num_bigint/biguint/

multiplication.rs

use super::addition::{__add2, add2};
use super::subtraction::sub2;
use super::{biguint_from_vec, cmp_slice, BigUint, IntDigits};

use crate::big_digit::{self, BigDigit, DoubleBigDigit};
use crate::Sign::{self, Minus, NoSign, Plus};
use crate::{BigInt, UsizePromotion};

use core::cmp::Ordering;
use core::iter::Product;
use core::ops::{Mul, MulAssign};
use num_traits::{CheckedMul, FromPrimitive, One, Zero};

#[inline]
pub(super) fn mac_with_carry(
    a: BigDigit,
    b: BigDigit,
    c: BigDigit,
    acc: &mut DoubleBigDigit,
) -> BigDigit {
    *acc += DoubleBigDigit::from(a);
    *acc += DoubleBigDigit::from(b) * DoubleBigDigit::from(c);
    let lo = *acc as BigDigit;
    *acc >>= big_digit::BITS;
    lo
}

#[inline]
fn mul_with_carry(a: BigDigit, b: BigDigit, acc: &mut DoubleBigDigit) -> BigDigit {
    *acc += DoubleBigDigit::from(a) * DoubleBigDigit::from(b);
    let lo = *acc as BigDigit;
    *acc >>= big_digit::BITS;
    lo
}

/// Three argument multiply accumulate:
/// acc += b * c
fn mac_digit(acc: &mut [BigDigit], b: &[BigDigit], c: BigDigit) {
    if c == 0 {
        return;
    }

    let mut carry = 0;
    let (a_lo, a_hi) = acc.split_at_mut(b.len());

    for (a, &b) in a_lo.iter_mut().zip(b) {
        *a = mac_with_carry(*a, b, c, &mut carry);
    }

    let (carry_hi, carry_lo) = big_digit::from_doublebigdigit(carry);

    let final_carry = if carry_hi == 0 {
        __add2(a_hi, &[carry_lo])
    } else {
        __add2(a_hi, &[carry_hi, carry_lo])
    };
    assert_eq!(final_carry, 0, "carry overflow during multiplication!");
}

fn bigint_from_slice(slice: &[BigDigit]) -> BigInt {
    BigInt::from(biguint_from_vec(slice.to_vec()))
}

/// Three argument multiply accumulate:
/// acc += b * c
#[allow(clippy::many_single_char_names)]
fn mac3(mut acc: &mut [BigDigit], mut b: &[BigDigit], mut c: &[BigDigit]) {
    // Least-significant zeros have no effect on the output.
    if let Some(&0) = b.first() {
        if let Some(nz) = b.iter().position(|&d| d != 0) {
            b = &b[nz..];
            acc = &mut acc[nz..];
        } else {
            return;
        }
    }
    if let Some(&0) = c.first() {
        if let Some(nz) = c.iter().position(|&d| d != 0) {
            c = &c[nz..];
            acc = &mut acc[nz..];
        } else {
            return;
        }
    }

    let acc = acc;
    let (x, y) = if b.len() < c.len() { (b, c) } else { (c, b) };

    // We use four algorithms for different input sizes.
    //
    // - For small inputs, long multiplication is fastest.
    // - If y is at least least twice as long as x, split using Half-Karatsuba.
    // - Next we use Karatsuba multiplication (Toom-2), which we have optimized
    //   to avoid unnecessary allocations for intermediate values.
    // - For the largest inputs we use Toom-3, which better optimizes the
    //   number of operations, but uses more temporary allocations.
    //
    // The thresholds are somewhat arbitrary, chosen by evaluating the results
    // of `cargo bench --bench bigint multiply`.

    if x.len() <= 32 {
        // Long multiplication:
        for (i, xi) in x.iter().enumerate() {
            mac_digit(&mut acc[i..], y, *xi);
        }
    } else if x.len() * 2 <= y.len() {
        // Karatsuba Multiplication for factors with significant length disparity.
        //
        // The Half-Karatsuba Multiplication Algorithm is a specialized case of
        // the normal Karatsuba multiplication algorithm, designed for the scenario
        // where y has at least twice as many base digits as x.
        //
        // In this case y (the longer input) is split into high2 and low2,
        // at m2 (half the length of y) and x (the shorter input),
        // is used directly without splitting.
        //
        // The algorithm then proceeds as follows:
        //
        // 1. Compute the product z0 = x * low2.
        // 2. Compute the product temp = x * high2.
        // 3. Adjust the weight of temp by adding m2 (* NBASE ^ m2)
        // 4. Add temp and z0 to obtain the final result.
        //
        // Proof:
        //
        // The algorithm can be derived from the original Karatsuba algorithm by
        // simplifying the formula when the shorter factor x is not split into
        // high and low parts, as shown below.
        //
        // Original Karatsuba formula:
        //
        //     result = (z2 * NBASE ^ (m2 × 2)) + ((z1 - z2 - z0) * NBASE ^ m2) + z0
        //
        // Substitutions:
        //
        //     low1 = x
        //     high1 = 0
        //
        // Applying substitutions:
        //
        //     z0 = (low1 * low2)
        //        = (x * low2)
        //
        //     z1 = ((low1 + high1) * (low2 + high2))
        //        = ((x + 0) * (low2 + high2))
        //        = (x * low2) + (x * high2)
        //
        //     z2 = (high1 * high2)
        //        = (0 * high2)
        //        = 0
        //
        // Simplified using the above substitutions:
        //
        //     result = (z2 * NBASE ^ (m2 × 2)) + ((z1 - z2 - z0) * NBASE ^ m2) + z0
        //            = (0 * NBASE ^ (m2 × 2)) + ((z1 - 0 - z0) * NBASE ^ m2) + z0
        //            = ((z1 - z0) * NBASE ^ m2) + z0
        //            = ((z1 - z0) * NBASE ^ m2) + z0
        //            = (x * high2) * NBASE ^ m2 + z0
        let m2 = y.len() / 2;
        let (low2, high2) = y.split_at(m2);

        // (x * high2) * NBASE ^ m2 + z0
        mac3(acc, x, low2);
        mac3(&mut acc[m2..], x, high2);
    } else if x.len() <= 256 {
        // Karatsuba multiplication:
        //
        // The idea is that we break x and y up into two smaller numbers that each have about half
        // as many digits, like so (note that multiplying by b is just a shift):
        //
        // x = x0 + x1 * b
        // y = y0 + y1 * b
        //
        // With some algebra, we can compute x * y with three smaller products, where the inputs to
        // each of the smaller products have only about half as many digits as x and y:
        //
        // x * y = (x0 + x1 * b) * (y0 + y1 * b)
        //
        // x * y = x0 * y0
        //       + x0 * y1 * b
        //       + x1 * y0 * b
        //       + x1 * y1 * b^2
        //
        // Let p0 = x0 * y0 and p2 = x1 * y1:
        //
        // x * y = p0
        //       + (x0 * y1 + x1 * y0) * b
        //       + p2 * b^2
        //
        // The real trick is that middle term:
        //
        //         x0 * y1 + x1 * y0
        //
        //       = x0 * y1 + x1 * y0 - p0 + p0 - p2 + p2
        //
        //       = x0 * y1 + x1 * y0 - x0 * y0 - x1 * y1 + p0 + p2
        //
        // Now we complete the square:
        //
        //       = -(x0 * y0 - x0 * y1 - x1 * y0 + x1 * y1) + p0 + p2
        //
        //       = -((x1 - x0) * (y1 - y0)) + p0 + p2
        //
        // Let p1 = (x1 - x0) * (y1 - y0), and substitute back into our original formula:
        //
        // x * y = p0
        //       + (p0 + p2 - p1) * b
        //       + p2 * b^2
        //
        // Where the three intermediate products are:
        //
        // p0 = x0 * y0
        // p1 = (x1 - x0) * (y1 - y0)
        // p2 = x1 * y1
        //
        // In doing the computation, we take great care to avoid unnecessary temporary variables
        // (since creating a BigUint requires a heap allocation): thus, we rearrange the formula a
        // bit so we can use the same temporary variable for all the intermediate products:
        //
        // x * y = p2 * b^2 + p2 * b
        //       + p0 * b + p0
        //       - p1 * b
        //
        // The other trick we use is instead of doing explicit shifts, we slice acc at the
        // appropriate offset when doing the add.

        // When x is smaller than y, it's significantly faster to pick b such that x is split in
        // half, not y:
        let b = x.len() / 2;
        let (x0, x1) = x.split_at(b);
        let (y0, y1) = y.split_at(b);

        // We reuse the same BigUint for all the intermediate multiplies and have to size p
        // appropriately here: x1.len() >= x0.len and y1.len() >= y0.len():
        let len = x1.len() + y1.len() + 1;
        let mut p = BigUint { data: vec![0; len] };

        // p2 = x1 * y1
        mac3(&mut p.data, x1, y1);

        // Not required, but the adds go faster if we drop any unneeded 0s from the end:
        p.normalize();

        add2(&mut acc[b..], &p.data);
        add2(&mut acc[b * 2..], &p.data);

        // Zero out p before the next multiply:
        p.data.truncate(0);
        p.data.resize(len, 0);

        // p0 = x0 * y0
        mac3(&mut p.data, x0, y0);
        p.normalize();

        add2(acc, &p.data);
        add2(&mut acc[b..], &p.data);

        // p1 = (x1 - x0) * (y1 - y0)
        // We do this one last, since it may be negative and acc can't ever be negative:
        let (j0_sign, j0) = sub_sign(x1, x0);
        let (j1_sign, j1) = sub_sign(y1, y0);

        match j0_sign * j1_sign {
            Plus => {
                p.data.truncate(0);
                p.data.resize(len, 0);

                mac3(&mut p.data, &j0.data, &j1.data);
                p.normalize();

                sub2(&mut acc[b..], &p.data);
            }
            Minus => {
                mac3(&mut acc[b..], &j0.data, &j1.data);
            }
            NoSign => (),
        }
    } else {
        // Toom-3 multiplication:
        //
        // Toom-3 is like Karatsuba above, but dividing the inputs into three parts.
        // Both are instances of Toom-Cook, using `k=3` and `k=2` respectively.
        //
        // The general idea is to treat the large integers digits as
        // polynomials of a certain degree and determine the coefficients/digits
        // of the product of the two via interpolation of the polynomial product.
        let i = y.len() / 3 + 1;

        let x0_len = Ord::min(x.len(), i);
        let x1_len = Ord::min(x.len() - x0_len, i);

        let y0_len = i;
        let y1_len = Ord::min(y.len() - y0_len, i);

        // Break x and y into three parts, representating an order two polynomial.
        // t is chosen to be the size of a digit so we can use faster shifts
        // in place of multiplications.
        //
        // x(t) = x2*t^2 + x1*t + x0
        let x0 = bigint_from_slice(&x[..x0_len]);
        let x1 = bigint_from_slice(&x[x0_len..x0_len + x1_len]);
        let x2 = bigint_from_slice(&x[x0_len + x1_len..]);

        // y(t) = y2*t^2 + y1*t + y0
        let y0 = bigint_from_slice(&y[..y0_len]);
        let y1 = bigint_from_slice(&y[y0_len..y0_len + y1_len]);
        let y2 = bigint_from_slice(&y[y0_len + y1_len..]);

        // Let w(t) = x(t) * y(t)
        //
        // This gives us the following order-4 polynomial.
        //
        // w(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
        //
        // We need to find the coefficients w4, w3, w2, w1 and w0. Instead
        // of simply multiplying the x and y in total, we can evaluate w
        // at 5 points. An n-degree polynomial is uniquely identified by (n + 1)
        // points.
        //
        // It is arbitrary as to what points we evaluate w at but we use the
        // following.
        //
        // w(t) at t = 0, 1, -1, -2 and inf
        //
        // The values for w(t) in terms of x(t)*y(t) at these points are:
        //
        // let a = w(0)   = x0 * y0
        // let b = w(1)   = (x2 + x1 + x0) * (y2 + y1 + y0)
        // let c = w(-1)  = (x2 - x1 + x0) * (y2 - y1 + y0)
        // let d = w(-2)  = (4*x2 - 2*x1 + x0) * (4*y2 - 2*y1 + y0)
        // let e = w(inf) = x2 * y2 as t -> inf

        // x0 + x2, avoiding temporaries
        let p = &x0 + &x2;

        // y0 + y2, avoiding temporaries
        let q = &y0 + &y2;

        // x2 - x1 + x0, avoiding temporaries
        let p2 = &p - &x1;

        // y2 - y1 + y0, avoiding temporaries
        let q2 = &q - &y1;

        // w(0)
        let r0 = &x0 * &y0;

        // w(inf)
        let r4 = &x2 * &y2;

        // w(1)
        let r1 = (p + x1) * (q + y1);

        // w(-1)
        let r2 = &p2 * &q2;

        // w(-2)
        let r3 = ((p2 + x2) * 2 - x0) * ((q2 + y2) * 2 - y0);

        // Evaluating these points gives us the following system of linear equations.
        //
        //  0  0  0  0  1 | a
        //  1  1  1  1  1 | b
        //  1 -1  1 -1  1 | c
        // 16 -8  4 -2  1 | d
        //  1  0  0  0  0 | e
        //
        // The solved equation (after gaussian elimination or similar)
        // in terms of its coefficients:
        //
        // w0 = w(0)
        // w1 = w(0)/2 + w(1)/3 - w(-1) + w(-2)/6 - 2*w(inf)
        // w2 = -w(0) + w(1)/2 + w(-1)/2 - w(inf)
        // w3 = -w(0)/2 + w(1)/6 + w(-1)/2 - w(-2)/6 + 2*w(inf)
        // w4 = w(inf)
        //
        // This particular sequence is given by Bodrato and is an interpolation
        // of the above equations.
        let mut comp3: BigInt = (r3 - &r1) / 3u32;
        let mut comp1: BigInt = (r1 - &r2) >> 1;
        let mut comp2: BigInt = r2 - &r0;
        comp3 = ((&comp2 - comp3) >> 1) + (&r4 << 1);
        comp2 += &comp1 - &r4;
        comp1 -= &comp3;

        // Recomposition. The coefficients of the polynomial are now known.
        //
        // Evaluate at w(t) where t is our given base to get the result.
        //
        //     let bits = u64::from(big_digit::BITS) * i as u64;
        //     let result = r0
        //         + (comp1 << bits)
        //         + (comp2 << (2 * bits))
        //         + (comp3 << (3 * bits))
        //         + (r4 << (4 * bits));
        //     let result_pos = result.to_biguint().unwrap();
        //     add2(&mut acc[..], &result_pos.data);
        //
        // But with less intermediate copying:
        for (j, result) in [&r0, &comp1, &comp2, &comp3, &r4].iter().enumerate().rev() {
            match result.sign() {
                Plus => add2(&mut acc[i * j..], result.digits()),
                Minus => sub2(&mut acc[i * j..], result.digits()),
                NoSign => {}
            }
        }
    }
}

fn mul3(x: &[BigDigit], y: &[BigDigit]) -> BigUint {
    let len = x.len() + y.len() + 1;
    let mut prod = BigUint { data: vec![0; len] };

    mac3(&mut prod.data, x, y);
    prod.normalized()
}

fn scalar_mul(a: &mut BigUint, b: BigDigit) {
    match b {
        0 => a.set_zero(),
        1 => {}
        _ => {
            if b.is_power_of_two() {
                *a <<= b.trailing_zeros();
            } else {
                let mut carry = 0;
                for a in a.data.iter_mut() {
                    *a = mul_with_carry(*a, b, &mut carry);
                }
                if carry != 0 {
                    a.data.push(carry as BigDigit);
                }
            }
        }
    }
}

fn sub_sign(mut a: &[BigDigit], mut b: &[BigDigit]) -> (Sign, BigUint) {
    // Normalize:
    if let Some(&0) = a.last() {
        a = &a[..a.iter().rposition(|&x| x != 0).map_or(0, |i| i + 1)];
    }
    if let Some(&0) = b.last() {
        b = &b[..b.iter().rposition(|&x| x != 0).map_or(0, |i| i + 1)];
    }

    match cmp_slice(a, b) {
        Ordering::Greater => {
            let mut a = a.to_vec();
            sub2(&mut a, b);
            (Plus, biguint_from_vec(a))
        }
        Ordering::Less => {
            let mut b = b.to_vec();
            sub2(&mut b, a);
            (Minus, biguint_from_vec(b))
        }
        Ordering::Equal => (NoSign, BigUint::ZERO),
    }
}

macro_rules! impl_mul {
    ($(impl Mul<$Other:ty> for $Self:ty;)*) => {$(
        impl Mul<$Other> for $Self {
            type Output = BigUint;

            #[inline]
            fn mul(self, other: $Other) -> BigUint {
                match (&*self.data, &*other.data) {
                    // multiply by zero
                    (&[], _) | (_, &[]) => BigUint::ZERO,
                    // multiply by a scalar
                    (_, &[digit]) => self * digit,
                    (&[digit], _) => other * digit,
                    // full multiplication
                    (x, y) => mul3(x, y),
                }
            }
        }
    )*}
}
impl_mul! {
    impl Mul<BigUint> for BigUint;
    impl Mul<BigUint> for &BigUint;
    impl Mul<&BigUint> for BigUint;
    impl Mul<&BigUint> for &BigUint;
}

macro_rules! impl_mul_assign {
    ($(impl MulAssign<$Other:ty> for BigUint;)*) => {$(
        impl MulAssign<$Other> for BigUint {
            #[inline]
            fn mul_assign(&mut self, other: $Other) {
                match (&*self.data, &*other.data) {
                    // multiply by zero
                    (&[], _) => {},
                    (_, &[]) => self.set_zero(),
                    // multiply by a scalar
                    (_, &[digit]) => *self *= digit,
                    (&[digit], _) => *self = other * digit,
                    // full multiplication
                    (x, y) => *self = mul3(x, y),
                }
            }
        }
    )*}
}
impl_mul_assign! {
    impl MulAssign<BigUint> for BigUint;
    impl MulAssign<&BigUint> for BigUint;
}

promote_unsigned_scalars!(impl Mul for BigUint, mul);
promote_unsigned_scalars_assign!(impl MulAssign for BigUint, mul_assign);
forward_all_scalar_binop_to_val_val_commutative!(impl Mul<u32> for BigUint, mul);
forward_all_scalar_binop_to_val_val_commutative!(impl Mul<u64> for BigUint, mul);
forward_all_scalar_binop_to_val_val_commutative!(impl Mul<u128> for BigUint, mul);

impl Mul<u32> for BigUint {
    type Output = BigUint;

    #[inline]
    fn mul(mut self, other: u32) -> BigUint {
        self *= other;
        self
    }
}
impl MulAssign<u32> for BigUint {
    #[inline]
    fn mul_assign(&mut self, other: u32) {
        scalar_mul(self, other as BigDigit);
    }
}

impl Mul<u64> for BigUint {
    type Output = BigUint;

    #[inline]
    fn mul(mut self, other: u64) -> BigUint {
        self *= other;
        self
    }
}
impl MulAssign<u64> for BigUint {
    cfg_digit!(
        #[inline]
        fn mul_assign(&mut self, other: u64) {
            if let Some(other) = BigDigit::from_u64(other) {
                scalar_mul(self, other);
            } else {
                let (hi, lo) = big_digit::from_doublebigdigit(other);
                *self = mul3(&self.data, &[lo, hi]);
            }
        }

        #[inline]
        fn mul_assign(&mut self, other: u64) {
            scalar_mul(self, other);
        }
    );
}

impl Mul<u128> for BigUint {
    type Output = BigUint;

    #[inline]
    fn mul(mut self, other: u128) -> BigUint {
        self *= other;
        self
    }
}

impl MulAssign<u128> for BigUint {
    cfg_digit!(
        #[inline]
        fn mul_assign(&mut self, other: u128) {
            if let Some(other) = BigDigit::from_u128(other) {
                scalar_mul(self, other);
            } else {
                *self = match super::u32_from_u128(other) {
                    (0, 0, c, d) => mul3(&self.data, &[d, c]),
                    (0, b, c, d) => mul3(&self.data, &[d, c, b]),
                    (a, b, c, d) => mul3(&self.data, &[d, c, b, a]),
                };
            }
        }

        #[inline]
        fn mul_assign(&mut self, other: u128) {
            if let Some(other) = BigDigit::from_u128(other) {
                scalar_mul(self, other);
            } else {
                let (hi, lo) = big_digit::from_doublebigdigit(other);
                *self = mul3(&self.data, &[lo, hi]);
            }
        }
    );
}

impl CheckedMul for BigUint {
    #[inline]
    fn checked_mul(&self, v: &BigUint) -> Option<BigUint> {
        Some(self.mul(v))
    }
}

impl_product_iter_type!(BigUint);

#[test]
fn test_sub_sign() {
    use crate::BigInt;
    use num_traits::Num;

    fn sub_sign_i(a: &[BigDigit], b: &[BigDigit]) -> BigInt {
        let (sign, val) = sub_sign(a, b);
        BigInt::from_biguint(sign, val)
    }

    let a = BigUint::from_str_radix("265252859812191058636308480000000", 10).unwrap();
    let b = BigUint::from_str_radix("26525285981219105863630848000000", 10).unwrap();
    let a_i = BigInt::from(a.clone());
    let b_i = BigInt::from(b.clone());

    assert_eq!(sub_sign_i(&a.data, &b.data), &a_i - &b_i);
    assert_eq!(sub_sign_i(&b.data, &a.data), &b_i - &a_i);
}