math: handle denormals in Frexp, Ilogb, Ldexp, and Logb

Also:
* document special cases for Frexp and Ldexp
* handle ±Inf in Ldexp
* correctly return -0 on underflow in Ldexp
* test special cases for Ldexp
* test boundary cases for Frexp, Ilogb, Ldexp, and Logb

R=rsc
CC=golang-dev
https://golang.org/cl/3676041

src/pkg/math/all_test.go

@@ -1112,6 +1112,33 @@ var jM3SC = []float64{
 	NaN(),
 }
 
+var vfldexpSC = []fi{
+	{0, 0},
+	{0, -1075},
+	{0, 1024},
+	{Copysign(0, -1), 0},
+	{Copysign(0, -1), -1075},
+	{Copysign(0, -1), 1024},
+	{Inf(1), 0},
+	{Inf(1), -1024},
+	{Inf(-1), 0},
+	{Inf(-1), -1024},
+	{NaN(), -1024},
+}
+var ldexpSC = []float64{
+	0,
+	0,
+	0,
+	Copysign(0, -1),
+	Copysign(0, -1),
+	Copysign(0, -1),
+	Inf(1),
+	Inf(1),
+	Inf(-1),
+	Inf(-1),
+	NaN(),
+}
+
 var vflgammaSC = []float64{
 	Inf(-1),
 	-3,
@@ -1440,6 +1467,65 @@ var yM3SC = []float64{
 	NaN(),
 }
 
+// arguments and expected results for boundary cases
+const (
+	SmallestNormalFloat64   = 2.2250738585072014e-308 // 2**-1022
+	LargestSubnormalFloat64 = SmallestNormalFloat64 - SmallestNonzeroFloat64
+)
+
+var vffrexpBC = []float64{
+	SmallestNormalFloat64,
+	LargestSubnormalFloat64,
+	SmallestNonzeroFloat64,
+	MaxFloat64,
+	-SmallestNormalFloat64,
+	-LargestSubnormalFloat64,
+	-SmallestNonzeroFloat64,
+	-MaxFloat64,
+}
+var frexpBC = []fi{
+	{0.5, -1021},
+	{0.99999999999999978, -1022},
+	{0.5, -1073},
+	{0.99999999999999989, 1024},
+	{-0.5, -1021},
+	{-0.99999999999999978, -1022},
+	{-0.5, -1073},
+	{-0.99999999999999989, 1024},
+}
+
+var vfldexpBC = []fi{
+	{SmallestNormalFloat64, -52},
+	{LargestSubnormalFloat64, -51},
+	{SmallestNonzeroFloat64, 1074},
+	{MaxFloat64, -(1023 + 1074)},
+	{1, -1075},
+	{-1, -1075},
+	{1, 1024},
+	{-1, 1024},
+}
+var ldexpBC = []float64{
+	SmallestNonzeroFloat64,
+	1e-323, // 2**-1073
+	1,
+	1e-323, // 2**-1073
+	0,
+	Copysign(0, -1),
+	Inf(1),
+	Inf(-1),
+}
+
+var logbBC = []float64{
+	-1022,
+	-1023,
+	-1074,
+	1023,
+	-1022,
+	-1023,
+	-1074,
+	1023,
+}
+
 func tolerance(a, b, e float64) bool {
 	d := a - b
 	if d < 0 {
@@ -1792,6 +1878,11 @@ func TestFrexp(t *testing.T) {
 			t.Errorf("Frexp(%g) = %g, %d, want %g, %d", vffrexpSC[i], f, j, frexpSC[i].f, frexpSC[i].i)
 		}
 	}
+	for i := 0; i < len(vffrexpBC); i++ {
+		if f, j := Frexp(vffrexpBC[i]); !alike(frexpBC[i].f, f) || frexpBC[i].i != j {
+			t.Errorf("Frexp(%g) = %g, %d, want %g, %d", vffrexpBC[i], f, j, frexpBC[i].f, frexpBC[i].i)
+		}
+	}
 }
 
 func TestGamma(t *testing.T) {
@@ -1833,6 +1924,11 @@ func TestIlogb(t *testing.T) {
 			t.Errorf("Ilogb(%g) = %d, want %d", vflogbSC[i], e, ilogbSC[i])
 		}
 	}
+	for i := 0; i < len(vffrexpBC); i++ {
+		if e := Ilogb(vffrexpBC[i]); int(logbBC[i]) != e {
+			t.Errorf("Ilogb(%g) = %d, want %d", vffrexpBC[i], e, int(logbBC[i]))
+		}
+	}
 }
 
 func TestJ0(t *testing.T) {
@@ -1891,6 +1987,21 @@ func TestLdexp(t *testing.T) {
 			t.Errorf("Ldexp(%g, %d) = %g, want %g", frexpSC[i].f, frexpSC[i].i, f, vffrexpSC[i])
 		}
 	}
+	for i := 0; i < len(vfldexpSC); i++ {
+		if f := Ldexp(vfldexpSC[i].f, vfldexpSC[i].i); !alike(ldexpSC[i], f) {
+			t.Errorf("Ldexp(%g, %d) = %g, want %g", vfldexpSC[i].f, vfldexpSC[i].i, f, ldexpSC[i])
+		}
+	}
+	for i := 0; i < len(vffrexpBC); i++ {
+		if f := Ldexp(frexpBC[i].f, frexpBC[i].i); !alike(vffrexpBC[i], f) {
+			t.Errorf("Ldexp(%g, %d) = %g, want %g", frexpBC[i].f, frexpBC[i].i, f, vffrexpBC[i])
+		}
+	}
+	for i := 0; i < len(vfldexpBC); i++ {
+		if f := Ldexp(vfldexpBC[i].f, vfldexpBC[i].i); !alike(ldexpBC[i], f) {
+			t.Errorf("Ldexp(%g, %d) = %g, want %g", vfldexpBC[i].f, vfldexpBC[i].i, f, ldexpBC[i])
+		}
+	}
 }
 
 func TestLgamma(t *testing.T) {
@@ -1934,6 +2045,11 @@ func TestLogb(t *testing.T) {
 			t.Errorf("Logb(%g) = %g, want %g", vflogbSC[i], f, logbSC[i])
 		}
 	}
+	for i := 0; i < len(vffrexpBC); i++ {
+		if e := Logb(vffrexpBC[i]); !alike(logbBC[i], e) {
+			t.Errorf("Ilogb(%g) = %g, want %g", vffrexpBC[i], e, logbBC[i])
+		}
+	}
 }
 
 func TestLog10(t *testing.T) {

src/pkg/math/bits.go

@@ -47,3 +47,13 @@ func IsInf(f float64, sign int) bool {
 	//	return sign >= 0 && x == uvinf || sign <= 0 && x == uvneginf;
 	return sign >= 0 && f > MaxFloat64 || sign <= 0 && f < -MaxFloat64
 }
+
+// normalize returns a normal number y and exponent exp
+// satisfying x == y × 2**exp. It assumes x is finite and non-zero.
+func normalize(x float64) (y float64, exp int) {
+	const SmallestNormal = 2.2250738585072014e-308 // 2**-1022
+	if Fabs(x) < SmallestNormal {
+		return x * (1 << 52), -52
+	}
+	return x, 0
+}

src/pkg/math/frexp.go

@@ -8,6 +8,11 @@ package math
 // and an integral power of two.
 // It returns frac and exp satisfying f == frac × 2**exp,
 // with the absolute value of frac in the interval [½, 1).
+//
+// Special cases are:
+//	Frexp(±0) = ±0, 0
+//	Frexp(±Inf) = ±Inf, 0
+//	Frexp(NaN) = NaN, 0
 func Frexp(f float64) (frac float64, exp int) {
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us
@@ -18,8 +23,9 @@ func Frexp(f float64) (frac float64, exp int) {
 	case f < -MaxFloat64 || f > MaxFloat64 || f != f: // IsInf(f, 0) || IsNaN(f):
 		return f, 0
 	}
+	f, exp = normalize(f)
 	x := Float64bits(f)
-	exp = int((x>>shift)&mask) - bias + 1
+	exp += int((x>>shift)&mask) - bias + 1
 	x &^= mask << shift
 	x |= (-1 + bias) << shift
 	frac = Float64frombits(x)

src/pkg/math/ldexp.go

@@ -6,6 +6,11 @@ package math
 
 // Ldexp is the inverse of Frexp.
 // It returns frac × 2**exp.
+//
+// Special cases are:
+//	Ldexp(±0, exp) = ±0
+//	Ldexp(±Inf, exp) = ±Inf
+//	Ldexp(NaN, exp) = NaN
 func Ldexp(frac float64, exp int) float64 {
 	// TODO(rsc): Remove manual inlining of IsNaN, IsInf
 	// when compiler does it for us
@@ -13,21 +18,28 @@ func Ldexp(frac float64, exp int) float64 {
 	switch {
 	case frac == 0:
 		return frac // correctly return -0
-	case frac != frac: // IsNaN(frac):
+	case frac < -MaxFloat64 || frac > MaxFloat64 || frac != frac: // IsInf(frac, 0) || IsNaN(frac):
-		return NaN()
+		return frac
 	}
+	frac, e := normalize(frac)
+	exp += e
 	x := Float64bits(frac)
-	exp += int(x>>shift) & mask
+	exp += int(x>>shift)&mask - bias
-	if exp <= 0 {
+	if exp < -1074 {
-		return 0 // underflow
+		return Copysign(0, frac) // underflow
 	}
-	if exp >= mask { // overflow
+	if exp > 1023 { // overflow
 		if frac < 0 {
 			return Inf(-1)
 		}
 		return Inf(1)
 	}
+	var m float64 = 1
+	if exp < -1022 { // denormal
+		exp += 52
+		m = 1.0 / (1 << 52) // 2**-52
+	}
 	x &^= mask << shift
-	x |= uint64(exp) << shift
+	x |= uint64(exp+bias) << shift
-	return Float64frombits(x)
+	return m * Float64frombits(x)
 }

src/pkg/math/logb.go

@@ -4,7 +4,7 @@
 
 package math
 
-// Logb(x) returns the binary exponent of non-zero x.
+// Logb(x) returns the binary exponent of x.
 //
 // Special cases are:
 //	Logb(±Inf) = +Inf
@@ -22,10 +22,10 @@ func Logb(x float64) float64 {
 	case x != x: // IsNaN(x):
 		return x
 	}
-	return float64(int((Float64bits(x)>>shift)&mask) - bias)
+	return float64(ilogb(x))
 }
 
-// Ilogb(x) returns the binary exponent of non-zero x as an integer.
+// Ilogb(x) returns the binary exponent of x as an integer.
 //
 // Special cases are:
 //	Ilogb(±Inf) = MaxInt32
@@ -43,5 +43,12 @@ func Ilogb(x float64) int {
 	case x < -MaxFloat64 || x > MaxFloat64: // IsInf(x, 0):
 		return MaxInt32
 	}
-	return int((Float64bits(x)>>shift)&mask) - bias
+	return ilogb(x)
+}
+
+// logb returns the binary exponent of x. It assumes x is finite and
+// non-zero.
+func ilogb(x float64) int {
+	x, exp := normalize(x)
+	return int((Float64bits(x)>>shift)&mask) - bias + exp
 }