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

src/pkg/big/calibrate_test.go

+// 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))
+	}
+}

src/pkg/big/int.go

@@ -230,7 +230,7 @@ Error:
 // 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 ProbablyPrime(z *Int, n int) bool { return !z.neg && z.abs.probablyPrime(n)
 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) Lsh(x *Int, n uint) *Int {
 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()

src/pkg/big/int_test.go

@@ -93,36 +93,55 @@ func TestProdZZ(t *testing.T) {
 }
 
 
-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))
 
+	// 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--
+	}
 
-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)
+	// normalize (remove leading 0's)
+	i := 0
+	for i < len(z) && z[i] == 0 {
+		i++
 	}
-	return &z
+
+	return z[i:]
 }
 
 
-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)
-		}
+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 checkSetBytes(b []byte) bool {
 
 
 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 checkBytes(b []byte) bool {
 
 
 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 @@ var divTests = []divTest{
 
 
 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 @@ var int64Tests = []int64{
 	-9223372036854775808,
 }
 
+
 func TestInt64(t *testing.T) {
 	for i, testVal := range int64Tests {
 		in := NewInt(testVal)

src/pkg/big/nat_test.go

@@ -111,6 +111,64 @@ func TestFunNN(t *testing.T) {
 }
 
 
+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