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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::num::arithmetic::traits::ModPowerOf2IsReduced;
use crate::num::basic::unsigneds::PrimitiveUnsigned;
use crate::num::conversion::traits::ExactFrom;
use crate::polynomial::{ModPowerOf2NthDerivative, ModPowerOf2NthDerivativeAssign};
use crate::unsigned_polynomial::UnsignedPolynomial;
fn assert_reduced<T: PrimitiveUnsigned>(p: &UnsignedPolynomial<T>, pow: u64) {
assert!(pow <= T::WIDTH);
assert!(
p.mod_power_of_2_is_reduced(pow),
"self must be reduced mod 2^pow, but {p} has a coefficient >= 2^{pow}"
);
}
// Multiplies the coefficient of x^i, for i at least n, by the falling factorial i(i - 1)...(i - n +
// 1) modulo 2^pow and moves it to x^(i - n). Each falling factorial is multiplied out from its n
// factors with wrapping arithmetic, which loses nothing, since 2^pow divides 2^WIDTH. They are all
// multiples of n!, so if 2^pow divides n!, which happens when n - n.count_ones() is at least pow,
// the result is zero. The products can be zero, so the result is trimmed.
fn mod_power_of_2_nth_derivative_in_place<T: PrimitiveUnsigned>(
p: &mut UnsignedPolynomial<T>,
n: u64,
pow: u64,
) {
assert_reduced(p, pow);
if n == 0 {
return;
}
if u64::exact_from(p.coefficients.len()) <= n || n - u64::from(n.count_ones()) >= pow {
p.coefficients.clear();
return;
}
let mut top = T::wrapping_from(n);
p.coefficients.drain(..usize::exact_from(n));
for c in &mut p.coefficients {
let mut falling = T::ONE;
let mut factor = top;
for _ in 0..n {
falling.wrapping_mul_assign(factor);
factor.wrapping_sub_assign(T::ONE);
}
*c = c.wrapping_mul(falling).mod_power_of_2(pow);
top.wrapping_add_assign(T::ONE);
}
p.trim();
}
impl<T: PrimitiveUnsigned> ModPowerOf2NthDerivative for UnsignedPolynomial<T> {
type Output = Self;
/// Computes the $n$th derivative of an [`UnsignedPolynomial`] modulo $2^k$, taking the
/// polynomial by value. The coefficients must already be reduced modulo $2^k$.
///
/// $$
/// f(p, n, k) = p^{(n)} \bmod 2^k.
/// $$
///
/// The coefficient of $x^i$ is multiplied by the falling factorial $i^{\underline n} =
/// i(i-1)\cdots(i-n+1)$, reduced, and moved to $x^{i-n}$. The products can be zero even when
/// the coefficients are not, so the derivative can lose any number of degrees; if $2^k$ divides
/// $n!$, every falling factorial is a multiple of the modulus, and the result is zero.
///
/// # Worst-case complexity
/// $T(k, n) = O(kn)$
///
/// $M(k) = O(k)$
///
/// where $T$ is time, $M$ is additional memory, $k$ is `self.len()`, and $n$ is `n`.
///
/// # Panics
/// Panics if `pow` is greater than `T::WIDTH`, or if any coefficient of `self` is greater than
/// or equal to $2^k$.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::polynomial::ModPowerOf2NthDerivative;
/// use malachite_base::unsigned_polynomial::UnsignedPolynomial;
///
/// let p = UnsignedPolynomial::<u8>::from_str("x^4+3*x^3+2*x+5").unwrap();
/// assert_eq!(
/// p.clone().mod_power_of_2_nth_derivative(2, 3).to_string(),
/// "4*x^2+2*x"
/// );
/// assert_eq!(
/// p.mod_power_of_2_nth_derivative(0, 3).to_string(),
/// "x^4+3*x^3+2*x+5"
/// );
///
/// // 2^3 divides 4!.
/// let p = UnsignedPolynomial::<u8>::from_str("x^5+x^4").unwrap();
/// assert_eq!(p.mod_power_of_2_nth_derivative(4, 3).to_string(), "0");
/// ```
///
/// FLINT has no `nmod_poly_nth_derivative`; this computes the same multipliers as
/// `fmpz_poly_nth_derivative` from `fmpz_poly/nth_derivative.c`, FLINT 3.6.0, directly modulo
/// $2^k$.
#[inline]
fn mod_power_of_2_nth_derivative(mut self, n: u64, pow: u64) -> Self {
mod_power_of_2_nth_derivative_in_place(&mut self, n, pow);
self
}
}
impl<T: PrimitiveUnsigned> ModPowerOf2NthDerivative for &UnsignedPolynomial<T> {
type Output = UnsignedPolynomial<T>;
/// Computes the $n$th derivative of an [`UnsignedPolynomial`] modulo $2^k$, taking the
/// polynomial by reference. The coefficients must already be reduced modulo $2^k$.
///
/// $$
/// f(p, n, k) = p^{(n)} \bmod 2^k.
/// $$
///
/// The coefficient of $x^i$ is multiplied by the falling factorial $i^{\underline n} =
/// i(i-1)\cdots(i-n+1)$, reduced, and moved to $x^{i-n}$. The products can be zero even when
/// the coefficients are not, so the derivative can lose any number of degrees; if $2^k$ divides
/// $n!$, every falling factorial is a multiple of the modulus, and the result is zero.
///
/// # Worst-case complexity
/// $T(k, n) = O(kn)$
///
/// $M(k) = O(k)$
///
/// where $T$ is time, $M$ is additional memory, $k$ is `self.len()`, and $n$ is `n`.
///
/// # Panics
/// Panics if `pow` is greater than `T::WIDTH`, or if any coefficient of `self` is greater than
/// or equal to $2^k$.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::polynomial::ModPowerOf2NthDerivative;
/// use malachite_base::unsigned_polynomial::UnsignedPolynomial;
///
/// let p = UnsignedPolynomial::<u8>::from_str("x^4+3*x^3+2*x+5").unwrap();
/// assert_eq!(
/// (&p).mod_power_of_2_nth_derivative(2, 3).to_string(),
/// "4*x^2+2*x"
/// );
/// assert_eq!(
/// (&p).mod_power_of_2_nth_derivative(0, 3).to_string(),
/// "x^4+3*x^3+2*x+5"
/// );
///
/// // 2^3 divides 4!.
/// let p = UnsignedPolynomial::<u8>::from_str("x^5+x^4").unwrap();
/// assert_eq!((&p).mod_power_of_2_nth_derivative(4, 3).to_string(), "0");
/// ```
///
/// FLINT has no `nmod_poly_nth_derivative`; this computes the same multipliers as
/// `fmpz_poly_nth_derivative` from `fmpz_poly/nth_derivative.c`, FLINT 3.6.0, directly modulo
/// $2^k$.
#[inline]
fn mod_power_of_2_nth_derivative(self, n: u64, pow: u64) -> UnsignedPolynomial<T> {
let mut p = self.clone();
mod_power_of_2_nth_derivative_in_place(&mut p, n, pow);
p
}
}
impl<T: PrimitiveUnsigned> ModPowerOf2NthDerivativeAssign for UnsignedPolynomial<T> {
/// Replaces an [`UnsignedPolynomial`] with its $n$th derivative modulo $2^k$, in place. The
/// coefficients must already be reduced modulo $2^k$.
///
/// $$
/// p \gets p^{(n)} \bmod 2^k.
/// $$
///
/// The coefficient of $x^i$ is multiplied by the falling factorial $i^{\underline n} =
/// i(i-1)\cdots(i-n+1)$, reduced, and moved to $x^{i-n}$. The products can be zero even when
/// the coefficients are not, so the derivative can lose any number of degrees; if $2^k$ divides
/// $n!$, every falling factorial is a multiple of the modulus, and the result is zero.
///
/// # Worst-case complexity
/// $T(k, n) = O(kn)$
///
/// $M(k) = O(k)$
///
/// where $T$ is time, $M$ is additional memory, $k$ is `self.len()`, and $n$ is `n`.
///
/// # Panics
/// Panics if `pow` is greater than `T::WIDTH`, or if any coefficient of `self` is greater than
/// or equal to $2^k$.
///
/// # Examples
/// ```
/// use core::str::FromStr;
/// use malachite_base::polynomial::ModPowerOf2NthDerivativeAssign;
/// use malachite_base::unsigned_polynomial::UnsignedPolynomial;
///
/// let mut p = UnsignedPolynomial::<u8>::from_str("x^4+3*x^3+2*x+5").unwrap();
/// p.mod_power_of_2_nth_derivative_assign(2, 3);
/// assert_eq!(p.to_string(), "4*x^2+2*x");
///
/// // 2^3 divides 4!.
/// let mut p = UnsignedPolynomial::<u8>::from_str("x^5+x^4").unwrap();
/// p.mod_power_of_2_nth_derivative_assign(4, 3);
/// assert_eq!(p.to_string(), "0");
/// ```
///
/// FLINT has no `nmod_poly_nth_derivative`; this computes the same multipliers as
/// `fmpz_poly_nth_derivative` from `fmpz_poly/nth_derivative.c`, FLINT 3.6.0, directly modulo
/// $2^k$.
#[inline]
fn mod_power_of_2_nth_derivative_assign(&mut self, n: u64, pow: u64) {
mod_power_of_2_nth_derivative_in_place(self, n, pow);
}
}