From 08b0022aab249d16f1b63fef51b89b2f5bfff931 Mon Sep 17 00:00:00 2001 From: Kent Overstreet Date: Thu, 10 Dec 2015 15:25:42 -0900 Subject: [PATCH] Improve multiply performance The main idea here is to do as much as possible with slices, instead of allocating new BigUints (= heap allocations). Current performance: multiply_0: 10,507 ns/iter (+/- 987) multiply_1: 2,788,734 ns/iter (+/- 100,079) multiply_2: 69,923,515 ns/iter (+/- 4,550,902) After this patch, we get: multiply_0: 364 ns/iter (+/- 62) multiply_1: 34,085 ns/iter (+/- 1,179) multiply_2: 3,753,883 ns/iter (+/- 46,876) --- src/bigint.rs | 289 ++++++++++++++++++++++++++++++++++++++------------ 1 file changed, 219 insertions(+), 70 deletions(-) diff --git a/src/bigint.rs b/src/bigint.rs index ad0c2ed..c47f99c 100644 --- a/src/bigint.rs +++ b/src/bigint.rs @@ -148,6 +148,16 @@ fn sbb(a: BigDigit, b: BigDigit, borrow: &mut BigDigit) -> BigDigit { lo } +#[inline] +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 +} + /// A big unsigned integer type. /// /// A `BigUint`-typed value `BigUint { data: vec!(a, b, c) }` represents a number @@ -172,18 +182,25 @@ impl PartialOrd for BigUint { } } +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; +} + impl Ord for BigUint { #[inline] fn cmp(&self, other: &BigUint) -> Ordering { - let (s_len, o_len) = (self.data.len(), other.data.len()); - if s_len < o_len { return Less; } - if s_len > o_len { return Greater; } - - for (&self_i, &other_i) in self.data.iter().rev().zip(other.data.iter().rev()) { - if self_i < other_i { return Less; } - if self_i > other_i { return Greater; } - } - return Equal; + cmp_slice(&self.data[..], &other.data[..]) } } @@ -608,80 +625,202 @@ impl<'a> Sub<&'a BigUint> for BigUint { } } +fn sub_sign(a: &[BigDigit], b: &[BigDigit]) -> BigInt { + // 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)]; -forward_all_binop_to_val_ref_commutative!(impl Mul for BigUint, mul); + match cmp_slice(a, b) { + Greater => { + let mut ret = BigUint::from_slice(a); + sub2(&mut ret.data[..], b); + BigInt::from_biguint(Plus, ret.normalize()) + }, + Less => { + let mut ret = BigUint::from_slice(b); + sub2(&mut ret.data[..], a); + BigInt::from_biguint(Minus, ret.normalize()) + }, + _ => Zero::zero(), + } +} -impl<'a> Mul<&'a BigUint> for BigUint { - type Output = BigUint; +forward_all_binop_to_ref_ref!(impl Mul for BigUint, mul); - fn mul(self, other: &BigUint) -> BigUint { - if self.is_zero() || other.is_zero() { return Zero::zero(); } +/// Three argument multiply accumulate: +/// acc += b * c +fn mac_digit(acc: &mut [BigDigit], b: &[BigDigit], c: BigDigit) { + if c == 0 { return; } - let (s_len, o_len) = (self.data.len(), other.data.len()); - if s_len == 1 { return mul_digit(other.clone(), self.data[0]); } - if o_len == 1 { return mul_digit(self, other.data[0]); } + let mut b_iter = b.iter(); + let mut carry = 0; - // Using Karatsuba multiplication - // (a1 * base + a0) * (b1 * base + b0) - // = a1*b1 * base^2 + - // (a1*b1 + a0*b0 - (a1-b0)*(b1-a0)) * base + - // a0*b0 - let half_len = cmp::max(s_len, o_len) / 2; - let (s_hi, s_lo) = cut_at(self, half_len); - let (o_hi, o_lo) = cut_at(other.clone(), half_len); - - let ll = &s_lo * &o_lo; - let hh = &s_hi * &o_hi; - let mm = { - let (s1, n1) = sub_sign(s_hi, s_lo); - let (s2, n2) = sub_sign(o_hi, o_lo); - match (s1, s2) { - (Equal, _) | (_, Equal) => &hh + &ll, - (Less, Greater) | (Greater, Less) => &hh + &ll + (n1 * n2), - (Less, Less) | (Greater, Greater) => &hh + &ll - (n1 * n2) - } - }; - - return ll + mm.shl_unit(half_len) + hh.shl_unit(half_len * 2); - - - fn mul_digit(a: BigUint, n: BigDigit) -> BigUint { - if n == 0 { return Zero::zero(); } - if n == 1 { return a; } - - let mut carry = 0; - let mut prod = a.data; - for a in &mut prod { - let d = (*a as DoubleBigDigit) - * (n as DoubleBigDigit) - + (carry as DoubleBigDigit); - let (hi, lo) = big_digit::from_doublebigdigit(d); - carry = hi; - *a = lo; - } - if carry != 0 { prod.push(carry); } - BigUint::new(prod) + for ai in acc.iter_mut() { + if let Some(bi) = b_iter.next() { + *ai = mac_with_carry(*ai, *bi, c, &mut carry); + } else if carry != 0 { + *ai = mac_with_carry(*ai, 0, c, &mut carry); + } else { + break; } + } - #[inline] - fn cut_at(mut a: BigUint, n: usize) -> (BigUint, BigUint) { - let mid = cmp::min(a.data.len(), n); - let hi = BigUint::from_slice(&a.data[mid ..]); - a.data.truncate(mid); - (hi, BigUint::new(a.data)) + assert!(carry == 0); +} + +/// 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) }; + + /* + * Karatsuba multiplication is slower than long multiplication for small x and y: + */ + if x.len() <= 4 { + for (i, xi) in x.iter().enumerate() { + mac_digit(&mut acc[i..], y, *xi); } + } else { + /* + * 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 * p0) * 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. + */ - #[inline] - fn sub_sign(a: BigUint, b: BigUint) -> (Ordering, BigUint) { - match a.cmp(&b) { - Less => (Less, b - a), - Greater => (Greater, a - b), - _ => (Equal, Zero::zero()) - } + /* + * 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: */ + + let len = y.len() + 1; + let mut p: BigUint = BigUint { data: Vec::with_capacity(len) }; + p.data.extend(repeat(0).take(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 = 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 = 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 = sub_sign(x1, x0); + let 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.data[..], &j1.data.data[..]); + p = p.normalize(); + + sub2(&mut acc[b..], &p.data[..]); + }, + Minus => { + mac3(&mut acc[b..], &j0.data.data[..], &j1.data.data[..]); + }, + NoSign => (), } } } +fn mul3(x: &[BigDigit], y: &[BigDigit]) -> BigUint { + let len = x.len() + y.len() + 1; + let mut prod: BigUint = BigUint { data: Vec::with_capacity(len) }; + + // resize isn't stable yet: + //prod.data.resize(len, 0); + prod.data.extend(repeat(0).take(len)); + + mac3(&mut prod.data[..], x, y); + prod.normalize() +} + +impl<'a, 'b> Mul<&'b BigUint> for &'a BigUint { + type Output = BigUint; + + #[inline] + fn mul(self, other: &BigUint) -> BigUint { + mul3(&self.data[..], &other.data[..]) + } +} forward_all_binop_to_ref_ref!(impl Div for BigUint, div); @@ -3131,6 +3270,16 @@ mod biguint_tests { // Switching u and l should fail: let _n: BigUint = rng.gen_biguint_range(&u, &l); } + + #[test] + fn test_sub_sign() { + use super::sub_sign; + let a = BigInt::from_str_radix("265252859812191058636308480000000", 10).unwrap(); + let b = BigInt::from_str_radix("26525285981219105863630848000000", 10).unwrap(); + + assert_eq!(sub_sign(&a.data.data[..], &b.data.data[..]), &a - &b); + assert_eq!(sub_sign(&b.data.data[..], &a.data.data[..]), &b - &a); + } } #[cfg(test)]