math/big: fix rounding to smallest denormal for Float.Float32/64
Converting a big.Float value x to a float32/64 value did not correctly
round x up to the smallest denormal float32/64 if x was smaller than the
smallest denormal float32/64, but larger than 0.5 of a smallest denormal
float32/64.
Handle this case explicitly and simplify some code in the turn.
For #14651.
Change-Id: I025e24bf8f0e671581a7de0abf7c1cd7e6403a6c
Reviewed-on: https://go-review.googlesource.com/20816
Run-TryBot: Robert Griesemer <gri@golang.org>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-by: Alan Donovan <adonovan@google.com>
diff --git a/src/math/big/float.go b/src/math/big/float.go
index eca85d4..f19f21f 100644
--- a/src/math/big/float.go
+++ b/src/math/big/float.go
@@ -874,21 +874,43 @@
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 +920,8 @@
// 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 +939,10 @@
// 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 +994,43 @@
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 +1040,13 @@
// 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 +1059,10 @@
// 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 {
diff --git a/src/math/big/float_test.go b/src/math/big/float_test.go
index 6fb4402..464619b 100644
--- a/src/math/big/float_test.go
+++ b/src/math/big/float_test.go
@@ -829,7 +829,7 @@
}{
{"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 @@
{"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},
+ // 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
- {"0x1.0p-149", math.Float32frombits(0x000000001), Exact},
+ {"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 @@
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 @@
}{
{"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 @@
{"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 @@
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
