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// Copyright © 2026 Mikhail Hogrefe
//
// Uses code adopted from the FLINT Library.
//
// Copyright © 2009, 2016 William Hart
//
// 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::num::arithmetic::traits::{ExtendedGcd, UnsignedAbs};
use crate::num::basic::signeds::PrimitiveSigned;
use crate::num::basic::unsigneds::PrimitiveUnsigned;
use crate::num::conversion::traits::WrappingFrom;
use crate::rounding_modes::RoundingMode::*;
use core::mem::swap;
fn extended_gcd_signed<
U: ExtendedGcd<Cofactor = S> + PrimitiveUnsigned,
S: PrimitiveSigned + UnsignedAbs<Output = U> + WrappingFrom<U>,
>(
a: S,
b: S,
) -> (U, S, S) {
let (gcd, mut x, mut y) = a.unsigned_abs().extended_gcd(b.unsigned_abs());
if a < S::ZERO {
x = x.checked_neg().unwrap();
}
if b < S::ZERO {
y = y.checked_neg().unwrap();
}
(gcd, x, y)
}
// This is equivalent to `n_xgcd` from `ulong_extras/xgcd.c`, FLINT 2.7.1, with an adjustment to
// find the minimal cofactors.
private_test_fn! {extended_gcd_unsigned_binary<
U: WrappingFrom<S> + PrimitiveUnsigned,
S: PrimitiveSigned + WrappingFrom<U>,
>(
mut a: U,
mut b: U,
) -> (U, S, S) {
if a == U::ZERO && b == U::ZERO {
return (U::ZERO, S::ZERO, S::ZERO);
} else if a == b || a == U::ZERO {
return (b, S::ZERO, S::ONE);
} else if b == U::ZERO {
return (a, S::ONE, S::ZERO);
}
let mut swapped = false;
if a < b {
swap(&mut a, &mut b);
swapped = true;
}
let mut u1 = S::ONE;
let mut v2 = S::ONE;
let mut u2 = S::ZERO;
let mut v1 = S::ZERO;
let mut u3 = a;
let mut v3 = b;
if (a & b).get_highest_bit() {
(u1, u2, u3, v1, v2, v3) = (u2, u1 - u2, v3, v2, v1 - v2, u3 - v3);
}
while v3.get_bit(U::WIDTH - 2) {
let d = u3 - v3;
(u1, u2, u3, v1, v2, v3) = if d < v3 {
// quot = 1
(u2, u1 - u2, v3, v2, v1 - v2, d)
} else if d < (v3 << 1) {
// quot = 2
(u2, u1 - (u2 << 1), v3, v2, v1 - (v2 << 1), d - v3)
} else {
// quot = 3
(
u2,
u1 - S::wrapping_from(3) * u2,
v3,
v2,
v1 - S::wrapping_from(3) * v2,
d - (v3 << 1),
)
};
}
while v3 != U::ZERO {
let d = u3 - v3;
// overflow not possible, top 2 bits of v3 not set
(u1, u2, u3, v1, v2, v3) = if u3 < (v3 << 2) {
if d < v3 {
// quot = 1
(u2, u1 - u2, v3, v2, v1 - v2, d)
} else if d < (v3 << 1) {
// quot = 2
(
u2,
u1.wrapping_sub(u2 << 1),
v3,
v2,
v1.wrapping_sub(v2 << 1),
d - v3,
)
} else {
// quot = 3
(
u2,
u1.wrapping_sub(S::wrapping_from(3).wrapping_mul(u2)),
v3,
v2,
v1.wrapping_sub(S::wrapping_from(3).wrapping_mul(v2)),
d.wrapping_sub(v3 << 1),
)
}
} else {
let (quot, rem) = u3.div_rem(v3);
(
u2,
u1.wrapping_sub(S::wrapping_from(quot).wrapping_mul(u2)),
v3,
v2,
v1.wrapping_sub(S::wrapping_from(quot).wrapping_mul(v2)),
rem,
)
};
}
// Remarkably, |u1| < x/2, thus comparison with 0 is valid
if u1 <= S::ZERO {
u1.wrapping_add_assign(S::wrapping_from(b));
v1.wrapping_sub_assign(S::wrapping_from(a));
}
// The cofactors at this point are not necessarily minimal, so we may need to adjust.
let gcd = u3;
let mut x = U::wrapping_from(u1);
let mut y = U::wrapping_from(v1);
let two_limit_a = a / gcd;
let two_limit_b = b / gcd;
let limit_b = two_limit_b >> 1;
if x > limit_b {
let k = (x - limit_b).div_round(two_limit_b, Ceiling).0;
x.wrapping_sub_assign(two_limit_b.wrapping_mul(k));
y.wrapping_add_assign(two_limit_a.wrapping_mul(k));
}
if swapped {
swap(&mut x, &mut y);
}
(gcd, S::wrapping_from(x), S::wrapping_from(y))
}}
macro_rules! impl_extended_gcd {
($u:ident, $s:ident) => {
impl ExtendedGcd<$u> for $u {
type Gcd = $u;
type Cofactor = $s;
/// Computes the GCD (greatest common divisor) of two numbers $a$ and $b$, and also the
/// coefficients $x$ and $y$ in Bézout's identity $ax+by=\gcd(a,b)$.
///
/// The are infinitely many $x$, $y$ that satisfy the identity for any $a$, $b$, so the
/// full specification is more detailed:
///
/// - $f(0, 0) = (0, 0, 0)$.
/// - $f(a, ak) = (a, 1, 0)$ if $a > 0$ and $k \neq 1$.
/// - $f(bk, b) = (b, 0, 1)$ if $b > 0$.
/// - $f(a, b) = (g, x, y)$ if $a \neq 0$ and $b \neq 0$ and $\gcd(a, b) \neq \min(a,
/// b)$, where $g = \gcd(a, b) \geq 0$, $ax + by = g$, $x \leq \lfloor b/g \rfloor$,
/// and $y \leq \lfloor a/g \rfloor$.
///
/// # Worst-case complexity
/// $T(n) = O(n)$
///
/// $M(n) = O(1)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is
/// `max(self.significant_bits(), other.significant_bits())`: this is the extended
/// Euclidean algorithm on words, whose $O(n)$ iterations (the remainder at least halves
/// every two division steps, and the subtract-only fast path is taken only when the
/// quotient is below 4) each cost $O(1)$ word operations, with no allocation.
///
/// # Examples
/// See [here](super::extended_gcd#extended_gcd).
#[inline]
fn extended_gcd(self, other: $u) -> ($u, $s, $s) {
extended_gcd_unsigned_binary(self, other)
}
}
impl ExtendedGcd<$s> for $s {
type Gcd = $u;
type Cofactor = $s;
/// Computes the GCD (greatest common divisor) of two numbers $a$ and $b$, and also the
/// coefficients $x$ and $y$ in Bézout's identity $ax+by=\gcd(a,b)$.
///
/// The are infinitely many $x$, $y$ that satisfy the identity for any $a$, $b$, so the
/// full specification is more detailed:
///
/// - $f(0, 0) = (0, 0, 0)$.
/// - $f(a, ak) = (a, 1, 0)$ if $a > 0$ and $k \neq 1$.
/// - $f(a, ak) = (-a, -1, 0)$ if $a < 0$ and $k \neq 1$.
/// - $f(bk, b) = (b, 0, 1)$ if $b > 0$.
/// - $f(bk, b) = (-b, 0, -1)$ if $b < 0$.
/// - $f(a, b) = (g, x, y)$ if $a \neq 0$ and $b \neq 0$ and $\gcd(a, b) \neq \min(|a|,
/// |b|)$, where $g = \gcd(a, b) \geq 0$, $ax + by = g$, $x \leq \lfloor b/g \rfloor$,
/// and $y \leq \lfloor a/g \rfloor$.
///
/// # Worst-case complexity
/// $T(n) = O(n)$
///
/// $M(n) = O(1)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is
/// `max(self.significant_bits(), other.significant_bits())`: this is the extended
/// Euclidean algorithm on words, whose $O(n)$ iterations (the remainder at least halves
/// every two division steps, and the subtract-only fast path is taken only when the
/// quotient is below 4) each cost $O(1)$ word operations, with no allocation.
///
/// # Examples
/// See [here](super::extended_gcd#extended_gcd).
#[inline]
fn extended_gcd(self, other: $s) -> ($u, $s, $s) {
extended_gcd_signed(self, other)
}
}
};
}
apply_to_unsigned_signed_pairs!(impl_extended_gcd);