strconv: parse hex floats

This CL updates ParseFloat to recognize
standard hexadecimal floating-point constants.
See golang.org/design/19308-number-literals for background.

For #29008.

Change-Id: I45f3b0c36b5d92c0e8a4b35c05443a83d7a6d4b3
Reviewed-on: https://go-review.googlesource.com/c/160241
Run-TryBot: Russ Cox <rsc@golang.org>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-by: Robert Griesemer <gri@golang.org>

3 files changed, 341 insertions, 48 deletions

diff --git a/src/strconv/atof.go b/src/strconv/atof.go
index ada85e9fed..3ced3c7167 100644
--- a/src/strconv/atof.go
+++ b/src/strconv/atof.go
@@ -12,7 +12,7 @@ package strconv
 
 import "math"
 
-var optimize = true // can change for testing
+var optimize = true // set to false to force slow-path conversions for testing
 
 func equalIgnoreCase(s1, s2 string) bool {
 	if len(s1) != len(s2) {
@@ -119,7 +119,7 @@ func (b *decimal) set(s string) (ok bool) {
 	// just be sure to move the decimal point by
 	// a lot (say, 100000).  it doesn't matter if it's
 	// not the exact number.
-	if i < len(s) && (s[i] == 'e' || s[i] == 'E') {
+	if i < len(s) && lower(s[i]) == 'e' {
 		i++
 		if i >= len(s) {
 			return
@@ -152,10 +152,9 @@ func (b *decimal) set(s string) (ok bool) {
 }
 
 // readFloat reads a decimal mantissa and exponent from a float
-// string representation. It sets ok to false if the number could
+// string representation. It returns ok==false if the number could
 // not fit return types or is invalid.
-func readFloat(s string) (mantissa uint64, exp int, neg, trunc, ok bool) {
-	const uint64digits = 19
+func readFloat(s string) (mantissa uint64, exp int, neg, trunc, hex, ok bool) {
 	i := 0
 
 	// optional sign
@@ -171,6 +170,16 @@ func readFloat(s string) (mantissa uint64, exp int, neg, trunc, ok bool) {
 	}
 
 	// digits
+	base := uint64(10)
+	maxMantDigits := 19 // 10^19 fits in uint64
+	expChar := byte('e')
+	if i+2 < len(s) && s[i] == '0' && lower(s[i+1]) == 'x' {
+		base = 16
+		maxMantDigits = 16 // 16^16 fits in uint64
+		i += 2
+		expChar = 'p'
+		hex = true
+	}
 	sawdot := false
 	sawdigits := false
 	nd := 0
@@ -193,11 +202,23 @@ func readFloat(s string) (mantissa uint64, exp int, neg, trunc, ok bool) {
 				continue
 			}
 			nd++
-			if ndMant < uint64digits {
-				mantissa *= 10
+			if ndMant < maxMantDigits {
+				mantissa *= base
 				mantissa += uint64(c - '0')
 				ndMant++
-			} else if s[i] != '0' {
+			} else if c != '0' {
+				trunc = true
+			}
+			continue
+
+		case base == 16 && 'a' <= lower(c) && lower(c) <= 'f':
+			sawdigits = true
+			nd++
+			if ndMant < maxMantDigits {
+				mantissa *= 16
+				mantissa += uint64(lower(c) - 'a' + 10)
+				ndMant++
+			} else {
 				trunc = true
 			}
 			continue
@@ -211,12 +232,17 @@ func readFloat(s string) (mantissa uint64, exp int, neg, trunc, ok bool) {
 		dp = nd
 	}
 
+	if base == 16 {
+		dp *= 4
+		ndMant *= 4
+	}
+
 	// optional exponent moves decimal point.
 	// if we read a very large, very long number,
 	// just be sure to move the decimal point by
 	// a lot (say, 100000).  it doesn't matter if it's
 	// not the exact number.
-	if i < len(s) && (s[i] == 'e' || s[i] == 'E') {
+	if i < len(s) && lower(s[i]) == expChar {
 		i++
 		if i >= len(s) {
 			return
@@ -238,6 +264,9 @@ func readFloat(s string) (mantissa uint64, exp int, neg, trunc, ok bool) {
 			}
 		}
 		dp += e * esign
+	} else if base == 16 {
+		// Must have exponent.
+		return
 	}
 
 	if i != len(s) {
@@ -249,7 +278,6 @@ func readFloat(s string) (mantissa uint64, exp int, neg, trunc, ok bool) {
 	}
 	ok = true
 	return
-
 }
 
 // decimal power of ten to binary power of two.
@@ -433,6 +461,76 @@ func atof32exact(mantissa uint64, exp int, neg bool) (f float32, ok bool) {
 	return
 }
 
+// atofHex converts the hex floating-point string s
+// to a rounded float32 or float64 value (depending on flt==&float32info or flt==&float64info)
+// and returns it as a float64.
+// The string s has already been parsed into a mantissa, exponent, and sign (neg==true for negative).
+// If trunc is true, trailing non-zero bits have been omitted from the mantissa.
+func atofHex(s string, flt *floatInfo, mantissa uint64, exp int, neg, trunc bool) (float64, error) {
+	maxExp := 1<<flt.expbits + flt.bias - 2
+	minExp := flt.bias + 1
+	exp += int(flt.mantbits) // mantissa now implicitly divided by 2^mantbits.
+
+	// Shift mantissa and exponent to bring representation into float range.
+	// Eventually we want a mantissa with a leading 1-bit followed by mantbits other bits.
+	// For rounding, we need two more, where the bottom bit represents
+	// whether that bit or any later bit was non-zero.
+	// (If the mantissa has already lost non-zero bits, trunc is true,
+	// and we OR in a 1 below after shifting left appropriately.)
+	for mantissa != 0 && mantissa>>(flt.mantbits+2) == 0 {
+		mantissa <<= 1
+		exp--
+	}
+	if trunc {
+		mantissa |= 1
+	}
+	for mantissa>>(1+flt.mantbits+2) != 0 {
+		mantissa = mantissa>>1 | mantissa&1
+		exp++
+	}
+
+	// If exponent is too negative,
+	// denormalize in hopes of making it representable.
+	// (The -2 is for the rounding bits.)
+	for mantissa > 1 && exp < minExp-2 {
+		mantissa = mantissa>>1 | mantissa&1
+		exp++
+	}
+
+	// Round using two bottom bits.
+	round := mantissa & 3
+	mantissa >>= 2
+	round |= mantissa & 1 // round to even (round up if mantissa is odd)
+	exp += 2
+	if round == 3 {
+		mantissa++
+		if mantissa == 1<<(1+flt.mantbits) {
+			mantissa >>= 1
+			exp++
+		}
+	}
+
+	if mantissa>>flt.mantbits == 0 { // Denormal or zero.
+		exp = flt.bias
+	}
+	var err error
+	if exp > maxExp { // infinity and range error
+		mantissa = 1 << flt.mantbits
+		exp = maxExp + 1
+		err = rangeError(fnParseFloat, s)
+	}
+
+	bits := mantissa & (1<<flt.mantbits - 1)
+	bits |= uint64((exp-flt.bias)&(1<<flt.expbits-1)) << flt.mantbits
+	if neg {
+		bits |= 1 << flt.mantbits << flt.expbits
+	}
+	if flt == &float32info {
+		return float64(math.Float32frombits(uint32(bits))), err
+	}
+	return math.Float64frombits(bits), err
+}
+
 const fnParseFloat = "ParseFloat"
 
 func atof32(s string) (f float32, err error) {
@@ -440,28 +538,32 @@ func atof32(s string) (f float32, err error) {
 		return float32(val), nil
 	}
 
-	if optimize {
-		// Parse mantissa and exponent.
-		mantissa, exp, neg, trunc, ok := readFloat(s)
-		if ok {
-			// Try pure floating-point arithmetic conversion.
-			if !trunc {
-				if f, ok := atof32exact(mantissa, exp, neg); ok {
-					return f, nil
-				}
+	mantissa, exp, neg, trunc, hex, ok := readFloat(s)
+	if hex && ok {
+		f, err := atofHex(s, &float32info, mantissa, exp, neg, trunc)
+		return float32(f), err
+	}
+
+	if optimize && ok {
+		// Try pure floating-point arithmetic conversion.
+		if !trunc {
+			if f, ok := atof32exact(mantissa, exp, neg); ok {
+				return f, nil
 			}
-			// Try another fast path.
-			ext := new(extFloat)
-			if ok := ext.AssignDecimal(mantissa, exp, neg, trunc, &float32info); ok {
-				b, ovf := ext.floatBits(&float32info)
-				f = math.Float32frombits(uint32(b))
-				if ovf {
-					err = rangeError(fnParseFloat, s)
-				}
-				return f, err
+		}
+		// Try another fast path.
+		ext := new(extFloat)
+		if ok := ext.AssignDecimal(mantissa, exp, neg, trunc, &float32info); ok {
+			b, ovf := ext.floatBits(&float32info)
+			f = math.Float32frombits(uint32(b))
+			if ovf {
+				err = rangeError(fnParseFloat, s)
 			}
+			return f, err
 		}
 	}
+
+	// Slow fallback.
 	var d decimal
 	if !d.set(s) {
 		return 0, syntaxError(fnParseFloat, s)
@@ -479,28 +581,31 @@ func atof64(s string) (f float64, err error) {
 		return val, nil
 	}
 
-	if optimize {
-		// Parse mantissa and exponent.
-		mantissa, exp, neg, trunc, ok := readFloat(s)
-		if ok {
-			// Try pure floating-point arithmetic conversion.
-			if !trunc {
-				if f, ok := atof64exact(mantissa, exp, neg); ok {
-					return f, nil
-				}
+	mantissa, exp, neg, trunc, hex, ok := readFloat(s)
+	if hex && ok {
+		return atofHex(s, &float64info, mantissa, exp, neg, trunc)
+	}
+
+	if optimize && ok {
+		// Try pure floating-point arithmetic conversion.
+		if !trunc {
+			if f, ok := atof64exact(mantissa, exp, neg); ok {
+				return f, nil
 			}
-			// Try another fast path.
-			ext := new(extFloat)
-			if ok := ext.AssignDecimal(mantissa, exp, neg, trunc, &float64info); ok {
-				b, ovf := ext.floatBits(&float64info)
-				f = math.Float64frombits(b)
-				if ovf {
-					err = rangeError(fnParseFloat, s)
-				}
-				return f, err
+		}
+		// Try another fast path.
+		ext := new(extFloat)
+		if ok := ext.AssignDecimal(mantissa, exp, neg, trunc, &float64info); ok {
+			b, ovf := ext.floatBits(&float64info)
+			f = math.Float64frombits(b)
+			if ovf {
+				err = rangeError(fnParseFloat, s)
 			}
+			return f, err
 		}
 	}
+
+	// Slow fallback.
 	var d decimal
 	if !d.set(s) {
 		return 0, syntaxError(fnParseFloat, s)
@@ -518,9 +623,13 @@ func atof64(s string) (f float64, err error) {
 // When bitSize=32, the result still has type float64, but it will be
 // convertible to float32 without changing its value.
 //
-// If s is well-formed and near a valid floating point number,
-// ParseFloat returns the nearest floating point number rounded
+// ParseFloat accepts decimal and hexadecimal floating-point number syntax.
+// If s is well-formed and near a valid floating-point number,
+// ParseFloat returns the nearest floating-point number rounded
 // using IEEE754 unbiased rounding.
+// (Parsing a hexadecimal floating-point value only rounds when
+// there are more bits in the hexadecimal representatiton than
+// will fit in the mantissa.)
 //
 // The errors that ParseFloat returns have concrete type *NumError
 // and include err.Num = s.
diff --git a/src/strconv/atof_test.go b/src/strconv/atof_test.go
index cf4d47c8b9..f67386a000 100644
--- a/src/strconv/atof_test.go
+++ b/src/strconv/atof_test.go
@@ -43,6 +43,20 @@ var atoftests = []atofTest{
 	{"1e-20", "1e-20", nil},
 	{"625e-3", "0.625", nil},
 
+	// Hexadecimal floating-point.
+	{"0x1p0", "1", nil},
+	{"0x1p1", "2", nil},
+	{"0x1p-1", "0.5", nil},
+	{"0x1p-200", "6.223015277861142e-61", nil},
+	{"0x1p200", "1.6069380442589903e+60", nil},
+	{"0x1fFe2.p0", "131042", nil},
+	{"0x1fFe2.P0", "131042", nil},
+	{"-0x2p3", "-16", nil},
+	{"0x0.fp4", "15", nil},
+	{"0x0.fp0", "0.9375", nil},
+	{"0x1e2", "0", ErrSyntax},
+	{"1p2", "0", ErrSyntax},
+
 	// zeros
 	{"0", "0", nil},
 	{"0e0", "0", nil},
@@ -58,6 +72,11 @@ var atoftests = []atofTest{
 	{"0.00e-01234567890123456789", "0", nil},
 	{"-0e+01234567890123456789", "-0", nil},
 	{"-0.00e-01234567890123456789", "-0", nil},
+	{"0x0p+01234567890123456789", "0", nil},
+	{"0x0.00p-01234567890123456789", "0", nil},
+	{"-0x0p+01234567890123456789", "-0", nil},
+	{"-0x0.00p-01234567890123456789", "-0", nil},
+
 	{"0e291", "0", nil}, // issue 15364
 	{"0e292", "0", nil}, // issue 15364
 	{"0e347", "0", nil}, // issue 15364
@@ -66,6 +85,26 @@ var atoftests = []atofTest{
 	{"-0e292", "-0", nil},
 	{"-0e347", "-0", nil},
 	{"-0e348", "-0", nil},
+	{"0x0p126", "0", nil},
+	{"0x0p127", "0", nil},
+	{"0x0p128", "0", nil},
+	{"0x0p129", "0", nil},
+	{"0x0p130", "0", nil},
+	{"0x0p1022", "0", nil},
+	{"0x0p1023", "0", nil},
+	{"0x0p1024", "0", nil},
+	{"0x0p1025", "0", nil},
+	{"0x0p1026", "0", nil},
+	{"-0x0p126", "-0", nil},
+	{"-0x0p127", "-0", nil},
+	{"-0x0p128", "-0", nil},
+	{"-0x0p129", "-0", nil},
+	{"-0x0p130", "-0", nil},
+	{"-0x0p1022", "-0", nil},
+	{"-0x0p1023", "-0", nil},
+	{"-0x0p1024", "-0", nil},
+	{"-0x0p1025", "-0", nil},
+	{"-0x0p1026", "-0", nil},
 
 	// NaNs
 	{"nan", "NaN", nil},
@@ -83,21 +122,46 @@ var atoftests = []atofTest{
 	// largest float64
 	{"1.7976931348623157e308", "1.7976931348623157e+308", nil},
 	{"-1.7976931348623157e308", "-1.7976931348623157e+308", nil},
+	{"0x1.fffffffffffffp1023", "1.7976931348623157e+308", nil},
+	{"-0x1.fffffffffffffp1023", "-1.7976931348623157e+308", nil},
+	{"0x1fffffffffffffp+971", "1.7976931348623157e+308", nil},
+	{"-0x1fffffffffffffp+971", "-1.7976931348623157e+308", nil},
+	{"0x.1fffffffffffffp1027", "1.7976931348623157e+308", nil},
+	{"-0x.1fffffffffffffp1027", "-1.7976931348623157e+308", nil},
+
 	// next float64 - too large
 	{"1.7976931348623159e308", "+Inf", ErrRange},
 	{"-1.7976931348623159e308", "-Inf", ErrRange},
+	{"0x1p1024", "+Inf", ErrRange},
+	{"-0x1p1024", "-Inf", ErrRange},
+	{"0x2p1023", "+Inf", ErrRange},
+	{"-0x2p1023", "-Inf", ErrRange},
+	{"0x.1p1028", "+Inf", ErrRange},
+	{"-0x.1p1028", "-Inf", ErrRange},
+	{"0x.2p1027", "+Inf", ErrRange},
+	{"-0x.2p1027", "-Inf", ErrRange},
+
 	// the border is ...158079
 	// borderline - okay
 	{"1.7976931348623158e308", "1.7976931348623157e+308", nil},
 	{"-1.7976931348623158e308", "-1.7976931348623157e+308", nil},
+	{"0x1.fffffffffffff7fffp1023", "1.7976931348623157e+308", nil},
+	{"-0x1.fffffffffffff7fffp1023", "-1.7976931348623157e+308", nil},
 	// borderline - too large
 	{"1.797693134862315808e308", "+Inf", ErrRange},
 	{"-1.797693134862315808e308", "-Inf", ErrRange},
+	{"0x1.fffffffffffff8p1023", "+Inf", ErrRange},
+	{"-0x1.fffffffffffff8p1023", "-Inf", ErrRange},
+	{"0x1fffffffffffff.8p+971", "+Inf", ErrRange},
+	{"-0x1fffffffffffff8p+967", "-Inf", ErrRange},
+	{"0x.1fffffffffffff8p1027", "+Inf", ErrRange},
+	{"-0x.1fffffffffffff9p1027", "-Inf", ErrRange},
 
 	// a little too large
 	{"1e308", "1e+308", nil},
 	{"2e308", "+Inf", ErrRange},
 	{"1e309", "+Inf", ErrRange},
+	{"0x1p1025", "+Inf", ErrRange},
 
 	// way too large
 	{"1e310", "+Inf", ErrRange},
@@ -106,6 +170,12 @@ var atoftests = []atofTest{
 	{"-1e400", "-Inf", ErrRange},
 	{"1e400000", "+Inf", ErrRange},
 	{"-1e400000", "-Inf", ErrRange},
+	{"0x1p1030", "+Inf", ErrRange},
+	{"0x1p2000", "+Inf", ErrRange},
+	{"0x1p2000000000", "+Inf", ErrRange},
+	{"-0x1p1030", "-Inf", ErrRange},
+	{"-0x1p2000", "-Inf", ErrRange},
+	{"-0x1p2000000000", "-Inf", ErrRange},
 
 	// denormalized
 	{"1e-305", "1e-305", nil},
@@ -125,17 +195,75 @@ var atoftests = []atofTest{
 	{"1e-350", "0", nil},
 	{"1e-400000", "0", nil},
 
+	// Near denormals and denormals.
+	{"0x2.00000000000000p-1010", "1.8227805048890994e-304", nil}, // 0x00e0000000000000
+	{"0x1.fffffffffffff0p-1010", "1.8227805048890992e-304", nil}, // 0x00dfffffffffffff
+	{"0x1.fffffffffffff7p-1010", "1.8227805048890992e-304", nil}, // rounded down
+	{"0x1.fffffffffffff8p-1010", "1.8227805048890994e-304", nil}, // rounded up
+	{"0x1.fffffffffffff9p-1010", "1.8227805048890994e-304", nil}, // rounded up
+
+	{"0x2.00000000000000p-1022", "4.450147717014403e-308", nil},  // 0x0020000000000000
+	{"0x1.fffffffffffff0p-1022", "4.4501477170144023e-308", nil}, // 0x001fffffffffffff
+	{"0x1.fffffffffffff7p-1022", "4.4501477170144023e-308", nil}, // rounded down
+	{"0x1.fffffffffffff8p-1022", "4.450147717014403e-308", nil},  // rounded up
+	{"0x1.fffffffffffff9p-1022", "4.450147717014403e-308", nil},  // rounded up
+
+	{"0x1.00000000000000p-1022", "2.2250738585072014e-308", nil}, // 0x0010000000000000
+	{"0x0.fffffffffffff0p-1022", "2.225073858507201e-308", nil},  // 0x000fffffffffffff
+	{"0x0.ffffffffffffe0p-1022", "2.2250738585072004e-308", nil}, // 0x000ffffffffffffe
+	{"0x0.ffffffffffffe7p-1022", "2.2250738585072004e-308", nil}, // rounded down
+	{"0x1.ffffffffffffe8p-1023", "2.225073858507201e-308", nil},  // rounded up
+	{"0x1.ffffffffffffe9p-1023", "2.225073858507201e-308", nil},  // rounded up
+
+	{"0x0.00000003fffff0p-1022", "2.072261e-317", nil},  // 0x00000000003fffff
+	{"0x0.00000003456780p-1022", "1.694649e-317", nil},  // 0x0000000000345678
+	{"0x0.00000003456787p-1022", "1.694649e-317", nil},  // rounded down
+	{"0x0.00000003456788p-1022", "1.694649e-317", nil},  // rounded down (half to even)
+	{"0x0.00000003456790p-1022", "1.6946496e-317", nil}, // 0x0000000000345679
+	{"0x0.00000003456789p-1022", "1.6946496e-317", nil}, // rounded up
+
+	{"0x0.0000000345678800000000000000000000000001p-1022", "1.6946496e-317", nil}, // rounded up
+
+	{"0x0.000000000000f0p-1022", "7.4e-323", nil}, // 0x000000000000000f
+	{"0x0.00000000000060p-1022", "3e-323", nil},   // 0x0000000000000006
+	{"0x0.00000000000058p-1022", "3e-323", nil},   // rounded up
+	{"0x0.00000000000057p-1022", "2.5e-323", nil}, // rounded down
+	{"0x0.00000000000050p-1022", "2.5e-323", nil}, // 0x0000000000000005
+
+	{"0x0.00000000000010p-1022", "5e-324", nil},  // 0x0000000000000001
+	{"0x0.000000000000081p-1022", "5e-324", nil}, // rounded up
+	{"0x0.00000000000008p-1022", "0", nil},       // rounded down
+	{"0x0.00000000000007fp-1022", "0", nil},      // rounded down
+
 	// try to overflow exponent
 	{"1e-4294967296", "0", nil},
 	{"1e+4294967296", "+Inf", ErrRange},
 	{"1e-18446744073709551616", "0", nil},
 	{"1e+18446744073709551616", "+Inf", ErrRange},
+	{"0x1p-4294967296", "0", nil},
+	{"0x1p+4294967296", "+Inf", ErrRange},
+	{"0x1p-18446744073709551616", "0", nil},
+	{"0x1p+18446744073709551616", "+Inf", ErrRange},
 
 	// Parse errors
 	{"1e", "0", ErrSyntax},
 	{"1e-", "0", ErrSyntax},
 	{".e-1", "0", ErrSyntax},
 	{"1\x00.2", "0", ErrSyntax},
+	{"0x", "0", ErrSyntax},
+	{"0x.", "0", ErrSyntax},
+	{"0x1", "0", ErrSyntax},
+	{"0x.1", "0", ErrSyntax},
+	{"0x1p", "0", ErrSyntax},
+	{"0x.1p", "0", ErrSyntax},
+	{"0x1p+", "0", ErrSyntax},
+	{"0x.1p+", "0", ErrSyntax},
+	{"0x1p-", "0", ErrSyntax},
+	{"0x.1p-", "0", ErrSyntax},
+	{"0x1p+2", "4", nil},
+	{"0x.1p+2", "0.25", nil},
+	{"0x1p-2", "0.25", nil},
+	{"0x.1p-2", "0.015625", nil},
 
 	// https://www.exploringbinary.com/java-hangs-when-converting-2-2250738585072012e-308/
 	{"2.2250738585072012e-308", "2.2250738585072014e-308", nil},
@@ -148,42 +276,75 @@ var atoftests = []atofTest{
 	// A different kind of very large number.
 	{"22.222222222222222", "22.22222222222222", nil},
 	{"2." + strings.Repeat("2", 4000) + "e+1", "22.22222222222222", nil},
+	{"0x1.1111111111111p222", "7.18931911124017e+66", nil},
+	{"0x2.2222222222222p221", "7.18931911124017e+66", nil},
+	{"0x2." + strings.Repeat("2", 4000) + "p221", "7.18931911124017e+66", nil},
 
 	// Exactly halfway between 1 and math.Nextafter(1, 2).
 	// Round to even (down).
 	{"1.00000000000000011102230246251565404236316680908203125", "1", nil},
+	{"0x1.00000000000008p0", "1", nil},
 	// Slightly lower; still round down.
 	{"1.00000000000000011102230246251565404236316680908203124", "1", nil},
+	{"0x1.00000000000007Fp0", "1", nil},
 	// Slightly higher; round up.
 	{"1.00000000000000011102230246251565404236316680908203126", "1.0000000000000002", nil},
+	{"0x1.000000000000081p0", "1.0000000000000002", nil},
+	{"0x1.00000000000009p0", "1.0000000000000002", nil},
 	// Slightly higher, but you have to read all the way to the end.
 	{"1.00000000000000011102230246251565404236316680908203125" + strings.Repeat("0", 10000) + "1", "1.0000000000000002", nil},
+	{"0x1.00000000000008" + strings.Repeat("0", 10000) + "1p0", "1.0000000000000002", nil},
+
+	// Halfway between x := math.Nextafter(1, 2) and math.Nextafter(x, 2)
+	// Round to even (up).
+	{"1.00000000000000033306690738754696212708950042724609375", "1.0000000000000004", nil},
+	{"0x1.00000000000018p0", "1.0000000000000004", nil},
 }
 
 var atof32tests = []atofTest{
+	// Hex
+	{"0x1p-100", "7.888609e-31", nil},
+	{"0x1p100", "1.2676506e+30", nil},
+
 	// Exactly halfway between 1 and the next float32.
 	// Round to even (down).
 	{"1.000000059604644775390625", "1", nil},
+	{"0x1.000001p0", "1", nil},
 	// Slightly lower.
 	{"1.000000059604644775390624", "1", nil},
+	{"0x1.0000008p0", "1", nil},
+	{"0x1.000000fp0", "1", nil},
 	// Slightly higher.
 	{"1.000000059604644775390626", "1.0000001", nil},
+	{"0x1.000002p0", "1.0000001", nil},
+	{"0x1.0000018p0", "1.0000001", nil},
+	{"0x1.0000011p0", "1.0000001", nil},
 	// Slightly higher, but you have to read all the way to the end.
 	{"1.000000059604644775390625" + strings.Repeat("0", 10000) + "1", "1.0000001", nil},
+	{"0x1.000001" + strings.Repeat("0", 10000) + "1p0", "1.0000001", nil},
 
 	// largest float32: (1<<128) * (1 - 2^-24)
 	{"340282346638528859811704183484516925440", "3.4028235e+38", nil},
 	{"-340282346638528859811704183484516925440", "-3.4028235e+38", nil},
+	{"0x.ffffffp128", "3.4028235e+38", nil},
+	{"-340282346638528859811704183484516925440", "-3.4028235e+38", nil},
+	{"-0x.ffffffp128", "-3.4028235e+38", nil},
 	// next float32 - too large
 	{"3.4028236e38", "+Inf", ErrRange},
 	{"-3.4028236e38", "-Inf", ErrRange},
+	{"0x1.0p128", "+Inf", ErrRange},
+	{"-0x1.0p128", "-Inf", ErrRange},
 	// the border is 3.40282356779...e+38
 	// borderline - okay
 	{"3.402823567e38", "3.4028235e+38", nil},
 	{"-3.402823567e38", "-3.4028235e+38", nil},
+	{"0x.ffffff7fp128", "3.4028235e+38", nil},
+	{"-0x.ffffff7fp128", "-3.4028235e+38", nil},
 	// borderline - too large
 	{"3.4028235678e38", "+Inf", ErrRange},
 	{"-3.4028235678e38", "-Inf", ErrRange},
+	{"0x.ffffff8p128", "+Inf", ErrRange},
+	{"-0x.ffffff8p128", "-Inf", ErrRange},
 
 	// Denormals: less than 2^-126
 	{"1e-38", "1e-38", nil},
@@ -195,9 +356,24 @@ var atof32tests = []atofTest{
 	{"1e-44", "1e-44", nil},
 	{"6e-45", "6e-45", nil}, // 4p-149 = 5.6e-45
 	{"5e-45", "6e-45", nil},
+
 	// Smallest denormal
 	{"1e-45", "1e-45", nil}, // 1p-149 = 1.4e-45
 	{"2e-45", "1e-45", nil},
+	{"3e-45", "3e-45", nil},
+
+	// Near denormals and denormals.
+	{"0x0.89aBcDp-125", "1.2643093e-38", nil},  // 0x0089abcd
+	{"0x0.8000000p-125", "1.1754944e-38", nil}, // 0x00800000
+	{"0x0.1234560p-125", "1.671814e-39", nil},  // 0x00123456
+	{"0x0.1234567p-125", "1.671814e-39", nil},  // rounded down
+	{"0x0.1234568p-125", "1.671814e-39", nil},  // rounded down
+	{"0x0.1234569p-125", "1.671815e-39", nil},  // rounded up
+	{"0x0.1234570p-125", "1.671815e-39", nil},  // 0x00123457
+	{"0x0.0000010p-125", "1e-45", nil},         // 0x00000001
+	{"0x0.00000081p-125", "1e-45", nil},        // rounded up
+	{"0x0.0000008p-125", "0", nil},             // rounded down
+	{"0x0.0000007p-125", "0", nil},             // rounded down
 
 	// 2^92 = 8388608p+69 = 4951760157141521099596496896 (4.9517602e27)
 	// is an exact power of two that needs 8 decimal digits to be correctly
diff --git a/src/strconv/atoi.go b/src/strconv/atoi.go
index ff33d555e4..186c9b3f86 100644
--- a/src/strconv/atoi.go
+++ b/src/strconv/atoi.go
@@ -6,6 +6,14 @@ package strconv
 
 import "errors"
 
+// lower(c) is a lower-case letter if and only if
+// c is either that lower-case letter or the equivalent upper-case letter.
+// Instead of writing c == 'x' || c == 'X' one can write lower(c) == 'x'.
+// Note that lower of non-letters can produce other non-letters.
+func lower(c byte) byte {
+	return c | ('x' - 'X')
+}
+
 // ErrRange indicates that a value is out of range for the target type.
 var ErrRange = errors.New("value out of range")