math/fixed: add Mul methods.
Change-Id: Id6f9f5825527b311b5b1aa4ae0923c9551fa076b
Reviewed-on: https://go-review.googlesource.com/27413
Reviewed-by: David Crawshaw <crawshaw@golang.org>
Run-TryBot: David Crawshaw <crawshaw@golang.org>
TryBot-Result: Gobot Gobot <gobot@golang.org>
diff --git a/math/fixed/fixed.go b/math/fixed/fixed.go
index df3540a..2c76ed4 100644
--- a/math/fixed/fixed.go
+++ b/math/fixed/fixed.go
@@ -56,6 +56,11 @@
// Its return type is int, not Int26_6.
func (x Int26_6) Ceil() int { return int((x + 0x3f) >> 6) }
+// Mul returns x*y in 26.6 fixed-point arithmetic.
+func (x Int26_6) Mul(y Int26_6) Int26_6 {
+ return Int26_6((int64(x)*int64(y) + 1<<5) >> 6)
+}
+
// Int52_12 is a signed 52.12 fixed-point number.
//
// The integer part ranges from -2251799813685248 to 2251799813685247,
@@ -95,6 +100,39 @@
// Its return type is int, not Int52_12.
func (x Int52_12) Ceil() int { return int((x + 0xfff) >> 12) }
+// Mul returns x*y in 52.12 fixed-point arithmetic.
+func (x Int52_12) Mul(y Int52_12) Int52_12 {
+ const M, N = 52, 12
+ lo, hi := muli64(int64(x), int64(y))
+ ret := Int52_12(hi<<M | lo>>N)
+ ret += Int52_12((lo >> (N - 1)) & 1) // Round to nearest, instead of rounding down.
+ return ret
+}
+
+// muli64 multiplies two int64 values, returning the 128-bit signed integer
+// result as two uint64 values.
+//
+// This implementation is similar to $GOROOT/src/runtime/softfloat64.go's mullu
+// function, which is in turn adapted from Hacker's Delight.
+func muli64(u, v int64) (lo, hi uint64) {
+ const (
+ s = 32
+ mask = 1<<s - 1
+ )
+
+ u1 := uint64(u >> s)
+ u0 := uint64(u & mask)
+ v1 := uint64(v >> s)
+ v0 := uint64(v & mask)
+
+ w0 := u0 * v0
+ t := u1*v0 + w0>>s
+ w1 := t & mask
+ w2 := uint64(int64(t) >> s)
+ w1 += u0 * v1
+ return uint64(u) * uint64(v), u1*v1 + w2 + uint64(int64(w1)>>s)
+}
+
// P returns the integer values x and y as a Point26_6.
//
// For example, passing the integer values (2, -3) yields Point26_6{128, -192}.
diff --git a/math/fixed/fixed_test.go b/math/fixed/fixed_test.go
index 065ab00..c81fb72 100644
--- a/math/fixed/fixed_test.go
+++ b/math/fixed/fixed_test.go
@@ -5,6 +5,8 @@
package fixed
import (
+ "math"
+ "math/rand"
"testing"
)
@@ -74,6 +76,7 @@
}}
func TestInt26_6(t *testing.T) {
+ const one = Int26_6(1 << 6)
for _, tc := range testCases {
x := Int26_6(tc.x * (1 << 6))
if got, want := x.String(), tc.s26_6; got != want {
@@ -88,10 +91,17 @@
if got, want := x.Ceil(), tc.ceil; got != want {
t.Errorf("tc.x=%v: Ceil: got %v, want %v", tc.x, got, want)
}
+ if got, want := x.Mul(one), x; got != want {
+ t.Errorf("tc.x=%v: Mul by one: got %v, want %v", tc.x, got, want)
+ }
+ if got, want := x.mul(one), x; got != want {
+ t.Errorf("tc.x=%v: mul by one: got %v, want %v", tc.x, got, want)
+ }
}
}
func TestInt52_12(t *testing.T) {
+ const one = Int52_12(1 << 12)
for _, tc := range testCases {
x := Int52_12(tc.x * (1 << 12))
if got, want := x.String(), tc.s52_12; got != want {
@@ -106,5 +116,324 @@
if got, want := x.Ceil(), tc.ceil; got != want {
t.Errorf("tc.x=%v: Ceil: got %v, want %v", tc.x, got, want)
}
+ if got, want := x.Mul(one), x; got != want {
+ t.Errorf("tc.x=%v: Mul by one: got %v, want %v", tc.x, got, want)
+ }
}
}
+
+var mulTestCases = []struct {
+ x float64
+ y float64
+ z26_6 float64 // Equals truncate26_6(x)*truncate26_6(y).
+ z52_12 float64 // Equals truncate52_12(x)*truncate52_12(y).
+ s26_6 string
+ s52_12 string
+}{{
+ x: 0,
+ y: 1.5,
+ z26_6: 0,
+ z52_12: 0,
+ s26_6: "0:00",
+ s52_12: "0:0000",
+}, {
+ x: +1.25,
+ y: +4,
+ z26_6: +5,
+ z52_12: +5,
+ s26_6: "5:00",
+ s52_12: "5:0000",
+}, {
+ x: +1.25,
+ y: -4,
+ z26_6: -5,
+ z52_12: -5,
+ s26_6: "-5:00",
+ s52_12: "-5:0000",
+}, {
+ x: -1.25,
+ y: +4,
+ z26_6: -5,
+ z52_12: -5,
+ s26_6: "-5:00",
+ s52_12: "-5:0000",
+}, {
+ x: -1.25,
+ y: -4,
+ z26_6: +5,
+ z52_12: +5,
+ s26_6: "5:00",
+ s52_12: "5:0000",
+}, {
+ x: 1.25,
+ y: 1.5,
+ z26_6: 1.875,
+ z52_12: 1.875,
+ s26_6: "1:56",
+ s52_12: "1:3584",
+}, {
+ x: 1234.5,
+ y: -8888.875,
+ z26_6: -10973316.1875,
+ z52_12: -10973316.1875,
+ s26_6: "-10973316:12",
+ s52_12: "-10973316:0768",
+}, {
+ x: 1.515625, // 1 + 33/64 = 97/64
+ y: 1.531250, // 1 + 34/64 = 98/64
+ z26_6: 2.32080078125, // 2 + 1314/4096 = 9506/4096
+ z52_12: 2.32080078125, // 2 + 1314/4096 = 9506/4096
+ s26_6: "2:21", // 2.32812500000, which is closer than 2:20 (in decimal, 2.3125)
+ s52_12: "2:1314", // 2.32080078125
+}, {
+ x: 0.500244140625, // 2049/4096, approximately 32/64
+ y: 0.500732421875, // 2051/4096, approximately 32/64
+ z26_6: 0.25, // 4194304/16777216, or 1024/4096
+ z52_12: 0.2504884600639343, // 4202499/16777216
+ s26_6: "0:16", // 0.25000000000
+ s52_12: "0:1026", // 0.25048828125, which is closer than 0:1027 (in decimal, 0.250732421875)
+}, {
+ x: 0.015625, // 1/64
+ y: 0.000244140625, // 1/4096, approximately 0/64
+ z26_6: 0.0, // 0
+ z52_12: 0.000003814697265625, // 1/262144
+ s26_6: "0:00", // 0
+ s52_12: "0:0000", // 0, which is closer than 0:0001 (in decimal, 0.000244140625)
+}, {
+ // Round the Int52_12 calculation down.
+ x: 1.44140625, // 1 + 1808/4096 = 5904/4096, approximately 92/64
+ y: 1.44140625, // 1 + 1808/4096 = 5904/4096, approximately 92/64
+ z26_6: 2.06640625, // 2 + 272/4096 = 8464/4096
+ z52_12: 2.0776519775390625, // 2 + 318/4096 + 256/16777216 = 34857216/16777216
+ s26_6: "2:04", // 2.06250000000, which is closer than 2:05 (in decimal, 2.078125000000)
+ s52_12: "2:0318", // 2.07763671875, which is closer than 2:0319 (in decimal, 2.077880859375)
+}, {
+ // Round the Int52_12 calculation up.
+ x: 1.44140625, // 1 + 1808/4096 = 5904/4096, approximately 92/64
+ y: 1.441650390625, // 1 + 1809/4096 = 5905/4096, approximately 92/64
+ z26_6: 2.06640625, // 2 + 272/4096 = 8464/4096
+ z52_12: 2.0780038833618164, // 2 + 319/4096 + 2064/16777216 = 34863120/16777216
+ s26_6: "2:04", // 2.06250000000, which is closer than 2:05 (in decimal, 2.078125000000)
+ s52_12: "2:0320", // 2.07812500000, which is closer than 2:0319 (in decimal, 2.077880859375)
+}}
+
+func TestInt26_6Mul(t *testing.T) {
+ for _, tc := range mulTestCases {
+ x := Int26_6(tc.x * (1 << 6))
+ y := Int26_6(tc.y * (1 << 6))
+ if z := float64(x) * float64(y) / (1 << 12); z != tc.z26_6 {
+ t.Errorf("tc.x=%v, tc.y=%v: z: got %v, want %v", tc.x, tc.y, z, tc.z26_6)
+ continue
+ }
+ if got, want := x.Mul(y).String(), tc.s26_6; got != want {
+ t.Errorf("tc.x=%v: Mul: got %q, want %q", tc.x, got, want)
+ }
+ }
+}
+
+func TestInt52_12Mul(t *testing.T) {
+ for _, tc := range mulTestCases {
+ x := Int52_12(tc.x * (1 << 12))
+ y := Int52_12(tc.y * (1 << 12))
+ if z := float64(x) * float64(y) / (1 << 24); z != tc.z52_12 {
+ t.Errorf("tc.x=%v, tc.y=%v: z: got %v, want %v", tc.x, tc.y, z, tc.z52_12)
+ continue
+ }
+ if got, want := x.Mul(y).String(), tc.s52_12; got != want {
+ t.Errorf("tc.x=%v: Mul: got %q, want %q", tc.x, got, want)
+ }
+ }
+}
+
+func TestInt26_6MulByOneMinusIota(t *testing.T) {
+ const (
+ totalBits = 32
+ fracBits = 6
+
+ oneMinusIota = Int26_6(1<<fracBits) - 1
+ oneMinusIotaF = float64(oneMinusIota) / (1 << fracBits)
+ )
+
+ for _, neg := range []bool{false, true} {
+ for i := uint(0); i < totalBits; i++ {
+ x := Int26_6(1 << i)
+ if neg {
+ x = -x
+ } else if i == totalBits-1 {
+ // A signed int32 can't represent 1<<31.
+ continue
+ }
+
+ // want equals x * oneMinusIota, rounded to nearest.
+ want := Int26_6(0)
+ if -1<<fracBits < x && x < 1<<fracBits {
+ // (x * oneMinusIota) isn't exactly representable as an
+ // Int26_6. Calculate the rounded value using float64 math.
+ xF := float64(x) / (1 << fracBits)
+ wantF := xF * oneMinusIotaF * (1 << fracBits)
+ want = Int26_6(math.Floor(wantF + 0.5))
+ } else {
+ // (x * oneMinusIota) is exactly representable.
+ want = oneMinusIota << (i - fracBits)
+ if neg {
+ want = -want
+ }
+ }
+
+ if got := x.Mul(oneMinusIota); got != want {
+ t.Errorf("neg=%t, i=%d, x=%v, Mul: got %v, want %v", neg, i, x, got, want)
+ }
+ if got := x.mul(oneMinusIota); got != want {
+ t.Errorf("neg=%t, i=%d, x=%v, mul: got %v, want %v", neg, i, x, got, want)
+ }
+ }
+ }
+}
+
+func TestInt52_12MulByOneMinusIota(t *testing.T) {
+ const (
+ totalBits = 64
+ fracBits = 12
+
+ oneMinusIota = Int52_12(1<<fracBits) - 1
+ oneMinusIotaF = float64(oneMinusIota) / (1 << fracBits)
+ )
+
+ for _, neg := range []bool{false, true} {
+ for i := uint(0); i < totalBits; i++ {
+ x := Int52_12(1 << i)
+ if neg {
+ x = -x
+ } else if i == totalBits-1 {
+ // A signed int64 can't represent 1<<63.
+ continue
+ }
+
+ // want equals x * oneMinusIota, rounded to nearest.
+ want := Int52_12(0)
+ if -1<<fracBits < x && x < 1<<fracBits {
+ // (x * oneMinusIota) isn't exactly representable as an
+ // Int52_12. Calculate the rounded value using float64 math.
+ xF := float64(x) / (1 << fracBits)
+ wantF := xF * oneMinusIotaF * (1 << fracBits)
+ want = Int52_12(math.Floor(wantF + 0.5))
+ } else {
+ // (x * oneMinusIota) is exactly representable.
+ want = oneMinusIota << (i - fracBits)
+ if neg {
+ want = -want
+ }
+ }
+
+ if got := x.Mul(oneMinusIota); got != want {
+ t.Errorf("neg=%t, i=%d, x=%v, Mul: got %v, want %v", neg, i, x, got, want)
+ }
+ }
+ }
+}
+
+func TestInt26_6MulVsMul(t *testing.T) {
+ rng := rand.New(rand.NewSource(1))
+ for i := 0; i < 10000; i++ {
+ u := Int26_6(rng.Uint32())
+ v := Int26_6(rng.Uint32())
+ Mul := u.Mul(v)
+ mul := u.mul(v)
+ if Mul != mul {
+ t.Errorf("u=%#08x, v=%#08x: Mul=%#08x and mul=%#08x differ",
+ uint32(u), uint32(v), uint32(Mul), uint32(mul))
+ }
+ }
+}
+
+func TestMuli32(t *testing.T) {
+ rng := rand.New(rand.NewSource(2))
+ for i := 0; i < 10000; i++ {
+ u := int32(rng.Uint32())
+ v := int32(rng.Uint32())
+ lo, hi := muli32(u, v)
+ got := uint64(lo) | uint64(hi)<<32
+ want := uint64(int64(u) * int64(v))
+ if got != want {
+ t.Errorf("u=%#08x, v=%#08x: got %#016x, want %#016x", uint32(u), uint32(v), got, want)
+ }
+ }
+}
+
+func TestMulu32(t *testing.T) {
+ rng := rand.New(rand.NewSource(3))
+ for i := 0; i < 10000; i++ {
+ u := rng.Uint32()
+ v := rng.Uint32()
+ lo, hi := mulu32(u, v)
+ got := uint64(lo) | uint64(hi)<<32
+ want := uint64(u) * uint64(v)
+ if got != want {
+ t.Errorf("u=%#08x, v=%#08x: got %#016x, want %#016x", u, v, got, want)
+ }
+ }
+}
+
+// mul (with a lower case 'm') is an alternative implementation of Int26_6.Mul
+// (with an upper case 'M'). It has the same structure as the Int52_12.Mul
+// implementation, but Int26_6.mul is easier to test since Go has built-in
+// 64-bit integers.
+func (x Int26_6) mul(y Int26_6) Int26_6 {
+ const M, N = 26, 6
+ lo, hi := muli32(int32(x), int32(y))
+ ret := Int26_6(hi<<M | lo>>N)
+ ret += Int26_6((lo >> (N - 1)) & 1) // Round to nearest, instead of rounding down.
+ return ret
+}
+
+// muli32 multiplies two int32 values, returning the 64-bit signed integer
+// result as two uint32 values.
+//
+// muli32 isn't used directly by this package, but it has the same structure as
+// muli64, and muli32 is easier to test since Go has built-in 64-bit integers.
+func muli32(u, v int32) (lo, hi uint32) {
+ const (
+ s = 16
+ mask = 1<<s - 1
+ )
+
+ u1 := uint32(u >> s)
+ u0 := uint32(u & mask)
+ v1 := uint32(v >> s)
+ v0 := uint32(v & mask)
+
+ w0 := u0 * v0
+ t := u1*v0 + w0>>s
+ w1 := t & mask
+ w2 := uint32(int32(t) >> s)
+ w1 += u0 * v1
+ return uint32(u) * uint32(v), u1*v1 + w2 + uint32(int32(w1)>>s)
+}
+
+// mulu32 is like muli32, except that it multiplies unsigned instead of signed
+// values.
+//
+// This implementation comes from $GOROOT/src/runtime/softfloat64.go's mullu
+// function, which is in turn adapted from Hacker's Delight.
+//
+// mulu32 (and its corresponding test, TestMulu32) isn't used directly by this
+// package. It is provided in this test file as a reference point to compare
+// the muli32 (and TestMuli32) implementations against.
+func mulu32(u, v uint32) (lo, hi uint32) {
+ const (
+ s = 16
+ mask = 1<<s - 1
+ )
+
+ u0 := u & mask
+ u1 := u >> s
+ v0 := v & mask
+ v1 := v >> s
+
+ w0 := u0 * v0
+ t := u1*v0 + w0>>s
+ w1 := t & mask
+ w2 := t >> s
+ w1 += u0 * v1
+ return u * v, u1*v1 + w2 + w1>>s
+}
