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// Copyright © 2026 Mikhail Hogrefe
//
// This file is part of Malachite.
//
// Malachite is free software: you can redistribute it and/or modify it under the terms of the GNU
// Lesser General Public License (LGPL) as published by the Free Software Foundation; either version
// 3 of the License, or (at your option) any later version. See <https://www.gnu.org/licenses/>.
use crate::Rational;
use crate::arithmetic::traits::{Approximate, ApproximateAssign};
use core::mem::swap;
use malachite_base::num::arithmetic::traits::{
AddMulAssign, DivMod, Floor, Parity, Reciprocal, ShrRound, UnsignedAbs,
};
use malachite_base::num::basic::traits::{One, Zero};
use malachite_base::num::comparison::traits::PartialOrdAbs;
use malachite_base::num::conversion::traits::RoundingFrom;
use malachite_base::rounding_modes::RoundingMode::*;
use malachite_nz::integer::Integer;
use malachite_nz::natural::Natural;
fn approximate_helper(q: &Rational, max_denominator: &Natural) -> Rational {
let floor = q.floor();
let mut x = (q - Rational::from(&floor)).reciprocal();
let mut previous_numerator = Integer::ONE;
let mut previous_denominator = Natural::ZERO;
let mut numerator = floor;
let mut denominator = Natural::ONE;
let mut result = None;
loop {
let n;
(n, x.numerator) = (&x.numerator).div_mod(&x.denominator);
swap(&mut x.numerator, &mut x.denominator);
let previous_previous_numerator = previous_numerator.clone();
let previous_previous_denominator = previous_denominator.clone();
previous_numerator.add_mul_assign(&numerator, Integer::from(&n));
previous_denominator.add_mul_assign(&denominator, &n);
if previous_denominator > *max_denominator {
previous_numerator = previous_previous_numerator;
previous_denominator = previous_previous_denominator;
// We need a term m such that previous_denominator + denominator * m is as large as
// possible without exceeding max_denominator.
let m = (max_denominator - &previous_denominator) / &denominator;
let half_n = (&n).shr_round(1, Ceiling).0;
if m < half_n {
} else if m == half_n && n.even() {
let previous_convergent = Rational {
sign: numerator >= 0u32,
numerator: (&numerator).unsigned_abs(),
denominator: denominator.clone(),
};
previous_numerator.add_mul_assign(&numerator, Integer::from(&m));
previous_denominator.add_mul_assign(&denominator, m);
let candidate = Rational {
sign: previous_numerator >= 0u32,
numerator: previous_numerator.unsigned_abs(),
denominator: previous_denominator,
};
result = Some(if (q - &previous_convergent).lt_abs(&(q - &candidate)) {
previous_convergent
} else {
candidate
});
} else {
numerator *= Integer::from(&m);
numerator += previous_numerator;
denominator *= m;
denominator += previous_denominator;
}
break;
}
swap(&mut numerator, &mut previous_numerator);
swap(&mut denominator, &mut previous_denominator);
}
let result = if let Some(result) = result {
result
} else {
Rational {
sign: numerator >= 0u32,
numerator: numerator.unsigned_abs(),
denominator,
}
};
// Suppose the input is (1/4, 2). The approximations 0 and 1/2 both satisfy the denominator
// limit and are equidistant from 1/4, but we prefer 0 because it has the smaller denominator.
// Unfortunately, the code above makes the wrong choice, so we need the following code to check
// whether the approximation on the opposite side of `self` is better.
let opposite: Rational = (q << 1) - &result;
if result.denominator_ref() <= opposite.denominator_ref() {
result
} else {
opposite
}
}
impl Approximate for Rational {
/// Finds the best approximation of a [`Rational`] using a denominator no greater than a
/// specified maximum, taking the [`Rational`] by value.
///
/// Let $f(x, d) = p/q$, with $p$ and $q$ relatively prime. Then the following properties hold:
/// - $q \leq d$
/// - For all $n \in \Z$ and all $m \in \Z$ with $0 < m \leq d$, $|x - p/q| \leq |x - n/m|$.
/// - If $|x - n/m| = |x - p/q|$, then $q \leq m$.
/// - If $|x - n/q| = |x - p/q|$, then $p$ is even and $n$ is either equal to $p$ or odd.
///
/// # Worst-case complexity
/// $T(n) = O(n^2 \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Panics
/// - If `max_denominator` is zero.
///
/// # Examples
/// ```
/// use malachite_base::num::conversion::traits::ExactFrom;
/// use malachite_nz::natural::Natural;
/// use malachite_q::arithmetic::traits::Approximate;
/// use malachite_q::Rational;
///
/// assert_eq!(
/// Rational::exact_from(std::f64::consts::PI)
/// .approximate(&Natural::from(1000u32))
/// .to_string(),
/// "355/113"
/// );
/// assert_eq!(
/// Rational::from_signeds(333i32, 1000)
/// .approximate(&Natural::from(100u32))
/// .to_string(),
/// "1/3"
/// );
/// ```
///
/// # Implementation notes
/// This algorithm follows the description in
/// <https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations>. One part of
/// the algorithm not mentioned in that article is that if the last term $n$ in the continued
/// fraction needs to be reduced, the optimal replacement term $m$ may be found using division.
fn approximate(self, max_denominator: &Natural) -> Rational {
assert_ne!(*max_denominator, 0);
if self.denominator_ref() <= max_denominator {
return self;
}
if *max_denominator == 1u32 {
return Self::from(Integer::rounding_from(self, Nearest).0);
}
approximate_helper(&self, max_denominator)
}
}
impl Approximate for &Rational {
/// Finds the best approximation of a [`Rational`] using a denominator no greater than a
/// specified maximum, taking the [`Rational`] by reference.
///
/// Let $f(x, d) = p/q$, with $p$ and $q$ relatively prime. Then the following properties hold:
/// - $q \leq d$
/// - For all $n \in \Z$ and all $m \in \Z$ with $0 < m \leq d$, $|x - p/q| \leq |x - n/m|$.
/// - If $|x - n/m| = |x - p/q|$, then $q \leq m$.
/// - If $|x - n/q| = |x - p/q|$, then $p$ is even and $n$ is either equal to $p$ or odd.
///
/// # Worst-case complexity
/// $T(n) = O(n^2 \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Panics
/// - If `max_denominator` is zero.
///
/// # Examples
/// ```
/// use malachite_base::num::conversion::traits::ExactFrom;
/// use malachite_nz::natural::Natural;
/// use malachite_q::arithmetic::traits::Approximate;
/// use malachite_q::Rational;
///
/// assert_eq!(
/// (&Rational::exact_from(std::f64::consts::PI))
/// .approximate(&Natural::from(1000u32))
/// .to_string(),
/// "355/113"
/// );
/// assert_eq!(
/// (&Rational::from_signeds(333i32, 1000))
/// .approximate(&Natural::from(100u32))
/// .to_string(),
/// "1/3"
/// );
/// ```
///
/// # Implementation notes
/// This algorithm follows the description in
/// <https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations>. One part of
/// the algorithm not mentioned in that article is that if the last term $n$ in the continued
/// fraction needs to be reduced, the optimal replacement term $m$ may be found using division.
fn approximate(self, max_denominator: &Natural) -> Rational {
assert_ne!(*max_denominator, 0);
if self.denominator_ref() <= max_denominator {
return self.clone();
}
if *max_denominator == 1u32 {
return Rational::from(Integer::rounding_from(self, Nearest).0);
}
approximate_helper(self, max_denominator)
}
}
impl ApproximateAssign for Rational {
/// Finds the best approximation of a [`Rational`] using a denominator no greater than a
/// specified maximum, mutating the [`Rational`] in place.
///
/// See [`Rational::approximate`] for more information.
///
/// # Worst-case complexity
/// $T(n) = O(n^2 \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `max(self.significant_bits(),
/// other.significant_bits())`.
///
/// # Panics
/// - If `max_denominator` is zero.
///
/// # Examples
/// ```
/// use malachite_base::num::conversion::traits::ExactFrom;
/// use malachite_nz::natural::Natural;
/// use malachite_q::arithmetic::traits::ApproximateAssign;
/// use malachite_q::Rational;
///
/// let mut x = Rational::exact_from(std::f64::consts::PI);
/// x.approximate_assign(&Natural::from(1000u32));
/// assert_eq!(x.to_string(), "355/113");
///
/// let mut x = Rational::from_signeds(333i32, 1000);
/// x.approximate_assign(&Natural::from(100u32));
/// assert_eq!(x.to_string(), "1/3");
/// ```
fn approximate_assign(&mut self, max_denominator: &Natural) {
assert_ne!(*max_denominator, 0);
if self.denominator_ref() <= max_denominator {
} else if *max_denominator == 1u32 {
*self = Self::from(Integer::rounding_from(&*self, Nearest).0);
} else {
*self = approximate_helper(&*self, max_denominator);
}
}
}