cmd/compile: update vendored math/big to latest version

This makes the rounding bug fix in math/big for issue 14651 available
to the compiler.

- changes to cmd/compile/internal/big fully automatic via script
- added test case for issue
- updated old test case with correct test data

Fixes #14651.

Change-Id: Iea37a2cd8d3a75f8c96193748b66156a987bbe40
Reviewed-on: https://go-review.googlesource.com/20818Reviewed-by: Brad Fitzpatrick <bradfitz@golang.org>

src/cmd/compile/internal/big/arith_test.go

@@ -158,21 +158,6 @@ var sumVW = []argVW{
 	{nat{585}, nat{314}, 271, 0},
 }
 
-var prodVW = []argVW{
-	{},
-	{nat{0}, nat{0}, 0, 0},
-	{nat{0}, nat{_M}, 0, 0},
-	{nat{0}, nat{0}, _M, 0},
-	{nat{1}, nat{1}, 1, 0},
-	{nat{22793}, nat{991}, 23, 0},
-	{nat{0, 0, 0, 22793}, nat{0, 0, 0, 991}, 23, 0},
-	{nat{0, 0, 0, 0}, nat{7893475, 7395495, 798547395, 68943}, 0, 0},
-	{nat{0, 0, 0, 0}, nat{0, 0, 0, 0}, 894375984, 0},
-	{nat{_M << 1 & _M}, nat{_M}, 1 << 1, _M >> (_W - 1)},
-	{nat{_M << 7 & _M}, nat{_M}, 1 << 7, _M >> (_W - 7)},
-	{nat{_M << 7 & _M, _M, _M, _M}, nat{_M, _M, _M, _M}, 1 << 7, _M >> (_W - 7)},
-}
-
 var lshVW = []argVW{
 	{},
 	{nat{0}, nat{0}, 0, 0},

src/cmd/compile/internal/big/float.go

@@ -392,15 +392,13 @@ func (z *Float) round(sbit uint) {
 	// m > 0 implies z.prec > 0 (checked by validate)
 
 	m := uint32(len(z.mant)) // present mantissa length in words
-	bits := m * _W           // present mantissa bits
+	bits := m * _W           // present mantissa bits; bits > 0
 	if bits <= z.prec {
 		// mantissa fits => nothing to do
 		return
 	}
 	// bits > z.prec
 
-	n := (z.prec + (_W - 1)) / _W // mantissa length in words for desired precision
-
 	// Rounding is based on two bits: the rounding bit (rbit) and the
 	// sticky bit (sbit). The rbit is the bit immediately before the
 	// z.prec leading mantissa bits (the "0.5"). The sbit is set if any
@@ -415,111 +413,77 @@ func (z *Float) round(sbit uint) {
 
 	// bits > z.prec: mantissa too large => round
 	r := uint(bits - z.prec - 1) // rounding bit position; r >= 0
-	rbit := z.mant.bit(r)        // rounding bit
+	rbit := z.mant.bit(r) & 1    // rounding bit; be safe and ensure it's a single bit
 	if sbit == 0 {
+		// TODO(gri) if rbit != 0 we don't need to compute sbit for some rounding modes (optimization)
 		sbit = z.mant.sticky(r)
 	}
-	if debugFloat && sbit&^1 != 0 {
-		panic(fmt.Sprintf("invalid sbit %#x", sbit))
-	}
-
-	// convert ToXInf rounding modes
-	mode := z.mode
-	switch mode {
-	case ToNegativeInf:
-		mode = ToZero
-		if z.neg {
-			mode = AwayFromZero
-		}
-	case ToPositiveInf:
-		mode = AwayFromZero
-		if z.neg {
-			mode = ToZero
-		}
-	}
+	sbit &= 1 // be safe and ensure it's a single bit
 
 	// cut off extra words
+	n := (z.prec + (_W - 1)) / _W // mantissa length in words for desired precision
 	if m > n {
 		copy(z.mant, z.mant[m-n:]) // move n last words to front
 		z.mant = z.mant[:n]
 	}
 
-	// determine number of trailing zero bits t
-	t := n*_W - z.prec // 0 <= t < _W
-	lsb := Word(1) << t
-
-	// make rounding decision
-	// TODO(gri) This can be simplified (see Bits.round in bits_test.go).
-	switch mode {
-	case ToZero:
-		// nothing to do
-	case ToNearestEven, ToNearestAway:
-		if rbit == 0 {
-			// rounding bits == 0b0x
-			mode = ToZero
-		} else if sbit == 1 {
-			// rounding bits == 0b11
-			mode = AwayFromZero
-		}
-	case AwayFromZero:
-		if rbit|sbit == 0 {
-			mode = ToZero
-		}
-	default:
-		// ToXInf modes have been converted to ToZero or AwayFromZero
-		panic("unreachable")
-	}
-
-	// round and determine accuracy
-	switch mode {
-	case ToZero:
-		if rbit|sbit != 0 {
-			z.acc = Below
+	// determine number of trailing zero bits (ntz) and compute lsb mask of mantissa's least-significant word
+	ntz := n*_W - z.prec // 0 <= ntz < _W
+	lsb := Word(1) << ntz
+
+	// round if result is inexact
+	if rbit|sbit != 0 {
+		// Make rounding decision: The result mantissa is truncated ("rounded down")
+		// by default. Decide if we need to increment, or "round up", the (unsigned)
+		// mantissa.
+		inc := false
+		switch z.mode {
+		case ToNegativeInf:
+			inc = z.neg
+		case ToZero:
+			// nothing to do
+		case ToNearestEven:
+			inc = rbit != 0 && (sbit != 0 || z.mant[0]&lsb != 0)
+		case ToNearestAway:
+			inc = rbit != 0
+		case AwayFromZero:
+			inc = true
+		case ToPositiveInf:
+			inc = !z.neg
+		default:
+			panic("unreachable")
 		}
 
-	case ToNearestEven, ToNearestAway:
-		if debugFloat && rbit != 1 {
-			panic("internal error in rounding")
-		}
-		if mode == ToNearestEven && sbit == 0 && z.mant[0]&lsb == 0 {
-			z.acc = Below
-			break
-		}
-		// mode == ToNearestAway || sbit == 1 || z.mant[0]&lsb != 0
-		fallthrough
-
-	case AwayFromZero:
-		// add 1 to mantissa
-		if addVW(z.mant, z.mant, lsb) != 0 {
-			// overflow => shift mantissa right by 1 and add msb
-			shrVU(z.mant, z.mant, 1)
-			z.mant[n-1] |= 1 << (_W - 1)
-			// adjust exponent
-			if z.exp < MaxExp {
+		// A positive result (!z.neg) is Above the exact result if we increment,
+		// and it's Below if we truncate (Exact results require no rounding).
+		// For a negative result (z.neg) it is exactly the opposite.
+		z.acc = makeAcc(inc != z.neg)
+
+		if inc {
+			// add 1 to mantissa
+			if addVW(z.mant, z.mant, lsb) != 0 {
+				// mantissa overflow => adjust exponent
+				if z.exp >= MaxExp {
+					// exponent overflow
+					z.form = inf
+					return
+				}
 				z.exp++
-			} else {
-				// exponent overflow
-				z.acc = makeAcc(!z.neg)
-				z.form = inf
-				return
+				// adjust mantissa: divide by 2 to compensate for exponent adjustment
+				shrVU(z.mant, z.mant, 1)
+				// set msb == carry == 1 from the mantissa overflow above
+				const msb = 1 << (_W - 1)
+				z.mant[n-1] |= msb
 			}
 		}
-		z.acc = Above
 	}
 
 	// zero out trailing bits in least-significant word
 	z.mant[0] &^= lsb - 1
 
-	// update accuracy
-	if z.acc != Exact && z.neg {
-		z.acc = -z.acc
-	}
-
 	if debugFloat {
 		z.validate()
 	}
-
-	return
 }
 
 func (z *Float) setBits64(neg bool, x uint64) *Float {
@@ -874,21 +838,43 @@ func (x *Float) Float32() (float32, Accuracy) {
 			emax  = bias              //   127  largest unbiased exponent (normal)
 		)
 
-		// Float mantissa m is 0.5 <= m < 1.0; compute exponent for float32 mantissa.
-		e := x.exp - 1 // exponent for mantissa m with 1.0 <= m < 2.0
-		p := mbits + 1 // precision of normal float
+		// Float mantissa m is 0.5 <= m < 1.0; compute exponent e for float32 mantissa.
+		e := x.exp - 1 // exponent for normal mantissa m with 1.0 <= m < 2.0
 
-		// If the exponent is too small, we may have a denormal number
-		// in which case we have fewer mantissa bits available: recompute
-		// precision.
+		// Compute precision p for float32 mantissa.
+		// If the exponent is too small, we have a denormal number before
+		// rounding and fewer than p mantissa bits of precision available
+		// (the exponent remains fixed but the mantissa gets shifted right).
+		p := mbits + 1 // precision of normal float
 		if e < emin {
+			// recompute precision
 			p = mbits + 1 - emin + int(e)
-			// Make sure we have at least 1 bit so that we don't
-			// lose numbers rounded up to the smallest denormal.
-			if p < 1 {
-				p = 1
+			// If p == 0, the mantissa of x is shifted so much to the right
+			// that its msb falls immediately to the right of the float32
+			// mantissa space. In other words, if the smallest denormal is
+			// considered "1.0", for p == 0, the mantissa value m is >= 0.5.
+			// If m > 0.5, it is rounded up to 1.0; i.e., the smallest denormal.
+			// If m == 0.5, it is rounded down to even, i.e., 0.0.
+			// If p < 0, the mantissa value m is <= "0.25" which is never rounded up.
+			if p < 0 /* m <= 0.25 */ || p == 0 && x.mant.sticky(uint(len(x.mant))*_W-1) == 0 /* m == 0.5 */ {
+				// underflow to ±0
+				if x.neg {
+					var z float32
+					return -z, Above
+				}
+				return 0.0, Below
+			}
+			// otherwise, round up
+			// We handle p == 0 explicitly because it's easy and because
+			// Float.round doesn't support rounding to 0 bits of precision.
+			if p == 0 {
+				if x.neg {
+					return -math.SmallestNonzeroFloat32, Below
+				}
+				return math.SmallestNonzeroFloat32, Above
 			}
 		}
+		// p > 0
 
 		// round
 		var r Float
@@ -898,12 +884,8 @@ func (x *Float) Float32() (float32, Accuracy) {
 
 		// Rounding may have caused r to overflow to ±Inf
 		// (rounding never causes underflows to 0).
-		if r.form == inf {
-			e = emax + 1 // cause overflow below
-		}
-
-		// If the exponent is too large, overflow to ±Inf.
-		if e > emax {
+		// If the exponent is too large, also overflow to ±Inf.
+		if r.form == inf || e > emax {
 			// overflow
 			if x.neg {
 				return float32(math.Inf(-1)), Below
@@ -921,17 +903,10 @@ func (x *Float) Float32() (float32, Accuracy) {
 		// Rounding may have caused a denormal number to
 		// become normal. Check again.
 		if e < emin {
-			// denormal number
-			if e < dmin {
-				// underflow to ±0
-				if x.neg {
-					var z float32
-					return -z, Above
-				}
-				return 0.0, Below
-			}
-			// bexp = 0
-			// recompute precision
+			// denormal number: recompute precision
+			// Since rounding may have at best increased precision
+			// and we have eliminated p <= 0 early, we know p > 0.
+			// bexp == 0 for denormals
 			p = mbits + 1 - emin + int(e)
 			mant = msb32(r.mant) >> uint(fbits-p)
 		} else {
@@ -983,21 +958,43 @@ func (x *Float) Float64() (float64, Accuracy) {
 			emax  = bias              //  1023  largest unbiased exponent (normal)
 		)
 
-		// Float mantissa m is 0.5 <= m < 1.0; compute exponent for float64 mantissa.
-		e := x.exp - 1 // exponent for mantissa m with 1.0 <= m < 2.0
-		p := mbits + 1 // precision of normal float
+		// Float mantissa m is 0.5 <= m < 1.0; compute exponent e for float64 mantissa.
+		e := x.exp - 1 // exponent for normal mantissa m with 1.0 <= m < 2.0
 
-		// If the exponent is too small, we may have a denormal number
-		// in which case we have fewer mantissa bits available: recompute
-		// precision.
+		// Compute precision p for float64 mantissa.
+		// If the exponent is too small, we have a denormal number before
+		// rounding and fewer than p mantissa bits of precision available
+		// (the exponent remains fixed but the mantissa gets shifted right).
+		p := mbits + 1 // precision of normal float
 		if e < emin {
+			// recompute precision
 			p = mbits + 1 - emin + int(e)
-			// Make sure we have at least 1 bit so that we don't
-			// lose numbers rounded up to the smallest denormal.
-			if p < 1 {
-				p = 1
+			// If p == 0, the mantissa of x is shifted so much to the right
+			// that its msb falls immediately to the right of the float64
+			// mantissa space. In other words, if the smallest denormal is
+			// considered "1.0", for p == 0, the mantissa value m is >= 0.5.
+			// If m > 0.5, it is rounded up to 1.0; i.e., the smallest denormal.
+			// If m == 0.5, it is rounded down to even, i.e., 0.0.
+			// If p < 0, the mantissa value m is <= "0.25" which is never rounded up.
+			if p < 0 /* m <= 0.25 */ || p == 0 && x.mant.sticky(uint(len(x.mant))*_W-1) == 0 /* m == 0.5 */ {
+				// underflow to ±0
+				if x.neg {
+					var z float64
+					return -z, Above
+				}
+				return 0.0, Below
+			}
+			// otherwise, round up
+			// We handle p == 0 explicitly because it's easy and because
+			// Float.round doesn't support rounding to 0 bits of precision.
+			if p == 0 {
+				if x.neg {
+					return -math.SmallestNonzeroFloat64, Below
+				}
+				return math.SmallestNonzeroFloat64, Above
 			}
 		}
+		// p > 0
 
 		// round
 		var r Float
@@ -1007,17 +1004,13 @@ func (x *Float) Float64() (float64, Accuracy) {
 
 		// Rounding may have caused r to overflow to ±Inf
 		// (rounding never causes underflows to 0).
-		if r.form == inf {
-			e = emax + 1 // cause overflow below
-		}
-
-		// If the exponent is too large, overflow to ±Inf.
-		if e > emax {
+		// If the exponent is too large, also overflow to ±Inf.
+		if r.form == inf || e > emax {
 			// overflow
 			if x.neg {
-				return math.Inf(-1), Below
+				return float64(math.Inf(-1)), Below
 			}
-			return math.Inf(+1), Above
+			return float64(math.Inf(+1)), Above
 		}
 		// e <= emax
 
@@ -1030,17 +1023,10 @@ func (x *Float) Float64() (float64, Accuracy) {
 		// Rounding may have caused a denormal number to
 		// become normal. Check again.
 		if e < emin {
-			// denormal number
-			if e < dmin {
-				// underflow to ±0
-				if x.neg {
-					var z float64
-					return -z, Above
-				}
-				return 0.0, Below
-			}
-			// bexp = 0
-			// recompute precision
+			// denormal number: recompute precision
+			// Since rounding may have at best increased precision
+			// and we have eliminated p <= 0 early, we know p > 0.
+			// bexp == 0 for denormals
 			p = mbits + 1 - emin + int(e)
 			mant = msb64(r.mant) >> uint(fbits-p)
 		} else {

src/cmd/compile/internal/big/float_test.go

@@ -829,7 +829,7 @@ func TestFloatFloat32(t *testing.T) {
 	}{
 		{"0", 0, Exact},
 
-		// underflow
+		// underflow to zero
 		{"1e-1000", 0, Below},
 		{"0x0.000002p-127", 0, Below},
 		{"0x.0000010p-126", 0, Below},
@@ -843,25 +843,39 @@ func TestFloatFloat32(t *testing.T) {
 		{"1p-149", math.SmallestNonzeroFloat32, Exact},
 		{"0x.fffffep-126", math.Float32frombits(0x7fffff), Exact}, // largest denormal
 
-		// special cases (see issue 14553)
-		{"0x0.bp-149", math.Float32frombits(0x000000000), Below}, // ToNearestEven rounds down (to even)
-		{"0x0.cp-149", math.Float32frombits(0x000000001), Above},
-
-		{"0x1.0p-149", math.Float32frombits(0x000000001), Exact},
+		// special denormal cases (see issues 14553, 14651)
+		{"0x0.0000001p-126", math.Float32frombits(0x00000000), Below}, // underflow to zero
+		{"0x0.0000008p-126", math.Float32frombits(0x00000000), Below}, // underflow to zero
+		{"0x0.0000010p-126", math.Float32frombits(0x00000000), Below}, // rounded down to even
+		{"0x0.0000011p-126", math.Float32frombits(0x00000001), Above}, // rounded up to smallest denormal
+		{"0x0.0000018p-126", math.Float32frombits(0x00000001), Above}, // rounded up to smallest denormal
+
+		{"0x1.0000000p-149", math.Float32frombits(0x00000001), Exact}, // smallest denormal
+		{"0x0.0000020p-126", math.Float32frombits(0x00000001), Exact}, // smallest denormal
+		{"0x0.fffffe0p-126", math.Float32frombits(0x007fffff), Exact}, // largest denormal
+		{"0x1.0000000p-126", math.Float32frombits(0x00800000), Exact}, // smallest normal
+
+		{"0x0.8p-149", math.Float32frombits(0x000000000), Below}, // rounded down to even
+		{"0x0.9p-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal
+		{"0x0.ap-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal
+		{"0x0.bp-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal
+		{"0x0.cp-149", math.Float32frombits(0x000000001), Above}, // rounded up to smallest denormal
+
+		{"0x1.0p-149", math.Float32frombits(0x000000001), Exact}, // smallest denormal
 		{"0x1.7p-149", math.Float32frombits(0x000000001), Below},
 		{"0x1.8p-149", math.Float32frombits(0x000000002), Above},
 		{"0x1.9p-149", math.Float32frombits(0x000000002), Above},
 
 		{"0x2.0p-149", math.Float32frombits(0x000000002), Exact},
-		{"0x2.8p-149", math.Float32frombits(0x000000002), Below}, // ToNearestEven rounds down (to even)
+		{"0x2.8p-149", math.Float32frombits(0x000000002), Below}, // rounded down to even
 		{"0x2.9p-149", math.Float32frombits(0x000000003), Above},
 
 		{"0x3.0p-149", math.Float32frombits(0x000000003), Exact},
 		{"0x3.7p-149", math.Float32frombits(0x000000003), Below},
-		{"0x3.8p-149", math.Float32frombits(0x000000004), Above}, // ToNearestEven rounds up (to even)
+		{"0x3.8p-149", math.Float32frombits(0x000000004), Above}, // rounded up to even
 
 		{"0x4.0p-149", math.Float32frombits(0x000000004), Exact},
-		{"0x4.8p-149", math.Float32frombits(0x000000004), Below}, // ToNearestEven rounds down (to even)
+		{"0x4.8p-149", math.Float32frombits(0x000000004), Below}, // rounded down to even
 		{"0x4.9p-149", math.Float32frombits(0x000000005), Above},
 
 		// specific case from issue 14553
@@ -907,7 +921,7 @@ func TestFloatFloat32(t *testing.T) {
 			x := makeFloat(tx)
 			out, acc := x.Float32()
 			if !alike32(out, tout) || acc != tacc {
-				t.Errorf("%s: got %g (%#x, %s); want %g (%#x, %s)", tx, out, math.Float32bits(out), acc, test.out, math.Float32bits(test.out), tacc)
+				t.Errorf("%s: got %g (%#08x, %s); want %g (%#08x, %s)", tx, out, math.Float32bits(out), acc, test.out, math.Float32bits(test.out), tacc)
 			}
 
 			// test that x.SetFloat64(float64(f)).Float32() == f
@@ -929,21 +943,30 @@ func TestFloatFloat64(t *testing.T) {
 	}{
 		{"0", 0, Exact},
 
-		// underflow
+		// underflow to zero
 		{"1e-1000", 0, Below},
 		{"0x0.0000000000001p-1023", 0, Below},
 		{"0x0.00000000000008p-1022", 0, Below},
 
 		// denormals
 		{"0x0.0000000000000cp-1022", math.SmallestNonzeroFloat64, Above}, // rounded up to smallest denormal
-		{"0x0.0000000000001p-1022", math.SmallestNonzeroFloat64, Exact},  // smallest denormal
+		{"0x0.00000000000010p-1022", math.SmallestNonzeroFloat64, Exact}, // smallest denormal
 		{"0x.8p-1073", math.SmallestNonzeroFloat64, Exact},
 		{"1p-1074", math.SmallestNonzeroFloat64, Exact},
 		{"0x.fffffffffffffp-1022", math.Float64frombits(0x000fffffffffffff), Exact}, // largest denormal
 
-		// special cases (see issue 14553)
-		{"0x0.bp-1074", math.Float64frombits(0x00000000000000000), Below}, // ToNearestEven rounds down (to even)
-		{"0x0.cp-1074", math.Float64frombits(0x00000000000000001), Above},
+		// special denormal cases (see issues 14553, 14651)
+		{"0x0.00000000000001p-1022", math.Float64frombits(0x00000000000000000), Below}, // underflow to zero
+		{"0x0.00000000000004p-1022", math.Float64frombits(0x00000000000000000), Below}, // underflow to zero
+		{"0x0.00000000000008p-1022", math.Float64frombits(0x00000000000000000), Below}, // rounded down to even
+		{"0x0.00000000000009p-1022", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
+		{"0x0.0000000000000ap-1022", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
+
+		{"0x0.8p-1074", math.Float64frombits(0x00000000000000000), Below}, // rounded down to even
+		{"0x0.9p-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
+		{"0x0.ap-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
+		{"0x0.bp-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
+		{"0x0.cp-1074", math.Float64frombits(0x00000000000000001), Above}, // rounded up to smallest denormal
 
 		{"0x1.0p-1074", math.Float64frombits(0x00000000000000001), Exact},
 		{"0x1.7p-1074", math.Float64frombits(0x00000000000000001), Below},
@@ -951,15 +974,15 @@ func TestFloatFloat64(t *testing.T) {
 		{"0x1.9p-1074", math.Float64frombits(0x00000000000000002), Above},
 
 		{"0x2.0p-1074", math.Float64frombits(0x00000000000000002), Exact},
-		{"0x2.8p-1074", math.Float64frombits(0x00000000000000002), Below}, // ToNearestEven rounds down (to even)
+		{"0x2.8p-1074", math.Float64frombits(0x00000000000000002), Below}, // rounded down to even
 		{"0x2.9p-1074", math.Float64frombits(0x00000000000000003), Above},
 
 		{"0x3.0p-1074", math.Float64frombits(0x00000000000000003), Exact},
 		{"0x3.7p-1074", math.Float64frombits(0x00000000000000003), Below},
-		{"0x3.8p-1074", math.Float64frombits(0x00000000000000004), Above}, // ToNearestEven rounds up (to even)
+		{"0x3.8p-1074", math.Float64frombits(0x00000000000000004), Above}, // rounded up to even
 
 		{"0x4.0p-1074", math.Float64frombits(0x00000000000000004), Exact},
-		{"0x4.8p-1074", math.Float64frombits(0x00000000000000004), Below}, // ToNearestEven rounds down (to even)
+		{"0x4.8p-1074", math.Float64frombits(0x00000000000000004), Below}, // rounded down to even
 		{"0x4.9p-1074", math.Float64frombits(0x00000000000000005), Above},
 
 		// normals
@@ -1005,7 +1028,7 @@ func TestFloatFloat64(t *testing.T) {
 			x := makeFloat(tx)
 			out, acc := x.Float64()
 			if !alike64(out, tout) || acc != tacc {
-				t.Errorf("%s: got %g (%#x, %s); want %g (%#x, %s)", tx, out, math.Float64bits(out), acc, test.out, math.Float64bits(test.out), tacc)
+				t.Errorf("%s: got %g (%#016x, %s); want %g (%#016x, %s)", tx, out, math.Float64bits(out), acc, test.out, math.Float64bits(test.out), tacc)
 			}
 
 			// test that x.SetFloat64(f).Float64() == f

src/cmd/compile/internal/big/ftoa.go

@@ -333,9 +333,9 @@ func (x *Float) fmtB(buf []byte) []byte {
 	return strconv.AppendInt(buf, e, 10)
 }
 
-// fmtP appends the string of x in the format 0x." mantissa "p" exponent
-// with a hexadecimal mantissa and a binary exponent, or 0" if x is zero,
-// ad returns the extended buffer.
+// fmtP appends the string of x in the format "0x." mantissa "p" exponent
+// with a hexadecimal mantissa and a binary exponent, or "0" if x is zero,
+// and returns the extended buffer.
 // The mantissa is normalized such that 0.5 <= 0.mantissa < 1.0.
 // The sign of x is ignored, and x must not be an Inf.
 func (x *Float) fmtP(buf []byte) []byte {
@@ -374,12 +374,11 @@ func min(x, y int) int {
 }
 
 // Format implements fmt.Formatter. It accepts all the regular
-// formats for floating-point numbers ('e', 'E', 'f', 'F', 'g',
-// 'G') as well as 'b', 'p', and 'v'. See (*Float).Text for the
-// interpretation of 'b' and 'p'. The 'v' format is handled like
-// 'g'.
+// formats for floating-point numbers ('b', 'e', 'E', 'f', 'F',
+// 'g', 'G') as well as 'p' and 'v'. See (*Float).Text for the
+// interpretation of 'p'. The 'v' format is handled like 'g'.
 // Format also supports specification of the minimum precision
-// in digits, the output field width, as well as the format verbs
+// in digits, the output field width, as well as the format flags
 // '+' and ' ' for sign control, '0' for space or zero padding,
 // and '-' for left or right justification. See the fmt package
 // for details.

src/cmd/compile/internal/big/intconv.go

@@ -52,16 +52,16 @@ func writeMultiple(s fmt.State, text string, count int) {
 	}
 }
 
-// Format is a support routine for fmt.Formatter. It accepts
-// the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x'
-// (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
+// Format implements fmt.Formatter. It accepts the formats
+// 'b' (binary), 'o' (octal), 'd' (decimal), 'x' (lowercase
+// hexadecimal), and 'X' (uppercase hexadecimal).
 // Also supported are the full suite of package fmt's format
-// verbs for integral types, including '+', '-', and ' '
-// for sign control, '#' for leading zero in octal and for
-// hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X"
-// respectively, specification of minimum digits precision,
-// output field width, space or zero padding, and left or
-// right justification.
+// flags for integral types, including '+' and ' ' for sign
+// control, '#' for leading zero in octal and for hexadecimal,
+// a leading "0x" or "0X" for "%#x" and "%#X" respectively,
+// specification of minimum digits precision, output field
+// width, space or zero padding, and '-' for left or right
+// justification.
 //
 func (x *Int) Format(s fmt.State, ch rune) {
 	// determine base

test/fixedbugs/issue14553.go

@@ -20,7 +20,10 @@ func main() {
 		bits  uint32
 	}{
 		{0e+00, 0x00000000},
-		{1e-45, 0x00000000},
+		{1e-46, 0x00000000},
+		{0.5e-45, 0x00000000},
+		{0.8e-45, 0x00000001},
+		{1e-45, 0x00000001},
 		{2e-45, 0x00000001},
 		{3e-45, 0x00000002},
 		{4e-45, 0x00000003},

test/fixedbugs/issue14651.go

+// run
+
+// Copyright 2016 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This test checks if the compiler's internal constant
+// arithmetic correctly rounds up floating-point values
+// that become the smallest denormal value.
+//
+// See also related issue 14553 and test issue14553.go.
+
+package main
+
+import (
+	"fmt"
+	"math"
+)
+
+const (
+	p149 = 1.0 / (1 << 149) // 1p-149
+	p500 = 1.0 / (1 << 500) // 1p-500
+	p1074 = p500 * p500 / (1<<74) // 1p-1074
+)
+
+const (
+	m0000p149 = 0x0 / 16.0 * p149 // = 0.0000p-149
+	m1000p149 = 0x8 / 16.0 * p149 // = 0.1000p-149
+	m1001p149 = 0x9 / 16.0 * p149 // = 0.1001p-149
+	m1011p149 = 0xb / 16.0 * p149 // = 0.1011p-149
+	m1100p149 = 0xc / 16.0 * p149 // = 0.1100p-149
+
+	m0000p1074 = 0x0 / 16.0 * p1074 // = 0.0000p-1074
+	m1000p1074 = 0x8 / 16.0 * p1074 // = 0.1000p-1074
+	m1001p1074 = 0x9 / 16.0 * p1074 // = 0.1001p-1074
+	m1011p1074 = 0xb / 16.0 * p1074 // = 0.1011p-1074
+	m1100p1074 = 0xc / 16.0 * p1074 // = 0.1100p-1074
+)
+
+func main() {
+	test32(float32(m0000p149), f32(m0000p149))
+	test32(float32(m1000p149), f32(m1000p149))
+	test32(float32(m1001p149), f32(m1001p149))
+	test32(float32(m1011p149), f32(m1011p149))
+	test32(float32(m1100p149), f32(m1100p149))
+
+	test64(float64(m0000p1074), f64(m0000p1074))
+	test64(float64(m1000p1074), f64(m1000p1074))
+	test64(float64(m1001p1074), f64(m1001p1074))
+	test64(float64(m1011p1074), f64(m1011p1074))
+	test64(float64(m1100p1074), f64(m1100p1074))
+}
+
+func f32(x float64) float32 { return float32(x) }
+func f64(x float64) float64 { return float64(x) }
+
+func test32(a, b float32) {
+	abits := math.Float32bits(a)
+	bbits := math.Float32bits(b)
+	if abits != bbits {
+		panic(fmt.Sprintf("%08x != %08x\n", abits, bbits))
+	}
+}
+
+func test64(a, b float64) {
+	abits := math.Float64bits(a)
+	bbits := math.Float64bits(b)
+	if abits != bbits {
+		panic(fmt.Sprintf("%016x != %016x\n", abits, bbits))
+	}
+}