// Licensed to the Apache Software Foundation (ASF) under one

// or more contributor license agreements.  See the NOTICE file

// distributed with this work for additional information

// regarding copyright ownership.  The ASF licenses this file

// to you under the Apache License, Version 2.0 (the

// "License"); you may not use this file except in compliance

// with the License.  You may obtain a copy of the License at

//

//   http://www.apache.org/licenses/LICENSE-2.0

//

// Unless required by applicable law or agreed to in writing,

// software distributed under the License is distributed on an

// "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY

// KIND, either express or implied.  See the License for the

// specific language governing permissions and limitations

// under the License.

//! N-digit division

//!

//! Implementation heavily inspired by [uint]

//!

//! [uint]: https://github.com/paritytech/parity-common/blob/d3a9327124a66e52ca1114bb8640c02c18c134b8/uint/src/uint.rs#L844

/// Unsigned, little-endian, n-digit division with remainder

///

/// # Panics

///

/// Panics if divisor is zero

pub fn div_rem<const N: usize>(numerator: &[u64; N], divisor: &[u64; N]) -> ([u64; N], [u64; N]) {

    let numerator_bits = bits(numerator);

    let divisor_bits = bits(divisor);

    assert_ne!(divisor_bits, 0, "division by zero");

    if numerator_bits < divisor_bits {

        return ([0; N], *numerator);

    }

    if divisor_bits <= 64 {

        return div_rem_small(numerator, divisor[0]);

    }

    let numerator_words = numerator_bits.div_ceil(64);

    let divisor_words = divisor_bits.div_ceil(64);

    let n = divisor_words;

    let m = numerator_words - divisor_words;

    div_rem_knuth(numerator, divisor, n, m)

}

/// Return the least number of bits needed to represent the number

fn bits(arr: &[u64]) -> usize {

    for (idx, v) in arr.iter().enumerate().rev() {

        if *v > 0 {

            return 64 - v.leading_zeros() as usize + 64 * idx;

        }

    }

    0

}

/// Division of numerator by a u64 divisor

fn div_rem_small<const N: usize>(numerator: &[u64; N], divisor: u64) -> ([u64; N], [u64; N]) {

    let mut rem = 0u64;

    let mut numerator = *numerator;

    numerator.iter_mut().rev().for_each(|d| {

        let (q, r) = div_rem_word(rem, *d, divisor);

        *d = q;

        rem = r;

    });

    let mut rem_padded = [0; N];

    rem_padded[0] = rem;

    (numerator, rem_padded)

}

/// Use Knuth Algorithm D to compute `numerator / divisor` returning the

/// quotient and remainder

///

/// `n` is the number of non-zero 64-bit words in `divisor`

/// `m` is the number of non-zero 64-bit words present in `numerator` beyond `divisor`, and

/// therefore the number of words in the quotient

///

/// A good explanation of the algorithm can be found [here](https://ridiculousfish.com/blog/posts/labor-of-division-episode-iv.html)

fn div_rem_knuth<const N: usize>(

    numerator: &[u64; N],

    divisor: &[u64; N],

    n: usize,

    m: usize,

) -> ([u64; N], [u64; N]) {

    assert!(n + m <= N);

    // The algorithm works by incrementally generating guesses `q_hat`, for the next digit

    // of the quotient, starting from the most significant digit.

    //

    // This relies on the property that for any `q_hat` where

    //

    //      (q_hat << (j * 64)) * divisor <= numerator`

    //

    // We can set

    //

    //      q += q_hat << (j * 64)

    //      numerator -= (q_hat << (j * 64)) * divisor

    //

    // And then iterate until `numerator < divisor`

    // We normalize the divisor so that the highest bit in the highest digit of the

    // divisor is set, this ensures our initial guess of `q_hat` is at most 2 off from

    // the correct value for q[j]

    let shift = divisor[n - 1].leading_zeros();

    // As the shift is computed based on leading zeros, don't need to perform full_shl

    let divisor = shl_word(divisor, shift);

    // numerator may have fewer leading zeros than divisor, so must add another digit

    let mut numerator = full_shl(numerator, shift);

    // The two most significant digits of the divisor

    let b0 = divisor[n - 1];

    let b1 = divisor[n - 2];

    let mut q = [0; N];

    for j in (0..=m).rev() {

        let a0 = numerator[j + n];

        let a1 = numerator[j + n - 1];

        let mut q_hat = if a0 < b0 {

            // The first estimate is [a1, a0] / b0, it may be too large by at most 2

            let (mut q_hat, mut r_hat) = div_rem_word(a0, a1, b0);

            // r_hat = [a1, a0] - q_hat * b0

            //

            // Now we want to compute a more precise estimate [a2,a1,a0] / [b1,b0]

            // which can only be less or equal to the current q_hat

            //

            // q_hat is too large if:

            // [a2,a1,a0] < q_hat * [b1,b0]

            // [a2,r_hat] < q_hat * b1

            let a2 = numerator[j + n - 2];

            loop {

                let r = u128::from(q_hat) * u128::from(b1);

                let (lo, hi) = (r as u64, (r >> 64) as u64);

                if (hi, lo) <= (r_hat, a2) {

                    break;

                }

                q_hat -= 1;

                let (new_r_hat, overflow) = r_hat.overflowing_add(b0);

                r_hat = new_r_hat;

                if overflow {

                    break;

                }

            }

            q_hat

        } else {

            u64::MAX

        };

        // q_hat is now either the correct quotient digit, or in rare cases 1 too large

        // Compute numerator -= (q_hat * divisor) << (j * 64)

        let q_hat_v = full_mul_u64(&divisor, q_hat);

        let c = sub_assign(&mut numerator[j..], &q_hat_v[..n + 1]);

        // If underflow, q_hat was too large by 1

        if c {

            // Reduce q_hat by 1

            q_hat -= 1;

            // Add back one multiple of divisor

            let c = add_assign(&mut numerator[j..], &divisor[..n]);

            numerator[j + n] = numerator[j + n].wrapping_add(u64::from(c));

        }

        // q_hat is the correct value for q[j]

        q[j] = q_hat;

    }

    // The remainder is what is left in numerator, with the initial normalization shl reversed

    let remainder = full_shr(&numerator, shift);

    (q, remainder)

}

/// Perform narrowing division of a u128 by a u64 divisor, returning the quotient and remainder

///

/// This method may trap or panic if hi >= divisor, i.e. the quotient would not fit

/// into a 64-bit integer

fn div_rem_word(hi: u64, lo: u64, divisor: u64) -> (u64, u64) {

    debug_assert!(hi < divisor);

    debug_assert_ne!(divisor, 0);

    // LLVM fails to use the div instruction as it is not able to prove

    // that hi < divisor, and therefore the result will fit into 64-bits

    #[cfg(all(target_arch = "x86_64", not(miri)))]

    unsafe {

        let mut quot = lo;

        let mut rem = hi;

        std::arch::asm!(

            "div {divisor}",

            divisor = in(reg) divisor,

            inout("rax") quot,

            inout("rdx") rem,

            options(pure, nomem, nostack)

        );

        (quot, rem)

    }

    #[cfg(any(not(target_arch = "x86_64"), miri))]

    {

        let x = (u128::from(hi) << 64) + u128::from(lo);

        let y = u128::from(divisor);

        ((x / y) as u64, (x % y) as u64)

    }

}

/// Perform `a += b`

fn add_assign(a: &mut [u64], b: &[u64]) -> bool {

    binop_slice(a, b, u64::overflowing_add)

}

/// Perform `a -= b`

fn sub_assign(a: &mut [u64], b: &[u64]) -> bool {

    binop_slice(a, b, u64::overflowing_sub)

}

/// Converts an overflowing binary operation on scalars to one on slices

fn binop_slice(a: &mut [u64], b: &[u64], binop: impl Fn(u64, u64) -> (u64, bool) + Copy) -> bool {

    let mut c = false;

    a.iter_mut().zip(b.iter()).for_each(|(x, y)| {

        let (res1, overflow1) = y.overflowing_add(u64::from(c));

        let (res2, overflow2) = binop(*x, res1);

        *x = res2;

        c = overflow1 || overflow2;

    });

    c

}

/// Widening multiplication of an N-digit array with a u64

fn full_mul_u64<const N: usize>(a: &[u64; N], b: u64) -> ArrayPlusOne<u64, N> {

    let mut carry = 0;

    let mut out = [0; N];

    out.iter_mut().zip(a).for_each(|(o, v)| {

        let r = *v as u128 * b as u128 + carry as u128;

        *o = r as u64;

        carry = (r >> 64) as u64;

    });

    ArrayPlusOne(out, carry)

}

/// Left shift of an N-digit array by at most 63 bits

fn shl_word<const N: usize>(v: &[u64; N], shift: u32) -> [u64; N] {

    full_shl(v, shift).0

}

/// Widening left shift of an N-digit array by at most 63 bits

fn full_shl<const N: usize>(v: &[u64; N], shift: u32) -> ArrayPlusOne<u64, N> {

    debug_assert!(shift < 64);

    if shift == 0 {

        return ArrayPlusOne(*v, 0);

    }

    let mut out = [0u64; N];

    out[0] = v[0] << shift;

    for i in 1..N {

        out[i] = (v[i - 1] >> (64 - shift)) | (v[i] << shift)

    }

    let carry = v[N - 1] >> (64 - shift);

    ArrayPlusOne(out, carry)

}

/// Narrowing right shift of an (N+1)-digit array by at most 63 bits

fn full_shr<const N: usize>(a: &ArrayPlusOne<u64, N>, shift: u32) -> [u64; N] {

    debug_assert!(shift < 64);

    if shift == 0 {

        return a.0;

    }

    let mut out = [0; N];

    for i in 0..N - 1 {

        out[i] = (a[i] >> shift) | (a[i + 1] << (64 - shift))

    }

    out[N - 1] = a[N - 1] >> shift;

    out

}

/// An array of N + 1 elements

///

/// This is a hack around lack of support for const arithmetic

#[repr(C)]

struct ArrayPlusOne<T, const N: usize>([T; N], T);

impl<T, const N: usize> std::ops::Deref for ArrayPlusOne<T, N> {

    type Target = [T];

    #[inline]

    fn deref(&self) -> &Self::Target {

        let x = self as *const Self;

        unsafe { std::slice::from_raw_parts(x as *const T, N + 1) }

    }

}

impl<T, const N: usize> std::ops::DerefMut for ArrayPlusOne<T, N> {

    fn deref_mut(&mut self) -> &mut Self::Target {

        let x = self as *mut Self;

        unsafe { std::slice::from_raw_parts_mut(x as *mut T, N + 1) }

    }

}