 1) != 0;
+    if ms_rounding_bit && (ls_bit || ls_rounding_bits || !remainder.is_zero()) {
+        quotient += 1u64 << n_rounding_bits;
+    }
+    quotient &= !rounding_bit_mask;
+
+    // The quotient is guaranteed to be exactly representable as it's now 53 bits + 2 or 3
+    // trailing zeros, so there is no risk of a rounding error here.
+    let q_float = quotient as f64 * flo_sign;
+    ldexp(q_float, shift as i32)
+}
+
+/// Multiply `x` by 2 to the power of `exp`. Returns an accurate result even if `2^exp` is not
+/// representable.
+fn ldexp(x: f64, exp: i32) -> f64 {
+    use core::f64::{INFINITY, MANTISSA_DIGITS, MAX_EXP, RADIX};
+
+    assert_eq!(
+        RADIX, 2,
+        "only floating point implementations with radix 2 are supported"
+    );
+
+    const EXPONENT_MASK: u64 = 0x7ff << 52;
+    const MAX_UNSIGNED_EXPONENT: i32 = 0x7fe;
+    const MIN_SUBNORMAL_POWER: i32 = MANTISSA_DIGITS as i32;
+
+    if x.is_zero() || x.is_infinite() || x.is_nan() {
+        return x;
+    }
+
+    // Filter out obvious over / underflows to make sure the resulting exponent fits in an isize.
+    if exp > 3 * MAX_EXP {
+        return INFINITY * x.signum();
+    } else if exp < -3 * MAX_EXP {
+        return 0.0 * x.signum();
+    }
+
+    // curr_exp is the x's *biased* exponent, and is in the [-54, MAX_UNSIGNED_EXPONENT] range.
+    let (bits, curr_exp) = if !x.is_normal() {
+        // If x is subnormal, we make it normal by multiplying by 2^53. This causes no loss of
+        // precision or rounding.
+        let normal_x = x * 2f64.powi(MIN_SUBNORMAL_POWER);
+        let bits = normal_x.to_bits();
+        // This cast is safe because the exponent is at most 0x7fe, which fits in an i32.
+        (
+            bits,
+            ((bits & EXPONENT_MASK) >> 52) as i32 - MIN_SUBNORMAL_POWER,
+        )
+    } else {
+        let bits = x.to_bits();
+        let curr_exp = (bits & EXPONENT_MASK) >> 52;
+        // This cast is safe because the exponent is at most 0x7fe, which fits in an i32.
+        (bits, curr_exp as i32)
+    };
+
+    // The addition can't overflow because exponent is between 0 and 0x7fe, and exp is between
+    // -2*MAX_EXP and 2*MAX_EXP.
+    let new_exp = curr_exp + exp;
+
+    if new_exp > MAX_UNSIGNED_EXPONENT {
+        INFINITY * x.signum()
+    } else if new_exp > 0 {
+        // Normal case: exponent is not too large nor subnormal.
+        let new_bits = (bits & !EXPONENT_MASK) | ((new_exp as u64) << 52);
+        f64::from_bits(new_bits)
+    } else if new_exp >= -(MANTISSA_DIGITS as i32) {
+        // Result is subnormal but may not be zero.
+        // In this case, we increase the exponent by 54 to make it normal, then multiply the end
+        // result by 2^-53. This results in a single multiplication with no prior rounding error,
+        // so there is no risk of double rounding.
+        let new_exp = new_exp + MIN_SUBNORMAL_POWER;
+        debug_assert!(new_exp >= 0);
+        let new_bits = (bits & !EXPONENT_MASK) | ((new_exp as u64) << 52);
+        f64::from_bits(new_bits) * 2f64.powi(-MIN_SUBNORMAL_POWER)
+    } else {
+        // Result is zero.
+        return 0.0 * x.signum();
+    }
+}
+
+#[cfg(test)]
+#[cfg(feature = "std")]
+fn hash<T: Hash>(x: &T) -> u64 {
+    use std::collections::hash_map::RandomState;
+    use std::hash::BuildHasher;
+    let mut hasher = <RandomState as BuildHasher>::Hasher::new();
+    x.hash(&mut hasher);
+    hasher.finish()
+}
+
+#[cfg(test)]
+mod test {
+    use super::ldexp;
+    #[cfg(all(feature = "num-bigint"))]
+    use super::BigInt;
+    #[cfg(feature = "num-bigint")]
+    use super::BigRational;
+    use super::{Ratio, Rational64};
+
+    use core::f64;
+    use core::i32;
+    use core::i64;
+    use core::str::FromStr;
+    use num_integer::Integer;
+    use num_traits::ToPrimitive;
+    use num_traits::{FromPrimitive, One, Pow, Signed, Zero};
+
+    pub const _0: Rational64 = Ratio { numer: 0, denom: 1 };
+    pub const _1: Rational64 = Ratio { numer: 1, denom: 1 };
+    pub const _2: Rational64 = Ratio { numer: 2, denom: 1 };
+    pub const _NEG2: Rational64 = Ratio {
+        numer: -2,
+        denom: 1,
+    };
+    pub const _8: Rational64 = Ratio { numer: 8, denom: 1 };
+    pub const _15: Rational64 = Ratio {
+        numer: 15,
+        denom: 1,
+    };
+    pub const _16: Rational64 = Ratio {
+        numer: 16,
+        denom: 1,
+    };
+
+    pub const _1_2: Rational64 = Ratio { numer: 1, denom: 2 };
+    pub const _1_8: Rational64 = Ratio { numer: 1, denom: 8 };
+    pub const _1_15: Rational64 = Ratio {
+        numer: 1,
+        denom: 15,
+    };
+    pub const _1_16: Rational64 = Ratio {
+        numer: 1,
+        denom: 16,
+    };
+    pub const _3_2: Rational64 = Ratio { numer: 3, denom: 2 };
+    pub const _5_2: Rational64 = Ratio { numer: 5, denom: 2 };
+    pub const _NEG1_2: Rational64 = Ratio {
+        numer: -1,
+        denom: 2,
+    };
+    pub const _1_NEG2: Rational64 = Ratio {
+        numer: 1,
+        denom: -2,
+    };
+    pub const _NEG1_NEG2: Rational64 = Ratio {
+        numer: -1,
+        denom: -2,
+    };
+    pub const _1_3: Rational64 = Ratio { numer: 1, denom: 3 };
+    pub const _NEG1_3: Rational64 = Ratio {
+        numer: -1,
+        denom: 3,
+    };
+    pub const _2_3: Rational64 = Ratio { numer: 2, denom: 3 };
+    pub const _NEG2_3: Rational64 = Ratio {
+        numer: -2,
+        denom: 3,
+    };
+    pub const _MIN: Rational64 = Ratio {
+        numer: i64::MIN,
+        denom: 1,
+    };
+    pub const _MIN_P1: Rational64 = Ratio {
+        numer: i64::MIN + 1,
+        denom: 1,
+    };
+    pub const _MAX: Rational64 = Ratio {
+        numer: i64::MAX,
+        denom: 1,
+    };
+    pub const _MAX_M1: Rational64 = Ratio {
+        numer: i64::MAX - 1,
+        denom: 1,
+    };
+    pub const _BILLION: Rational64 = Ratio {
+        numer: 1_000_000_000,
+        denom: 1,
+    };
+
+    #[cfg(feature = "num-bigint")]
+    pub fn to_big(n: Rational64) -> BigRational {
+        Ratio::new(
+            FromPrimitive::from_i64(n.numer).unwrap(),
+            FromPrimitive::from_i64(n.denom).unwrap(),
+        )
+    }
+    #[cfg(not(feature = "num-bigint"))]
+    pub fn to_big(n: Rational64) -> Rational64 {
+        Ratio::new(
+            FromPrimitive::from_i64(n.numer).unwrap(),
+            FromPrimitive::from_i64(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() {
+        assert_eq!(Ratio::new(2, 2), One::one());
+        assert_eq!(Ratio::new(0, i32::MIN), Zero::zero());
+        assert_eq!(Ratio::new(i32::MIN, i32::MIN), 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]
+    #[allow(clippy::eq_op)]
+    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));
+
+        let _0_2: Rational64 = Ratio::new_raw(0, 2);
+        assert_eq!(_0, _0_2);
+    }
+
+    #[test]
+    fn test_cmp_overflow() {
+        use core::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 = [
+            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) {
+            #[cfg(feature = "std")]
+            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());
+    }
+
+    #[cfg(not(feature = "std"))]
+    use core::fmt::{self, Write};
+    #[cfg(not(feature = "std"))]
+    #[derive(Debug)]
+    struct NoStdTester {
+        cursor: usize,
+        buf: [u8; NoStdTester::BUF_SIZE],
+    }
+
+    #[cfg(not(feature = "std"))]
+    impl NoStdTester {
+        fn new() -> NoStdTester {
+            NoStdTester {
+                buf: [0; Self::BUF_SIZE],
+                cursor: 0,
+            }
+        }
+
+        fn clear(&mut self) {
+            self.buf = [0; Self::BUF_SIZE];
+            self.cursor = 0;
+        }
+
+        const WRITE_ERR: &'static str = "Formatted output too long";
+        const BUF_SIZE: usize = 32;
+    }
+
+    #[cfg(not(feature = "std"))]
+    impl Write for NoStdTester {
+        fn write_str(&mut self, s: &str) -> fmt::Result {
+            for byte in s.bytes() {
+                self.buf[self.cursor] = byte;
+                self.cursor += 1;
+                if self.cursor >= self.buf.len() {
+                    return Err(fmt::Error {});
+                }
+            }
+            Ok(())
+        }
+    }
+
+    #[cfg(not(feature = "std"))]
+    impl PartialEq<str> for NoStdTester {
+        fn eq(&self, other: &str) -> bool {
+            let other = other.as_bytes();
+            for index in 0..self.cursor {
+                if self.buf.get(index) != other.get(index) {
+                    return false;
+                }
+            }
+            true
+        }
+    }
+
+    macro_rules! assert_fmt_eq {
+        ($fmt_args:expr, $string:expr) => {
+            #[cfg(not(feature = "std"))]
+            {
+                let mut tester = NoStdTester::new();
+                write!(tester, "{}", $fmt_args).expect(NoStdTester::WRITE_ERR);
+                assert_eq!(tester, *$string);
+                tester.clear();
+            }
+            #[cfg(feature = "std")]
+            {
+                assert_eq!(std::fmt::format($fmt_args), $string);
+            }
+        };
+    }
+
+    #[test]
+    fn test_show() {
+        // Test:
+        // :b :o :x, :X, :?
+        // alternate or not (#)
+        // positive and negative
+        // padding
+        // does not test precision (i.e. truncation)
+        assert_fmt_eq!(format_args!("{}", _2), "2");
+        assert_fmt_eq!(format_args!("{:+}", _2), "+2");
+        assert_fmt_eq!(format_args!("{:-}", _2), "2");
+        assert_fmt_eq!(format_args!("{}", _1_2), "1/2");
+        assert_fmt_eq!(format_args!("{}", -_1_2), "-1/2"); // test negatives
+        assert_fmt_eq!(format_args!("{}", _0), "0");
+        assert_fmt_eq!(format_args!("{}", -_2), "-2");
+        assert_fmt_eq!(format_args!("{:+}", -_2), "-2");
+        assert_fmt_eq!(format_args!("{:b}", _2), "10");
+        assert_fmt_eq!(format_args!("{:#b}", _2), "0b10");
+        assert_fmt_eq!(format_args!("{:b}", _1_2), "1/10");
+        assert_fmt_eq!(format_args!("{:+b}", _1_2), "+1/10");
+        assert_fmt_eq!(format_args!("{:-b}", _1_2), "1/10");
+        assert_fmt_eq!(format_args!("{:b}", _0), "0");
+        assert_fmt_eq!(format_args!("{:#b}", _1_2), "0b1/0b10");
+        // no std does not support padding
+        #[cfg(feature = "std")]
+        assert_eq!(&format!("{:010b}", _1_2), "0000001/10");
+        #[cfg(feature = "std")]
+        assert_eq!(&format!("{:#010b}", _1_2), "0b001/0b10");
+        let half_i8: Ratio<i8> = Ratio::new(1_i8, 2_i8);
+        assert_fmt_eq!(format_args!("{:b}", -half_i8), "11111111/10");
+        assert_fmt_eq!(format_args!("{:#b}", -half_i8), "0b11111111/0b10");
+        #[cfg(feature = "std")]
+        assert_eq!(&format!("{:05}", Ratio::new(-1_i8, 1_i8)), "-0001");
+
+        assert_fmt_eq!(format_args!("{:o}", _8), "10");
+        assert_fmt_eq!(format_args!("{:o}", _1_8), "1/10");
+        assert_fmt_eq!(format_args!("{:o}", _0), "0");
+        assert_fmt_eq!(format_args!("{:#o}", _1_8), "0o1/0o10");
+        #[cfg(feature = "std")]
+        assert_eq!(&format!("{:010o}", _1_8), "0000001/10");
+        #[cfg(feature = "std")]
+        assert_eq!(&format!("{:#010o}", _1_8), "0o001/0o10");
+        assert_fmt_eq!(format_args!("{:o}", -half_i8), "377/2");
+        assert_fmt_eq!(format_args!("{:#o}", -half_i8), "0o377/0o2");
+
+        assert_fmt_eq!(format_args!("{:x}", _16), "10");
+        assert_fmt_eq!(format_args!("{:x}", _15), "f");
+        assert_fmt_eq!(format_args!("{:x}", _1_16), "1/10");
+        assert_fmt_eq!(format_args!("{:x}", _1_15), "1/f");
+        assert_fmt_eq!(format_args!("{:x}", _0), "0");
+        assert_fmt_eq!(format_args!("{:#x}", _1_16), "0x1/0x10");
+        #[cfg(feature = "std")]
+        assert_eq!(&format!("{:010x}", _1_16), "0000001/10");
+        #[cfg(feature = "std")]
+        assert_eq!(&format!("{:#010x}", _1_16), "0x001/0x10");
+        assert_fmt_eq!(format_args!("{:x}", -half_i8), "ff/2");
+        assert_fmt_eq!(format_args!("{:#x}", -half_i8), "0xff/0x2");
+
+        assert_fmt_eq!(format_args!("{:X}", _16), "10");
+        assert_fmt_eq!(format_args!("{:X}", _15), "F");
+        assert_fmt_eq!(format_args!("{:X}", _1_16), "1/10");
+        assert_fmt_eq!(format_args!("{:X}", _1_15), "1/F");
+        assert_fmt_eq!(format_args!("{:X}", _0), "0");
+        assert_fmt_eq!(format_args!("{:#X}", _1_16), "0x1/0x10");
+        #[cfg(feature = "std")]
+        assert_eq!(format!("{:010X}", _1_16), "0000001/10");
+        #[cfg(feature = "std")]
+        assert_eq!(format!("{:#010X}", _1_16), "0x001/0x10");
+        assert_fmt_eq!(format_args!("{:X}", -half_i8), "FF/2");
+        assert_fmt_eq!(format_args!("{:#X}", -half_i8), "0xFF/0x2");
+
+        #[cfg(has_int_exp_fmt)]
+        {
+            assert_fmt_eq!(format_args!("{:e}", -_2), "-2e0");
+            assert_fmt_eq!(format_args!("{:#e}", -_2), "-2e0");
+            assert_fmt_eq!(format_args!("{:+e}", -_2), "-2e0");
+            assert_fmt_eq!(format_args!("{:e}", _BILLION), "1e9");
+            assert_fmt_eq!(format_args!("{:+e}", _BILLION), "+1e9");
+            assert_fmt_eq!(format_args!("{:e}", _BILLION.recip()), "1e0/1e9");
+            assert_fmt_eq!(format_args!("{:+e}", _BILLION.recip()), "+1e0/1e9");
+
+            assert_fmt_eq!(format_args!("{:E}", -_2), "-2E0");
+            assert_fmt_eq!(format_args!("{:#E}", -_2), "-2E0");
+            assert_fmt_eq!(format_args!("{:+E}", -_2), "-2E0");
+            assert_fmt_eq!(format_args!("{:E}", _BILLION), "1E9");
+            assert_fmt_eq!(format_args!("{:+E}", _BILLION), "+1E9");
+            assert_fmt_eq!(format_args!("{:E}", _BILLION.recip()), "1E0/1E9");
+            assert_fmt_eq!(format_args!("{:+E}", _BILLION.recip()), "+1E0/1E9");
+        }
+    }
+
+    mod arith {
+        use super::super::{Ratio, Rational64};
+        use super::{to_big, _0, _1, _1_2, _2, _3_2, _5_2, _MAX, _MAX_M1, _MIN, _MIN_P1, _NEG1_2};
+        use core::fmt::Debug;
+        use num_integer::Integer;
+        use num_traits::{Bounded, CheckedAdd, CheckedDiv, CheckedMul, CheckedSub, NumAssign};
+
+        #[test]
+        fn test_add() {
+            fn test(a: Rational64, b: Rational64, c: Rational64) {
+                assert_eq!(a + b, c);
+                assert_eq!(
+                    {
+                        let mut x = a;
+                        x += b;
+                        x
+                    },
+                    c
+                );
+                assert_eq!(to_big(a) + to_big(b), to_big(c));
+                assert_eq!(a.checked_add(&b), Some(c));
+                assert_eq!(to_big(a).checked_add(&to_big(b)), Some(to_big(c)));
+            }
+            fn test_assign(a: Rational64, b: i64, c: Rational64) {
+                assert_eq!(a + b, c);
+                assert_eq!(
+                    {
+                        let mut x = a;
+                        x += b;
+                        x
+                    },
+                    c
+                );
+            }
+
+            test(_1, _1_2, _3_2);
+            test(_1, _1, _2);
+            test(_1_2, _3_2, _2);
+            test(_1_2, _NEG1_2, _0);
+            test_assign(_1_2, 1, _3_2);
+        }
+
+        #[test]
+        fn test_add_overflow() {
+            // compares Ratio(1, T::max_value()) + Ratio(1, T::max_value())
+            // to Ratio(1+1, T::max_value()) for each integer type.
+            // Previously, this calculation would overflow.
+            fn test_add_typed_overflow<T>()
+            where
+                T: Integer + Bounded + Clone + Debug + NumAssign,
+            {
+                let _1_max = Ratio::new(T::one(), T::max_value());
+                let _2_max = Ratio::new(T::one() + T::one(), T::max_value());
+                assert_eq!(_1_max.clone() + _1_max.clone(), _2_max);
+                assert_eq!(
+                    {
+                        let mut tmp = _1_max.clone();
+                        tmp += _1_max;
+                        tmp
+                    },
+                    _2_max
+                );
+            }
+            test_add_typed_overflow::<u8>();
+            test_add_typed_overflow::<u16>();
+            test_add_typed_overflow::<u32>();
+            test_add_typed_overflow::<u64>();
+            test_add_typed_overflow::<usize>();
+            test_add_typed_overflow::<u128>();
+
+            test_add_typed_overflow::<i8>();
+            test_add_typed_overflow::<i16>();
+            test_add_typed_overflow::<i32>();
+            test_add_typed_overflow::<i64>();
+            test_add_typed_overflow::<isize>();
+            test_add_typed_overflow::<i128>();
+        }
+
+        #[test]
+        fn test_sub() {
+            fn test(a: Rational64, b: Rational64, c: Rational64) {
+                assert_eq!(a - b, c);
+                assert_eq!(
+                    {
+                        let mut x = a;
+                        x -= b;
+                        x
+                    },
+                    c
+                );
+                assert_eq!(to_big(a) - to_big(b), to_big(c));
+                assert_eq!(a.checked_sub(&b), Some(c));
+                assert_eq!(to_big(a).checked_sub(&to_big(b)), Some(to_big(c)));
+            }
+            fn test_assign(a: Rational64, b: i64, c: Rational64) {
+                assert_eq!(a - b, c);
+                assert_eq!(
+                    {
+                        let mut x = a;
+                        x -= b;
+                        x
+                    },
+                    c
+                );
+            }
+
+            test(_1, _1_2, _1_2);
+            test(_3_2, _1_2, _1);
+            test(_1, _NEG1_2, _3_2);
+            test_assign(_1_2, 1, _NEG1_2);
+        }
+
+        #[test]
+        fn test_sub_overflow() {
+            // compares Ratio(1, T::max_value()) - Ratio(1, T::max_value()) to T::zero()
+            // for each integer type. Previously, this calculation would overflow.
+            fn test_sub_typed_overflow<T>()
+            where
+                T: Integer + Bounded + Clone + Debug + NumAssign,
+            {
+                let _1_max: Ratio<T> = Ratio::new(T::one(), T::max_value());
+                assert!(T::is_zero(&(_1_max.clone() - _1_max.clone()).numer));
+                {
+                    let mut tmp: Ratio<T> = _1_max.clone();
+                    tmp -= _1_max;
+                    assert!(T::is_zero(&tmp.numer));
+                }
+            }
+            test_sub_typed_overflow::<u8>();
+            test_sub_typed_overflow::<u16>();
+            test_sub_typed_overflow::<u32>();
+            test_sub_typed_overflow::<u64>();
+            test_sub_typed_overflow::<usize>();
+            test_sub_typed_overflow::<u128>();
+
+            test_sub_typed_overflow::<i8>();
+            test_sub_typed_overflow::<i16>();
+            test_sub_typed_overflow::<i32>();
+            test_sub_typed_overflow::<i64>();
+            test_sub_typed_overflow::<isize>();
+            test_sub_typed_overflow::<i128>();
+        }
+
+        #[test]
+        fn test_mul() {
+            fn test(a: Rational64, b: Rational64, c: Rational64) {
+                assert_eq!(a * b, c);
+                assert_eq!(
+                    {
+                        let mut x = a;
+                        x *= b;
+                        x
+                    },
+                    c
+                );
+                assert_eq!(to_big(a) * to_big(b), to_big(c));
+                assert_eq!(a.checked_mul(&b), Some(c));
+                assert_eq!(to_big(a).checked_mul(&to_big(b)), Some(to_big(c)));
+            }
+            fn test_assign(a: Rational64, b: i64, c: Rational64) {
+                assert_eq!(a * b, c);
+                assert_eq!(
+                    {
+                        let mut x = a;
+                        x *= b;
+                        x
+                    },
+                    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_assign(_1_2, 2, _1);
+        }
+
+        #[test]
+        fn test_mul_overflow() {
+            fn test_mul_typed_overflow<T>()
+            where
+                T: Integer + Bounded + Clone + Debug + NumAssign + CheckedMul,
+            {
+                let two = T::one() + T::one();
+                let _3 = T::one() + T::one() + T::one();
+
+                // 1/big * 2/3 = 1/(max/4*3), where big is max/2
+                // make big = max/2, but also divisible by 2
+                let big = T::max_value() / two.clone() / two.clone() * two.clone();
+                let _1_big: Ratio<T> = Ratio::new(T::one(), big.clone());
+                let _2_3: Ratio<T> = Ratio::new(two.clone(), _3.clone());
+                assert_eq!(None, big.clone().checked_mul(&_3.clone()));
+                let expected = Ratio::new(T::one(), big / two.clone() * _3.clone());
+                assert_eq!(expected.clone(), _1_big.clone() * _2_3.clone());
+                assert_eq!(
+                    Some(expected.clone()),
+                    _1_big.clone().checked_mul(&_2_3.clone())
+                );
+                assert_eq!(expected, {
+                    let mut tmp = _1_big;
+                    tmp *= _2_3;
+                    tmp
+                });
+
+                // big/3 * 3 = big/1
+                // make big = max/2, but make it indivisible by 3
+                let big = T::max_value() / two / _3.clone() * _3.clone() + T::one();
+                assert_eq!(None, big.clone().checked_mul(&_3.clone()));
+                let big_3 = Ratio::new(big.clone(), _3.clone());
+                let expected = Ratio::new(big, T::one());
+                assert_eq!(expected, big_3.clone() * _3.clone());
+                assert_eq!(expected, {
+                    let mut tmp = big_3;
+                    tmp *= _3;
+                    tmp
+                });
+            }
+            test_mul_typed_overflow::<u16>();
+            test_mul_typed_overflow::<u8>();
+            test_mul_typed_overflow::<u32>();
+            test_mul_typed_overflow::<u64>();
+            test_mul_typed_overflow::<usize>();
+            test_mul_typed_overflow::<u128>();
+
+            test_mul_typed_overflow::<i8>();
+            test_mul_typed_overflow::<i16>();
+            test_mul_typed_overflow::<i32>();
+            test_mul_typed_overflow::<i64>();
+            test_mul_typed_overflow::<isize>();
+            test_mul_typed_overflow::<i128>();
+        }
+
+        #[test]
+        fn test_div() {
+            fn test(a: Rational64, b: Rational64, c: Rational64) {
+                assert_eq!(a / b, c);
+                assert_eq!(
+                    {
+                        let mut x = a;
+                        x /= b;
+                        x
+                    },
+                    c
+                );
+                assert_eq!(to_big(a) / to_big(b), to_big(c));
+                assert_eq!(a.checked_div(&b), Some(c));
+                assert_eq!(to_big(a).checked_div(&to_big(b)), Some(to_big(c)));
+            }
+            fn test_assign(a: Rational64, b: i64, c: Rational64) {
+                assert_eq!(a / b, c);
+                assert_eq!(
+                    {
+                        let mut x = a;
+                        x /= b;
+                        x
+                    },
+                    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_assign(_1, 2, _1_2);
+        }
+
+        #[test]
+        fn test_div_overflow() {
+            fn test_div_typed_overflow<T>()
+            where
+                T: Integer + Bounded + Clone + Debug + NumAssign + CheckedMul,
+            {
+                let two = T::one() + T::one();
+                let _3 = T::one() + T::one() + T::one();
+
+                // 1/big / 3/2 = 1/(max/4*3), where big is max/2
+                // big ~ max/2, and big is divisible by 2
+                let big = T::max_value() / two.clone() / two.clone() * two.clone();
+                assert_eq!(None, big.clone().checked_mul(&_3.clone()));
+                let _1_big: Ratio<T> = Ratio::new(T::one(), big.clone());
+                let _3_two: Ratio<T> = Ratio::new(_3.clone(), two.clone());
+                let expected = Ratio::new(T::one(), big / two.clone() * _3.clone());
+                assert_eq!(expected.clone(), _1_big.clone() / _3_two.clone());
+                assert_eq!(
+                    Some(expected.clone()),
+                    _1_big.clone().checked_div(&_3_two.clone())
+                );
+                assert_eq!(expected, {
+                    let mut tmp = _1_big;
+                    tmp /= _3_two;
+                    tmp
+                });
+
+                // 3/big / 3 = 1/big where big is max/2
+                // big ~ max/2, and big is not divisible by 3
+                let big = T::max_value() / two / _3.clone() * _3.clone() + T::one();
+                assert_eq!(None, big.clone().checked_mul(&_3.clone()));
+                let _3_big = Ratio::new(_3.clone(), big.clone());
+                let expected = Ratio::new(T::one(), big);
+                assert_eq!(expected, _3_big.clone() / _3.clone());
+                assert_eq!(expected, {
+                    let mut tmp = _3_big;
+                    tmp /= _3;
+                    tmp
+                });
+            }
+            test_div_typed_overflow::<u8>();
+            test_div_typed_overflow::<u16>();
+            test_div_typed_overflow::<u32>();
+            test_div_typed_overflow::<u64>();
+            test_div_typed_overflow::<usize>();
+            test_div_typed_overflow::<u128>();
+
+            test_div_typed_overflow::<i8>();
+            test_div_typed_overflow::<i16>();
+            test_div_typed_overflow::<i32>();
+            test_div_typed_overflow::<i64>();
+            test_div_typed_overflow::<isize>();
+            test_div_typed_overflow::<i128>();
+        }
+
+        #[test]
+        fn test_rem() {
+            fn test(a: Rational64, b: Rational64, c: Rational64) {
+                assert_eq!(a % b, c);
+                assert_eq!(
+                    {
+                        let mut x = a;
+                        x %= b;
+                        x
+                    },
+                    c
+                );
+                assert_eq!(to_big(a) % to_big(b), to_big(c))
+            }
+            fn test_assign(a: Rational64, b: i64, c: Rational64) {
+                assert_eq!(a % b, c);
+                assert_eq!(
+                    {
+                        let mut x = a;
+                        x %= b;
+                        x
+                    },
+                    c
+                );
+            }
+
+            test(_3_2, _1, _1_2);
+            test(_3_2, _1_2, _0);
+            test(_5_2, _3_2, _1);
+            test(_2, _NEG1_2, _0);
+            test(_1_2, _2, _1_2);
+            test_assign(_3_2, 1, _1_2);
+        }
+
+        #[test]
+        fn test_rem_overflow() {
+            // tests that Ratio(1,2) % Ratio(1, T::max_value()) equals 0
+            // for each integer type. Previously, this calculation would overflow.
+            fn test_rem_typed_overflow<T>()
+            where
+                T: Integer + Bounded + Clone + Debug + NumAssign,
+            {
+                let two = T::one() + T::one();
+                // value near to maximum, but divisible by two
+                let max_div2 = T::max_value() / two.clone() * two.clone();
+                let _1_max: Ratio<T> = Ratio::new(T::one(), max_div2);
+                let _1_two: Ratio<T> = Ratio::new(T::one(), two);
+                assert!(T::is_zero(&(_1_two.clone() % _1_max.clone()).numer));
+                {
+                    let mut tmp: Ratio<T> = _1_two;
+                    tmp %= _1_max;
+                    assert!(T::is_zero(&tmp.numer));
+                }
+            }
+            test_rem_typed_overflow::<u8>();
+            test_rem_typed_overflow::<u16>();
+            test_rem_typed_overflow::<u32>();
+            test_rem_typed_overflow::<u64>();
+            test_rem_typed_overflow::<usize>();
+            test_rem_typed_overflow::<u128>();
+
+            test_rem_typed_overflow::<i8>();
+            test_rem_typed_overflow::<i16>();
+            test_rem_typed_overflow::<i32>();
+            test_rem_typed_overflow::<i64>();
+            test_rem_typed_overflow::<isize>();
+            test_rem_typed_overflow::<i128>();
+        }
+
+        #[test]
+        fn test_neg() {
+            fn test(a: Rational64, b: Rational64) {
+                assert_eq!(-a, b);
+                assert_eq!(-to_big(a), to_big(b))
+            }
+
+            test(_0, _0);
+            test(_1_2, _NEG1_2);
+            test(-_1, _1);
+        }
+        #[test]
+        #[allow(clippy::eq_op)]
+        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_checked_failures() {
+            let big = Ratio::new(128u8, 1);
+            let small = Ratio::new(1, 128u8);
+            assert_eq!(big.checked_add(&big), None);
+            assert_eq!(small.checked_sub(&big), None);
+            assert_eq!(big.checked_mul(&big), None);
+            assert_eq!(small.checked_div(&big), None);
+            assert_eq!(_1.checked_div(&_0), None);
+        }
+
+        #[test]
+        fn test_checked_zeros() {
+            assert_eq!(_0.checked_add(&_0), Some(_0));
+            assert_eq!(_0.checked_sub(&_0), Some(_0));
+            assert_eq!(_0.checked_mul(&_0), Some(_0));
+            assert_eq!(_0.checked_div(&_0), None);
+        }
+
+        #[test]
+        fn test_checked_min() {
+            assert_eq!(_MIN.checked_add(&_MIN), None);
+            assert_eq!(_MIN.checked_sub(&_MIN), Some(_0));
+            assert_eq!(_MIN.checked_mul(&_MIN), None);
+            assert_eq!(_MIN.checked_div(&_MIN), Some(_1));
+            assert_eq!(_0.checked_add(&_MIN), Some(_MIN));
+            assert_eq!(_0.checked_sub(&_MIN), None);
+            assert_eq!(_0.checked_mul(&_MIN), Some(_0));
+            assert_eq!(_0.checked_div(&_MIN), Some(_0));
+            assert_eq!(_1.checked_add(&_MIN), Some(_MIN_P1));
+            assert_eq!(_1.checked_sub(&_MIN), None);
+            assert_eq!(_1.checked_mul(&_MIN), Some(_MIN));
+            assert_eq!(_1.checked_div(&_MIN), None);
+            assert_eq!(_MIN.checked_add(&_0), Some(_MIN));
+            assert_eq!(_MIN.checked_sub(&_0), Some(_MIN));
+            assert_eq!(_MIN.checked_mul(&_0), Some(_0));
+            assert_eq!(_MIN.checked_div(&_0), None);
+            assert_eq!(_MIN.checked_add(&_1), Some(_MIN_P1));
+            assert_eq!(_MIN.checked_sub(&_1), None);
+            assert_eq!(_MIN.checked_mul(&_1), Some(_MIN));
+            assert_eq!(_MIN.checked_div(&_1), Some(_MIN));
+        }
+
+        #[test]
+        fn test_checked_max() {
+            assert_eq!(_MAX.checked_add(&_MAX), None);
+            assert_eq!(_MAX.checked_sub(&_MAX), Some(_0));
+            assert_eq!(_MAX.checked_mul(&_MAX), None);
+            assert_eq!(_MAX.checked_div(&_MAX), Some(_1));
+            assert_eq!(_0.checked_add(&_MAX), Some(_MAX));
+            assert_eq!(_0.checked_sub(&_MAX), Some(_MIN_P1));
+            assert_eq!(_0.checked_mul(&_MAX), Some(_0));
+            assert_eq!(_0.checked_div(&_MAX), Some(_0));
+            assert_eq!(_1.checked_add(&_MAX), None);
+            assert_eq!(_1.checked_sub(&_MAX), Some(-_MAX_M1));
+            assert_eq!(_1.checked_mul(&_MAX), Some(_MAX));
+            assert_eq!(_1.checked_div(&_MAX), Some(_MAX.recip()));
+            assert_eq!(_MAX.checked_add(&_0), Some(_MAX));
+            assert_eq!(_MAX.checked_sub(&_0), Some(_MAX));
+            assert_eq!(_MAX.checked_mul(&_0), Some(_0));
+            assert_eq!(_MAX.checked_div(&_0), None);
+            assert_eq!(_MAX.checked_add(&_1), None);
+            assert_eq!(_MAX.checked_sub(&_1), Some(_MAX_M1));
+            assert_eq!(_MAX.checked_mul(&_1), Some(_MAX));
+            assert_eq!(_MAX.checked_div(&_1), Some(_MAX));
+        }
+
+        #[test]
+        fn test_checked_min_max() {
+            assert_eq!(_MIN.checked_add(&_MAX), Some(-_1));
+            assert_eq!(_MIN.checked_sub(&_MAX), None);
+            assert_eq!(_MIN.checked_mul(&_MAX), None);
+            assert_eq!(
+                _MIN.checked_div(&_MAX),
+                Some(Ratio::new(_MIN.numer, _MAX.numer))
+            );
+            assert_eq!(_MAX.checked_add(&_MIN), Some(-_1));
+            assert_eq!(_MAX.checked_sub(&_MIN), None);
+            assert_eq!(_MAX.checked_mul(&_MIN), None);
+            assert_eq!(_MAX.checked_div(&_MIN), None);
+        }
+    }
+
+    #[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 = "division by zero")]
+    fn test_recip_fail() {
+        let _a = Ratio::new(0, 1).recip();
+    }
+
+    #[test]
+    fn test_pow() {
+        fn test(r: Rational64, e: i32, expected: Rational64) {
+            assert_eq!(r.pow(e), expected);
+            assert_eq!(Pow::pow(r, e), expected);
+            assert_eq!(Pow::pow(r, &e), expected);
+            assert_eq!(Pow::pow(&r, e), expected);
+            assert_eq!(Pow::pow(&r, &e), expected);
+            #[cfg(feature = "num-bigint")]
+            test_big(r, e, expected);
+        }
+
+        #[cfg(feature = "num-bigint")]
+        fn test_big(r: Rational64, e: i32, expected: Rational64) {
+            let r = BigRational::new_raw(r.numer.into(), r.denom.into());
+            let expected = BigRational::new_raw(expected.numer.into(), expected.denom.into());
+            assert_eq!((&r).pow(e), expected);
+            assert_eq!(Pow::pow(r.clone(), e), expected);
+            assert_eq!(Pow::pow(r.clone(), &e), expected);
+            assert_eq!(Pow::pow(&r, e), expected);
+            assert_eq!(Pow::pow(&r, &e), expected);
+        }
+
+        test(_1_2, 2, Ratio::new(1, 4));
+        test(_1_2, -2, Ratio::new(4, 1));
+        test(_1, 1, _1);
+        test(_1, i32::MAX, _1);
+        test(_1, i32::MIN, _1);
+        test(_NEG1_2, 2, _1_2.pow(2i32));
+        test(_NEG1_2, 3, -_1_2.pow(3i32));
+        test(_3_2, 0, _1);
+        test(_3_2, -1, _3_2.recip());
+        test(_3_2, 3, Ratio::new(27, 8));
+    }
+
+    #[test]
+    #[cfg(feature = "std")]
+    fn test_to_from_str() {
+        use std::string::{String, ToString};
+        fn test(r: Rational64, 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<Rational64, _> = 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() {
+        use num_traits::float::FloatCore;
+        fn test<T: FloatCore>(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(core::f32::consts::PI, ("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(
+            core::f64::consts::PI,
+            ("884279719003555", "281474976710656"),
+        );
+        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 core::{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<i64>>::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]
+    #[cfg(feature = "std")]
+    fn test_hash() {
+        assert!(crate::hash(&_0) != crate::hash(&_1));
+        assert!(crate::hash(&_0) != crate::hash(&_3_2));
+
+        // a == b -> hash(a) == hash(b)
+        let a = Rational64::new_raw(4, 2);
+        let b = Rational64::new_raw(6, 3);
+        assert_eq!(a, b);
+        assert_eq!(crate::hash(&a), crate::hash(&b));
+
+        let a = Rational64::new_raw(123456789, 1000);
+        let b = Rational64::new_raw(123456789 * 5, 5000);
+        assert_eq!(a, b);
+        assert_eq!(crate::hash(&a), crate::hash(&b));
+    }
+
+    #[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)));
+    }
+
+    #[test]
+    fn ratio_iter_sum() {
+        // generic function to assure the iter method can be called
+        // for any Iterator with Item = Ratio<impl Integer> or Ratio<&impl Integer>
+        fn iter_sums<T: Integer + Clone>(slice: &[Ratio<T>]) -> [Ratio<T>; 3] {
+            let mut manual_sum = Ratio::new(T::zero(), T::one());
+            for ratio in slice {
+                manual_sum = manual_sum + ratio;
+            }
+            [manual_sum, slice.iter().sum(), slice.iter().cloned().sum()]
+        }
+        // collect into array so test works on no_std
+        let mut nums = [Ratio::new(0, 1); 1000];
+        for (i, r) in (0..1000).map(|n| Ratio::new(n, 500)).enumerate() {
+            nums[i] = r;
+        }
+        let sums = iter_sums(&nums[..]);
+        assert_eq!(sums[0], sums[1]);
+        assert_eq!(sums[0], sums[2]);
+    }
+
+    #[test]
+    fn ratio_iter_product() {
+        // generic function to assure the iter method can be called
+        // for any Iterator with Item = Ratio<impl Integer> or Ratio<&impl Integer>
+        fn iter_products<T: Integer + Clone>(slice: &[Ratio<T>]) -> [Ratio<T>; 3] {
+            let mut manual_prod = Ratio::new(T::one(), T::one());
+            for ratio in slice {
+                manual_prod = manual_prod * ratio;
+            }
+            [
+                manual_prod,
+                slice.iter().product(),
+                slice.iter().cloned().product(),
+            ]
+        }
+
+        // collect into array so test works on no_std
+        let mut nums = [Ratio::new(0, 1); 1000];
+        for (i, r) in (0..1000).map(|n| Ratio::new(n, 500)).enumerate() {
+            nums[i] = r;
+        }
+        let products = iter_products(&nums[..]);
+        assert_eq!(products[0], products[1]);
+        assert_eq!(products[0], products[2]);
+    }
+
+    #[test]
+    fn test_num_zero() {
+        let zero = Rational64::zero();
+        assert!(zero.is_zero());
+
+        let mut r = Rational64::new(123, 456);
+        assert!(!r.is_zero());
+        assert_eq!(r + zero, r);
+
+        r.set_zero();
+        assert!(r.is_zero());
+    }
+
+    #[test]
+    fn test_num_one() {
+        let one = Rational64::one();
+        assert!(one.is_one());
+
+        let mut r = Rational64::new(123, 456);
+        assert!(!r.is_one());
+        assert_eq!(r * one, r);
+
+        r.set_one();
+        assert!(r.is_one());
+    }
+
+    #[test]
+    fn test_const() {
+        const N: Ratio<i32> = Ratio::new_raw(123, 456);
+        const N_NUMER: &i32 = N.numer();
+        const N_DENOM: &i32 = N.denom();
+
+        assert_eq!(N_NUMER, &123);
+        assert_eq!(N_DENOM, &456);
+
+        let r = N.reduced();
+        assert_eq!(r.numer(), &(123 / 3));
+        assert_eq!(r.denom(), &(456 / 3));
+    }
+
+    #[test]
+    fn test_ratio_to_i64() {
+        assert_eq!(5, Rational64::new(70, 14).to_u64().unwrap());
+        assert_eq!(-3, Rational64::new(-31, 8).to_i64().unwrap());
+        assert_eq!(None, Rational64::new(-31, 8).to_u64());
+    }
+
+    #[test]
+    #[cfg(feature = "num-bigint")]
+    fn test_ratio_to_i128() {
+        assert_eq!(
+            1i128 << 70,
+            Ratio::<i128>::new(1i128 << 77, 1i128 << 7)
+                .to_i128()
+                .unwrap()
+        );
+    }
+
+    #[test]
+    #[cfg(feature = "num-bigint")]
+    fn test_big_ratio_to_f64() {
+        assert_eq!(
+            BigRational::new(
+                "1234567890987654321234567890987654321234567890"
+                    .parse()
+                    .unwrap(),
+                "3".parse().unwrap()
+            )
+            .to_f64(),
+            Some(411522630329218100000000000000000000000000000f64)
+        );
+        assert_eq!(Ratio::from_float(5e-324).unwrap().to_f64(), Some(5e-324));
+        assert_eq!(
+            // subnormal
+            BigRational::new(BigInt::one(), BigInt::one() << 1050).to_f64(),
+            Some(2.0f64.powi(-50).powi(21))
+        );
+        assert_eq!(
+            // definite underflow
+            BigRational::new(BigInt::one(), BigInt::one() << 1100).to_f64(),
+            Some(0.0)
+        );
+        assert_eq!(
+            BigRational::from(BigInt::one() << 1050).to_f64(),
+            Some(core::f64::INFINITY)
+        );
+        assert_eq!(
+            BigRational::from((-BigInt::one()) << 1050).to_f64(),
+            Some(core::f64::NEG_INFINITY)
+        );
+        assert_eq!(
+            BigRational::new(
+                "1234567890987654321234567890".parse().unwrap(),
+                "987654321234567890987654321".parse().unwrap()
+            )
+            .to_f64(),
+            Some(1.2499999893125f64)
+        );
+        assert_eq!(
+            BigRational::new_raw(BigInt::one(), BigInt::zero()).to_f64(),
+            Some(core::f64::INFINITY)
+        );
+        assert_eq!(
+            BigRational::new_raw(-BigInt::one(), BigInt::zero()).to_f64(),
+            Some(core::f64::NEG_INFINITY)
+        );
+        assert_eq!(
+            BigRational::new_raw(BigInt::zero(), BigInt::zero()).to_f64(),
+            None
+        );
+    }
+
+    #[test]
+    fn test_ratio_to_f64() {
+        assert_eq!(Ratio::<u8>::new(1, 2).to_f64(), Some(0.5f64));
+        assert_eq!(Rational64::new(1, 2).to_f64(), Some(0.5f64));
+        assert_eq!(Rational64::new(1, -2).to_f64(), Some(-0.5f64));
+        assert_eq!(Rational64::new(0, 2).to_f64(), Some(0.0f64));
+        assert_eq!(Rational64::new(0, -2).to_f64(), Some(-0.0f64));
+        assert_eq!(Rational64::new((1 << 57) + 1, 1 << 54).to_f64(), Some(8f64));
+        assert_eq!(
+            Rational64::new((1 << 52) + 1, 1 << 52).to_f64(),
+            Some(1.0000000000000002f64),
+        );
+        assert_eq!(
+            Rational64::new((1 << 60) + (1 << 8), 1 << 60).to_f64(),
+            Some(1.0000000000000002f64),
+        );
+        assert_eq!(
+            Ratio::<i32>::new_raw(1, 0).to_f64(),
+            Some(core::f64::INFINITY)
+        );
+        assert_eq!(
+            Ratio::<i32>::new_raw(-1, 0).to_f64(),
+            Some(core::f64::NEG_INFINITY)
+        );
+        assert_eq!(Ratio::<i32>::new_raw(0, 0).to_f64(), None);
+    }
+
+    #[test]
+    fn test_ldexp() {
+        use core::f64::{INFINITY, MAX_EXP, MIN_EXP, NAN, NEG_INFINITY};
+        assert_eq!(ldexp(1.0, 0), 1.0);
+        assert_eq!(ldexp(1.0, 1), 2.0);
+        assert_eq!(ldexp(0.0, 1), 0.0);
+        assert_eq!(ldexp(-0.0, 1), -0.0);
+
+        // Cases where ldexp is equivalent to multiplying by 2^exp because there's no over- or
+        // underflow.
+        assert_eq!(ldexp(3.5, 5), 3.5 * 2f64.powi(5));
+        assert_eq!(ldexp(1.0, MAX_EXP - 1), 2f64.powi(MAX_EXP - 1));
+        assert_eq!(ldexp(2.77, MIN_EXP + 3), 2.77 * 2f64.powi(MIN_EXP + 3));
+
+        // Case where initial value is subnormal
+        assert_eq!(ldexp(5e-324, 4), 5e-324 * 2f64.powi(4));
+        assert_eq!(ldexp(5e-324, 200), 5e-324 * 2f64.powi(200));
+
+        // Near underflow (2^exp is too small to represent, but not x*2^exp)
+        assert_eq!(ldexp(4.0, MIN_EXP - 3), 2f64.powi(MIN_EXP - 1));
+
+        // Near overflow
+        assert_eq!(ldexp(0.125, MAX_EXP + 3), 2f64.powi(MAX_EXP));
+
+        // Overflow and underflow cases
+        assert_eq!(ldexp(1.0, MIN_EXP - 54), 0.0);
+        assert_eq!(ldexp(-1.0, MIN_EXP - 54), -0.0);
+        assert_eq!(ldexp(1.0, MAX_EXP), INFINITY);
+        assert_eq!(ldexp(-1.0, MAX_EXP), NEG_INFINITY);
+
+        // Special values
+        assert_eq!(ldexp(INFINITY, 1), INFINITY);
+        assert_eq!(ldexp(NEG_INFINITY, 1), NEG_INFINITY);
+        assert!(ldexp(NAN, 1).is_nan());
+    }
+}