math/big: use correct precision in Float.Float32/64 for denormals

When a big.Float is converted to a denormal float32/64, the rounding
precision depends on the size of the denormal. Rounding may round up
and thus change the size (exponent) of the denormal. Recompute the
correct precision again for correct placement of the mantissa.

Fixes #14553.

Change-Id: Iedab5810a2d2a405cc5da28c6de7be34cb035b86
Reviewed-on: https://go-review.googlesource.com/20198Reviewed-by: Alan Donovan <adonovan@google.com>

src/math/big/float.go

@@ -874,15 +874,15 @@ 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 floatxx mantissa.
+		// 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
 
 		// If the exponent is too small, we may have a denormal number
-		// in which case we have fewer mantissa bits available: reduce
-		// precision accordingly.
+		// in which case we have fewer mantissa bits available: recompute
+		// precision.
 		if e < emin {
-			p -= emin - int(e)
+			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 {
@@ -931,7 +931,9 @@ func (x *Float) Float32() (float32, Accuracy) {
 				return 0.0, Below
 			}
 			// bexp = 0
-			mant = msb32(r.mant) >> (fbits - r.prec)
+			// recompute precision
+			p = mbits + 1 - emin + int(e)
+			mant = msb32(r.mant) >> uint(fbits-p)
 		} else {
 			// normal number: emin <= e <= emax
 			bexp = uint32(e+bias) << mbits
@@ -981,15 +983,15 @@ 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 floatxx mantissa.
+		// 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
 
 		// If the exponent is too small, we may have a denormal number
-		// in which case we have fewer mantissa bits available: reduce
-		// precision accordingly.
+		// in which case we have fewer mantissa bits available: recompute
+		// precision.
 		if e < emin {
-			p -= emin - int(e)
+			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 {
@@ -1038,7 +1040,9 @@ func (x *Float) Float64() (float64, Accuracy) {
 				return 0.0, Below
 			}
 			// bexp = 0
-			mant = msb64(r.mant) >> (fbits - r.prec)
+			// recompute precision
+			p = mbits + 1 - emin + int(e)
+			mant = msb64(r.mant) >> uint(fbits-p)
 		} else {
 			// normal number: emin <= e <= emax
 			bexp = uint64(e+bias) << mbits

src/math/big/float_test.go

@@ -843,6 +843,32 @@ 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},
+		{"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)
+		{"0x2.9p-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)
+
+		{"0x4.0p-149", math.Float32frombits(0x000000004), Exact},
+		{"0x4.8p-149", math.Float32frombits(0x000000004), Below}, // ToNearestEven rounds down (to even)
+		{"0x4.9p-149", math.Float32frombits(0x000000005), Above},
+
+		// specific case from issue 14553
+		{"0x7.7p-149", math.Float32frombits(0x000000007), Below},
+		{"0x7.8p-149", math.Float32frombits(0x000000008), Above},
+		{"0x7.9p-149", math.Float32frombits(0x000000008), Above},
+
 		// normals
 		{"0x.ffffffp-126", math.Float32frombits(0x00800000), Above}, // rounded up to smallest normal
 		{"1p-126", math.Float32frombits(0x00800000), Exact},         // smallest normal
@@ -915,6 +941,27 @@ func TestFloatFloat64(t *testing.T) {
 		{"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},
+
+		{"0x1.0p-1074", math.Float64frombits(0x00000000000000001), Exact},
+		{"0x1.7p-1074", math.Float64frombits(0x00000000000000001), Below},
+		{"0x1.8p-1074", math.Float64frombits(0x00000000000000002), Above},
+		{"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)
+		{"0x2.9p-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)
+
+		{"0x4.0p-1074", math.Float64frombits(0x00000000000000004), Exact},
+		{"0x4.8p-1074", math.Float64frombits(0x00000000000000004), Below}, // ToNearestEven rounds down (to even)
+		{"0x4.9p-1074", math.Float64frombits(0x00000000000000005), Above},
+
 		// normals
 		{"0x.fffffffffffff8p-1022", math.Float64frombits(0x0010000000000000), Above}, // rounded up to smallest normal
 		{"1p-1022", math.Float64frombits(0x0010000000000000), Exact},                 // smallest normal