From 08b0022aab249d16f1b63fef51b89b2f5bfff931 Mon Sep 17 00:00:00 2001
From: Kent Overstreet <kent.overstreet@gmail.com>
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)]