math: speed up and improve accuracy of Pow10

Removes init function from the math package.

Allows stripping of arrays with pre-computed values
used for Pow10 from binaries if Pow10 is not used.
cmd/go shrinks by 128 bytes.

Fixed small values like 10**-323 being 0 instead of 1e-323.

Overall precision is increased but still not as good as
predefined constants for some inputs.

Samples:

Pow10(208)
before: 1.0000000000000006662e+208
after:  1.0000000000000000959e+208

Pow10(202)
before 1.0000000000000009895e+202
after  1.0000000000000001193e+202

Pow10(60)
before 1.0000000000000001278e+60
after  0.9999999999999999494e+60

Pow10(-100)
before 0.99999999999999938551e-100
after  0.99999999999999989309e-100

Pow10(-200)
before 0.9999999999999988218e-200
after  1.0000000000000001271e-200

name        old time/op  new time/op  delta
Pow10Pos-4  44.6ns ± 2%   1.2ns ± 1%  -97.39%  (p=0.000 n=19+17)
Pow10Neg-4  50.8ns ± 1%   4.1ns ± 2%  -92.02%  (p=0.000 n=17+19)

Change-Id: If094034286b8ac64be3a95fd9e8ffa3d4ad39b31
Reviewed-on: https://go-review.googlesource.com/36331
Reviewed-by: Robert Griesemer <gri@golang.org>
Run-TryBot: Martin Möhrmann <moehrmann@google.com>
TryBot-Result: Gobot Gobot <gobot@golang.org>

src/math/all_test.go

@@ -1646,16 +1646,38 @@ var powSC = []float64{
 
 var vfpow10SC = []int{
 	MinInt32,
-	MaxInt32,
-	-325,
+	-324,
+	-323,
+	-50,
+	-22,
+	-1,
+	0,
+	1,
+	22,
+	50,
+	100,
+	200,
+	308,
 	309,
+	MaxInt32,
 }
 
 var pow10SC = []float64{
-	0,      // pow10(MinInt32)
-	Inf(1), // pow10(MaxInt32)
-	0,      // pow10(-325)
-	Inf(1), // pow10(309)
+	0,        // pow10(MinInt32)
+	0,        // pow10(-324)
+	1.0e-323, // pow10(-323)
+	1.0e-50,  // pow10(-50)
+	1.0e-22,  // pow10(-22)
+	1.0e-1,   // pow10(-1)
+	1.0e0,    // pow10(0)
+	1.0e1,    // pow10(1)
+	1.0e22,   // pow10(22)
+	1.0e50,   // pow10(50)
+	1.0e100,  // pow10(100)
+	1.0e200,  // pow10(200)
+	1.0e308,  // pow10(308)
+	Inf(1),   // pow10(309)
+	Inf(1),   // pow10(MaxInt32)
 }
 
 var vfsignbitSC = []float64{
@@ -3179,18 +3201,22 @@ func BenchmarkPowFrac(b *testing.B) {
 	GlobalF = x
 }
 
+var pow10pos = int(300)
+
 func BenchmarkPow10Pos(b *testing.B) {
 	x := 0.0
 	for i := 0; i < b.N; i++ {
-		x = Pow10(300)
+		x = Pow10(pow10pos)
 	}
 	GlobalF = x
 }
 
+var pow10neg = int(-300)
+
 func BenchmarkPow10Neg(b *testing.B) {
 	x := 0.0
 	for i := 0; i < b.N; i++ {
-		x = Pow10(-300)
+		x = Pow10(pow10neg)
 	}
 	GlobalF = x
 }

src/math/pow10.go

@@ -4,37 +4,43 @@
 
 package math
 
-// This table might overflow 127-bit exponent representations.
-// In that case, truncate it after 1.0e38.
-var pow10tab [70]float64
+// pow10tab stores the pre-computed values 10**i for i < 32.
+var pow10tab = [...]float64{
+	1e00, 1e01, 1e02, 1e03, 1e04, 1e05, 1e06, 1e07, 1e08, 1e09,
+	1e10, 1e11, 1e12, 1e13, 1e14, 1e15, 1e16, 1e17, 1e18, 1e19,
+	1e20, 1e21, 1e22, 1e23, 1e24, 1e25, 1e26, 1e27, 1e28, 1e29,
+	1e30, 1e31,
+}
 
-// Pow10 returns 10**e, the base-10 exponential of e.
+// pow10postab32 stores the pre-computed value for 10**(i*32) at index i.
+var pow10postab32 = [...]float64{
+	1e00, 1e32, 1e64, 1e96, 1e128, 1e160, 1e192, 1e224, 1e256, 1e288,
+}
+
+// pow10negtab32 stores the pre-computed value for 10**(-i*32) at index i.
+var pow10negtab32 = [...]float64{
+	1e-00, 1e-32, 1e-64, 1e-96, 1e-128, 1e-160, 1e-192, 1e-224, 1e-256, 1e-288, 1e-320,
+}
+
+// Pow10 returns 10**n, the base-10 exponential of n.
 //
 // Special cases are:
-//	Pow10(e) = +Inf for e > 309
-//	Pow10(e) = 0 for e < -324
-func Pow10(e int) float64 {
-	if e <= -325 {
-		return 0
-	} else if e > 309 {
+//	Pow10(n) =    0 for n < -323
+//	Pow10(n) = +Inf for n > 308
+func Pow10(n int) float64 {
+	if 0 <= n && n <= 308 {
+		return pow10postab32[uint(n)/32] * pow10tab[uint(n)%32]
+	}
+
+	if -323 <= n && n <= 0 {
+		return pow10negtab32[uint(-n)/32] / pow10tab[uint(-n)%32]
+	}
+
+	// n < -323 || 308 < n
+	if n > 0 {
 		return Inf(1)
 	}
 
-	if e < 0 {
-		return 1 / Pow10(-e)
-	}
-	if e < len(pow10tab) {
-		return pow10tab[e]
-	}
-	m := e / 2
-	return Pow10(m) * Pow10(e-m)
-}
-
-func init() {
-	pow10tab[0] = 1.0e0
-	pow10tab[1] = 1.0e1
-	for i := 2; i < len(pow10tab); i++ {
-		m := i / 2
-		pow10tab[i] = pow10tab[m] * pow10tab[i-m]
-	}
+	// n < -323
+	return 0
 }