resource: optimize scale function

The original scale function takes around 800ns/op with more
than 10 allocations. It significantly slow down scheduler
and other components that heavily relys on resource pkg.
For more information see #18126.

This pull request tries to optimize scale function. It takes
two approach:

1. when the value is small, only use normal math ops.
2. when the value is large, use math.Big with buffer pool.

The final result is:

BenchmarkScaledValueSmall-4	20000000	        66.9 ns/op	       0 B/op	       0 allocs/op
BenchmarkScaledValueLarge-4	 2000000	       711 ns/op	      48 B/op	       1 allocs/op

I also run the scheduler benchmark again. It doubles the throughput of
scheduler for 1000 nodes case.

pkg/api/resource/quantity.go

@@ -377,8 +377,7 @@ func (q *Quantity) Value() int64 {
 	if q.Amount == nil {
 		return 0
 	}
-	tmp := &inf.Dec{}
+	return scaledValue(q.Amount.UnscaledBig(), int(q.Amount.Scale()), 0)
-	return tmp.Round(q.Amount, 0, inf.RoundUp).UnscaledBig().Int64()
 }
 
 // MilliValue returns the value of q * 1000; this could overflow an int64;
@@ -387,8 +386,7 @@ func (q *Quantity) MilliValue() int64 {
 	if q.Amount == nil {
 		return 0
 	}
-	tmp := &inf.Dec{}
+	return scaledValue(q.Amount.UnscaledBig(), int(q.Amount.Scale()), 3)
-	return tmp.Round(tmp.Mul(q.Amount, decThousand), 0, inf.RoundUp).UnscaledBig().Int64()
 }
 
 // Set sets q's value to be value.

pkg/api/resource/scale_int.go

@@ -0,0 +1,95 @@
+/*
+Copyright 2015 The Kubernetes Authors All rights reserved.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+package resource
+
+import (
+	"math"
+	"math/big"
+	"sync"
+)
+
+var (
+	// A sync pool to reduce allocation.
+	intPool  sync.Pool
+	maxInt64 = big.NewInt(math.MaxInt64)
+)
+
+func init() {
+	intPool.New = func() interface{} {
+		return &big.Int{}
+	}
+}
+
+// scaledValue scales given unscaled value from scale to new Scale and returns
+// an int64. It ALWAYS rounds up the result when scale down. The final result might
+// overflow.
+//
+// scale, newScale represents the scale of the unscaled decimal.
+// The mathematical value of the decimal is unscaled * 10**(-scale).
+func scaledValue(unscaled *big.Int, scale, newScale int) int64 {
+	dif := scale - newScale
+	if dif == 0 {
+		return unscaled.Int64()
+	}
+
+	// Handle scale up
+	// This is an easy case, we do not need to care about rounding and overflow.
+	// If any intermediate operation causes overflow, the result will overflow.
+	if dif < 0 {
+		return unscaled.Int64() * int64(math.Pow10(-dif))
+	}
+
+	// Handle scale down
+	// We have to be careful about the intermediate operations.
+
+	// fast path when unscaled < max.Int64 and exp(10,dif) < max.Int64
+	const log10MaxInt64 = 19
+	if unscaled.Cmp(maxInt64) < 0 && dif < log10MaxInt64 {
+		divide := int64(math.Pow10(dif))
+		result := unscaled.Int64() / divide
+		mod := unscaled.Int64() % divide
+		if mod != 0 {
+			return result + 1
+		}
+		return result
+	}
+
+	// We should only convert back to int64 when getting the result.
+	divisor := intPool.Get().(*big.Int)
+	exp := intPool.Get().(*big.Int)
+	result := intPool.Get().(*big.Int)
+	defer func() {
+		intPool.Put(divisor)
+		intPool.Put(exp)
+		intPool.Put(result)
+	}()
+
+	// divisor = 10^(dif)
+	// TODO: create loop up table if exp costs too much.
+	divisor.Exp(bigTen, exp.SetInt64(int64(dif)), nil)
+	// reuse exp
+	remainder := exp
+
+	// result = unscaled / divisor
+	// remainder = unscaled % divisor
+	result.DivMod(unscaled, divisor, remainder)
+	if remainder.Sign() != 0 {
+		return result.Int64() + 1
+	}
+
+	return result.Int64()
+}

pkg/api/resource/scale_int_test.go

@@ -0,0 +1,85 @@
+/*
+Copyright 2015 The Kubernetes Authors All rights reserved.
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+package resource
+
+import (
+	"math"
+	"math/big"
+	"testing"
+)
+
+func TestScaledValue(t *testing.T) {
+	tests := []struct {
+		unscaled *big.Int
+		scale    int
+		newScale int
+
+		want int64
+	}{
+		// remain scale
+		{big.NewInt(1000), 0, 0, 1000},
+
+		// scale down
+		{big.NewInt(1000), 0, -3, 1},
+		{big.NewInt(1000), 3, 0, 1},
+		{big.NewInt(0), 3, 0, 0},
+
+		// always round up
+		{big.NewInt(999), 3, 0, 1},
+		{big.NewInt(500), 3, 0, 1},
+		{big.NewInt(499), 3, 0, 1},
+		{big.NewInt(1), 3, 0, 1},
+		// large scaled value does not lose precision
+		{big.NewInt(0).Sub(maxInt64, bigOne), 1, 0, (math.MaxInt64-1)/10 + 1},
+		// large intermidiate result.
+		{big.NewInt(1).Exp(big.NewInt(10), big.NewInt(100), nil), 100, 0, 1},
+
+		// scale up
+		{big.NewInt(0), 0, 3, 0},
+		{big.NewInt(1), 0, 3, 1000},
+		{big.NewInt(1), -3, 0, 1000},
+		{big.NewInt(1000), -3, 2, 100000000},
+		{big.NewInt(0).Div(big.NewInt(math.MaxInt64), bigThousand), 0, 3,
+			(math.MaxInt64 / 1000) * 1000},
+	}
+
+	for i, tt := range tests {
+		old := (&big.Int{}).Set(tt.unscaled)
+		got := scaledValue(tt.unscaled, tt.scale, tt.newScale)
+		if got != tt.want {
+			t.Errorf("#%d: got = %v, want %v", i, got, tt.want)
+		}
+		if tt.unscaled.Cmp(old) != 0 {
+			t.Errorf("#%d: unscaled = %v, want %v", i, tt.unscaled, old)
+		}
+	}
+}
+
+func BenchmarkScaledValueSmall(b *testing.B) {
+	s := big.NewInt(1000)
+	for i := 0; i < b.N; i++ {
+		scaledValue(s, 3, 0)
+	}
+}
+
+func BenchmarkScaledValueLarge(b *testing.B) {
+	s := big.NewInt(math.MaxInt64)
+	s.Mul(s, big.NewInt(1000))
+	for i := 0; i < b.N; i++ {
+		scaledValue(s, 10, 0)
+	}
+}