big: implemented Karatsuba multiplication
Plus:
- calibration "test" - include in tests with gotest -calibrate
- basic Mul benchmark
- extra multiplication tests
- various cleanups
This change improves multiplication speed of numbers >= 30 words
in length (current threshold; found empirically with calibrate):
The multiplication benchmark (multiplication of a variety of long numbers)
improves by ~35%, individual multiplies can be significantly faster.
gotest -benchmarks=Mul
big.BenchmarkMul 500 6829290 ns/op (w/ Karatsuba)
big.BenchmarkMul 100 10600760 ns/op
There's no impact on pidigits for -n=10000 or -n=20000
because the operands are are too small.
R=rsc
CC=golang-dev
https://golang.org/cl/1004042
diff --git a/src/pkg/big/calibrate_test.go b/src/pkg/big/calibrate_test.go
new file mode 100644
index 0000000..04da8af
--- /dev/null
+++ b/src/pkg/big/calibrate_test.go
@@ -0,0 +1,91 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file computes the Karatsuba threshold as a "test".
+// Usage: gotest -calibrate
+
+package big
+
+import (
+ "flag"
+ "fmt"
+ "testing"
+ "time"
+ "unsafe" // for Sizeof
+)
+
+
+var calibrate = flag.Bool("calibrate", false, "run calibration test")
+
+
+// makeNumber creates an n-word number 0xffff...ffff
+func makeNumber(n int) *Int {
+ var w Word
+ b := make([]byte, n*unsafe.Sizeof(w))
+ for i := range b {
+ b[i] = 0xff
+ }
+ var x Int
+ x.SetBytes(b)
+ return &x
+}
+
+
+// measure returns the time to compute x*x in nanoseconds
+func measure(f func()) int64 {
+ const N = 100
+ start := time.Nanoseconds()
+ for i := N; i > 0; i-- {
+ f()
+ }
+ stop := time.Nanoseconds()
+ return (stop - start) / N
+}
+
+
+func computeThreshold(t *testing.T) int {
+ // use a mix of numbers as work load
+ x := make([]*Int, 20)
+ for i := range x {
+ x[i] = makeNumber(10 * (i + 1))
+ }
+
+ threshold := -1
+ for n := 8; threshold < 0 || n <= threshold+20; n += 2 {
+ // set work load
+ f := func() {
+ var t Int
+ for _, x := range x {
+ t.Mul(x, x)
+ }
+ }
+
+ karatsubaThreshold = 1e9 // disable karatsuba
+ t1 := measure(f)
+
+ karatsubaThreshold = n // enable karatsuba
+ t2 := measure(f)
+
+ c := '<'
+ mark := ""
+ if t1 > t2 {
+ c = '>'
+ if threshold < 0 {
+ threshold = n
+ mark = " *"
+ }
+ }
+
+ fmt.Printf("%4d: %8d %c %8d%s\n", n, t1, c, t2, mark)
+ }
+ return threshold
+}
+
+
+func TestCalibrate(t *testing.T) {
+ if *calibrate {
+ fmt.Printf("Computing Karatsuba threshold\n")
+ fmt.Printf("threshold = %d\n", computeThreshold(t))
+ }
+}
diff --git a/src/pkg/big/int.go b/src/pkg/big/int.go
index 6b570a0..e5e589a 100644
--- a/src/pkg/big/int.go
+++ b/src/pkg/big/int.go
@@ -230,7 +230,7 @@
// sets z to that value.
func (z *Int) SetBytes(b []byte) *Int {
s := int(_S)
- z.abs = z.abs.make((len(b)+s-1)/s, false)
+ z.abs = z.abs.make((len(b) + s - 1) / s)
z.neg = false
j := 0
@@ -386,7 +386,7 @@
func (z *Int) Lsh(x *Int, n uint) *Int {
addedWords := int(n) / _W
// Don't assign z.abs yet, in case z == x
- znew := z.abs.make(len(x.abs)+addedWords+1, false)
+ znew := z.abs.make(len(x.abs) + addedWords + 1)
z.neg = x.neg
znew[addedWords:].shiftLeft(x.abs, n%_W)
for i := range znew[0:addedWords] {
@@ -401,7 +401,7 @@
func (z *Int) Rsh(x *Int, n uint) *Int {
removedWords := int(n) / _W
// Don't assign z.abs yet, in case z == x
- znew := z.abs.make(len(x.abs)-removedWords, false)
+ znew := z.abs.make(len(x.abs) - removedWords)
z.neg = x.neg
znew.shiftRight(x.abs[removedWords:], n%_W)
z.abs = znew.norm()
diff --git a/src/pkg/big/int_test.go b/src/pkg/big/int_test.go
index bb42f81..cdcd28e 100644
--- a/src/pkg/big/int_test.go
+++ b/src/pkg/big/int_test.go
@@ -93,36 +93,55 @@
}
-var facts = map[int]string{
- 0: "1",
- 1: "1",
- 2: "2",
- 10: "3628800",
- 20: "2432902008176640000",
- 100: "933262154439441526816992388562667004907159682643816214685929" +
- "638952175999932299156089414639761565182862536979208272237582" +
- "51185210916864000000000000000000000000",
-}
+// mulBytes returns x*y via grade school multiplication. Both inputs
+// and the result are assumed to be in big-endian representation (to
+// match the semantics of Int.Bytes and Int.SetBytes).
+func mulBytes(x, y []byte) []byte {
+ z := make([]byte, len(x)+len(y))
-
-func fact(n int) *Int {
- var z Int
- z.New(1)
- for i := 2; i <= n; i++ {
- var t Int
- t.New(int64(i))
- z.Mul(&z, &t)
- }
- return &z
-}
-
-
-func TestFact(t *testing.T) {
- for n, s := range facts {
- f := fact(n).String()
- if f != s {
- t.Errorf("%d! = %s; want %s", n, f, s)
+ // multiply
+ k0 := len(z) - 1
+ for j := len(y) - 1; j >= 0; j-- {
+ d := int(y[j])
+ if d != 0 {
+ k := k0
+ carry := 0
+ for i := len(x) - 1; i >= 0; i-- {
+ t := int(z[k]) + int(x[i])*d + carry
+ z[k], carry = byte(t), t>>8
+ k--
+ }
+ z[k] = byte(carry)
}
+ k0--
+ }
+
+ // normalize (remove leading 0's)
+ i := 0
+ for i < len(z) && z[i] == 0 {
+ i++
+ }
+
+ return z[i:]
+}
+
+
+func checkMul(a, b []byte) bool {
+ var x, y, z1 Int
+ x.SetBytes(a)
+ y.SetBytes(b)
+ z1.Mul(&x, &y)
+
+ var z2 Int
+ z2.SetBytes(mulBytes(a, b))
+
+ return z1.Cmp(&z2) == 0
+}
+
+
+func TestMul(t *testing.T) {
+ if err := quick.Check(checkMul, nil); err != nil {
+ t.Error(err)
}
}
@@ -235,8 +254,7 @@
func TestSetBytes(t *testing.T) {
- err := quick.Check(checkSetBytes, nil)
- if err != nil {
+ if err := quick.Check(checkSetBytes, nil); err != nil {
t.Error(err)
}
}
@@ -249,8 +267,7 @@
func TestBytes(t *testing.T) {
- err := quick.Check(checkSetBytes, nil)
- if err != nil {
+ if err := quick.Check(checkSetBytes, nil); err != nil {
t.Error(err)
}
}
@@ -302,8 +319,7 @@
func TestDiv(t *testing.T) {
- err := quick.Check(checkDiv, nil)
- if err != nil {
+ if err := quick.Check(checkDiv, nil); err != nil {
t.Error(err)
}
@@ -676,6 +692,7 @@
-9223372036854775808,
}
+
func TestInt64(t *testing.T) {
for i, testVal := range int64Tests {
in := NewInt(testVal)
diff --git a/src/pkg/big/nat.go b/src/pkg/big/nat.go
index 2c8f837..0675416 100644
--- a/src/pkg/big/nat.go
+++ b/src/pkg/big/nat.go
@@ -36,6 +36,20 @@
type nat []Word
+var (
+ natOne = nat{1}
+ natTwo = nat{2}
+)
+
+
+func (z nat) clear() nat {
+ for i := range z {
+ z[i] = 0
+ }
+ return z
+}
+
+
func (z nat) norm() nat {
i := len(z)
for i > 0 && z[i-1] == 0 {
@@ -46,15 +60,9 @@
}
-func (z nat) make(m int, clear bool) nat {
+func (z nat) make(m int) nat {
if cap(z) > m {
- z = z[0:m] // reuse z - has at least one extra word for a carry, if any
- if clear {
- for i := range z {
- z[i] = 0
- }
- }
- return z
+ return z[0:m] // reuse z - has at least one extra word for a carry, if any
}
c := 4 // minimum capacity
@@ -67,12 +75,12 @@
func (z nat) new(x uint64) nat {
if x == 0 {
- return z.make(0, false)
+ return z.make(0)
}
// single-digit values
if x == uint64(Word(x)) {
- z = z.make(1, false)
+ z = z.make(1)
z[0] = Word(x)
return z
}
@@ -84,7 +92,7 @@
}
// split x into n words
- z = z.make(n, false)
+ z = z.make(n)
for i := 0; i < n; i++ {
z[i] = Word(x & _M)
x >>= _W
@@ -95,7 +103,7 @@
func (z nat) set(x nat) nat {
- z = z.make(len(x), false)
+ z = z.make(len(x))
for i, d := range x {
z[i] = d
}
@@ -112,14 +120,14 @@
return z.add(y, x)
case m == 0:
// n == 0 because m >= n; result is 0
- return z.make(0, false)
+ return z.make(0)
case n == 0:
// result is x
return z.set(x)
}
// m > 0
- z = z.make(m, false)
+ z = z.make(m)
c := addVV(&z[0], &x[0], &y[0], n)
if m > n {
c = addVW(&z[n], &x[n], c, m-n)
@@ -142,14 +150,14 @@
panic("underflow")
case m == 0:
// n == 0 because m >= n; result is 0
- return z.make(0, false)
+ return z.make(0)
case n == 0:
// result is x
return z.set(x)
}
// m > 0
- z = z.make(m, false)
+ z = z.make(m)
c := subVV(&z[0], &x[0], &y[0], n)
if m > n {
c = subVW(&z[n], &x[n], c, m-n)
@@ -198,7 +206,7 @@
}
// m > 0
- z = z.make(m, false)
+ z = z.make(m)
c := mulAddVWW(&z[0], &x[0], y, r, m)
if c > 0 {
z = z[0 : m+1]
@@ -209,6 +217,173 @@
}
+// basicMul multiplies x and y and leaves the result in z.
+// The (non-normalized) result is placed in z[0 : len(x) + len(y)].
+func basicMul(z, x, y nat) {
+ // initialize z
+ for i := range z[0 : len(x)+len(y)] {
+ z[i] = 0
+ }
+ // multiply
+ for i, d := range y {
+ if d != 0 {
+ z[len(x)+i] = addMulVVW(&z[i], &x[0], d, len(x))
+ }
+ }
+}
+
+
+// Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks.
+// Factored out for readability - do not use outside karatsuba.
+func karatsubaAdd(z, x nat, n int) {
+ if c := addVV(&z[0], &z[0], &x[0], n); c != 0 {
+ addVW(&z[n], &z[n], c, n>>1)
+ }
+}
+
+
+// Like karatsubaAdd, but does subtract.
+func karatsubaSub(z, x nat, n int) {
+ if c := subVV(&z[0], &z[0], &x[0], n); c != 0 {
+ subVW(&z[n], &z[n], c, n>>1)
+ }
+}
+
+
+// Operands that are shorter than karatsubaThreshold are multiplied using
+// "grade school" multiplication; for longer operands the Karatsuba algorithm
+// is used.
+var karatsubaThreshold int = 30 // modified by calibrate.go
+
+// karatsuba multiplies x and y and leaves the result in z.
+// Both x and y must have the same length n and n must be a
+// power of 2. The result vector z must have len(z) >= 6*n.
+// The (non-normalized) result is placed in z[0 : 2*n].
+func karatsuba(z, x, y nat) {
+ n := len(y)
+
+ // Switch to basic multiplication if numbers are odd or small.
+ // (n is always even if karatsubaThreshold is even, but be
+ // conservative)
+ if n&1 != 0 || n < karatsubaThreshold || n < 2 {
+ basicMul(z, x, y)
+ return
+ }
+ // n&1 == 0 && n >= karatsubaThreshold && n >= 2
+
+ // Karatsuba multiplication is based on the observation that
+ // for two numbers x and y with:
+ //
+ // x = x1*b + x0
+ // y = y1*b + y0
+ //
+ // the product x*y can be obtained with 3 products z2, z1, z0
+ // instead of 4:
+ //
+ // x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0
+ // = z2*b*b + z1*b + z0
+ //
+ // with:
+ //
+ // xd = x1 - x0
+ // yd = y0 - y1
+ //
+ // z1 = xd*yd + z1 + z0
+ // = (x1-x0)*(y0 - y1) + z1 + z0
+ // = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z1 + z0
+ // = x1*y0 - z1 - z0 + x0*y1 + z1 + z0
+ // = x1*y0 + x0*y1
+
+ // split x, y into "digits"
+ n2 := n >> 1 // n2 >= 1
+ x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0
+ y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0
+
+ // z is used for the result and temporary storage:
+ //
+ // 6*n 5*n 4*n 3*n 2*n 1*n 0*n
+ // z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ]
+ //
+ // For each recursive call of karatsuba, an unused slice of
+ // z is passed in that has (at least) half the length of the
+ // caller's z.
+
+ // compute z0 and z2 with the result "in place" in z
+ karatsuba(z, x0, y0) // z0 = x0*y0
+ karatsuba(z[n:], x1, y1) // z2 = x1*y1
+
+ // compute xd (or the negative value if underflow occurs)
+ s := 1 // sign of product xd*yd
+ xd := z[2*n : 2*n+n2]
+ if subVV(&xd[0], &x1[0], &x0[0], n2) != 0 { // x1-x0
+ s = -s
+ subVV(&xd[0], &x0[0], &x1[0], n2) // x0-x1
+ }
+
+ // compute yd (or the negative value if underflow occurs)
+ yd := z[2*n+n2 : 3*n]
+ if subVV(&yd[0], &y0[0], &y1[0], n2) != 0 { // y0-y1
+ s = -s
+ subVV(&yd[0], &y1[0], &y0[0], n2) // y1-y0
+ }
+
+ // p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0
+ // p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0
+ p := z[n*3:]
+ karatsuba(p, xd, yd)
+
+ // save original z2:z0
+ // (ok to use upper half of z since we're done recursing)
+ r := z[n*4:]
+ copy(r, z)
+
+ // add up all partial products
+ //
+ // 2*n n 0
+ // z = [ z2 | z0 ]
+ // + [ z0 ]
+ // + [ z2 ]
+ // + [ p ]
+ //
+ karatsubaAdd(z[n2:], r, n)
+ karatsubaAdd(z[n2:], r[n:], n)
+ if s > 0 {
+ karatsubaAdd(z[n2:], p, n)
+ } else {
+ karatsubaSub(z[n2:], p, n)
+ }
+}
+
+
+// alias returns true if x and y share the same base array.
+func alias(x, y nat) bool {
+ return &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
+}
+
+
+// addAt implements z += x*(1<<(_W*i)); z must be long enough.
+// (we don't use nat.add because we need z to stay the same
+// slice, and we don't need to normalize z after each addition)
+func addAt(z, x nat, i int) {
+ if n := len(x); n > 0 {
+ if c := addVV(&z[i], &z[i], &x[0], n); c != 0 {
+ j := i + n
+ if j < len(z) {
+ addVW(&z[j], &z[j], c, len(z)-j)
+ }
+ }
+ }
+}
+
+
+func max(x, y int) int {
+ if x > y {
+ return x
+ }
+ return y
+}
+
+
func (z nat) mul(x, y nat) nat {
m := len(x)
n := len(y)
@@ -217,25 +392,86 @@
case m < n:
return z.mul(y, x)
case m == 0 || n == 0:
- return z.make(0, false)
+ return z.make(0)
case n == 1:
return z.mulAddWW(x, y[0], 0)
}
- // m >= n && m > 1 && n > 1
+ // m >= n > 1
- if z == nil || &z[0] == &x[0] || &z[0] == &y[0] {
- z = nat(nil).make(m+n, true) // z is an alias for x or y - cannot reuse
- } else {
- z = z.make(m+n, true)
+ // determine if z can be reused
+ if len(z) > 0 && (alias(z, x) || alias(z, y)) {
+ z = nil // z is an alias for x or y - cannot reuse
}
- for i := 0; i < n; i++ {
- if f := y[i]; f != 0 {
- z[m+i] = addMulVVW(&z[i], &x[0], f, m)
- }
- }
- z = z.norm()
- return z
+ // use basic multiplication if the numbers are small
+ if n < karatsubaThreshold || n < 2 {
+ z = z.make(m + n)
+ basicMul(z, x, y)
+ return z.norm()
+ }
+ // m >= n && n >= karatsubaThreshold && n >= 2
+
+ // determine largest k such that
+ //
+ // x = x1*b + x0
+ // y = y1*b + y0 (and k <= len(y), which implies k <= len(x))
+ // b = 1<<(_W*k) ("base" of digits xi, yi)
+ //
+ // and k is karatsubaThreshold multiplied by a power of 2
+ k := max(karatsubaThreshold, 2)
+ for k*2 <= n {
+ k *= 2
+ }
+ // k <= n
+
+ // multiply x0 and y0 via Karatsuba
+ x0 := x[0:k] // x0 is not normalized
+ y0 := y[0:k] // y0 is not normalized
+ z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y
+ karatsuba(z, x0, y0)
+ z = z[0 : m+n] // z has final length but may be incomplete, upper portion is garbage
+
+ // If x1 and/or y1 are not 0, add missing terms to z explicitly:
+ //
+ // m+n 2*k 0
+ // z = [ ... | x0*y0 ]
+ // + [ x1*y1 ]
+ // + [ x1*y0 ]
+ // + [ x0*y1 ]
+ //
+ if k < n || m != n {
+ x1 := x[k:] // x1 is normalized because x is
+ y1 := y[k:] // y1 is normalized because y is
+ var t nat
+ t = t.mul(x1, y1)
+ copy(z[2*k:], t)
+ z[2*k+len(t):].clear() // upper portion of z is garbage
+ t = t.mul(x1, y0.norm())
+ addAt(z, t, k)
+ t = t.mul(x0.norm(), y1)
+ addAt(z, t, k)
+ }
+
+ return z.norm()
+}
+
+
+// mulRange computes the product of all the unsigned integers in the
+// range [a, b] inclusively. If a > b (empty range), the result is 1.
+func (z nat) mulRange(a, b uint64) nat {
+ switch {
+ case a == 0:
+ // cut long ranges short (optimization)
+ return z.new(0)
+ case a > b:
+ return z.new(1)
+ case a == b:
+ return z.new(a)
+ case a+1 == b:
+ return z.mul(nat(nil).new(a), nat(nil).new(b))
+ }
+ m := (a + b) / 2
+ return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b))
}
@@ -253,7 +489,7 @@
return
}
// m > 0
- z = z.make(m, false)
+ z = z.make(m)
r = divWVW(&z[0], 0, &x[0], y, m)
q = z.norm()
return
@@ -266,7 +502,7 @@
}
if u.cmp(v) < 0 {
- q = z.make(0, false)
+ q = z.make(0)
r = z2.set(u)
return
}
@@ -275,10 +511,10 @@
var rprime Word
q, rprime = z.divW(u, v[0])
if rprime > 0 {
- r = z2.make(1, false)
+ r = z2.make(1)
r[0] = rprime
} else {
- r = z2.make(0, false)
+ r = z2.make(0)
}
return
}
@@ -299,12 +535,12 @@
var u nat
if z2 == nil || &z2[0] == &uIn[0] {
- u = u.make(len(uIn)+1, true) // uIn is an alias for z2
+ u = u.make(len(uIn) + 1).clear() // uIn is an alias for z2
} else {
- u = z2.make(len(uIn)+1, true)
+ u = z2.make(len(uIn) + 1).clear()
}
qhatv := make(nat, len(v)+1)
- q = z.make(m+1, false)
+ q = z.make(m + 1)
// D1.
shift := uint(leadingZeroBits(v[n-1]))
@@ -363,11 +599,11 @@
// The result is the integer n for which 2^n <= x < 2^(n+1).
// If x == 0, the result is -1.
func log2(x Word) int {
- n := 0
+ n := -1
for ; x > 0; x >>= 1 {
n++
}
- return n - 1
+ return n
}
@@ -375,9 +611,8 @@
// The result is the integer n for which 2^n <= x < 2^(n+1).
// If x == 0, the result is -1.
func (x nat) log2() int {
- m := len(x)
- if m > 0 {
- return (m-1)*_W + log2(x[m-1])
+ if i := len(x) - 1; i >= 0 {
+ return i*_W + log2(x[i])
}
return -1
}
@@ -535,6 +770,9 @@
}
+// TODO(gri) Make the shift routines faster.
+// Use pidigits.go benchmark as a test case.
+
// To avoid losing the top n bits, z should be sized so that
// len(z) == len(x) + 1.
func (z nat) shiftLeft(x nat, n uint) nat {
@@ -582,7 +820,7 @@
func (x nat) modW(d Word) (r Word) {
// TODO(agl): we don't actually need to store the q value.
var q nat
- q = q.make(len(x), false)
+ q = q.make(len(x))
return divWVW(&q[0], 0, &x[0], d, len(x))
}
@@ -601,7 +839,7 @@
// zeroWords < len(n).
x := trailingZeroBits(n[zeroWords])
- q = q.make(len(n)-zeroWords, false)
+ q = q.make(len(n) - zeroWords)
q.shiftRight(n[zeroWords:], uint(x))
q = q.norm()
@@ -618,7 +856,7 @@
bitLengthOfMSW = _W
}
mask := Word((1 << bitLengthOfMSW) - 1)
- z = z.make(len(limit), false)
+ z = z.make(len(limit))
for {
for i := range z {
@@ -645,14 +883,14 @@
// reuses the storage of z if possible.
func (z nat) expNN(x, y, m nat) nat {
if len(y) == 0 {
- z = z.make(1, false)
+ z = z.make(1)
z[0] = 1
return z
}
if m != nil {
// We likely end up being as long as the modulus.
- z = z.make(len(m), false)
+ z = z.make(len(m))
}
z = z.set(x)
v := y[len(y)-1]
@@ -715,14 +953,6 @@
}
-const (
- primesProduct32 = 0xC0CFD797 // Π {p ∈ primes, 2 < p <= 29}
- primesProduct64 = 0xE221F97C30E94E1D // Π {p ∈ primes, 2 < p <= 53}
-)
-
-var bigOne = nat{1}
-var bigTwo = nat{2}
-
// probablyPrime performs reps Miller-Rabin tests to check whether n is prime.
// If it returns true, n is prime with probability 1 - 1/4^reps.
// If it returns false, n is not prime.
@@ -750,6 +980,9 @@
}
}
+ const primesProduct32 = 0xC0CFD797 // Π {p ∈ primes, 2 < p <= 29}
+ const primesProduct64 = 0xE221F97C30E94E1D // Π {p ∈ primes, 2 < p <= 53}
+
var r Word
switch _W {
case 32:
@@ -770,11 +1003,11 @@
return false
}
- nm1 := nat(nil).sub(n, bigOne)
+ nm1 := nat(nil).sub(n, natOne)
// 1<<k * q = nm1;
q, k := nm1.powersOfTwoDecompose()
- nm3 := nat(nil).sub(nm1, bigTwo)
+ nm3 := nat(nil).sub(nm1, natTwo)
rand := rand.New(rand.NewSource(int64(n[0])))
var x, y, quotient nat
@@ -783,9 +1016,9 @@
NextRandom:
for i := 0; i < reps; i++ {
x = x.random(rand, nm3, nm3Len)
- x = x.add(x, bigTwo)
+ x = x.add(x, natTwo)
y = y.expNN(x, q, n)
- if y.cmp(bigOne) == 0 || y.cmp(nm1) == 0 {
+ if y.cmp(natOne) == 0 || y.cmp(nm1) == 0 {
continue
}
for j := Word(1); j < k; j++ {
@@ -794,7 +1027,7 @@
if y.cmp(nm1) == 0 {
continue NextRandom
}
- if y.cmp(bigOne) == 0 {
+ if y.cmp(natOne) == 0 {
return false
}
}
diff --git a/src/pkg/big/nat_test.go b/src/pkg/big/nat_test.go
index ec24d61..52f712f 100644
--- a/src/pkg/big/nat_test.go
+++ b/src/pkg/big/nat_test.go
@@ -111,6 +111,64 @@
}
+type mulRange struct {
+ a, b uint64
+ prod string
+}
+
+
+var mulRanges = []mulRange{
+ mulRange{0, 0, "0"},
+ mulRange{1, 1, "1"},
+ mulRange{1, 2, "2"},
+ mulRange{1, 3, "6"},
+ mulRange{1, 3, "6"},
+ mulRange{10, 10, "10"},
+ mulRange{0, 100, "0"},
+ mulRange{0, 1e9, "0"},
+ mulRange{100, 1, "1"}, // empty range
+ mulRange{1, 10, "3628800"}, // 10!
+ mulRange{1, 20, "2432902008176640000"}, // 20!
+ mulRange{1, 100,
+ "933262154439441526816992388562667004907159682643816214685929" +
+ "638952175999932299156089414639761565182862536979208272237582" +
+ "51185210916864000000000000000000000000", // 100!
+ },
+}
+
+
+func TestMulRange(t *testing.T) {
+ for i, r := range mulRanges {
+ prod := nat(nil).mulRange(r.a, r.b).string(10)
+ if prod != r.prod {
+ t.Errorf("%d: got %s; want %s", i, prod, r.prod)
+ }
+ }
+}
+
+
+var mulArg nat
+
+func init() {
+ const n = 1000
+ mulArg = make(nat, n)
+ for i := 0; i < n; i++ {
+ mulArg[i] = _M
+ }
+}
+
+
+func BenchmarkMul(b *testing.B) {
+ for i := 0; i < b.N; i++ {
+ var t nat
+ for j := 1; j <= 10; j++ {
+ x := mulArg[0 : j*100]
+ t.mul(x, x)
+ }
+ }
+}
+
+
type strN struct {
x nat
b int
