# algorithms.rs.html -- source
Source to the Rust file `/Users/travis/.cargo/registry/src/github.com-1ecc6299db9ec823/num-bigint-0.1.43/src/algorithms.rs`.
Keywords: rust, rustlang, rust-lang

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    use std::borrow::Cow;
    use std::cmp;
    use std::cmp::Ordering::{self, Less, Greater, Equal};
    use std::iter::repeat;
    use std::mem;
    use traits;
    use traits::{Zero, One};

    use biguint::BigUint;

    use bigint::BigInt;
    use bigint::Sign;
    use bigint::Sign::{Minus, NoSign, Plus};

    #[allow(non_snake_case)]
    pub mod big_digit {
        /// A `BigDigit` is a `BigUint`'s composing element.
        pub type BigDigit = u32;

        /// A `DoubleBigDigit` is the internal type used to do the computations.  Its
        /// size is the double of the size of `BigDigit`.
        pub type DoubleBigDigit = u64;

        pub const ZERO_BIG_DIGIT: BigDigit = 0;

        // `DoubleBigDigit` size dependent
        pub const BITS: usize = 32;

        pub const BASE: DoubleBigDigit = 1 << BITS;
        const LO_MASK: DoubleBigDigit = (-1i32 as DoubleBigDigit) >> BITS;

        #[inline]
        fn get_hi(n: DoubleBigDigit) -> BigDigit {
            (n >> BITS) as BigDigit
        }
        #[inline]
        fn get_lo(n: DoubleBigDigit) -> BigDigit {
            (n & LO_MASK) as BigDigit
        }

        /// Split one `DoubleBigDigit` into two `BigDigit`s.
        #[inline]
        pub fn from_doublebigdigit(n: DoubleBigDigit) -> (BigDigit, BigDigit) {
            (get_hi(n), get_lo(n))
        }

        /// Join two `BigDigit`s into one `DoubleBigDigit`
        #[inline]
        pub fn to_doublebigdigit(hi: BigDigit, lo: BigDigit) -> DoubleBigDigit {
            (lo as DoubleBigDigit) | ((hi as DoubleBigDigit) << BITS)
        }
    }

    use big_digit::{BigDigit, DoubleBigDigit};

    // Generic functions for add/subtract/multiply with carry/borrow:

    // Add with carry:
    #[inline]
    fn adc(a: BigDigit, b: BigDigit, carry: &mut BigDigit) -> BigDigit {
        let (hi, lo) = big_digit::from_doublebigdigit((a as DoubleBigDigit) + (b as DoubleBigDigit) +
                                                      (*carry as DoubleBigDigit));

        *carry = hi;
        lo
    }

    // Subtract with borrow:
    #[inline]
    fn sbb(a: BigDigit, b: BigDigit, borrow: &mut BigDigit) -> BigDigit {
        let (hi, lo) = big_digit::from_doublebigdigit(big_digit::BASE + (a as DoubleBigDigit) -
                                                      (b as DoubleBigDigit) -
                                                      (*borrow as DoubleBigDigit));
        // hi * (base) + lo == 1*(base) + ai - bi - borrow
        // => ai - bi - borrow < 0 <=> hi == 0
        *borrow = (hi == 0) as BigDigit;
        lo
    }

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

    #[inline]
    pub fn mul_with_carry(a: BigDigit, b: BigDigit, carry: &mut BigDigit) -> BigDigit {
        let (hi, lo) = big_digit::from_doublebigdigit((a as DoubleBigDigit) * (b as DoubleBigDigit) +
                                                      (*carry as DoubleBigDigit));

        *carry = hi;
        lo
    }

    /// Divide a two digit numerator by a one digit divisor, returns quotient and remainder:
    ///
    /// Note: the caller must ensure that both the quotient and remainder will fit into a single digit.
    /// This is _not_ true for an arbitrary numerator/denominator.
    ///
    /// (This function also matches what the x86 divide instruction does).
    #[inline]
    fn div_wide(hi: BigDigit, lo: BigDigit, divisor: BigDigit) -> (BigDigit, BigDigit) {
        debug_assert!(hi < divisor);

        let lhs = big_digit::to_doublebigdigit(hi, lo);
        let rhs = divisor as DoubleBigDigit;
        ((lhs / rhs) as BigDigit, (lhs % rhs) as BigDigit)
    }

    pub fn div_rem_digit(mut a: BigUint, b: BigDigit) -> (BigUint, BigDigit) {
        let mut rem = 0;

        for d in a.data.iter_mut().rev() {
            let (q, r) = div_wide(rem, *d, b);
            *d = q;
            rem = r;
        }

        (a.normalized(), rem)
    }

    // Only for the Add impl:
    #[inline]
    pub fn __add2(a: &mut [BigDigit], b: &[BigDigit]) -> BigDigit {
        debug_assert!(a.len() >= b.len());

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

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

        if carry != 0 {
            for a in a_hi {
                *a = adc(*a, 0, &mut carry);
                if carry == 0 { break }
            }
        }

        carry
    }

    /// /Two argument addition of raw slices:
    /// a += b
    ///
    /// The caller _must_ ensure that a is big enough to store the result - typically this means
    /// resizing a to max(a.len(), b.len()) + 1, to fit a possible carry.
    pub fn add2(a: &mut [BigDigit], b: &[BigDigit]) {
        let carry = __add2(a, b);

        debug_assert!(carry == 0);
    }

    pub fn sub2(a: &mut [BigDigit], b: &[BigDigit]) {
        let mut borrow = 0;

        let len = cmp::min(a.len(), b.len());
        let (a_lo, a_hi) = a.split_at_mut(len);
        let (b_lo, b_hi) = b.split_at(len);

        for (a, b) in a_lo.iter_mut().zip(b_lo) {
            *a = sbb(*a, *b, &mut borrow);
        }

        if borrow != 0 {
            for a in a_hi {
                *a = sbb(*a, 0, &mut borrow);
                if borrow == 0 { break }
            }
        }

        // note: we're _required_ to fail on underflow
        assert!(borrow == 0 && b_hi.iter().all(|x| *x == 0),
                "Cannot subtract b from a because b is larger than a.");
    }

    pub fn sub2rev(a: &[BigDigit], b: &mut [BigDigit]) {
        debug_assert!(b.len() >= a.len());

        let mut borrow = 0;

        let len = cmp::min(a.len(), b.len());
        let (a_lo, a_hi) = a.split_at(len);
        let (b_lo, b_hi) = b.split_at_mut(len);

        for (a, b) in a_lo.iter().zip(b_lo) {
            *b = sbb(*a, *b, &mut borrow);
        }

        assert!(a_hi.is_empty());

        // note: we're _required_ to fail on underflow
        assert!(borrow == 0 && b_hi.iter().all(|x| *x == 0),
                "Cannot subtract b from a because b is larger than a.");
    }

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

        match cmp_slice(a, b) {
            Greater => {
                let mut a = a.to_vec();
                sub2(&mut a, b);
                (Plus, BigUint::new(a))
            }
            Less => {
                let mut b = b.to_vec();
                sub2(&mut b, a);
                (Minus, BigUint::new(b))
            }
            _ => (NoSign, Zero::zero()),
        }
    }

    /// Three argument multiply accumulate:
    /// acc += b * c
    pub 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 mut a = a_hi.iter_mut();
        while carry != 0 {
            let a = a.next().expect("carry overflow during multiplication!");
            *a = adc(*a, 0, &mut carry);
        }
    }

    /// Three argument multiply accumulate:
    /// acc += b * c
    fn mac3(acc: &mut [BigDigit], b: &[BigDigit], c: &[BigDigit]) {
        let (x, y) = if b.len() < c.len() {
            (b, c)
        } else {
            (c, b)
        };

        // We use three algorithms for different input sizes.
        //
        // - For small inputs, long multiplication is fastest.
        // - 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() <= 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.extend(repeat(0).take(len));

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

            add2(&mut 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.extend(repeat(0).take(len));

                    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.
            //
            // FIXME: It would be nice to have comments breaking down the operations below.

            let i = y.len()/3 + 1;

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

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

            let x0 = BigInt::from_slice(Plus, &x[..x0_len]);
            let x1 = BigInt::from_slice(Plus, &x[x0_len..x0_len + x1_len]);
            let x2 = BigInt::from_slice(Plus, &x[x0_len + x1_len..]);

            let y0 = BigInt::from_slice(Plus, &y[..y0_len]);
            let y1 = BigInt::from_slice(Plus, &y[y0_len..y0_len + y1_len]);
            let y2 = BigInt::from_slice(Plus, &y[y0_len + y1_len..]);

            let p = &x0 + &x2;
            let q = &y0 + &y2;

            let p2 = &p - &x1;
            let q2 = &q - &y1;

            let r0 = &x0 * &y0;
            let r4 = &x2 * &y2;
            let r1 = (p + x1) * (q + y1);
            let r2 = &p2 * &q2;
            let r3 = ((p2 + x2)*2 - x0) * ((q2 + y2)*2 - y0);

            let mut comp3: BigInt = (r3 - &r1) / 3;
            let mut comp1: BigInt = (r1 - &r2) / 2;
            let mut comp2: BigInt = r2 - &r0;
            comp3 = (&comp2 - comp3)/2 + &r4*2;
            comp2 = comp2 + &comp1 - &r4;
            comp1 = comp1 - &comp3;

            let result = r0 + (comp1 << 32*i) + (comp2 << 2*32*i) + (comp3 << 3*32*i) + (r4 << 4*32*i);
            let result_pos = result.to_biguint().unwrap();
            add2(&mut acc[..], &result_pos.data);
        }
    }

    pub 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()
    }

    pub fn scalar_mul(a: &mut [BigDigit], b: BigDigit) -> BigDigit {
        let mut carry = 0;
        for a in a.iter_mut() {
            *a = mul_with_carry(*a, b, &mut carry);
        }
        carry
    }

    pub fn div_rem(u: &BigUint, d: &BigUint) -> (BigUint, BigUint) {
        if d.is_zero() {
            panic!()
        }
        if u.is_zero() {
            return (Zero::zero(), Zero::zero());
        }
        if *d == One::one() {
            return (u.clone(), Zero::zero());
        }

        // Required or the q_len calculation below can underflow:
        match u.cmp(d) {
            Less => return (Zero::zero(), u.clone()),
            Equal => return (One::one(), Zero::zero()),
            Greater => {} // Do nothing
        }

        // This algorithm is from Knuth, TAOCP vol 2 section 4.3, algorithm D:
        //
        // First, normalize the arguments so the highest bit in the highest digit of the divisor is
        // set: the main loop uses the highest digit of the divisor for generating guesses, so we
        // want it to be the largest number we can efficiently divide by.
        //
        let shift = d.data.last().unwrap().leading_zeros() as usize;
        let mut a = u << shift;
        let b = d << shift;

        // The algorithm works by incrementally calculating "guesses", q0, for part of the
        // remainder. Once we have any number q0 such that q0 * b <= a, we can set
        //
        //     q += q0
        //     a -= q0 * b
        //
        // and then iterate until a < b. Then, (q, a) will be our desired quotient and remainder.
        //
        // q0, our guess, is calculated by dividing the last few digits of a by the last digit of b
        // - this should give us a guess that is "close" to the actual quotient, but is possibly
        // greater than the actual quotient. If q0 * b > a, we simply use iterated subtraction
        // until we have a guess such that q0 * b <= a.
        //

        let bn = *b.data.last().unwrap();
        let q_len = a.data.len() - b.data.len() + 1;
        let mut q = BigUint { data: vec![0; q_len] };

        // We reuse the same temporary to avoid hitting the allocator in our inner loop - this is
        // sized to hold a0 (in the common case; if a particular digit of the quotient is zero a0
        // can be bigger).
        //
        let mut tmp = BigUint { data: Vec::with_capacity(2) };

        for j in (0..q_len).rev() {
            /*
             * When calculating our next guess q0, we don't need to consider the digits below j
             * + b.data.len() - 1: we're guessing digit j of the quotient (i.e. q0 << j) from
             * digit bn of the divisor (i.e. bn << (b.data.len() - 1) - so the product of those
             * two numbers will be zero in all digits up to (j + b.data.len() - 1).
             */
            let offset = j + b.data.len() - 1;
            if offset >= a.data.len() {
                continue;
            }

            /* just avoiding a heap allocation: */
            let mut a0 = tmp;
            a0.data.truncate(0);
            a0.data.extend(a.data[offset..].iter().cloned());

            /*
             * q0 << j * big_digit::BITS is our actual quotient estimate - we do the shifts
             * implicitly at the end, when adding and subtracting to a and q. Not only do we
             * save the cost of the shifts, the rest of the arithmetic gets to work with
             * smaller numbers.
             */
            let (mut q0, _) = div_rem_digit(a0, bn);
            let mut prod = &b * &q0;

            while cmp_slice(&prod.data[..], &a.data[j..]) == Greater {
                let one: BigUint = One::one();
                q0 = q0 - one;
                prod = prod - &b;
            }

            add2(&mut q.data[j..], &q0.data[..]);
            sub2(&mut a.data[j..], &prod.data[..]);
            a.normalize();

            tmp = q0;
        }

        debug_assert!(a < b);

        (q.normalized(), a >> shift)
    }

    /// Find last set bit
    /// fls(0) == 0, fls(u32::MAX) == 32
    pub fn fls<T: traits::PrimInt>(v: T) -> usize {
        mem::size_of::<T>() * 8 - v.leading_zeros() as usize
    }

    pub fn ilog2<T: traits::PrimInt>(v: T) -> usize {
        fls(v) - 1
    }

    #[inline]
    pub fn biguint_shl(n: Cow<BigUint>, bits: usize) -> BigUint {
        let n_unit = bits / big_digit::BITS;
        let mut data = match n_unit {
            0 => n.into_owned().data,
            _ => {
                let len = n_unit + n.data.len() + 1;
                let mut data = Vec::with_capacity(len);
                data.extend(repeat(0).take(n_unit));
                data.extend(n.data.iter().cloned());
                data
            }
        };

        let n_bits = bits % big_digit::BITS;
        if n_bits > 0 {
            let mut carry = 0;
            for elem in data[n_unit..].iter_mut() {
                let new_carry = *elem >> (big_digit::BITS - n_bits);
                *elem = (*elem << n_bits) | carry;
                carry = new_carry;
            }
            if carry != 0 {
                data.push(carry);
            }
        }

        BigUint::new(data)
    }

    #[inline]
    pub fn biguint_shr(n: Cow<BigUint>, bits: usize) -> BigUint {
        let n_unit = bits / big_digit::BITS;
        if n_unit >= n.data.len() {
            return Zero::zero();
        }
        let mut data = match n {
            Cow::Borrowed(n) => n.data[n_unit..].to_vec(),
            Cow::Owned(mut n) => {
                n.data.drain(..n_unit);
                n.data
            }
        };

        let n_bits = bits % big_digit::BITS;
        if n_bits > 0 {
            let mut borrow = 0;
            for elem in data.iter_mut().rev() {
                let new_borrow = *elem << (big_digit::BITS - n_bits);
                *elem = (*elem >> n_bits) | borrow;
                borrow = new_borrow;
            }
        }

        BigUint::new(data)
    }

    pub fn cmp_slice(a: &[BigDigit], b: &[BigDigit]) -> Ordering {
        debug_assert!(a.last() != Some(&0));
        debug_assert!(b.last() != Some(&0));

        let (a_len, b_len) = (a.len(), b.len());
        if a_len < b_len {
            return Less;
        }
        if a_len > b_len {
            return Greater;
        }

        for (&ai, &bi) in a.iter().rev().zip(b.iter().rev()) {
            if ai < bi {
                return Less;
            }
            if ai > bi {
                return Greater;
            }
        }
        return Equal;
    }

    #[cfg(test)]
    mod algorithm_tests {
        use {BigDigit, BigUint, BigInt};
        use Sign::Plus;
        use traits::Num;

        #[test]
        fn test_sub_sign() {
            use super::sub_sign;

            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_biguint(Plus, a.clone());
            let b_i = BigInt::from_biguint(Plus, 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);
        }
    }

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