math/big: fix superpolynomial complexity in Karatsuba algorithm.

benchmark                     old ns/op    new ns/op    delta
BenchmarkExp3Power0x10              732          734   +0.27%
BenchmarkExp3Power0x40              834          836   +0.24%
BenchmarkExp3Power0x100            1600         1579   -1.31%
BenchmarkExp3Power0x400            3478         3417   -1.75%
BenchmarkExp3Power0x1000          19388        19229   -0.82%
BenchmarkExp3Power0x4000         160274       156881   -2.12%
BenchmarkExp3Power0x10000       1552050      1372058  -11.60%
BenchmarkExp3Power0x40000      27328710     15216920  -44.32%
BenchmarkExp3Power0x100000    612349000    131407100  -78.54%
BenchmarkExp3Power0x400000  44073524000   1122195000  -97.45%

R=golang-dev, mtj, gri, rsc
CC=golang-dev, remy
https://golang.org/cl/6176043

src/pkg/math/big/nat.go

@@ -271,10 +271,10 @@ func karatsuba(z, x, y nat) {
 	//   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
+	//   z1 =      xd*yd                    + z2 + z0
+	//      = (x1-x0)*(y0 - y1)             + z2 + z0
+	//      = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z2 + z0
+	//      = x1*y0 -    z2 -    z0 + x0*y1 + z2 + z0
 	//      = x1*y0                 + x0*y1
 
 	// split x, y into "digits"
@@ -318,7 +318,7 @@ func karatsuba(z, x, y nat) {
 	// save original z2:z0
 	// (ok to use upper half of z since we're done recursing)
 	r := z[n*4:]
-	copy(r, z)
+	copy(r, z[:n*2])
 
 	// add up all partial products
 	//

src/pkg/math/big/nat_test.go

@@ -661,3 +661,21 @@ func TestExpNN(t *testing.T) {
 		}
 	}
 }
+
+func ExpHelper(b *testing.B, x, y Word) {
+	var z nat
+	for i := 0; i < b.N; i++ {
+		z.expWW(x, y)
+	}
+}
+
+func BenchmarkExp3Power0x10(b *testing.B)     { ExpHelper(b, 3, 0x10) }
+func BenchmarkExp3Power0x40(b *testing.B)     { ExpHelper(b, 3, 0x40) }
+func BenchmarkExp3Power0x100(b *testing.B)    { ExpHelper(b, 3, 0x100) }
+func BenchmarkExp3Power0x400(b *testing.B)    { ExpHelper(b, 3, 0x400) }
+func BenchmarkExp3Power0x1000(b *testing.B)   { ExpHelper(b, 3, 0x1000) }
+func BenchmarkExp3Power0x4000(b *testing.B)   { ExpHelper(b, 3, 0x4000) }
+func BenchmarkExp3Power0x10000(b *testing.B)  { ExpHelper(b, 3, 0x10000) }
+func BenchmarkExp3Power0x40000(b *testing.B)  { ExpHelper(b, 3, 0x40000) }
+func BenchmarkExp3Power0x100000(b *testing.B) { ExpHelper(b, 3, 0x100000) }
+func BenchmarkExp3Power0x400000(b *testing.B) { ExpHelper(b, 3, 0x400000) }