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use malachite_base::num::arithmetic::traits::{
CeilingDivAssignNegMod, CeilingDivNegMod, DivAssignMod, DivAssignRem, DivMod, DivRem,
WrappingAddAssign, WrappingSub, WrappingSubAssign, XMulYToZZ, XXDivModYToQR,
};
use malachite_base::num::basic::integers::PrimitiveInt;
use malachite_base::num::basic::traits::{Iverson, One, Zero};
use malachite_base::num::conversion::traits::{JoinHalves, SplitInHalf};
use malachite_base::num::logic::traits::LeadingZeros;
use malachite_base::slices::{slice_move_left, slice_set_zero};
use natural::arithmetic::add::{
limbs_add_limb_to_out, limbs_add_same_length_to_out,
limbs_add_same_length_with_carry_in_in_place_left, limbs_add_same_length_with_carry_in_to_out,
limbs_slice_add_limb_in_place, limbs_slice_add_same_length_in_place_left,
};
use natural::arithmetic::div::{limbs_div_divide_and_conquer_approx, limbs_div_schoolbook_approx};
use natural::arithmetic::mul::mul_mod::{
limbs_mul_mod_base_pow_n_minus_1, limbs_mul_mod_base_pow_n_minus_1_next_size,
limbs_mul_mod_base_pow_n_minus_1_scratch_len,
};
use natural::arithmetic::mul::{
limbs_mul_greater_to_out, limbs_mul_same_length_to_out, limbs_mul_to_out,
};
use natural::arithmetic::shl::{limbs_shl_to_out, limbs_slice_shl_in_place};
use natural::arithmetic::shr::{limbs_shr_to_out, limbs_slice_shr_in_place};
use natural::arithmetic::sub::{
limbs_sub_greater_in_place_left, limbs_sub_limb_in_place, limbs_sub_same_length_in_place_left,
limbs_sub_same_length_in_place_right, limbs_sub_same_length_to_out,
limbs_sub_same_length_with_borrow_in_in_place_left,
limbs_sub_same_length_with_borrow_in_in_place_right,
limbs_sub_same_length_with_borrow_in_to_out,
};
use natural::arithmetic::sub_mul::limbs_sub_mul_limb_same_length_in_place_left;
use natural::comparison::cmp::limbs_cmp_same_length;
use natural::logic::not::limbs_not_to_out;
use natural::InnerNatural::{Large, Small};
use natural::Natural;
use platform::{
DoubleLimb, Limb, DC_DIVAPPR_Q_THRESHOLD, DC_DIV_QR_THRESHOLD, INV_MULMOD_BNM1_THRESHOLD,
INV_NEWTON_THRESHOLD, MAYBE_DCP1_DIVAPPR, MU_DIV_QR_SKEW_THRESHOLD, MU_DIV_QR_THRESHOLD,
};
use std::cmp::{min, Ordering};
use std::mem::swap;
// The highest bit of the input must be set.
//
// # Worst-case complexity
// Constant time and additional memory.
//
// # Panics
// Panics if `d` is zero.
//
// This is equivalent to `mpn_invert_limb`, or `invert_limb`, from `gmp-impl.h`, GMP 6.2.1.
pub_crate_test! {limbs_invert_limb(d: Limb) -> Limb {
(DoubleLimb::join_halves(!d, Limb::MAX) / DoubleLimb::from(d)).lower_half()
}}
// # Worst-case complexity
// Constant time and additional memory.
//
// This is equivalent to `udiv_qrnnd_preinv` from `gmp-impl.h`, GMP 6.2.1.
pub_crate_test! {div_mod_by_preinversion(
n_high: Limb,
n_low: Limb,
d: Limb,
d_inv: Limb
) -> (Limb, Limb) {
let (mut q_high, q_low) = (DoubleLimb::from(n_high) * DoubleLimb::from(d_inv))
.wrapping_add(DoubleLimb::join_halves(n_high.wrapping_add(1), n_low))
.split_in_half();
let mut r = n_low.wrapping_sub(q_high.wrapping_mul(d));
if r > q_low {
let (r_plus_d, overflow) = r.overflowing_add(d);
if overflow {
q_high.wrapping_sub_assign(1);
r = r_plus_d;
}
} else if r >= d {
q_high.wrapping_add_assign(1);
r -= d;
}
(q_high, r)
}}
// Interpreting a slice of `Limb`s as the limbs (in ascending order) of a `Natural`, returns the
// quotient limbs and remainder of the `Natural` divided by a `Limb`. The divisor limb cannot be
// zero and the limb slice must have at least two elements.
//
// # Worst-case complexity
// $T(n) = O(n)$
//
// $M(n) = O(n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
//
// # Panics
// Panics if the length of `ns` is less than 2 or if `d` is zero.
//
// This is equivalent to `mpn_divrem_1` from `mpn/generic/divrem_1.c`, GMP 6.2.1, where `qxn == 0`,
// `un > 1`, and both results are returned. Experiments show that `DIVREM_1_NORM_THRESHOLD` and
// `DIVREM_1_UNNORM_THRESHOLD` are unnecessary (they would always be 0).
pub_test! {limbs_div_limb_mod(ns: &[Limb], d: Limb) -> (Vec<Limb>, Limb) {
let mut qs = vec![0; ns.len()];
let r = limbs_div_limb_to_out_mod(&mut qs, ns, d);
(qs, r)
}}
// Interpreting a slice of `Limb`s as the limbs (in ascending order) of a `Natural`, writes the
// limbs of the quotient of the `Natural` and a `Limb` to an output slice, and returns the
// remainder. The output slice must be at least as long as the input slice. The divisor limb cannot
// be zero and the input limb slice must have at least two elements.
//
// # Worst-case complexity
// $T(n) = O(n)$
//
// $M(n) = O(1)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
//
// # Panics
// Panics if `out` is shorter than `ns`, the length of `ns` is less than 2, or if `d` is zero.
//
// This is equivalent to `mpn_divrem_1` from `mpn/generic/divrem_1.c`, GMP 6.2.1, where `qxn == 0`
// and `un > 1`. Experiments show that `DIVREM_1_NORM_THRESHOLD` and `DIVREM_1_UNNORM_THRESHOLD`
// are unnecessary (they would always be 0).
pub_crate_test! {limbs_div_limb_to_out_mod(out: &mut [Limb], ns: &[Limb], d: Limb) -> Limb {
assert_ne!(d, 0);
let len = ns.len();
assert!(len > 1);
let out = &mut out[..len];
let bits = LeadingZeros::leading_zeros(d);
if bits == 0 {
// High quotient limb is 0 or 1, skip a divide step.
let (r, ns_init) = ns.split_last().unwrap();
let mut r = *r;
let (out_last, out_init) = out.split_last_mut().unwrap();
let adjust = r >= d;
if adjust {
r -= d;
}
*out_last = Limb::iverson(adjust);
// Multiply-by-inverse, divisor already normalized.
let d_inv = limbs_invert_limb(d);
for (out_q, &n) in out_init.iter_mut().zip(ns_init.iter()).rev() {
(*out_q, r) = div_mod_by_preinversion(r, n, d, d_inv);
}
r
} else {
// Skip a division if high < divisor (high quotient 0). Testing here before normalizing will
// still skip as often as possible.
let (ns_last, ns_init) = ns.split_last().unwrap();
let (ns, mut r) = if *ns_last < d {
*out.last_mut().unwrap() = 0;
(ns_init, *ns_last)
} else {
(ns, 0)
};
let d = d << bits;
r <<= bits;
let d_inv = limbs_invert_limb(d);
let (previous_n, ns_init) = ns.split_last().unwrap();
let mut previous_n = *previous_n;
let cobits = Limb::WIDTH - bits;
r |= previous_n >> cobits;
let (out_head, out_tail) = out.split_first_mut().unwrap();
for (out_q, &n) in out_tail.iter_mut().zip(ns_init.iter()).rev() {
let n_shifted = (previous_n << bits) | (n >> cobits);
(*out_q, r) = div_mod_by_preinversion(r, n_shifted, d, d_inv);
previous_n = n;
}
let out_r;
(*out_head, out_r) = div_mod_by_preinversion(r, previous_n << bits, d, d_inv);
out_r >> bits
}
}}
// Interpreting a slice of `Limb`s as the limbs (in ascending order) of a `Natural`, writes the
// limbs of the quotient of the `Natural` and a `Limb` to the input slice and returns the
// remainder. The divisor limb cannot be zero and the input limb slice must have at least two
// elements.
//
// # Worst-case complexity
// $T(n) = O(n)$
//
// $M(n) = O(1)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
//
// # Panics
// Panics if the length of `ns` is less than 2 or if `d` is zero.
//
// This is equivalent to `mpn_divrem_1` from `mpn/generic/divrem_1.c`, GMP 6.2.1, where `qp == up`,
// `qxn == 0`, and `un > 1`. Experiments show that `DIVREM_1_NORM_THRESHOLD` and
// `DIVREM_1_UNNORM_THRESHOLD` are unnecessary (they would always be 0).
pub_crate_test! {limbs_div_limb_in_place_mod(ns: &mut [Limb], d: Limb) -> Limb {
assert_ne!(d, 0);
let len = ns.len();
assert!(len > 1);
let bits = LeadingZeros::leading_zeros(d);
let (ns_last, ns_init) = ns.split_last_mut().unwrap();
if bits == 0 {
// High quotient limb is 0 or 1, skip a divide step.
let mut r = *ns_last;
let adjust = r >= d;
if adjust {
r -= d;
}
*ns_last = Limb::iverson(adjust);
// Multiply-by-inverse, divisor already normalized.
let d_inv = limbs_invert_limb(d);
for n in ns_init.iter_mut().rev() {
(*n, r) = div_mod_by_preinversion(r, *n, d, d_inv);
}
r
} else {
// Skip a division if high < divisor (high quotient 0). Testing here before normalizing will
// still skip as often as possible.
let (ns, mut r) = if *ns_last < d {
let r = *ns_last;
*ns_last = 0;
(ns_init, r)
} else {
(ns, 0)
};
let d = d << bits;
r <<= bits;
let d_inv = limbs_invert_limb(d);
let last_index = ns.len() - 1;
let mut previous_n = ns[last_index];
let cobits = Limb::WIDTH - bits;
r |= previous_n >> cobits;
for i in (0..last_index).rev() {
let n = ns[i];
let shifted_n = (previous_n << bits) | (n >> cobits);
(ns[i + 1], r) = div_mod_by_preinversion(r, shifted_n, d, d_inv);
previous_n = n;
}
let out_r;
(ns[0], out_r) = div_mod_by_preinversion(r, previous_n << bits, d, d_inv);
out_r >> bits
}
}}
// Let `ns` be the limbs of a `Natural` $n$, and let $f$ be `fraction_len`. This function performs
// the integer division $B^fn / d$, writing the `ns.len() + fraction_len` limbs of the quotient to
// `out` and returning the remainder.
//
// `shift` must be the number of leading zeros of `d`, and `d_inv` must be
// `limbs_invert_limb(d << shift)`.
//
// # Worst-case complexity
// $T(n) = O(n)$
//
// $M(n) = O(1)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len() + fraction_len`.
//
// # Panics
// Panics if `out` is shorter than `ns.len()` + `fraction_len`, if `ns` is empty, or if `d` is
// zero.
//
// This is equivalent to `mpn_preinv_divrem_1` from `mpn/generic/pre_divrem_1.c`, GMP 6.2.1, where
// `qp != ap`.
pub_test! {limbs_div_mod_extra(
out: &mut [Limb],
fraction_len: usize,
mut ns: &[Limb],
d: Limb,
d_inv: Limb,
shift: u64,
) -> Limb {
assert!(!ns.is_empty());
assert_ne!(d, 0);
let (ns_last, ns_init) = ns.split_last().unwrap();
let ns_last = *ns_last;
let d_norm = d << shift;
let (fraction_out, integer_out) = out.split_at_mut(fraction_len);
let mut integer_out = &mut integer_out[..ns.len()];
let mut r;
if shift == 0 {
r = ns_last;
let q_high = r >= d_norm;
if r >= d_norm {
r -= d_norm;
}
let (integer_out_last, integer_out_init) = integer_out.split_last_mut().unwrap();
*integer_out_last = Limb::iverson(q_high);
for (q, &n) in integer_out_init.iter_mut().zip(ns_init.iter()).rev() {
(*q, r) = div_mod_by_preinversion(r, n, d_norm, d_inv);
}
} else {
r = 0;
if ns_last < d {
r = ns_last << shift;
let integer_out_last;
(integer_out_last, integer_out) = integer_out.split_last_mut().unwrap();
*integer_out_last = 0;
ns = ns_init;
}
if !ns.is_empty() {
let co_shift = Limb::WIDTH - shift;
let (ns_last, ns_init) = ns.split_last().unwrap();
let mut previous_n = *ns_last;
r |= previous_n >> co_shift;
let (integer_out_head, integer_out_tail) = integer_out.split_first_mut().unwrap();
for (q, &n) in integer_out_tail.iter_mut().zip(ns_init.iter()).rev() {
assert!(r < d_norm);
(*q, r) = div_mod_by_preinversion(
r,
(previous_n << shift) | (n >> co_shift),
d_norm,
d_inv,
);
previous_n = n;
}
(*integer_out_head, r) = div_mod_by_preinversion(r, previous_n << shift, d_norm, d_inv);
}
}
for q in fraction_out.iter_mut().rev() {
(*q, r) = div_mod_by_preinversion(r, 0, d_norm, d_inv);
}
r >> shift
}}
// Let `&ns[fraction_len..]` be the limbs of a `Natural` $n$, and let $f$ be `fraction_len`. This
// function performs the integer division $B^fn / d$, writing the `ns.len() + fraction_len` limbs
// of the quotient to `ns` and returning the remainder.
//
// `shift` must be the number of leading zeros of `d`, and `d_inv` must be
// `limbs_invert_limb(d << shift)`.
//
// # Worst-case complexity
// $T(n) = O(n)$
//
// $M(n) = O(1)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len() + fraction_len`.
//
// # Panics
// Panics if `ns` is empty, if `ns.len()` is less than `fraction_len`, or if `d` is zero.
//
// This is equivalent to `mpn_preinv_divrem_1` from `mpn/generic/pre_divrem_1.c`, GMP 6.2.1, where
// `qp == ap`.
pub_crate_test! {limbs_div_mod_extra_in_place(
ns: &mut [Limb],
fraction_len: usize,
d: Limb,
d_inv: Limb,
shift: u64,
) -> Limb {
assert_ne!(d, 0);
let (fraction_ns, mut integer_ns) = ns.split_at_mut(fraction_len);
let ns_last = *integer_ns.last().unwrap();
let d_norm = d << shift;
let mut r;
if shift == 0 {
r = ns_last;
let q_high = r >= d_norm;
if r >= d_norm {
r -= d_norm;
}
let (integer_ns_last, integer_ns_init) = integer_ns.split_last_mut().unwrap();
*integer_ns_last = Limb::iverson(q_high);
for q in integer_ns_init.iter_mut().rev() {
(*q, r) = div_mod_by_preinversion(r, *q, d_norm, d_inv);
}
} else {
r = 0;
if ns_last < d {
r = ns_last << shift;
let integer_ns_last;
(integer_ns_last, integer_ns) = integer_ns.split_last_mut().unwrap();
*integer_ns_last = 0;
}
if !integer_ns.is_empty() {
let co_shift = Limb::WIDTH - shift;
let mut previous_n = *integer_ns.last().unwrap();
r |= previous_n >> co_shift;
for i in (1..integer_ns.len()).rev() {
assert!(r < d_norm);
let n = integer_ns[i - 1];
(integer_ns[i], r) = div_mod_by_preinversion(
r,
(previous_n << shift) | (n >> co_shift),
d_norm,
d_inv,
);
previous_n = n;
}
(integer_ns[0], r) = div_mod_by_preinversion(r, previous_n << shift, d_norm, d_inv);
}
}
for q in fraction_ns.iter_mut().rev() {
(*q, r) = div_mod_by_preinversion(r, 0, d_norm, d_inv);
}
r >> shift
}}
// Computes floor((B ^ 3 - 1) / (`hi` * B + `lo`)) - B, where B = 2 ^ `Limb::WIDTH`, assuming the
// highest bit of `hi` is set.
//
// # Worst-case complexity
// Constant time and additional memory.
//
// # Panics
// Panics if `hi` is zero.
//
// This is equivalent to `invert_pi1` from `gmp-impl.h`, GMP 6.2.1, where the result is returned
// instead of being written to `dinv`.
pub_crate_test! {limbs_two_limb_inverse_helper(hi: Limb, lo: Limb) -> Limb {
let mut d_inv = limbs_invert_limb(hi);
let mut hi_product = hi.wrapping_mul(d_inv);
hi_product.wrapping_add_assign(lo);
if hi_product < lo {
d_inv.wrapping_sub_assign(1);
if hi_product >= hi {
hi_product.wrapping_sub_assign(hi);
d_inv.wrapping_sub_assign(1);
}
hi_product.wrapping_sub_assign(hi);
}
let (lo_product_hi, lo_product_lo) = Limb::x_mul_y_to_zz(lo, d_inv);
hi_product.wrapping_add_assign(lo_product_hi);
if hi_product < lo_product_hi {
d_inv.wrapping_sub_assign(1);
if hi_product > hi || hi_product == hi && lo_product_lo >= lo {
d_inv.wrapping_sub_assign(1);
}
}
d_inv
}}
// Computes the quotient and remainder of `[n_2, n_1, n_0]` / `[d_1, d_0]`. Requires the highest
// bit of `d_1` to be set, and `[n_2, n_1]` < `[d_1, d_0]`. `d_inv` is the inverse of `[d_1, d_0]`
// computed by `limbs_two_limb_inverse_helper`.
//
// # Worst-case complexity
// Constant time and additional memory.
//
// This is equivalent to `udiv_qr_3by2` from `gmp-impl.h`, GMP 6.2.1.
pub_crate_test! {limbs_div_mod_three_limb_by_two_limb(
n_2: Limb,
n_1: Limb,
n_0: Limb,
d_1: Limb,
d_0: Limb,
d_inv: Limb,
) -> (Limb, DoubleLimb) {
let (mut q, q_lo) = (DoubleLimb::from(n_2) * DoubleLimb::from(d_inv))
.wrapping_add(DoubleLimb::join_halves(n_2, n_1))
.split_in_half();
let d = DoubleLimb::join_halves(d_1, d_0);
// Compute the two most significant limbs of n - q * d
let mut r = DoubleLimb::join_halves(n_1.wrapping_sub(d_1.wrapping_mul(q)), n_0)
.wrapping_sub(d)
.wrapping_sub(DoubleLimb::from(d_0) * DoubleLimb::from(q));
q.wrapping_add_assign(1);
// Conditionally adjust q and the remainder
if r.upper_half() >= q_lo {
let (r_plus_d, overflow) = r.overflowing_add(d);
if overflow {
q.wrapping_sub_assign(1);
r = r_plus_d;
}
} else if r >= d {
q.wrapping_add_assign(1);
r.wrapping_sub_assign(d);
}
(q, r)
}}
// Divides `ns` by `ds` and writes the `ns.len()` - 2 least-significant quotient limbs to `qs` and
// the 2-long remainder to `ns`. Returns the most significant limb of the quotient; `true` means 1
// and `false` means 0. `ds` must have length 2, `ns` must have length at least 2, and the most
// significant bit of `ds[1]` must be set.
//
// # Worst-case complexity
// $T(n) = O(n)$
//
// $M(n) = O(1)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
//
// # Panics
// Panics if `ds` does not have length 2, `ns` has length less than 2, `qs` has length less than
// `ns.len() - 2`, or `ds[1]` does not have its highest bit set.
//
// This is equivalent to `mpn_divrem_2` from `mpn/generic/divrem_2.c`, GMP 6.2.1.
pub_crate_test! {limbs_div_mod_by_two_limb_normalized(
qs: &mut [Limb],
ns: &mut [Limb],
ds: &[Limb]
) -> bool {
assert_eq!(ds.len(), 2);
let n_len = ns.len();
assert!(n_len >= 2);
let n_limit = n_len - 2;
assert!(ds[1].get_highest_bit());
let d_1 = ds[1];
let d_0 = ds[0];
let d = DoubleLimb::join_halves(d_1, d_0);
let mut r = DoubleLimb::join_halves(ns[n_limit + 1], ns[n_limit]);
let highest_q = r >= d;
if highest_q {
r.wrapping_sub_assign(d);
}
let (mut r_1, mut r_0) = r.split_in_half();
let d_inv = limbs_two_limb_inverse_helper(d_1, d_0);
for (&n, q) in ns[..n_limit].iter().zip(qs[..n_limit].iter_mut()).rev() {
let r;
(*q, r) = limbs_div_mod_three_limb_by_two_limb(r_1, r_0, n, d_1, d_0, d_inv);
(r_1, r_0) = r.split_in_half();
}
ns[1] = r_1;
ns[0] = r_0;
highest_q
}}
// Schoolbook division using the Möller-Granlund 3/2 division algorithm.
//
// Divides `ns` by `ds` and writes the `ns.len()` - `ds.len()` least-significant quotient limbs to
// `qs` and the `ds.len()` limbs of the remainder to `ns`. Returns the most significant limb of the
// quotient; `true` means 1 and `false` means 0. `ds` must have length greater than 2, `ns` must be
// at least as long as `ds`, and the most significant bit of `ds` must be set. `d_inv` should be
// the result of `limbs_two_limb_inverse_helper` applied to the two highest limbs of the
// denominator.
//
// # Worst-case complexity
// $T(n, d) = O(d(n - d + 1)) = O(n^2)$
//
// $M(n) = O(1)$
//
// where $T$ is time, $M$ is additional memory, $n$ is `ns.len()`, and $d$ is `ds.len()`.
//
// # Panics
// Panics if `ds` has length smaller than 3, `ns` is shorter than `ds`, `qs` has length less than
// `ns.len()` - `ds.len()`, or the last limb of `ds` does not have its highest bit set.
//
// This is equivalent to `mpn_sbpi1_div_qr` from `mpn/generic/sbpi1_div_qr.c`, GMP 6.2.1.
pub_crate_test! {limbs_div_mod_schoolbook(
qs: &mut [Limb],
ns: &mut [Limb],
ds: &[Limb],
d_inv: Limb,
) -> bool {
let d_len = ds.len();
assert!(d_len > 2);
let n_len = ns.len();
assert!(n_len >= d_len);
let d_1 = ds[d_len - 1];
assert!(d_1.get_highest_bit());
let d_0 = ds[d_len - 2];
let ds_except_last = &ds[..d_len - 1];
let ds_except_last_two = &ds[..d_len - 2];
let ns_hi = &mut ns[n_len - d_len..];
let highest_q = limbs_cmp_same_length(ns_hi, ds) >= Ordering::Equal;
if highest_q {
limbs_sub_same_length_in_place_left(ns_hi, ds);
}
let mut n_1 = ns[n_len - 1];
for i in (d_len..n_len).rev() {
let j = i - d_len;
let mut q;
if n_1 == d_1 && ns[i - 1] == d_0 {
q = Limb::MAX;
limbs_sub_mul_limb_same_length_in_place_left(&mut ns[j..i], ds, q);
n_1 = ns[i - 1]; // update n_1, last loop's value will now be invalid
} else {
let (ns_lo, ns_hi) = ns.split_at_mut(i - 2);
let n;
(q, n) = limbs_div_mod_three_limb_by_two_limb(n_1, ns_hi[1], ns_hi[0], d_1, d_0, d_inv);
let mut n_0;
(n_1, n_0) = n.split_in_half();
let local_carry_1 = limbs_sub_mul_limb_same_length_in_place_left(
&mut ns_lo[j..],
ds_except_last_two,
q,
);
let local_carry_2 = n_0 < local_carry_1;
n_0.wrapping_sub_assign(local_carry_1);
let carry = local_carry_2 && n_1 == 0;
if local_carry_2 {
n_1.wrapping_sub_assign(1);
}
ns_hi[0] = n_0;
if carry {
n_1.wrapping_add_assign(d_1);
if limbs_slice_add_same_length_in_place_left(&mut ns[j..i - 1], ds_except_last) {
n_1.wrapping_add_assign(1);
}
q.wrapping_sub_assign(1);
}
}
qs[j] = q;
}
ns[d_len - 1] = n_1;
highest_q
}}
// # Worst-case complexity
// $T(n) = O(n (\log n)^2 \log \log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ds.len()`.
//
// This is equivalent to `mpn_dcpi1_div_qr_n` from `mpn/generic/dcpi1_div_qr.c`, GMP 6.2.1.
pub(crate) fn limbs_div_mod_divide_and_conquer_helper(
qs: &mut [Limb],
ns: &mut [Limb],
ds: &[Limb],
d_inv: Limb,
scratch: &mut [Limb],
) -> bool {
let n = ds.len();
let lo = n >> 1; // floor(n / 2)
let hi = n - lo; // ceil(n / 2)
let qs_hi = &mut qs[lo..];
let (ds_lo, ds_hi) = ds.split_at(lo);
let mut highest_q = if hi < DC_DIV_QR_THRESHOLD {
limbs_div_mod_schoolbook(qs_hi, &mut ns[lo << 1..n << 1], ds_hi, d_inv)
} else {
limbs_div_mod_divide_and_conquer_helper(qs_hi, &mut ns[lo << 1..], ds_hi, d_inv, scratch)
};
let qs_hi = &mut qs_hi[..hi];
limbs_mul_greater_to_out(scratch, qs_hi, ds_lo);
let ns_lo = &mut ns[..n + lo];
let mut carry = Limb::iverson(limbs_sub_same_length_in_place_left(
&mut ns_lo[lo..],
&scratch[..n],
));
if highest_q && limbs_sub_same_length_in_place_left(&mut ns_lo[n..], ds_lo) {
carry += 1;
}
while carry != 0 {
if limbs_sub_limb_in_place(qs_hi, 1) {
assert!(highest_q);
highest_q = false;
}
if limbs_slice_add_same_length_in_place_left(&mut ns_lo[lo..], ds) {
carry -= 1;
}
}
let (ds_lo, ds_hi) = ds.split_at(hi);
let q_lo = if lo < DC_DIV_QR_THRESHOLD {
limbs_div_mod_schoolbook(qs, &mut ns[hi..n + lo], ds_hi, d_inv)
} else {
limbs_div_mod_divide_and_conquer_helper(qs, &mut ns[hi..], ds_hi, d_inv, scratch)
};
let qs_lo = &mut qs[..lo];
let ns_lo = &mut ns[..n];
limbs_mul_greater_to_out(scratch, ds_lo, qs_lo);
let mut carry = Limb::iverson(limbs_sub_same_length_in_place_left(ns_lo, &scratch[..n]));
if q_lo && limbs_sub_same_length_in_place_left(&mut ns_lo[lo..], ds_lo) {
carry += 1;
}
while carry != 0 {
limbs_sub_limb_in_place(qs_lo, 1);
if limbs_slice_add_same_length_in_place_left(ns_lo, ds) {
carry -= 1;
}
}
highest_q
}
// # Worst-case complexity
// $T(n) = O(n (\log n)^2 \log \log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ds.len()`.
pub_test! {limbs_div_dc_helper(
qs: &mut [Limb],
ns: &mut [Limb],
ds: &[Limb],
d_inv: Limb,
scratch: &mut [Limb],
) -> bool {
if qs.len() < DC_DIV_QR_THRESHOLD {
limbs_div_mod_schoolbook(qs, ns, ds, d_inv)
} else {
limbs_div_mod_divide_and_conquer_helper(qs, ns, ds, d_inv, scratch)
}
}}
// Recursive divide-and-conquer division.
//
// # Worst-case complexity
// $T(n) = O(n (\log n)^2 \log \log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ds.len()`.
//
// # Panics
// Panics if `ds` has length smaller than 6, `ns.len()` is less than `ds.len()` + 3, `qs` has
// length less than `ns.len()` - `ds.len()`, or the last limb of `ds` does not have its highest bit
// set.
//
// This is equivalent to `mpn_dcpi1_div_qr` from `mpn/generic/dcpi1_div_qr.c`, GMP 6.2.1.
pub_test! {limbs_div_mod_divide_and_conquer(
qs: &mut [Limb],
ns: &mut [Limb],
ds: &[Limb],
d_inv: Limb,
) -> bool {
let n_len = ns.len();
let d_len = ds.len();
assert!(d_len >= 6); // to adhere to limbs_div_mod_schoolbook's limits
assert!(n_len >= d_len + 3); // to adhere to limbs_div_mod_schoolbook's limits
let a = d_len - 1;
let d_1 = ds[a];
let b = d_len - 2;
assert!(d_1.get_highest_bit());
let mut scratch = vec![0; d_len];
let mut highest_q;
let q_len = n_len - d_len;
if q_len > d_len {
let q_len_mod_d_len = {
let mut m = q_len % d_len;
if m == 0 {
m = d_len;
}
m
};
// Perform the typically smaller block first.
// point at low limb of next quotient block
let qs_alt = &mut qs[q_len - q_len_mod_d_len..q_len];
if q_len_mod_d_len == 1 {
// Handle highest_q up front, for simplicity.
let ns_hi = &mut ns[q_len - 1..];
let ns_hi_hi = &mut ns_hi[1..];
highest_q = limbs_cmp_same_length(ns_hi_hi, ds) >= Ordering::Equal;
if highest_q {
assert!(!limbs_sub_same_length_in_place_left(ns_hi_hi, ds));
}
// A single iteration of schoolbook: One 3/2 division, followed by the bignum update
// and adjustment.
let (last_n, ns) = ns_hi.split_last_mut().unwrap();
let n_2 = *last_n;
let mut n_1 = ns[a];
let mut n_0 = ns[b];
let d_0 = ds[b];
assert!(n_2 < d_1 || n_2 == d_1 && n_1 <= d_0);
let mut q;
if n_2 == d_1 && n_1 == d_0 {
q = Limb::MAX;
assert_eq!(limbs_sub_mul_limb_same_length_in_place_left(ns, ds, q), n_2);
} else {
let n;
(q, n) = limbs_div_mod_three_limb_by_two_limb(n_2, n_1, n_0, d_1, d_0, d_inv);
(n_1, n_0) = n.split_in_half();
// d_len > 2 because of precondition. No need to check
let local_carry_1 =
limbs_sub_mul_limb_same_length_in_place_left(&mut ns[..b], &ds[..b], q);
let local_carry_2 = n_0 < local_carry_1;
n_0.wrapping_sub_assign(local_carry_1);
let carry = local_carry_2 && n_1 == 0;
if local_carry_2 {
n_1.wrapping_sub_assign(1);
}
ns[b] = n_0;
let (last_n, ns) = ns.split_last_mut().unwrap();
if carry {
n_1.wrapping_add_assign(d_1);
if limbs_slice_add_same_length_in_place_left(ns, &ds[..a]) {
n_1.wrapping_add_assign(1);
}
if q == 0 {
assert!(highest_q);
highest_q = false;
}
q.wrapping_sub_assign(1);
}
*last_n = n_1;
}
qs_alt[0] = q;
} else {
// Do a 2 * q_len_mod_d_len / q_len_mod_d_len division
let (ds_lo, ds_hi) = ds.split_at(d_len - q_len_mod_d_len);
highest_q = {
let ns = &mut ns[n_len - (q_len_mod_d_len << 1)..];
if q_len_mod_d_len == 2 {
limbs_div_mod_by_two_limb_normalized(qs_alt, ns, ds_hi)
} else {
limbs_div_dc_helper(qs_alt, ns, ds_hi, d_inv, &mut scratch)
}
};
if q_len_mod_d_len != d_len {
limbs_mul_to_out(&mut scratch, qs_alt, ds_lo);
let ns = &mut ns[q_len - q_len_mod_d_len..n_len - q_len_mod_d_len];
let mut carry = Limb::iverson(limbs_sub_same_length_in_place_left(ns, &scratch));
if highest_q
&& limbs_sub_same_length_in_place_left(&mut ns[q_len_mod_d_len..], ds_lo)
{
carry += 1;
}
while carry != 0 {
if limbs_sub_limb_in_place(qs_alt, 1) {
assert!(highest_q);
highest_q = false;
}
if limbs_slice_add_same_length_in_place_left(ns, ds) {
carry -= 1;
}
}
}
}
// offset is a multiple of d_len
let mut offset = n_len.checked_sub(d_len + q_len_mod_d_len).unwrap();
while offset != 0 {
offset -= d_len;
limbs_div_mod_divide_and_conquer_helper(
&mut qs[offset..],
&mut ns[offset..],
ds,
d_inv,
&mut scratch,
);
}
} else {
let m = d_len - q_len;
let (ds_lo, ds_hi) = ds.split_at(m);
highest_q = if q_len < DC_DIV_QR_THRESHOLD {
limbs_div_mod_schoolbook(qs, &mut ns[m..], ds_hi, d_inv)
} else {
limbs_div_mod_divide_and_conquer_helper(qs, &mut ns[m..], ds_hi, d_inv, &mut scratch)
};
if m != 0 {
let qs = &mut qs[..q_len];
let ns = &mut ns[..d_len];
limbs_mul_to_out(&mut scratch, qs, ds_lo);
let mut carry = Limb::iverson(limbs_sub_same_length_in_place_left(ns, &scratch));
if highest_q && limbs_sub_same_length_in_place_left(&mut ns[q_len..], ds_lo) {
carry += 1;
}
while carry != 0 {
if limbs_sub_limb_in_place(qs, 1) {
assert!(highest_q);
highest_q = false;
}
if limbs_slice_add_same_length_in_place_left(ns, ds) {
carry -= 1;
}
}
}
}
highest_q
}}
pub_test! {limbs_div_approx_helper(qs: &mut [Limb], ns: &mut [Limb], ds: &[Limb], d_inv: Limb) {
if ds.len() < DC_DIVAPPR_Q_THRESHOLD {
limbs_div_schoolbook_approx(qs, ns, ds, d_inv);
} else {
limbs_div_divide_and_conquer_approx(qs, ns, ds, d_inv);
}
}}
// Takes the strictly normalized value ds (i.e., most significant bit must be set) as an input, and
// computes the approximate reciprocal of `ds`, with the same length as `ds`. See documentation for
// `limbs_invert_approx` for an explanation of the return value.
//
// # Worst-case complexity
// $T(n) = O(n (\log n)^2 \log \log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ds.len()`.
//
// # Panics
// Panics if `ds` is empty, `is` is shorter than `ds`, `scratch` is shorter than twice the length
// of `ds`, or the last limb of `ds` does not have its highest bit set.
//
// This is equivalent to `mpn_bc_invertappr` from `mpn/generic/invertappr.c`, GMP 6.2.1, where the
// return value is `true` iff the return value of `mpn_bc_invertappr` would be 0.
pub_test! {limbs_invert_basecase_approx(
is: &mut [Limb],
ds: &[Limb],
scratch: &mut [Limb]
) -> bool {
let d_len = ds.len();
assert_ne!(d_len, 0);
let highest_d = ds[d_len - 1];
assert!(highest_d.get_highest_bit());
if d_len == 1 {
let d = ds[0];
is[0] = limbs_invert_limb(d);
} else {
let scratch = &mut scratch[..d_len << 1];
let (scratch_lo, scratch_hi) = scratch.split_at_mut(d_len);
for s in scratch_lo.iter_mut() {
*s = Limb::MAX;
}
limbs_not_to_out(scratch_hi, ds);
// Now scratch contains 2 ^ (2 * d_len * Limb::WIDTH) - d * 2 ^ (d_len * Limb::WIDTH) - 1
if d_len == 2 {
limbs_div_mod_by_two_limb_normalized(is, scratch, ds);
} else {
let d_inv = limbs_two_limb_inverse_helper(highest_d, ds[d_len - 2]);
if !MAYBE_DCP1_DIVAPPR {
limbs_div_schoolbook_approx(is, scratch, ds, d_inv);
} else {
limbs_div_approx_helper(is, scratch, ds, d_inv);
}
assert!(!limbs_sub_limb_in_place(&mut is[..d_len], 1));
return false;
}
}
true
}}
// Takes the strictly normalized value ds (i.e., most significant bit must be set) as an input, and
// computes the approximate reciprocal of `ds`, with the same length as `ds`. See documentation for
// `limbs_invert_approx` for an explanation of the return value.
//
// Uses Newton's iterations (at least one). Inspired by Algorithm "ApproximateReciprocal",
// published in "Modern Computer Arithmetic" by Richard P. Brent and Paul Zimmermann, algorithm
// 3.5, page 121 in version 0.4 of the book.
//
// Some adaptations were introduced, to allow product mod B ^ m - 1 and return the value e.
//
// We introduced a correction in such a way that "the value of B ^ {n + h} - T computed at step 8
// cannot exceed B ^ n - 1" (the book reads "2 * B ^ n - 1").
//
// Maximum scratch needed by this branch <= 2 * n, but have to fit 3 * rn in the scratch, i.e.
// 3 * rn <= 2 * n: we require n > 4.
//
// We use a wrapped product modulo B ^ m - 1.
//
// # Worst-case complexity
// $T(n) = O(n \log n \log \log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `is.len()`.
//
// # Panics
// Panics if `ds` has length less than 5, `is` is shorter than `ds`, `scratch` is shorter than
// twice the length of `ds`, or the last limb of `ds` does not have its highest bit set.
//
// This is equivalent to `mpn_ni_invertappr` from `mpn/generic/invertappr.c`, GMP 6.2.1, where the
// return value is `true` iff the return value of `mpn_ni_invertappr` would be 0.
pub_test! {limbs_invert_newton_approx(is: &mut [Limb], ds: &[Limb], scratch: &mut [Limb]) -> bool {
let d_len = ds.len();
assert!(d_len > 4);
assert!(ds[d_len - 1].get_highest_bit());
let is = &mut is[..d_len];
// Compute the computation precisions from highest to lowest, leaving the base case size in
// 'previous_d'.
let mut size = d_len;
let mut sizes = vec![size];
size = (size >> 1) + 1;
let mut scratch2 = vec![];
let mut mul_size = 0;
if d_len >= INV_MULMOD_BNM1_THRESHOLD {
mul_size = limbs_mul_mod_base_pow_n_minus_1_next_size(d_len + 1);
scratch2 = vec![0; limbs_mul_mod_base_pow_n_minus_1_scratch_len(mul_size, d_len, size)];
}
while size >= INV_NEWTON_THRESHOLD {
sizes.push(size);
size = (size >> 1) + 1;
}
// We compute the inverse of 0.ds as 1.is.
// Compute a base value of previous_d limbs.
limbs_invert_basecase_approx(&mut is[d_len - size..], &ds[d_len - size..], scratch);
let mut previous_size = size;
let mut a = 0;
// Use Newton's iterations to get the desired precision.
for &size in sizes.iter().rev() {
// v d v
// +----+-------+
// ^ previous_d ^
//
// Compute i_j * d
let ds_hi = &ds[d_len - size..];
let condition = size < INV_MULMOD_BNM1_THRESHOLD || {
mul_size = limbs_mul_mod_base_pow_n_minus_1_next_size(size + 1);
mul_size > size + previous_size
};
let diff = size - previous_size;
let is_hi = &mut is[d_len - previous_size..];
if condition {
limbs_mul_greater_to_out(scratch, ds_hi, is_hi);
limbs_slice_add_same_length_in_place_left(
&mut scratch[previous_size..size + 1],
&ds_hi[..diff + 1],
);
} else {
// Remember we truncated mod B ^ (d + 1)
// We computed (truncated) xp of length d + 1 <- 1.is * 0.ds
// Use B ^ mul_size - 1 wraparound
limbs_mul_mod_base_pow_n_minus_1(scratch, mul_size, ds_hi, is_hi, &mut scratch2);
let scratch = &mut scratch[..mul_size + 1];
// We computed {xp, mul_size} <- {is, previous_d} * {ds, d} mod (B ^ mul_size - 1)
// We know that 2 * |is * ds + ds * B ^ previous_d - B ^ {previous_d + d}| <
// B ^ mul_size - 1
// Add ds * B ^ previous_d mod (B ^ mul_size - 1)
let mul_diff = mul_size - previous_size;
assert!(size >= mul_diff);
let (ds_hi_lo, ds_hi_hi) = ds_hi.split_at(mul_diff);
let carry = limbs_slice_add_same_length_in_place_left(
&mut scratch[previous_size..mul_size],
ds_hi_lo,
);
// Subtract B ^ {previous_d + d}, maybe only compensate the carry
scratch[mul_size] = 1; // set a limit for decrement
let (scratch_lo, scratch_hi) = scratch.split_at_mut(size - mul_diff);
if !limbs_add_same_length_with_carry_in_in_place_left(scratch_lo, ds_hi_hi, carry) {
assert!(!limbs_sub_limb_in_place(scratch_hi, 1));
}
// if decrement eroded xp[mul_size]
let (scratch_last, scratch_init) = scratch.split_last_mut().unwrap();
assert!(!limbs_sub_limb_in_place(
scratch_init,
1.wrapping_sub(*scratch_last)
));
// Remember we are working mod B ^ mul_size - 1
}
if scratch[size] < 2 {
// "positive" residue class
let (scratch_lo, scratch_hi) = scratch.split_at_mut(size);
let mut carry = scratch_hi[0] + 1; // 1 <= carry <= 2 here.
if carry == 2 && !limbs_sub_same_length_in_place_left(scratch_lo, ds_hi) {
carry = 3;
assert!(limbs_sub_same_length_in_place_left(scratch_lo, ds_hi));
}
// 1 <= carry <= 3 here.
if limbs_cmp_same_length(scratch_lo, ds_hi) == Ordering::Greater {
assert!(!limbs_sub_same_length_in_place_left(scratch_lo, ds_hi));
carry += 1;
}
let (scratch_lo, scratch_mid) = scratch_lo.split_at_mut(diff);
let (ds_hi_lo, ds_hi_hi) = ds_hi.split_at(diff);
let borrow = limbs_cmp_same_length(scratch_lo, ds_hi_lo) == Ordering::Greater;
assert!(!limbs_sub_same_length_with_borrow_in_to_out(
&mut scratch_hi[diff..],
ds_hi_hi,
scratch_mid,
borrow
));
assert!(!limbs_sub_limb_in_place(is_hi, carry)); // 1 <= carry <= 4 here
} else {
// "negative" residue class
assert!(scratch[size] >= Limb::MAX - 1);
if condition {
assert!(!limbs_sub_limb_in_place(&mut scratch[..size + 1], 1));
}
let (scratch_lo, scratch_hi) = scratch.split_at_mut(size);
if scratch_hi[0] != Limb::MAX {
assert!(!limbs_slice_add_limb_in_place(is_hi, 1));
assert!(limbs_slice_add_same_length_in_place_left(scratch_lo, ds_hi));
}
limbs_not_to_out(&mut scratch_hi[diff..size], &scratch_lo[diff..]);
}
// Compute x_j * u_j
let (scratch_lo, scratch_hi) = scratch.split_at_mut(size + diff);
limbs_mul_same_length_to_out(scratch_lo, &scratch_hi[..previous_size], is_hi);
a = (previous_size << 1) - diff;
let carry = {
let (scratch_lo, scratch_hi) = scratch.split_at_mut(a);
limbs_slice_add_same_length_in_place_left(
&mut scratch_lo[previous_size..],
&scratch_hi[3 * diff - previous_size..diff << 1],
)
};
if limbs_add_same_length_with_carry_in_to_out(
&mut is[d_len - size..],
&scratch[a..previous_size << 1],
&scratch[size + previous_size..size << 1],
carry,
) {
assert!(!limbs_slice_add_limb_in_place(
&mut is[d_len - previous_size..],
1
));
}
previous_size = size;
}
// Check for possible carry propagation from below. Be conservative.
scratch[a - 1] <= Limb::MAX - 7
}}
// Takes the strictly normalized value ds (i.e., most significant bit must be set) as an input, and
// computes the approximate reciprocal of `ds`, with the same length as `ds`.
//
// Let result_definitely_exact = limbs_invert_basecase_approx(is, ds, scratch) be the returned
// value. If result_definitely_exact is `true`, the error e is 0; otherwise, it may be 0 or 1. The
// following condition is satisfied by the output:
//
// ds * (2 ^ (n * Limb::WIDTH) + is) < 2 ^ (2 * n * Limb::WIDTH) <=
// ds * (2 ^ (n * Limb::WIDTH) + is + 1 + e),
// where n = `ds.len()`.
//
// When the strict result is needed, i.e., e = 0 in the relation above, the function `mpn_invert`
// (TODO!) should be used instead.
//
// # Worst-case complexity
// $T(n) = O(n \log n \log \log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `is.len()`.
//
// # Panics
// Panics if `ds` is empty, `is` is shorter than `ds`, `scratch` is shorter than twice the length
// of `ds`, or the last limb of `ds` does not have its highest bit set.
//
// This is equivalent to `mpn_invertappr` from `mpn/generic/invertappr.c`, GMP 6.2.1, where the
// return value is `true` iff the return value of `mpn_invertappr` would be 0.
pub_crate_test! {limbs_invert_approx(is: &mut [Limb], ds: &[Limb], scratch: &mut [Limb]) -> bool {
if ds.len() < INV_NEWTON_THRESHOLD {
limbs_invert_basecase_approx(is, ds, scratch)
} else {
limbs_invert_newton_approx(is, ds, scratch)
}
}}
//TODO tune
pub(crate) const MUL_TO_MULMOD_BNM1_FOR_2NXN_THRESHOLD: usize = INV_MULMOD_BNM1_THRESHOLD >> 1;
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ds.len()`.
//
// ds.len() >= 2
// n_len >= 3
// n_len >= ds.len()
// i_len == limbs_div_mod_barrett_is_len(n_len - ds.len(), ds.len())
// qs.len() == i_len
// scratch_len == limbs_mul_mod_base_pow_n_minus_1_next_size(ds.len() + 1)
// scratch.len() == limbs_div_mod_barrett_scratch_len(n_len, d_len) - i_len
// rs_hi.len() == i_len
pub_crate_test! {limbs_div_barrett_large_product(
scratch: &mut [Limb],
ds: &[Limb],
qs: &[Limb],
rs_hi: &[Limb],
scratch_len: usize,
i_len: usize,
) {
let d_len = ds.len();
let (scratch, scratch_out) = scratch.split_at_mut(scratch_len);
limbs_mul_mod_base_pow_n_minus_1(scratch, scratch_len, ds, qs, scratch_out);
if d_len + i_len > scratch_len {
let (rs_hi_lo, rs_hi_hi) = rs_hi.split_at(scratch_len - d_len);
let carry_1 = limbs_sub_greater_in_place_left(scratch, rs_hi_hi);
let carry_2 = limbs_cmp_same_length(rs_hi_lo, &scratch[d_len..]) == Ordering::Less;
if !carry_1 && carry_2 {
assert!(!limbs_slice_add_limb_in_place(scratch, 1));
} else {
assert_eq!(carry_1, carry_2);
}
}
}}
// # Worst-case complexity
// $T(n, d) = O(n \log d \log\log d)$
//
// $M(n) = O(d \log d)$
//
// where $T$ is time, $M$ is additional memory, $n$ is `ns.len()`, and $d$ is `ds.len()`.
//
// This is equivalent to `mpn_preinv_mu_div_qr` from `mpn/generic/mu_div_qr.c`, GMP 6.2.1.
fn limbs_div_mod_barrett_preinverted(
qs: &mut [Limb],
rs: &mut [Limb],
ns: &[Limb],
ds: &[Limb],
mut is: &[Limb],
scratch: &mut [Limb],
) -> bool {
let n_len = ns.len();
let d_len = ds.len();
assert_eq!(rs.len(), d_len);
let mut i_len = is.len();
let q_len = n_len - d_len;
let qs = &mut qs[..q_len];
let (ns_lo, ns_hi) = ns.split_at(q_len);
let highest_q = limbs_cmp_same_length(ns_hi, ds) >= Ordering::Equal;
if highest_q {
limbs_sub_same_length_to_out(rs, ns_hi, ds);
} else {
rs.copy_from_slice(ns_hi);
}
let scratch_len = if i_len < MUL_TO_MULMOD_BNM1_FOR_2NXN_THRESHOLD {
0
} else {
limbs_mul_mod_base_pow_n_minus_1_next_size(d_len + 1)
};
let mut n = d_len - i_len;
for (ns, qs) in ns_lo.rchunks(i_len).zip(qs.rchunks_mut(i_len)) {
let chunk_len = ns.len();
if i_len != chunk_len {
// last iteration
is = &is[i_len - chunk_len..];
i_len = chunk_len;
n = d_len - i_len;
}
let (rs_lo, rs_hi) = rs.split_at_mut(n);
// Compute the next block of quotient limbs by multiplying the inverse by the upper part of
// the partial remainder.
limbs_mul_same_length_to_out(scratch, rs_hi, is);
// The inverse's most significant bit is implicit.
assert!(!limbs_add_same_length_to_out(
qs,
&scratch[i_len..i_len << 1],
rs_hi,
));
// Compute the product of the quotient block and the divisor, to be subtracted from the
// partial remainder combined with new limbs from the dividend. We only really need the low
// d_len + 1 limbs.
if i_len < MUL_TO_MULMOD_BNM1_FOR_2NXN_THRESHOLD {
limbs_mul_greater_to_out(scratch, ds, qs);
} else {
limbs_div_barrett_large_product(scratch, ds, qs, rs_hi, scratch_len, i_len)
}
let mut r = rs_hi[0].wrapping_sub(scratch[d_len]);
// Subtract the product from the partial remainder combined with new limbs from the
// dividend, generating a new partial remainder.
let scratch = &mut scratch[..d_len];
let carry = if n == 0 {
// Get next i_len limbs from n.
limbs_sub_same_length_to_out(rs, ns, scratch)
} else {
let (scratch_lo, scratch_hi) = scratch.split_at_mut(i_len);
// Get next i_len limbs from n.
let carry = limbs_sub_same_length_with_borrow_in_in_place_right(
rs_lo,
scratch_hi,
limbs_sub_same_length_in_place_right(ns, scratch_lo),
);
rs.copy_from_slice(scratch);
carry
};
// Check the remainder and adjust the quotient as needed.
if carry {
r.wrapping_sub_assign(1);
}
while r != 0 {
// We loop 0 times with about 69% probability, 1 time with about 31% probability, and 2
// times with about 0.6% probability, if the inverse is computed as recommended.
assert!(!limbs_slice_add_limb_in_place(qs, 1));
if limbs_sub_same_length_in_place_left(rs, ds) {
r -= 1;
}
}
if limbs_cmp_same_length(rs, ds) >= Ordering::Equal {
// This is executed with about 76% probability.
assert!(!limbs_slice_add_limb_in_place(qs, 1));
limbs_sub_same_length_in_place_left(rs, ds);
}
}
highest_q
}
// We distinguish 3 cases:
//
// (a) d_len < q_len: i_len = ceil(q_len / ceil(q_len / d_len))
// (b) d_len / 3 < q_len <= d_len: i_len = ceil(q_len / 2)
// (c) q_len < d_len / 3: i_len = q_len
//
// In all cases we have i_len <= d_len.
//
// # Worst-case complexity
// Constant time and additional memory.
//
// Result is O(`q_len`)
//
// This is equivalent to `mpn_mu_div_qr_choose_in` from `mpn/generic/mu_div_qr.c`, GMP 6.2.1, where
// `k == 0`.
pub_const_crate_test! {limbs_div_mod_barrett_is_len(q_len: usize, d_len: usize) -> usize {
let q_len_minus_1 = q_len - 1;
if q_len > d_len {
// Compute an inverse size that is a nice partition of the quotient.
let b = q_len_minus_1 / d_len + 1; // ceil(q_len / d_len), number of blocks
q_len_minus_1 / b + 1 // ceil(q_len / b) = ceil(q_len / ceil(q_len / d_len))
} else if 3 * q_len > d_len {
(q_len_minus_1 >> 1) + 1 // b = 2
} else {
q_len_minus_1 + 1 // b = 1
}
}}
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
//
// This is equivalent to `mpn_mu_div_qr2` from `mpn/generic/mu_div_qr.c`, GMP 6.2.1.
pub_crate_test! {limbs_div_mod_barrett_helper(
qs: &mut [Limb],
rs: &mut [Limb],
ns: &[Limb],
ds: &[Limb],
scratch: &mut [Limb],
) -> bool {
let n_len = ns.len();
let d_len = ds.len();
assert_eq!(rs.len(), d_len);
assert!(d_len > 1);
assert!(n_len > d_len);
let q_len = n_len - d_len;
// Compute the inverse size.
let i_len = limbs_div_mod_barrett_is_len(q_len, d_len);
assert!(i_len <= d_len);
let i_len_plus_1 = i_len + 1;
let (is, scratch_hi) = scratch.split_at_mut(i_len_plus_1);
// compute an approximate inverse on i_len + 1 limbs
if d_len == i_len {
let (scratch_lo, scratch_hi) = scratch_hi.split_at_mut(i_len_plus_1);
let (scratch_first, scratch_lo_tail) = scratch_lo.split_first_mut().unwrap();
scratch_lo_tail.copy_from_slice(&ds[..i_len]);
*scratch_first = 1;
limbs_invert_approx(is, scratch_lo, scratch_hi);
slice_move_left(is, 1);
} else if limbs_add_limb_to_out(scratch_hi, &ds[d_len - i_len_plus_1..], 1) {
slice_set_zero(&mut is[..i_len]);
} else {
let (scratch_lo, scratch_hi) = scratch_hi.split_at_mut(i_len_plus_1);
limbs_invert_approx(is, scratch_lo, scratch_hi);
slice_move_left(is, 1);
}
let (scratch_lo, scratch_hi) = scratch.split_at_mut(i_len);
limbs_div_mod_barrett_preinverted(qs, rs, ns, ds, scratch_lo, scratch_hi)
}}
// # Worst-case complexity
// Constant time and additional memory.
//
// Result is O(`d_len`)
//
// This is equivalent to `mpn_preinv_mu_div_qr_itch` from `mpn/generic/mu_div_qr.c`, GMP 6.2.1, but
// `nn` is omitted from the arguments as it is unused.
fn limbs_div_mod_barrett_preinverse_scratch_len(d_len: usize, is_len: usize) -> usize {
let itch_local = limbs_mul_mod_base_pow_n_minus_1_next_size(d_len + 1);
let itch_out = limbs_mul_mod_base_pow_n_minus_1_scratch_len(itch_local, d_len, is_len);
itch_local + itch_out
}
// # Worst-case complexity
// Constant time and additional memory.
//
// This is equivalent to `mpn_invertappr_itch` from `gmp-impl.h`, GMP 6.2.1.
pub(crate) const fn limbs_invert_approx_scratch_len(is_len: usize) -> usize {
is_len << 1
}
// # Worst-case complexity
// Constant time and additional memory.
//
// Result is O(`n_len`)
//
// This is equivalent to `mpn_mu_div_qr_itch` from `mpn/generic/mu_div_qr.c`, GMP 6.2.1, where
// `mua_k == 0`.
pub_crate_test! {limbs_div_mod_barrett_scratch_len(n_len: usize, d_len: usize) -> usize {
let is_len = limbs_div_mod_barrett_is_len(n_len - d_len, d_len);
let preinverse_len = limbs_div_mod_barrett_preinverse_scratch_len(d_len, is_len);
// 3 * is_len + 4
let inv_approx_len = limbs_invert_approx_scratch_len(is_len + 1) + is_len + 2;
assert!(preinverse_len >= inv_approx_len);
is_len + preinverse_len
}}
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ds.len()`.
pub_test! {limbs_div_mod_barrett_large_helper(
qs: &mut [Limb],
rs: &mut [Limb],
ns: &[Limb],
ds: &[Limb],
scratch: &mut [Limb],
) -> bool {
let n_len = ns.len();
let d_len = ds.len();
let q_len = qs.len();
let q_len_plus_one = q_len + 1;
let n = n_len - q_len - q_len_plus_one; // 2 * d_len - n_len - 1
let (ns_lo, ns_hi) = ns.split_at(n);
let (ds_lo, ds_hi) = ds.split_at(d_len - q_len_plus_one);
let (rs_lo, rs_hi) = rs.split_at_mut(n);
let rs_hi = &mut rs_hi[..q_len_plus_one];
let mut highest_q = limbs_div_mod_barrett_helper(qs, rs_hi, ns_hi, ds_hi, scratch);
// Multiply the quotient by the divisor limbs ignored above.
// The product is d_len - 1 limbs long.
limbs_mul_to_out(scratch, ds_lo, qs);
let (scratch_last, scratch_init) = scratch[..d_len].split_last_mut().unwrap();
*scratch_last = Limb::iverson(
highest_q && limbs_slice_add_same_length_in_place_left(&mut scratch_init[q_len..], ds_lo),
);
let (scratch_lo, scratch_hi) = scratch.split_at(n);
let scratch_hi = &scratch_hi[..q_len_plus_one];
if limbs_sub_same_length_with_borrow_in_in_place_left(
rs_hi,
scratch_hi,
limbs_sub_same_length_to_out(rs_lo, ns_lo, scratch_lo),
) {
if limbs_sub_limb_in_place(qs, 1) {
assert!(highest_q);
highest_q = false;
}
limbs_slice_add_same_length_in_place_left(&mut rs[..d_len], ds);
}
highest_q
}}
// Block-wise Barrett division. The idea of the algorithm used herein is to compute a smaller
// inverted value than used in the standard Barrett algorithm, and thus save time in the Newton
// iterations, and pay just a small price when using the inverted value for developing quotient
// bits. This algorithm was presented at ICMS 2006.
//
// `ns` must have length at least 3, `ds` must have length at least 2 and be no longer than `ns`,
// and the most significant bit of `ds` must be set.
//
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
//
// # Panics
// Panics if `ds` has length smaller than 2, `ns.len()` is less than `ds.len()`, `qs` has length
// less than `ns.len()` - `ds.len()`, `scratch` is too short, or the last limb of `ds` does not
// have its highest bit set.
//
// This is equivalent to `mpn_mu_div_qr` from `mpn/generic/mu_div_qr.c`, GMP 6.2.1.
pub_test! {limbs_div_mod_barrett(
qs: &mut [Limb],
rs: &mut [Limb],
ns: &[Limb],
ds: &[Limb],
scratch: &mut [Limb],
) -> bool {
let n_len = ns.len();
let d_len = ds.len();
let q_len = n_len - d_len;
let qs = &mut qs[..q_len];
// Test whether 2 * d_len - n_len > MU_DIV_QR_SKEW_THRESHOLD
if d_len <= q_len + MU_DIV_QR_SKEW_THRESHOLD {
limbs_div_mod_barrett_helper(qs, &mut rs[..d_len], ns, ds, scratch)
} else {
limbs_div_mod_barrett_large_helper(qs, rs, ns, ds, scratch)
}
}}
// `ds` must have length 2, `ns` must have length at least 2, `qs` must have length at least
// `ns.len() - 2`, `rs` must have length at least 2, and the most-significant limb of `ds` must be
// nonzero.
//
// # Worst-case complexity
// $T(n) = O(n)$
//
// $M(n) = O(n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
fn limbs_div_mod_by_two_limb(qs: &mut [Limb], rs: &mut [Limb], ns: &[Limb], ds: &[Limb]) {
let n_len = ns.len();
let ds_1 = ds[1];
let bits = LeadingZeros::leading_zeros(ds_1);
if bits == 0 {
let mut ns = ns.to_vec();
// always store n_len - 1 quotient limbs
qs[n_len - 2] = Limb::iverson(limbs_div_mod_by_two_limb_normalized(qs, &mut ns, ds));
rs[0] = ns[0];
rs[1] = ns[1];
} else {
let ds_0 = ds[0];
let cobits = Limb::WIDTH - bits;
let mut ns_shifted = vec![0; n_len + 1];
let ns_shifted = &mut ns_shifted;
let carry = limbs_shl_to_out(ns_shifted, ns, bits);
let ds_shifted = &mut [ds_0 << bits, (ds_1 << bits) | (ds_0 >> cobits)];
if carry == 0 {
// always store n_len - 1 quotient limbs
qs[n_len - 2] = Limb::iverson(limbs_div_mod_by_two_limb_normalized(
qs,
&mut ns_shifted[..n_len],
ds_shifted,
));
} else {
ns_shifted[n_len] = carry;
limbs_div_mod_by_two_limb_normalized(qs, ns_shifted, ds_shifted);
}
let ns_shifted_1 = ns_shifted[1];
rs[0] = (ns_shifted[0] >> bits) | (ns_shifted_1 << cobits);
rs[1] = ns_shifted_1 >> bits;
}
}
//TODO tune
pub(crate) const MUPI_DIV_QR_THRESHOLD: usize = 74;
// # Worst-case complexity
// Constant time and additional memory.
fn limbs_div_mod_dc_condition(n_len: usize, d_len: usize) -> bool {
let n_64 = n_len as f64;
let d_64 = d_len as f64;
d_len < MUPI_DIV_QR_THRESHOLD
|| n_len < MU_DIV_QR_THRESHOLD << 1
|| (((MU_DIV_QR_THRESHOLD - MUPI_DIV_QR_THRESHOLD) << 1) as f64)
.mul_add(d_64, MUPI_DIV_QR_THRESHOLD as f64 * n_64)
> d_64 * n_64
}
// This function is optimized for the case when the numerator has at least twice the length of the
// denominator.
//
// `ds` must have length at least 3, `ns` must be at least as long as `ds`, `qs` must have length
// at least `ns.len() - ds.len() + 1`, `rs` must have the same length as `ds`, and the most-
// significant limb of `ds` must be nonzero.
//
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
fn limbs_div_mod_unbalanced(
qs: &mut [Limb],
rs: &mut [Limb],
ns: &[Limb],
ds: &[Limb],
adjusted_n_len: usize,
) {
let mut n_len = ns.len();
let d_len = ds.len();
qs[n_len - d_len] = 0; // zero high quotient limb
let mut ds_shifted_vec;
let ds_shifted: &[Limb];
let mut ns_shifted_vec = vec![0; n_len + 1];
let ns_shifted = &mut ns_shifted_vec;
let bits = LeadingZeros::leading_zeros(*ds.last().unwrap());
if bits == 0 {
ds_shifted = ds;
ns_shifted[..n_len].copy_from_slice(ns);
} else {
// normalize divisor
ds_shifted_vec = vec![0; d_len];
limbs_shl_to_out(&mut ds_shifted_vec, ds, bits);
ds_shifted = &ds_shifted_vec;
let (ns_shifted_last, ns_shifted_init) = ns_shifted.split_last_mut().unwrap();
*ns_shifted_last = limbs_shl_to_out(ns_shifted_init, ns, bits);
}
n_len = adjusted_n_len;
let d_inv = limbs_two_limb_inverse_helper(ds_shifted[d_len - 1], ds_shifted[d_len - 2]);
let ns_shifted = &mut ns_shifted[..n_len];
if d_len < DC_DIV_QR_THRESHOLD {
limbs_div_mod_schoolbook(qs, ns_shifted, ds_shifted, d_inv);
let ns_shifted = &ns_shifted[..d_len];
if bits == 0 {
rs.copy_from_slice(ns_shifted);
} else {
limbs_shr_to_out(rs, ns_shifted, bits);
}
} else if limbs_div_mod_dc_condition(n_len, d_len) {
limbs_div_mod_divide_and_conquer(qs, ns_shifted, ds_shifted, d_inv);
let ns_shifted = &ns_shifted[..d_len];
if bits == 0 {
rs.copy_from_slice(ns_shifted);
} else {
limbs_shr_to_out(rs, ns_shifted, bits);
}
} else {
let scratch_len = limbs_div_mod_barrett_scratch_len(n_len, d_len);
let mut scratch = vec![0; scratch_len];
limbs_div_mod_barrett(qs, rs, ns_shifted, ds_shifted, &mut scratch);
if bits != 0 {
limbs_slice_shr_in_place(rs, bits);
}
}
}
// The numerator must have less than twice the length of the denominator.
//
// Problem:
//
// Divide a numerator N with `n_len` limbs by a denominator D with `d_len` limbs, forming a
// quotient of `q_len` = `n_len` - `d_len` + 1 limbs. When `q_len` is small compared to `d_len`,
// conventional division algorithms perform poorly. We want an algorithm that has an expected
// running time that is dependent only on `q_len`.
//
// Algorithm (very informally stated):
//
// 1) Divide the 2 * `q_len` most significant limbs from the numerator by the `q_len` most-
// significant limbs from the denominator. Call the result `qest`. This is either the correct
// quotient, or 1 or 2 too large. Compute the remainder from the division.
//
// 2) Is the most significant limb from the remainder < p, where p is the product of the most-
// significant limb from the quotient and the next(d)? (Next(d) denotes the next ignored limb from
// the denominator.) If it is, decrement `qest`, and adjust the remainder accordingly.
//
// 3) Is the remainder >= `qest`? If it is, `qest` is the desired quotient. The algorithm
// terminates.
//
// 4) Subtract `qest` * next(d) from the remainder. If there is borrow out, decrement `qest`, and
// adjust the remainder accordingly.
//
// 5) Skip one word from the denominator (i.e., let next(d) denote the next less significant limb).
//
// `ds` must have length at least 3, `ns` must be at least as long as `ds` but no more than twice
// as long, `qs` must have length at least `ns.len() - ds.len() + 1`,`rs` must have the same length
// as `ds`, and the most-significant limb of `ds` must be nonzero.
//
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
pub(crate) fn limbs_div_mod_balanced(
qs: &mut [Limb],
rs: &mut [Limb],
ns: &[Limb],
ds: &[Limb],
adjust: bool,
) {
let n_len = ns.len();
let d_len = ds.len();
let mut q_len = n_len - d_len;
assert!(d_len >= q_len);
qs[q_len] = 0; // zero high quotient limb
if adjust {
q_len += 1;
} else if q_len == 0 {
rs.copy_from_slice(&ns[..d_len]);
return;
}
let q_len = q_len;
// `i_len` is the (at least partially) ignored number of limbs.
let i_len = d_len - q_len;
// Normalize the denominator by shifting it to the left such that its most significant bit is
// set. Then shift the numerator the same amount, to mathematically preserve the quotient.
let bits = LeadingZeros::leading_zeros(ds[d_len - 1]);
let cobits = Limb::WIDTH - bits;
let q_len_2 = q_len << 1;
let m = n_len - q_len_2;
let mut ns_shifted_vec = vec![0; q_len_2 + 1];
let mut ds_shifted_vec;
let ds_shifted: &[Limb];
let ds_hi = &ds[i_len..];
let ds_lo_last = ds[i_len - 1];
let carry = if bits == 0 {
ds_shifted = ds_hi;
ns_shifted_vec[..q_len_2].copy_from_slice(&ns[m..]);
0
} else {
ds_shifted_vec = vec![0; q_len];
limbs_shl_to_out(&mut ds_shifted_vec, ds_hi, bits);
ds_shifted_vec[0] |= ds_lo_last >> cobits;
ds_shifted = &ds_shifted_vec;
let carry = limbs_shl_to_out(&mut ns_shifted_vec, &ns[m..], bits);
if !adjust {
ns_shifted_vec[0] |= ns[m - 1] >> cobits;
}
carry
};
let ns_shifted = if adjust {
ns_shifted_vec[q_len_2] = carry;
&mut ns_shifted_vec[1..]
} else {
&mut ns_shifted_vec
};
// Get an approximate quotient using the extracted operands.
if q_len == 1 {
(qs[0], ns_shifted[0]) =
Limb::xx_div_mod_y_to_qr(ns_shifted[1], ns_shifted[0], ds_shifted[0]);
} else if q_len == 2 {
limbs_div_mod_by_two_limb_normalized(qs, ns_shifted, ds_shifted);
} else {
let ns_shifted = &mut ns_shifted[..q_len_2];
let d_inv = limbs_two_limb_inverse_helper(ds_shifted[q_len - 1], ds_shifted[q_len - 2]);
if q_len < DC_DIV_QR_THRESHOLD {
limbs_div_mod_schoolbook(qs, ns_shifted, ds_shifted, d_inv);
} else if q_len < MU_DIV_QR_THRESHOLD {
limbs_div_mod_divide_and_conquer(qs, ns_shifted, ds_shifted, d_inv);
} else {
let mut scratch = vec![0; limbs_div_mod_barrett_scratch_len(q_len_2, q_len)];
limbs_div_mod_barrett(qs, rs, ns_shifted, ds_shifted, &mut scratch);
ns_shifted[..q_len].copy_from_slice(&rs[..q_len]);
}
}
// Multiply the first ignored divisor limb by the most significant quotient limb. If that
// product is > the partial remainder's most significant limb, we know the quotient is too
// large. This test quickly catches most cases where the quotient is too large; it catches all
// cases where the quotient is 2 too large.
let mut r_len = q_len;
let mut x = ds_lo_last << bits;
if i_len >= 2 {
x |= ds[i_len - 2] >> 1 >> (!bits & Limb::WIDTH_MASK);
}
if ns_shifted[q_len - 1] < (DoubleLimb::from(x) * DoubleLimb::from(qs[q_len - 1])).upper_half()
{
assert!(!limbs_sub_limb_in_place(qs, 1));
let carry = limbs_slice_add_same_length_in_place_left(&mut ns_shifted[..q_len], ds_shifted);
if carry {
// The partial remainder is safely large.
ns_shifted[q_len] = 1;
r_len += 1;
}
}
let mut q_too_large = false;
let mut do_extra_cleanup = true;
let mut scratch = vec![0; d_len];
let mut i_len_alt = i_len;
let qs_lo = &mut qs[..q_len];
if bits != 0 {
// Append the partially used numerator limb to the partial remainder.
let carry_1 = limbs_slice_shl_in_place(&mut ns_shifted[..r_len], cobits);
let mask = Limb::MAX >> bits;
ns_shifted[0] |= ns[i_len - 1] & mask;
// Update partial remainder with partially used divisor limb.
let (ns_shifted_last, ns_shifted_init) = ns_shifted[..q_len + 1].split_last_mut().unwrap();
let carry_2 = limbs_sub_mul_limb_same_length_in_place_left(
ns_shifted_init,
qs_lo,
ds[i_len - 1] & mask,
);
if q_len != r_len {
assert!(*ns_shifted_last >= carry_2);
ns_shifted_last.wrapping_sub_assign(carry_2);
} else {
(*ns_shifted_last, q_too_large) = carry_1.overflowing_sub(carry_2);
r_len += 1;
}
i_len_alt -= 1;
}
// True: partial remainder now is neutral, i.e., it is not shifted up.
if i_len_alt == 0 {
rs.copy_from_slice(&ns_shifted[..r_len]);
do_extra_cleanup = false;
} else {
limbs_mul_to_out(&mut scratch, qs_lo, &ds[..i_len_alt]);
}
if do_extra_cleanup {
let (scratch_lo, scratch_hi) = scratch.split_at_mut(i_len_alt);
q_too_large |=
limbs_sub_greater_in_place_left(&mut ns_shifted[..r_len], &scratch_hi[..q_len]);
let (rs_lo, rs_hi) = rs.split_at_mut(i_len_alt);
let rs_hi_len = rs_hi.len();
rs_hi.copy_from_slice(&ns_shifted[..rs_hi_len]);
q_too_large |= limbs_sub_same_length_to_out(rs_lo, &ns[..i_len_alt], scratch_lo)
&& limbs_sub_limb_in_place(&mut rs_hi[..min(rs_hi_len, r_len)], 1);
}
if q_too_large {
assert!(!limbs_sub_limb_in_place(qs, 1));
limbs_slice_add_same_length_in_place_left(rs, ds);
}
}
// Interpreting two slices of `Limb`s, `ns` and `ds`, as the limbs (in ascending order) of two
// `Natural`s, divides them, returning the quotient and remainder. The quotient has
// `ns.len() - ds.len() + 1` limbs and the remainder `ds.len()` limbs.
//
// `ns` must be at least as long as `ds` and `ds` must have length at least 2 and its most
// significant limb must be greater than zero.
//
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
//
// # Panics
// Panics if `ns` is shorter than `ds`, `ds` has length less than 2, or the most-significant limb
// of `ds` is zero.
//
// This is equivalent to `mpn_tdiv_qr` from `mpn/generic/tdiv_qr.c`, GMP 6.2.1, where `dn > 1` and
// `qp` and `rp` are returned.
pub_test! {limbs_div_mod(ns: &[Limb], ds: &[Limb]) -> (Vec<Limb>, Vec<Limb>) {
let d_len = ds.len();
let mut qs = vec![0; ns.len() - d_len + 1];
let mut rs = vec![0; d_len];
limbs_div_mod_to_out(&mut qs, &mut rs, ns, ds);
(qs, rs)
}}
// Interpreting two slices of `Limb`s, `ns` and `ds`, as the limbs (in ascending order) of two
// `Natural`s, divides them, writing the `ns.len() - ds.len() + 1` limbs of the quotient to `qs`
// and the `ds.len()` limbs of the remainder to `rs`.
//
// `ns` must be at least as long as `ds`, `qs` must have length at least `ns.len() - ds.len() + 1`,
// `rs` must be at least as long as `ds`, and `ds` must have length at least 2 and its most
// significant limb must be greater than zero.
//
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
//
// # Panics
// Panics if `qs` or `rs` are too short, `ns` is shorter than `ds`, `ds` has length less than 2, or
// the most-significant limb of `ds` is zero.
//
// This is equivalent to `mpn_tdiv_qr` from `mpn/generic/tdiv_qr.c`, GMP 6.2.1, where `dn > 1`.
pub_test! {limbs_div_mod_to_out(qs: &mut [Limb], rs: &mut [Limb], ns: &[Limb], ds: &[Limb]) {
let n_len = ns.len();
let d_len = ds.len();
assert!(d_len > 1);
assert!(n_len >= d_len);
assert!(qs.len() > n_len - d_len);
let rs = &mut rs[..d_len];
let ds_last = *ds.last().unwrap();
assert!(ds_last != 0);
if d_len == 2 {
limbs_div_mod_by_two_limb(qs, rs, ns, ds);
} else {
// conservative tests for quotient size
let adjust = ns[n_len - 1] >= ds_last;
let adjusted_n_len = if adjust { n_len + 1 } else { n_len };
if adjusted_n_len < d_len << 1 {
limbs_div_mod_balanced(qs, rs, ns, ds, adjust);
} else {
limbs_div_mod_unbalanced(qs, rs, ns, ds, adjusted_n_len);
}
}
}}
// TODO improve!
//
// # Worst-case complexity
// $T(n) = O(n \log n \log\log n)$
//
// $M(n) = O(n \log n)$
//
// where $T$ is time, $M$ is additional memory, and $n$ is `ns.len()`.
pub_crate_test! {limbs_div_mod_qs_to_out_rs_to_ns(qs: &mut [Limb], ns: &mut [Limb], ds: &[Limb]) {
let ns_copy = ns.to_vec();
limbs_div_mod_to_out(qs, ns, &ns_copy, ds);
}}
impl Natural {
fn div_mod_limb_ref(&self, other: Limb) -> (Natural, Limb) {
match (self, other) {
(_, 0) => panic!("division by zero"),
(n, 1) => (n.clone(), 0),
(Natural(Small(small)), other) => {
let (q, r) = small.div_rem(other);
(Natural(Small(q)), r)
}
(Natural(Large(ref limbs)), other) => {
let (qs, r) = limbs_div_limb_mod(limbs, other);
(Natural::from_owned_limbs_asc(qs), r)
}
}
}
pub_test! {div_assign_mod_limb(&mut self, other: Limb) -> Limb {
match (&mut *self, other) {
(_, 0) => panic!("division by zero"),
(_, 1) => 0,
(Natural(Small(ref mut small)), other) => small.div_assign_rem(other),
(Natural(Large(ref mut limbs)), other) => {
let r = limbs_div_limb_in_place_mod(limbs, other);
self.trim();
r
}
}
}}
}
impl DivMod<Natural> for Natural {
type DivOutput = Natural;
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking both by value and returning the
/// quotient and remainder. The quotient is rounded towards negative infinity.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lfloor \frac{x}{y} \right \rfloor, \space
/// x - y\left \lfloor \frac{x}{y} \right \rfloor \right ).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivMod;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// assert_eq!(Natural::from(23u32).div_mod(Natural::from(10u32)).to_debug_string(), "(2, 3)");
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// assert_eq!(
/// Natural::from_str("1000000000000000000000000").unwrap()
/// .div_mod(Natural::from_str("1234567890987").unwrap()).to_debug_string(),
/// "(810000006723, 530068894399)"
/// );
/// ```
#[inline]
fn div_mod(mut self, other: Natural) -> (Natural, Natural) {
let r = self.div_assign_mod(other);
(self, r)
}
}
impl<'a> DivMod<&'a Natural> for Natural {
type DivOutput = Natural;
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking the first by value and the second by
/// reference and returning the quotient and remainder. The quotient is rounded towards
/// negative infinity.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lfloor \frac{x}{y} \right \rfloor, \space
/// x - y\left \lfloor \frac{x}{y} \right \rfloor \right ).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivMod;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// assert_eq!(
/// Natural::from(23u32).div_mod(&Natural::from(10u32)).to_debug_string(),
/// "(2, 3)"
/// );
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// assert_eq!(
/// Natural::from_str("1000000000000000000000000").unwrap()
/// .div_mod(&Natural::from_str("1234567890987").unwrap()).to_debug_string(),
/// "(810000006723, 530068894399)"
/// );
/// ```
#[inline]
fn div_mod(mut self, other: &'a Natural) -> (Natural, Natural) {
let r = self.div_assign_mod(other);
(self, r)
}
}
impl<'a> DivMod<Natural> for &'a Natural {
type DivOutput = Natural;
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking the first by reference and the second
/// by value and returning the quotient and remainder. The quotient is rounded towards negative
/// infinity.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lfloor \frac{x}{y} \right \rfloor, \space
/// x - y\left \lfloor \frac{x}{y} \right \rfloor \right ).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivMod;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// assert_eq!(
/// (&Natural::from(23u32)).div_mod(Natural::from(10u32)).to_debug_string(),
/// "(2, 3)"
/// );
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// assert_eq!(
/// (&Natural::from_str("1000000000000000000000000").unwrap())
/// .div_mod(Natural::from_str("1234567890987").unwrap()).to_debug_string(),
/// "(810000006723, 530068894399)"
/// );
/// ```
fn div_mod(self, mut other: Natural) -> (Natural, Natural) {
if *self == other {
return (Natural::ONE, Natural::ZERO);
}
match (self, &mut other) {
(_, natural_zero!()) => panic!("division by zero"),
(n, natural_one!()) => (n.clone(), Natural::ZERO),
(n, &mut Natural(Small(d))) => {
let (q, r) = n.div_mod_limb_ref(d);
(q, Natural(Small(r)))
}
(Natural(Small(_)), _) => (Natural::ZERO, self.clone()),
(&Natural(Large(ref ns)), &mut Natural(Large(ref mut ds))) => {
if ns.len() < ds.len() {
(Natural::ZERO, self.clone())
} else {
let (qs, mut rs) = limbs_div_mod(ns, ds);
swap(&mut rs, ds);
other.trim();
(Natural::from_owned_limbs_asc(qs), other)
}
}
}
}
}
impl<'a, 'b> DivMod<&'b Natural> for &'a Natural {
type DivOutput = Natural;
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking both by reference and returning the
/// quotient and remainder. The quotient is rounded towards negative infinity.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lfloor \frac{x}{y} \right \rfloor, \space
/// x - y\left \lfloor \frac{x}{y} \right \rfloor \right ).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivMod;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// assert_eq!(
/// (&Natural::from(23u32)).div_mod(&Natural::from(10u32)).to_debug_string(),
/// "(2, 3)"
/// );
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// assert_eq!(
/// (&Natural::from_str("1000000000000000000000000").unwrap())
/// .div_mod(&Natural::from_str("1234567890987").unwrap()).to_debug_string(),
/// "(810000006723, 530068894399)"
/// );
/// ```
fn div_mod(self, other: &'b Natural) -> (Natural, Natural) {
if self == other {
return (Natural::ONE, Natural::ZERO);
}
match (self, other) {
(_, natural_zero!()) => panic!("division by zero"),
(n, natural_one!()) => (n.clone(), Natural::ZERO),
(n, Natural(Small(d))) => {
let (q, r) = n.div_mod_limb_ref(*d);
(q, Natural(Small(r)))
}
(Natural(Small(_)), _) => (Natural::ZERO, self.clone()),
(&Natural(Large(ref ns)), Natural(Large(ref ds))) => {
if ns.len() < ds.len() {
(Natural::ZERO, self.clone())
} else {
let (qs, rs) = limbs_div_mod(ns, ds);
(
Natural::from_owned_limbs_asc(qs),
Natural::from_owned_limbs_asc(rs),
)
}
}
}
}
}
impl DivAssignMod<Natural> for Natural {
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`] in place, taking the [`Natural`] on the
/// right-hand side by value and returning the remainder. The quotient is rounded towards
/// negative infinity.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = x - y\left \lfloor \frac{x}{y} \right \rfloor,
/// $$
/// $$
/// x \gets \left \lfloor \frac{x}{y} \right \rfloor.
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivAssignMod;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// let mut x = Natural::from(23u32);
/// assert_eq!(x.div_assign_mod(Natural::from(10u32)), 3);
/// assert_eq!(x, 2);
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// let mut x = Natural::from_str("1000000000000000000000000").unwrap();
/// assert_eq!(x.div_assign_mod(Natural::from_str("1234567890987").unwrap()), 530068894399u64);
/// assert_eq!(x, 810000006723u64);
/// ```
fn div_assign_mod(&mut self, mut other: Natural) -> Natural {
if *self == other {
*self = Natural::ONE;
return Natural::ZERO;
}
match (&mut *self, &mut other) {
(_, natural_zero!()) => panic!("division by zero"),
(_, natural_one!()) => Natural::ZERO,
(n, &mut Natural(Small(d))) => Natural(Small(n.div_assign_mod_limb(d))),
(Natural(Small(_)), _) => {
let mut r = Natural::ZERO;
swap(self, &mut r);
r
}
(&mut Natural(Large(ref mut ns)), &mut Natural(Large(ref mut ds))) => {
if ns.len() < ds.len() {
let mut r = Natural::ZERO;
swap(self, &mut r);
r
} else {
let (mut qs, mut rs) = limbs_div_mod(ns, ds);
swap(&mut qs, ns);
swap(&mut rs, ds);
self.trim();
other.trim();
other
}
}
}
}
}
impl<'a> DivAssignMod<&'a Natural> for Natural {
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`] in place, taking the [`Natural`] on the
/// right-hand side by value and returning the remainder. The quotient is rounded towards
/// negative infinity.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = x - y\left \lfloor \frac{x}{y} \right \rfloor,
/// $$
/// $$
/// x \gets \left \lfloor \frac{x}{y} \right \rfloor.
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivAssignMod;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// let mut x = Natural::from(23u32);
/// assert_eq!(x.div_assign_mod(&Natural::from(10u32)), 3);
/// assert_eq!(x, 2);
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// let mut x = Natural::from_str("1000000000000000000000000").unwrap();
/// assert_eq!(
/// x.div_assign_mod(&Natural::from_str("1234567890987").unwrap()),
/// 530068894399u64
/// );
/// assert_eq!(x, 810000006723u64);
/// ```
fn div_assign_mod(&mut self, other: &'a Natural) -> Natural {
if self == other {
*self = Natural::ONE;
return Natural::ZERO;
}
match (&mut *self, other) {
(_, natural_zero!()) => panic!("division by zero"),
(_, natural_one!()) => Natural::ZERO,
(_, Natural(Small(d))) => Natural(Small(self.div_assign_mod_limb(*d))),
(Natural(Small(_)), _) => {
let mut r = Natural::ZERO;
swap(self, &mut r);
r
}
(&mut Natural(Large(ref mut ns)), Natural(Large(ref ds))) => {
if ns.len() < ds.len() {
let mut r = Natural::ZERO;
swap(self, &mut r);
r
} else {
let (mut qs, rs) = limbs_div_mod(ns, ds);
swap(&mut qs, ns);
self.trim();
Natural::from_owned_limbs_asc(rs)
}
}
}
}
}
impl DivRem<Natural> for Natural {
type DivOutput = Natural;
type RemOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking both by value and returning the
/// quotient and remainder. The quotient is rounded towards zero.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lfloor \frac{x}{y} \right \rfloor, \space
/// x - y\left \lfloor \frac{x}{y} \right \rfloor \right ).
/// $$
///
/// For [`Natural`]s, `div_rem` is equivalent to
/// [`div_mod`](malachite_base::num::arithmetic::traits::DivMod::div_mod).
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivRem;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// assert_eq!(Natural::from(23u32).div_rem(Natural::from(10u32)).to_debug_string(), "(2, 3)");
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// assert_eq!(
/// Natural::from_str("1000000000000000000000000").unwrap()
/// .div_rem(Natural::from_str("1234567890987").unwrap()).to_debug_string(),
/// "(810000006723, 530068894399)"
/// );
/// ```
#[inline]
fn div_rem(self, other: Natural) -> (Natural, Natural) {
self.div_mod(other)
}
}
impl<'a> DivRem<&'a Natural> for Natural {
type DivOutput = Natural;
type RemOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking the first by value and the second by
/// reference and returning the quotient and remainder. The quotient is rounded towards zero.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lfloor \frac{x}{y} \right \rfloor, \space
/// x - y\left \lfloor \frac{x}{y} \right \rfloor \right ).
/// $$
///
/// For [`Natural`]s, `div_rem` is equivalent to
/// [`div_mod`](malachite_base::num::arithmetic::traits::DivMod::div_mod).
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivRem;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// assert_eq!(
/// Natural::from(23u32).div_rem(&Natural::from(10u32)).to_debug_string(),
/// "(2, 3)"
/// );
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// assert_eq!(
/// Natural::from_str("1000000000000000000000000").unwrap()
/// .div_rem(&Natural::from_str("1234567890987").unwrap()).to_debug_string(),
/// "(810000006723, 530068894399)"
/// );
/// ```
#[inline]
fn div_rem(self, other: &'a Natural) -> (Natural, Natural) {
self.div_mod(other)
}
}
impl<'a> DivRem<Natural> for &'a Natural {
type DivOutput = Natural;
type RemOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking the first by reference and the second
/// by value and returning the quotient and remainder. The quotient is rounded towards zero.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lfloor \frac{x}{y} \right \rfloor, \space
/// x - y\left \lfloor \frac{x}{y} \right \rfloor \right ).
/// $$
///
/// For [`Natural`]s, `div_rem` is equivalent to
/// [`div_mod`](malachite_base::num::arithmetic::traits::DivMod::div_mod).
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivRem;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// assert_eq!(
/// (&Natural::from(23u32)).div_rem(Natural::from(10u32)).to_debug_string(),
/// "(2, 3)"
/// );
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// assert_eq!(
/// (&Natural::from_str("1000000000000000000000000").unwrap())
/// .div_rem(Natural::from_str("1234567890987").unwrap()).to_debug_string(),
/// "(810000006723, 530068894399)"
/// );
/// ```
#[inline]
fn div_rem(self, other: Natural) -> (Natural, Natural) {
self.div_mod(other)
}
}
impl<'a, 'b> DivRem<&'b Natural> for &'a Natural {
type DivOutput = Natural;
type RemOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking both by reference and returning the
/// quotient and remainder. The quotient is rounded towards zero.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lfloor \frac{x}{y} \right \rfloor, \space
/// x - y\left \lfloor \frac{x}{y} \right \rfloor \right ).
/// $$
///
/// For [`Natural`]s, `div_rem` is equivalent to
/// [`div_mod`](malachite_base::num::arithmetic::traits::DivMod::div_mod).
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivRem;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// assert_eq!(
/// (&Natural::from(23u32)).div_rem(&Natural::from(10u32)).to_debug_string(),
/// "(2, 3)"
/// );
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// assert_eq!(
/// (&Natural::from_str("1000000000000000000000000").unwrap())
/// .div_rem(&Natural::from_str("1234567890987").unwrap()).to_debug_string(),
/// "(810000006723, 530068894399)"
/// );
/// ```
#[inline]
fn div_rem(self, other: &'b Natural) -> (Natural, Natural) {
self.div_mod(other)
}
}
impl DivAssignRem<Natural> for Natural {
type RemOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`] in place, taking the [`Natural`] on the
/// right-hand side by value and returning the remainder. The quotient is rounded towards zero.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = x - y\left \lfloor \frac{x}{y} \right \rfloor,
/// $$
/// $$
/// x \gets \left \lfloor \frac{x}{y} \right \rfloor.
/// $$
///
/// For [`Natural`]s, `div_assign_rem` is equivalent to
/// [`div_assign_mod`](malachite_base::num::arithmetic::traits::DivAssignMod::div_assign_mod).
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivAssignRem;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// let mut x = Natural::from(23u32);
/// assert_eq!(x.div_assign_rem(Natural::from(10u32)), 3);
/// assert_eq!(x, 2);
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// let mut x = Natural::from_str("1000000000000000000000000").unwrap();
/// assert_eq!(x.div_assign_rem(Natural::from_str("1234567890987").unwrap()), 530068894399u64);
/// assert_eq!(x, 810000006723u64);
/// ```
#[inline]
fn div_assign_rem(&mut self, other: Natural) -> Natural {
self.div_assign_mod(other)
}
}
impl<'a> DivAssignRem<&'a Natural> for Natural {
type RemOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`] in place, taking the [`Natural`] on the
/// right-hand side by reference and returning the remainder. The quotient is rounded towards
/// zero.
///
/// The quotient and remainder satisfy $x = qy + r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = x - y\left \lfloor \frac{x}{y} \right \rfloor,
/// $$
/// $$
/// x \gets \left \lfloor \frac{x}{y} \right \rfloor.
/// $$
///
/// For [`Natural`]s, `div_assign_rem` is equivalent to
/// [`div_assign_mod`](malachite_base::num::arithmetic::traits::DivAssignMod::div_assign_mod).
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::DivAssignRem;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 2 * 10 + 3 = 23
/// let mut x = Natural::from(23u32);
/// assert_eq!(x.div_assign_rem(&Natural::from(10u32)), 3);
/// assert_eq!(x, 2);
///
/// // 810000006723 * 1234567890987 + 530068894399 = 1000000000000000000000000
/// let mut x = Natural::from_str("1000000000000000000000000").unwrap();
/// assert_eq!(
/// x.div_assign_rem(&Natural::from_str("1234567890987").unwrap()),
/// 530068894399u64
/// );
/// assert_eq!(x, 810000006723u64);
/// ```
#[inline]
fn div_assign_rem(&mut self, other: &'a Natural) -> Natural {
self.div_assign_mod(other)
}
}
impl CeilingDivNegMod<Natural> for Natural {
type DivOutput = Natural;
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking both by value and returning the
/// ceiling of the quotient and the remainder of the negative of the first [`Natural`] divided
/// by the second.
///
/// The quotient and remainder satisfy $x = qy - r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lceil \frac{x}{y} \right \rceil, \space
/// y\left \lceil \frac{x}{y} \right \rceil - x \right ).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::CeilingDivNegMod;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 3 * 10 - 7 = 23
/// assert_eq!(
/// Natural::from(23u32).ceiling_div_neg_mod(Natural::from(10u32)).to_debug_string(),
/// "(3, 7)"
/// );
///
/// // 810000006724 * 1234567890987 - 704498996588 = 1000000000000000000000000
/// assert_eq!(
/// Natural::from_str("1000000000000000000000000").unwrap()
/// .ceiling_div_neg_mod(Natural::from_str("1234567890987").unwrap())
/// .to_debug_string(),
/// "(810000006724, 704498996588)"
/// );
/// ```
#[inline]
fn ceiling_div_neg_mod(mut self, other: Natural) -> (Natural, Natural) {
let r = self.ceiling_div_assign_neg_mod(other);
(self, r)
}
}
impl<'a> CeilingDivNegMod<&'a Natural> for Natural {
type DivOutput = Natural;
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking the first by value and the second by
/// reference and returning the ceiling of the quotient and the remainder of the negative of
/// the first [`Natural`] divided by the second.
///
/// The quotient and remainder satisfy $x = qy - r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lceil \frac{x}{y} \right \rceil, \space
/// y\left \lceil \frac{x}{y} \right \rceil - x \right ).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::CeilingDivNegMod;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 3 * 10 - 7 = 23
/// assert_eq!(
/// Natural::from(23u32).ceiling_div_neg_mod(&Natural::from(10u32)).to_debug_string(),
/// "(3, 7)"
/// );
///
/// // 810000006724 * 1234567890987 - 704498996588 = 1000000000000000000000000
/// assert_eq!(
/// Natural::from_str("1000000000000000000000000").unwrap()
/// .ceiling_div_neg_mod(&Natural::from_str("1234567890987").unwrap())
/// .to_debug_string(),
/// "(810000006724, 704498996588)"
/// );
/// ```
#[inline]
fn ceiling_div_neg_mod(mut self, other: &'a Natural) -> (Natural, Natural) {
let r = self.ceiling_div_assign_neg_mod(other);
(self, r)
}
}
impl<'a> CeilingDivNegMod<Natural> for &'a Natural {
type DivOutput = Natural;
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking the first by reference and the second
/// by value and returning the ceiling of the quotient and the remainder of the negative of the
/// first [`Natural`] divided by the second.
///
/// The quotient and remainder satisfy $x = qy - r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lceil \frac{x}{y} \right \rceil, \space
/// y\left \lceil \frac{x}{y} \right \rceil - x \right ).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::CeilingDivNegMod;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 3 * 10 - 7 = 23
/// assert_eq!(
/// (&Natural::from(23u32)).ceiling_div_neg_mod(Natural::from(10u32)).to_debug_string(),
/// "(3, 7)"
/// );
///
/// // 810000006724 * 1234567890987 - 704498996588 = 1000000000000000000000000
/// assert_eq!(
/// (&Natural::from_str("1000000000000000000000000").unwrap())
/// .ceiling_div_neg_mod(Natural::from_str("1234567890987").unwrap())
/// .to_debug_string(),
/// "(810000006724, 704498996588)"
/// );
/// ```
fn ceiling_div_neg_mod(self, other: Natural) -> (Natural, Natural) {
let (q, r) = self.div_mod(&other);
if r == 0 {
(q, r)
} else {
(q.add_limb(1), other - r)
}
}
}
impl<'a, 'b> CeilingDivNegMod<&'b Natural> for &'a Natural {
type DivOutput = Natural;
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`], taking both by reference and returning the
/// ceiling of the quotient and the remainder of the negative of the first [`Natural`] divided
/// by the second.
///
/// The quotient and remainder satisfy $x = qy - r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = \left ( \left \lceil \frac{x}{y} \right \rceil, \space
/// y\left \lceil \frac{x}{y} \right \rceil - x \right ).
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::CeilingDivNegMod;
/// use malachite_base::strings::ToDebugString;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 3 * 10 - 7 = 23
/// assert_eq!(
/// (&Natural::from(23u32)).ceiling_div_neg_mod(&Natural::from(10u32)).to_debug_string(),
/// "(3, 7)"
/// );
///
/// // 810000006724 * 1234567890987 - 704498996588 = 1000000000000000000000000
/// assert_eq!(
/// (&Natural::from_str("1000000000000000000000000").unwrap())
/// .ceiling_div_neg_mod(&Natural::from_str("1234567890987").unwrap())
/// .to_debug_string(),
/// "(810000006724, 704498996588)"
/// );
/// ```
fn ceiling_div_neg_mod(self, other: &'b Natural) -> (Natural, Natural) {
let (q, r) = self.div_mod(other);
if r == 0 {
(q, r)
} else {
(q.add_limb(1), other - r)
}
}
}
impl CeilingDivAssignNegMod<Natural> for Natural {
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`] in place, taking the [`Natural`] on the
/// right-hand side by value and returning the remainder of the negative of the first number
/// divided by the second.
///
/// The quotient and remainder satisfy $x = qy - r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = y\left \lceil \frac{x}{y} \right \rceil - x,
/// $$
/// $$
/// x \gets \left \lceil \frac{x}{y} \right \rceil.
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::CeilingDivAssignNegMod;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 3 * 10 - 7 = 23
/// let mut x = Natural::from(23u32);
/// assert_eq!(x.ceiling_div_assign_neg_mod(Natural::from(10u32)), 7);
/// assert_eq!(x, 3);
///
/// // 810000006724 * 1234567890987 - 704498996588 = 1000000000000000000000000
/// let mut x = Natural::from_str("1000000000000000000000000").unwrap();
/// assert_eq!(
/// x.ceiling_div_assign_neg_mod(Natural::from_str("1234567890987").unwrap()),
/// 704498996588u64,
/// );
/// assert_eq!(x, 810000006724u64);
/// ```
fn ceiling_div_assign_neg_mod(&mut self, other: Natural) -> Natural {
let r = self.div_assign_mod(&other);
if r == 0 {
Natural::ZERO
} else {
*self += Natural::ONE;
other - r
}
}
}
impl<'a> CeilingDivAssignNegMod<&'a Natural> for Natural {
type ModOutput = Natural;
/// Divides a [`Natural`] by another [`Natural`] in place, taking the [`Natural`] on the
/// right-hand side by reference and returning the remainder of the negative of the first
/// number divided by the second.
///
/// The quotient and remainder satisfy $x = qy - r$ and $0 \leq r < y$.
///
/// $$
/// f(x, y) = y\left \lceil \frac{x}{y} \right \rceil - x,
/// $$
/// $$
/// x \gets \left \lceil \frac{x}{y} \right \rceil.
/// $$
///
/// # Worst-case complexity
/// $T(n) = O(n \log n \log\log n)$
///
/// $M(n) = O(n \log n)$
///
/// where $T$ is time, $M$ is additional memory, and $n$ is `self.significant_bits()`.
///
/// # Panics
/// Panics if `other` is zero.
///
/// # Examples
/// ```
/// extern crate malachite_base;
///
/// use malachite_base::num::arithmetic::traits::CeilingDivAssignNegMod;
/// use malachite_nz::natural::Natural;
/// use std::str::FromStr;
///
/// // 3 * 10 - 7 = 23
/// let mut x = Natural::from(23u32);
/// assert_eq!(x.ceiling_div_assign_neg_mod(&Natural::from(10u32)), 7);
/// assert_eq!(x, 3);
///
/// // 810000006724 * 1234567890987 - 704498996588 = 1000000000000000000000000
/// let mut x = Natural::from_str("1000000000000000000000000").unwrap();
/// assert_eq!(
/// x.ceiling_div_assign_neg_mod(&Natural::from_str("1234567890987").unwrap()),
/// 704498996588u64,
/// );
/// assert_eq!(x, 810000006724u64);
/// ```
fn ceiling_div_assign_neg_mod(&mut self, other: &'a Natural) -> Natural {
let r = self.div_assign_mod(other);
if r == 0 {
Natural::ZERO
} else {
*self += Natural::ONE;
other - r
}
}
}