math/bits: faster OnesCount

Using some additional suggestions per "Hacker's Delight".
Added documentation and extra tests.

Measured on 1.7 GHz Intel Core i7, running macOS 10.12.3.

benchmark                  old ns/op     new ns/op     delta
BenchmarkOnesCount-4       7.34          5.38          -26.70%
BenchmarkOnesCount8-4      2.03          1.98          -2.46%
BenchmarkOnesCount16-4     2.56          2.50          -2.34%
BenchmarkOnesCount32-4     2.98          2.39          -19.80%
BenchmarkOnesCount64-4     4.22          2.96          -29.86%

Change-Id: I566b0ef766e55cf5776b1662b6016024ebe5d878
Reviewed-on: https://go-review.googlesource.com/37223Reviewed-by: Matthew Dempsky <mdempsky@google.com>
Run-TryBot: Matthew Dempsky <mdempsky@google.com>
TryBot-Result: Gobot Gobot <gobot@golang.org>

src/math/bits/bits.go

@@ -50,7 +50,6 @@ const m1 = 0x3333333333333333 // 00110011 ...
 const m2 = 0x0f0f0f0f0f0f0f0f // 00001111 ...
 const m3 = 0x00ff00ff00ff00ff // etc.
 const m4 = 0x0000ffff0000ffff
-const m5 = 0x00000000ffffffff
 
 // OnesCount returns the number of one bits ("population count") in x.
 func OnesCount(x uint) int {
@@ -65,7 +64,8 @@ func OnesCount8(x uint8) int {
 	const m = 1<<8 - 1
 	x = x>>1&(m0&m) + x&(m0&m)
 	x = x>>2&(m1&m) + x&(m1&m)
-	return int(x>>4 + x&(m2&m))
+	x += x >> 4
+	return int(x) & (1<<4 - 1)
 }
 
 // OnesCount16 returns the number of one bits ("population count") in x.
@@ -73,8 +73,9 @@ func OnesCount16(x uint16) int {
 	const m = 1<<16 - 1
 	x = x>>1&(m0&m) + x&(m0&m)
 	x = x>>2&(m1&m) + x&(m1&m)
-	x = x>>4&(m2&m) + x&(m2&m)
-	return int(x>>8 + x&(m3&m))
+	x = (x>>4 + x) & (m2 & m)
+	x += x >> 8
+	return int(x) & (1<<5 - 1)
 }
 
 // OnesCount32 returns the number of one bits ("population count") in x.
@@ -82,20 +83,41 @@ func OnesCount32(x uint32) int {
 	const m = 1<<32 - 1
 	x = x>>1&(m0&m) + x&(m0&m)
 	x = x>>2&(m1&m) + x&(m1&m)
-	x = x>>4&(m2&m) + x&(m2&m)
-	x = x>>8&(m3&m) + x&(m3&m)
-	return int(x>>16 + x&(m4&m))
+	x = (x>>4 + x) & (m2 & m)
+	x += x >> 8
+	x += x >> 16
+	return int(x) & (1<<6 - 1)
 }
 
 // OnesCount64 returns the number of one bits ("population count") in x.
 func OnesCount64(x uint64) int {
+	// Implementation: Parallel summing of adjacent bits.
+	// See "Hacker's Delight", Chap. 5: Counting Bits.
+	// The following pattern shows the general approach:
+	//
+	//   x = x>>1&(m0&m) + x&(m0&m)
+	//   x = x>>2&(m1&m) + x&(m1&m)
+	//   x = x>>4&(m2&m) + x&(m2&m)
+	//   x = x>>8&(m3&m) + x&(m3&m)
+	//   x = x>>16&(m4&m) + x&(m4&m)
+	//   x = x>>32&(m5&m) + x&(m5&m)
+	//   return int(x)
+	//
+	// Masking (& operations) can be left away when there's no
+	// danger that a field's sum will carry over into the next
+	// field: Since the result cannot be > 64, 8 bits is enough
+	// and we can ignore the masks for the shifts by 8 and up.
+	// Per "Hacker's Delight", the first line can be simplified
+	// more, but it saves at best one instruction, so we leave
+	// it alone for clarity.
 	const m = 1<<64 - 1
 	x = x>>1&(m0&m) + x&(m0&m)
 	x = x>>2&(m1&m) + x&(m1&m)
-	x = x>>4&(m2&m) + x&(m2&m)
-	x = x>>8&(m3&m) + x&(m3&m)
-	x = x>>16&(m4&m) + x&(m4&m)
-	return int(x>>32 + x&(m5&m))
+	x = (x>>4 + x) & (m2 & m)
+	x += x >> 8
+	x += x >> 16
+	x += x >> 32
+	return int(x) & (1<<7 - 1)
 }
 
 // --- RotateLeft ---

src/math/bits/bits_test.go

@@ -232,10 +232,25 @@ func BenchmarkTrailingZeros64(b *testing.B) {
 }
 
 func TestOnesCount(t *testing.T) {
+	var x uint64
+	for i := 0; i <= 64; i++ {
+		testOnesCount(t, x, i)
+		x = x<<1 | 1
+	}
+
+	for i := 64; i >= 0; i-- {
+		testOnesCount(t, x, i)
+		x = x << 1
+	}
+
 	for i := 0; i < 256; i++ {
-		want := tab[i].pop
 		for k := 0; k < 64-8; k++ {
-			x := uint64(i) << uint(k)
+			testOnesCount(t, uint64(i)<<uint(k), tab[i].pop)
+		}
+	}
+}
+
+func testOnesCount(t *testing.T, x uint64, want int) {
 	if x <= 1<<8-1 {
 		got := OnesCount8(uint8(x))
 		if got != want {
@@ -275,8 +290,6 @@ func TestOnesCount(t *testing.T) {
 			}
 		}
 	}
-		}
-	}
 }
 
 func BenchmarkOnesCount(b *testing.B) {