math/big: added nat.trailingZeroBits, simplified some code

Will simplify implementation of binaryGCD.

R=rsc, cswenson
CC=golang-dev
https://golang.org/cl/6299064

src/pkg/math/big/nat.go

@@ -740,7 +740,7 @@ func (x nat) string(charset string) string {
 	// convert power of two and non power of two bases separately
 	if b == b&-b {
 		// shift is base-b digit size in bits
-		shift := uint(trailingZeroBits(b)) // shift > 0 because b >= 2
+		shift := trailingZeroBits(b) // shift > 0 because b >= 2
 		mask := Word(1)<<shift - 1
 		w := x[0]
 		nbits := uint(_W) // number of unprocessed bits in w
@@ -993,10 +993,9 @@ var deBruijn64Lookup = []byte{
 	54, 26, 40, 15, 34, 20, 31, 10, 25, 14, 19, 9, 13, 8, 7, 6,
 }
 
-// trailingZeroBits returns the number of consecutive zero bits on the right
-// side of the given Word.
-// See Knuth, volume 4, section 7.3.1
-func trailingZeroBits(x Word) int {
+// trailingZeroBits returns the number of consecutive least significant zero
+// bits of x.
+func trailingZeroBits(x Word) uint {
 	// x & -x leaves only the right-most bit set in the word. Let k be the
 	// index of that bit. Since only a single bit is set, the value is two
 	// to the power of k. Multiplying by a power of two is equivalent to
@@ -1005,11 +1004,12 @@ func trailingZeroBits(x Word) int {
 	// Therefore, if we have a left shifted version of this constant we can
 	// find by how many bits it was shifted by looking at which six bit
 	// substring ended up at the top of the word.
+	// (Knuth, volume 4, section 7.3.1)
 	switch _W {
 	case 32:
-		return int(deBruijn32Lookup[((x&-x)*deBruijn32)>>27])
+		return uint(deBruijn32Lookup[((x&-x)*deBruijn32)>>27])
 	case 64:
-		return int(deBruijn64Lookup[((x&-x)*(deBruijn64&_M))>>58])
+		return uint(deBruijn64Lookup[((x&-x)*(deBruijn64&_M))>>58])
 	default:
 		panic("Unknown word size")
 	}
@@ -1017,6 +1017,20 @@ func trailingZeroBits(x Word) int {
 	return 0
 }
 
+// trailingZeroBits returns the number of consecutive least significant zero
+// bits of x.
+func (x nat) trailingZeroBits() uint {
+	if len(x) == 0 {
+		return 0
+	}
+	var i uint
+	for x[i] == 0 {
+		i++
+	}
+	// x[i] != 0
+	return i*_W + trailingZeroBits(x[i])
+}
+
 // z = x << s
 func (z nat) shl(x nat, s uint) nat {
 	m := len(x)
@@ -1169,29 +1183,6 @@ func (x nat) modW(d Word) (r Word) {
 	return divWVW(q, 0, x, d)
 }
 
-// powersOfTwoDecompose finds q and k with x = q * 1<<k and q is odd, or q and k are 0.
-func (x nat) powersOfTwoDecompose() (q nat, k int) {
-	if len(x) == 0 {
-		return x, 0
-	}
-
-	// One of the words must be non-zero by definition,
-	// so this loop will terminate with i < len(x), and
-	// i is the number of 0 words.
-	i := 0
-	for x[i] == 0 {
-		i++
-	}
-	n := trailingZeroBits(x[i]) // x[i] != 0
-
-	q = make(nat, len(x)-i)
-	shrVU(q, x[i:], uint(n))
-
-	q = q.norm()
-	k = i*_W + n
-	return
-}
-
 // random creates a random integer in [0..limit), using the space in z if
 // possible. n is the bit length of limit.
 func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
@@ -1343,8 +1334,9 @@ func (n nat) probablyPrime(reps int) bool {
 	}
 
 	nm1 := nat(nil).sub(n, natOne)
-	// 1<<k * q = nm1;
-	q, k := nm1.powersOfTwoDecompose()
+	// determine q, k such that nm1 = q << k
+	k := nm1.trailingZeroBits()
+	q := nat(nil).shr(nm1, k)
 
 	nm3 := nat(nil).sub(nm1, natTwo)
 	rand := rand.New(rand.NewSource(int64(n[0])))
@@ -1360,7 +1352,7 @@ NextRandom:
 		if y.cmp(natOne) == 0 || y.cmp(nm1) == 0 {
 			continue
 		}
-		for j := 1; j < k; j++ {
+		for j := uint(1); j < k; j++ {
 			y = y.mul(y, y)
 			quotient, y = quotient.div(y, y, n)
 			if y.cmp(nm1) == 0 {

src/pkg/math/big/nat_test.go

@@ -613,14 +613,23 @@ func TestModW(t *testing.T) {
 }
 
 func TestTrailingZeroBits(t *testing.T) {
-	var x Word
-	x--
-	for i := 0; i < _W; i++ {
-		if trailingZeroBits(x) != i {
-			t.Errorf("Failed at step %d: x: %x got: %d", i, x, trailingZeroBits(x))
+	x := Word(1)
+	for i := uint(0); i <= _W; i++ {
+		n := trailingZeroBits(x)
+		if n != i%_W {
+			t.Errorf("got trailingZeroBits(%#x) = %d; want %d", x, n, i%_W)
 		}
 		x <<= 1
 	}
+
+	y := nat(nil).set(natOne)
+	for i := uint(0); i <= 3*_W; i++ {
+		n := y.trailingZeroBits()
+		if n != i {
+			t.Errorf("got 0x%s.trailingZeroBits() = %d; want %d", y.string(lowercaseDigits[0:16]), n, i)
+		}
+		y = y.shl(y, 1)
+	}
 }
 
 var expNNTests = []struct {