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use crate::{
div_exact::{hensel_div_odd_in_place, hensel_is_multiple_of},
math::inv_mod_pow2,
primitive::{extend_word, shrink_dword, WORD_BITS, WORD_BITS_USIZE},
repr::{Repr, TypedReprRef},
shift,
ubig::UBig,
Word,
};
use alloc::vec;
use dashu_base::{DivRem, PowerOfTwo};
impl UBig {
/// Divide out all multiples of the factor from the integer,
/// returns the exponent of the removed factor.
///
/// For self = 0 or factor = 0 or 1, this method returns None.
///
/// # Examples
///
/// ```
/// use dashu_int::UBig;
///
/// let mut a = UBig::from(8u32) * 3u32;
/// assert_eq!(a.remove(&UBig::from(2u32)), Some(3));
/// assert_eq!(a, UBig::from(3u32));
/// ```
pub fn remove(&mut self, factor: &UBig) -> Option<usize> {
// A factor that fits in a single word is handled by the faster `remove_word`.
if let TypedReprRef::RefSmall(dword) = factor.repr() {
if let Some(word) = shrink_dword(dword) {
return self.remove_word(word);
}
}
if self.is_zero() || factor.is_zero() || factor.is_one() {
return None;
}
// shortcut for power of 2
if factor.is_power_of_two() {
let bits = factor.trailing_zeros().unwrap();
let exp = self.trailing_zeros().unwrap() / bits;
*self >>= exp * bits;
return Some(exp);
}
let (mut q, r) = (&*self).div_rem(factor);
if !r.is_zero() {
return Some(0);
}
// first stage, division with exponentially growing factors
let mut exp = 1;
let mut pows = vec![factor.sqr()];
loop {
let last = pows.last().unwrap();
let (new_q, r) = (&q).div_rem(last);
if !r.is_zero() {
break;
}
exp += 1 << pows.len();
q = new_q;
let next_sq = last.sqr();
pows.push(next_sq);
}
// second stage, division from highest power to the lowest
while let Some(last) = pows.pop() {
let (new_q, r) = (&q).div_rem(last);
if r.is_zero() {
exp += 1 << (pows.len() + 1);
q = new_q;
}
}
// last division
let (new_q, r) = (&q).div_rem(factor);
if r.is_zero() {
exp += 1;
q = new_q;
}
*self = q;
Some(exp)
}
/// Divide out all multiples of a single-word factor from the integer,
/// returns the exponent of the removed factor.
///
/// The single-word specialization of [`remove`](Self::remove): the factor's power-of-two part is
/// stripped by the 2-valuation, and the odd part is divided out by **Hensel (2-adic) exact
/// division** — the modular inverse of the factor is precomputed by Newton iteration, and each
/// quotient limb is computed as `(word − carry) · d^{-1} mod 2^WORD_BITS`, i.e. only multiplies
/// and subtracts with no division or normalization. Powers `d², d⁴, …` (which exceed a single
/// word) use the general division, mirroring the binary-splitting in [`remove`](Self::remove).
///
/// For self = 0 or factor = 0 or 1, this method returns None.
///
/// # Examples
///
/// ```
/// use dashu_int::UBig;
///
/// let mut a = UBig::from(10u32).pow(8) * 7u32;
/// assert_eq!(a.remove_word(10), Some(8));
/// assert_eq!(a, UBig::from(7u32));
/// ```
pub fn remove_word(&mut self, factor: Word) -> Option<usize> {
if self.is_zero() || factor == 0 || factor == 1 {
return None;
}
// factor = 2^s · d_odd: the power-of-two part and the odd part are removed separately.
let trailing = factor.trailing_zeros();
let s = trailing as usize;
let d_odd = factor >> trailing;
if d_odd == 1 {
// A pure power of two: the exponent is bounded by the 2-valuation of self.
let exp = self.trailing_zeros().unwrap() / s;
*self >>= exp * s;
return Some(exp);
}
if s == 0 {
// An odd factor: no power-of-two part to reconcile.
return Some(remove_odd_powers(self, d_odd, usize::MAX));
}
// A mixed factor (e.g. 10 = 2·5): strip the trailing 2s once, divide out the odd part (each
// full `factor`-power consumes `s` of them, so the odd-part count is capped by `tz/s`), then
// reassemble the cofactor `self / factor^exp = (odd part) << (tz − s·exp)` (the shift amount
// is non-negative because `exp ≤ cap = tz/s`). `self` is taken and its own buffer is shifted
// right by `tz` in place (the low `tz` bits are zero by construction) — no allocation for the
// working value; `*self` is rebuilt from the result.
let tz = self.trailing_zeros().unwrap();
let cap = tz / s;
let mut odd_buf = core::mem::take(self).0.into_buffer();
odd_buf.erase_front(tz / WORD_BITS_USIZE);
if tz % WORD_BITS_USIZE != 0 {
shift::shr_in_place(&mut odd_buf, (tz % WORD_BITS_USIZE) as u32);
}
odd_buf.pop_zeros();
let mut odd = UBig(Repr::from_buffer(odd_buf));
let exp = remove_odd_powers(&mut odd, d_odd, cap);
*self = odd << (tz - s * exp);
Some(exp)
}
}
/// Remove powers of the odd single-word `d` from `n` (in place), at most `cap` of them, and return
/// the number removed. `n` is left unchanged when `d` does not divide it.
///
/// The first division by `d` uses the Hensel (2-adic) exact division
/// ([`crate::div_exact::hensel_div_odd_in_place`]), which computes the quotient with only multiplies
/// and subtracts; the powers `d², d⁴, …` that follow exceed a single word and use the general
/// division, mirroring the binary-splitting of [`UBig::remove`].
fn remove_odd_powers(n: &mut UBig, d: Word, cap: usize) -> usize {
if cap == 0 {
return 0;
}
let di = inv_mod_pow2(extend_word(d), WORD_BITS) as Word;
// A read-only divisibility probe first leaves `n` untouched when `d` does not divide it, so the
// in-place Hensel division that follows is guaranteed to succeed and can consume `n`'s own
// buffer — no scratch allocation.
if !hensel_is_multiple_of(n.as_words(), d, di) {
return 0;
}
let mut q = core::mem::take(n).0.into_buffer();
debug_assert!(
hensel_div_odd_in_place(&mut q, d, di),
"the probe passed, so the division is exact"
);
*n = UBig(Repr::from_buffer(q));
let mut exp = 1;
// Grow the powers d², d⁴, … while each divides the current quotient exactly (and stays within
// `cap`), then refine downward with the collected powers, and finally divide out one last single
// `d` (the count may be odd, leaving a leftover below d²).
let mut power = UBig::from_word(d).sqr();
let mut power_exp = 2usize;
let mut pows = vec![(power.clone(), power_exp)];
while exp + power_exp <= cap && *n >= power {
let (new_q, r) = (&*n).div_rem(&power);
if r.is_zero() {
*n = new_q;
exp += power_exp;
power = power.sqr();
power_exp <<= 1;
pows.push((power.clone(), power_exp));
} else {
break;
}
}
while let Some((p, e)) = pows.pop() {
if exp + e > cap {
continue;
}
let (new_q, r) = (&*n).div_rem(&p);
if r.is_zero() {
*n = new_q;
exp += e;
}
}
if exp < cap {
let (new_q, r) = (&*n).div_rem(&UBig::from_word(d));
if r.is_zero() {
*n = new_q;
exp += 1;
}
}
exp
}
#[cfg(test)]
mod tests {
use super::*;
use crate::UBig;
/// The Hensel kernel must agree with the general division on a sweep of small values — both the
/// exactness test (remainder == 0) and the quotient itself.
#[test]
fn test_hensel_div_matches_div_rem() {
for d in [3u16, 5, 7, 9, 15, 21, 255, 1001] {
let d = d as Word;
let d = d as Word;
for lo in [0u8, 1, 2, 5, 200, 255] {
let lo = lo as Word;
for hi in [0u8, 1, 2, 7, 199] {
let hi = hi as Word;
let u = UBig::from_words(&[lo, hi]);
if u.is_zero() {
continue;
}
let mut q: alloc::vec::Vec<Word> = u.as_words().to_vec();
let di = inv_mod_pow2(extend_word(d), WORD_BITS) as Word;
let exact = hensel_div_odd_in_place(&mut q, d, di);
let (qr, r) = (&u).div_rem(&UBig::from_word(d));
assert_eq!(exact, r.is_zero(), "d={d} u={u:?}");
if r.is_zero() {
assert_eq!(UBig::from_words(&q), qr, "quotient d={d} u={u:?}");
}
}
}
}
}
/// The Newton inverse must satisfy `d · d^{-1} ≡ 1 (mod 2^WORD_BITS)` for odd `d`.
#[test]
fn test_inv_mod_pow2() {
for d in [3u8, 5, 7, 9, 11, 15, 31, 127, 255] {
let d = d as Word;
let inv = inv_mod_pow2(extend_word(d), WORD_BITS) as Word;
assert_eq!(d.wrapping_mul(inv), 1, "d·d^-1 ≡ 1 for d={d}",);
}
}
/// `remove_word` must agree with `remove` for every single-word factor, on both odd and even
/// factors and with extra 2s mixed in.
#[test]
fn test_remove_word_matches_remove() {
for d in [2u8, 3, 5, 7, 10, 12, 16, 25] {
let d = d as Word;
for i in 0..10usize {
for rest in [1u8, 5, 7, 11] {
let base = UBig::from(d).pow(i) * rest;
for extra_twos in 0..3usize {
let n = base.clone() << extra_twos;
let mut a = n.clone();
let exp_a = a.remove_word(d);
let mut b = n.clone();
let exp_b = b.remove(&UBig::from(d));
assert_eq!(exp_a, exp_b, "d={d} i={i} rest={rest} twos={extra_twos}");
assert_eq!(a, b, "d={d} i={i} rest={rest} twos={extra_twos}");
}
}
}
}
}
/// A not-divisible factor must leave the value unchanged (`remove_word` computes the quotient in
/// a scratch and only commits it on an exact division).
#[test]
fn test_remove_word_noop_when_not_divisible() {
for n in [3u32, 7, 9, 11, 13, 25, 100, 1000] {
for d in [2u8, 3, 5, 7, 10] {
let d = d as Word;
let original = UBig::from(n);
let mut a = original.clone();
if a.remove_word(d) == Some(0) {
assert_eq!(a, original, "not-divisible must leave n unchanged: d={d} n={n}");
}
}
}
}
/// Multi-word operands, including a pure power of two, a mixed factor, and a large odd factor.
#[test]
fn test_remove_word_large() {
let n = UBig::from(10u8).pow(200);
let mut a = n.clone();
assert_eq!(a.remove_word(10), Some(200));
assert_eq!(a, UBig::ONE);
// 5^200 · 2^100 · 7: the odd factor's 2-valuation is irrelevant to it.
let mut b = UBig::from(5u8).pow(200) * (UBig::ONE << 100) * 7u8;
assert_eq!(b.remove_word(5), Some(200));
assert_eq!(b, (UBig::ONE << 100) * 7u8);
// A pure power of two.
let mut c = (UBig::ONE << 1000) * 3u8;
assert_eq!(c.remove_word(4), Some(500));
assert_eq!(c, UBig::from(3u8));
}
}