import (
"fmt"
+ "math"
)
+// The largest exact float32 is f₁ = (1+(1-2²³))×2¹²⁷ = (1-2²⁴)×2¹²⁸ = 2¹²⁸ - 2¹⁰⁴.
+// The next float32 would be f₂ = (1+1)×2¹²⁷ = 1×2¹²⁸, except that exponent is out of range.
+// Float32 conversion rounds to the nearest float32, rounding to even mantissa:
+// between f₁ and f₂, values closer to f₁ round to f₁and values closer to f₂ are rejected as out of range.
+// f₁ is an odd mantissa, so the halfway point (f₁+f₂)/2 rounds to f₂ and is rejected.
+// The halfway point (f₁+f₂)/2 = 2¹²⁸ - 2¹⁰⁵.
+//
+// The same is true of float64, with different constants: s/24/53/ and s/128/1024/.
+
const (
- m32bits = 23 // number of float32 mantissa bits
- e32max = 127 // max. float32 exponent
- maxExp32 = e32max - m32bits
- maxMant32 = 1<<(m32bits+1) - 1
-
- maxFloat32_0 = (maxMant32 - 0) << maxExp32
- maxFloat32_1 = (maxMant32 - 1) << maxExp32
- maxFloat32_2 = (maxMant32 - 2) << maxExp32
+ two24 = 1.0 * (1 << 24)
+ two53 = 1.0 * (1 << 53)
+ two64 = 1.0 * (1 << 64)
+ two128 = two64 * two64
+ two256 = two128 * two128
+ two512 = two256 * two256
+ two768 = two512 * two256
+ two1024 = two512 * two512
+
+ ulp32 = two128 / two24
+ max32 = two128 - ulp32
+
+ ulp64 = two1024 / two53
+ max64 = two1024 - ulp64
)
-func init() {
- if maxExp32 != 104 {
- panic("incorrect maxExp32")
- }
- if maxMant32 != 16777215 {
- panic("incorrect maxMant32")
- }
- if maxFloat32_0 != 340282346638528859811704183484516925440 {
- panic("incorrect maxFloat32_0")
- }
+var cvt = []struct {
+ bits uint64 // keep us honest
+ exact interface{}
+ approx interface{}
+ text string
+}{
+ // 0
+ {0x7f7ffffe, float32(max32 - ulp32), float32(max32 - ulp32 - ulp32/2), "max32 - ulp32 - ulp32/2"},
+ {0x7f7ffffe, float32(max32 - ulp32), float32(max32 - ulp32), "max32 - ulp32"},
+ {0x7f7ffffe, float32(max32 - ulp32), float32(max32 - ulp32/2), "max32 - ulp32/2"},
+ {0x7f7ffffe, float32(max32 - ulp32), float32(max32 - ulp32 + ulp32/2), "max32 - ulp32 + ulp32/2"},
+ {0x7f7fffff, float32(max32), float32(max32 - ulp32 + ulp32/2 + ulp32/two64), "max32 - ulp32 + ulp32/2 + ulp32/two64"},
+ {0x7f7fffff, float32(max32), float32(max32 - ulp32/2 + ulp32/two64), "max32 - ulp32/2 + ulp32/two64"},
+ {0x7f7fffff, float32(max32), float32(max32), "max32"},
+ {0x7f7fffff, float32(max32), float32(max32 + ulp32/2 - ulp32/two64), "max32 + ulp32/2 - ulp32/two64"},
+
+ {0xff7ffffe, float32(-(max32 - ulp32)), float32(-(max32 - ulp32 - ulp32/2)), "-(max32 - ulp32 - ulp32/2)"},
+ {0xff7ffffe, float32(-(max32 - ulp32)), float32(-(max32 - ulp32)), "-(max32 - ulp32)"},
+ {0xff7ffffe, float32(-(max32 - ulp32)), float32(-(max32 - ulp32/2)), "-(max32 - ulp32/2)"},
+ {0xff7ffffe, float32(-(max32 - ulp32)), float32(-(max32 - ulp32 + ulp32/2)), "-(max32 - ulp32 + ulp32/2)"},
+ {0xff7fffff, float32(-(max32)), float32(-(max32 - ulp32 + ulp32/2 + ulp32/two64)), "-(max32 - ulp32 + ulp32/2 + ulp32/two64)"},
+ {0xff7fffff, float32(-(max32)), float32(-(max32 - ulp32/2 + ulp32/two64)), "-(max32 - ulp32/2 + ulp32/two64)"},
+ {0xff7fffff, float32(-(max32)), float32(-(max32)), "-(max32)"},
+ {0xff7fffff, float32(-(max32)), float32(-(max32 + ulp32/2 - ulp32/two64)), "-(max32 + ulp32/2 - ulp32/two64)"},
+
+ // These are required to work: according to the Go spec, the internal float mantissa must be at least 256 bits,
+ // and these expressions can be represented exactly with a 256-bit mantissa.
+ {0x7f7fffff, float32(max32), float32(max32 - ulp32 + ulp32/2 + 1), "max32 - ulp32 + ulp32/2 + 1"},
+ {0x7f7fffff, float32(max32), float32(max32 - ulp32/2 + 1), "max32 - ulp32/2 + 1"},
+ {0x7f7fffff, float32(max32), float32(max32 + ulp32/2 - 1), "max32 + ulp32/2 - 1"},
+ {0xff7fffff, float32(-(max32)), float32(-(max32 - ulp32 + ulp32/2 + 1)), "-(max32 - ulp32 + ulp32/2 + 1)"},
+ {0xff7fffff, float32(-(max32)), float32(-(max32 - ulp32/2 + 1)), "-(max32 - ulp32/2 + 1)"},
+ {0xff7fffff, float32(-(max32)), float32(-(max32 + ulp32/2 - 1)), "-(max32 + ulp32/2 - 1)"},
+
+ {0x7f7fffff, float32(max32), float32(max32 - ulp32 + ulp32/2 + 1/two128), "max32 - ulp32 + ulp32/2 + 1/two128"},
+ {0x7f7fffff, float32(max32), float32(max32 - ulp32/2 + 1/two128), "max32 - ulp32/2 + 1/two128"},
+ {0x7f7fffff, float32(max32), float32(max32 + ulp32/2 - 1/two128), "max32 + ulp32/2 - 1/two128"},
+ {0xff7fffff, float32(-(max32)), float32(-(max32 - ulp32 + ulp32/2 + 1/two128)), "-(max32 - ulp32 + ulp32/2 + 1/two128)"},
+ {0xff7fffff, float32(-(max32)), float32(-(max32 - ulp32/2 + 1/two128)), "-(max32 - ulp32/2 + 1/two128)"},
+ {0xff7fffff, float32(-(max32)), float32(-(max32 + ulp32/2 - 1/two128)), "-(max32 + ulp32/2 - 1/two128)"},
+
+ {0x7feffffffffffffe, float64(max64 - ulp64), float64(max64 - ulp64 - ulp64/2), "max64 - ulp64 - ulp64/2"},
+ {0x7feffffffffffffe, float64(max64 - ulp64), float64(max64 - ulp64), "max64 - ulp64"},
+ {0x7feffffffffffffe, float64(max64 - ulp64), float64(max64 - ulp64/2), "max64 - ulp64/2"},
+ {0x7feffffffffffffe, float64(max64 - ulp64), float64(max64 - ulp64 + ulp64/2), "max64 - ulp64 + ulp64/2"},
+ {0x7fefffffffffffff, float64(max64), float64(max64 - ulp64 + ulp64/2 + ulp64/two64), "max64 - ulp64 + ulp64/2 + ulp64/two64"},
+ {0x7fefffffffffffff, float64(max64), float64(max64 - ulp64/2 + ulp64/two64), "max64 - ulp64/2 + ulp64/two64"},
+ {0x7fefffffffffffff, float64(max64), float64(max64), "max64"},
+ {0x7fefffffffffffff, float64(max64), float64(max64 + ulp64/2 - ulp64/two64), "max64 + ulp64/2 - ulp64/two64"},
+
+ {0xffeffffffffffffe, float64(-(max64 - ulp64)), float64(-(max64 - ulp64 - ulp64/2)), "-(max64 - ulp64 - ulp64/2)"},
+ {0xffeffffffffffffe, float64(-(max64 - ulp64)), float64(-(max64 - ulp64)), "-(max64 - ulp64)"},
+ {0xffeffffffffffffe, float64(-(max64 - ulp64)), float64(-(max64 - ulp64/2)), "-(max64 - ulp64/2)"},
+ {0xffeffffffffffffe, float64(-(max64 - ulp64)), float64(-(max64 - ulp64 + ulp64/2)), "-(max64 - ulp64 + ulp64/2)"},
+ {0xffefffffffffffff, float64(-(max64)), float64(-(max64 - ulp64 + ulp64/2 + ulp64/two64)), "-(max64 - ulp64 + ulp64/2 + ulp64/two64)"},
+ {0xffefffffffffffff, float64(-(max64)), float64(-(max64 - ulp64/2 + ulp64/two64)), "-(max64 - ulp64/2 + ulp64/two64)"},
+ {0xffefffffffffffff, float64(-(max64)), float64(-(max64)), "-(max64)"},
+ {0xffefffffffffffff, float64(-(max64)), float64(-(max64 + ulp64/2 - ulp64/two64)), "-(max64 + ulp64/2 - ulp64/two64)"},
+
+ // These are required to work.
+ // The mantissas are exactly 256 bits.
+ // max64 is just below 2¹⁰²⁴ so the bottom bit we can use is 2⁷⁶⁸.
+ {0x7fefffffffffffff, float64(max64), float64(max64 - ulp64 + ulp64/2 + two768), "max64 - ulp64 + ulp64/2 + two768"},
+ {0x7fefffffffffffff, float64(max64), float64(max64 - ulp64/2 + two768), "max64 - ulp64/2 + two768"},
+ {0x7fefffffffffffff, float64(max64), float64(max64 + ulp64/2 - two768), "max64 + ulp64/2 - two768"},
+ {0xffefffffffffffff, float64(-(max64)), float64(-(max64 - ulp64 + ulp64/2 + two768)), "-(max64 - ulp64 + ulp64/2 + two768)"},
+ {0xffefffffffffffff, float64(-(max64)), float64(-(max64 - ulp64/2 + two768)), "-(max64 - ulp64/2 + two768)"},
+ {0xffefffffffffffff, float64(-(max64)), float64(-(max64 + ulp64/2 - two768)), "-(max64 + ulp64/2 - two768)"},
}
-const (
- m64bits = 52 // number of float64 mantissa bits
- e64max = 1023 // max. float64 exponent
- maxExp64 = e64max - m64bits
- maxMant64 = 1<<(m64bits+1) - 1
-
- // These expressions are not permitted due to implementation restrictions.
- // maxFloat64_0 = (maxMant64-0) << maxExp64
- // maxFloat64_1 = (maxMant64-1) << maxExp64
- // maxFloat64_2 = (maxMant64-2) << maxExp64
-
- // These equivalent values were computed using math/big.
- maxFloat64_0 = 1.7976931348623157e308
- maxFloat64_1 = 1.7976931348623155e308
- maxFloat64_2 = 1.7976931348623153e308
-)
+var bugged = false
-func init() {
- if maxExp64 != 971 {
- panic("incorrect maxExp64")
- }
- if maxMant64 != 9007199254740991 {
- panic("incorrect maxMant64")
+func bug() {
+ if !bugged {
+ bugged = true
+ fmt.Println("BUG")
}
}
-var cvt = []struct {
- val interface{}
- binary string
-}{
-
- {float32(maxFloat32_0), fmt.Sprintf("%dp+%d", int32(maxMant32-0), maxExp32)},
- {float32(maxFloat32_1), fmt.Sprintf("%dp+%d", int32(maxMant32-1), maxExp32)},
- {float32(maxFloat32_2), fmt.Sprintf("%dp+%d", int32(maxMant32-2), maxExp32)},
+func main() {
+ u64 := math.Float64frombits(0x7fefffffffffffff) - math.Float64frombits(0x7feffffffffffffe)
+ if ulp64 != u64 {
+ bug()
+ fmt.Printf("ulp64=%g, want %g", ulp64, u64)
+ }
- {float64(maxFloat64_0), fmt.Sprintf("%dp+%d", int64(maxMant64-0), maxExp64)},
- {float64(maxFloat64_1), fmt.Sprintf("%dp+%d", int64(maxMant64-1), maxExp64)},
- {float64(maxFloat64_2), fmt.Sprintf("%dp+%d", int64(maxMant64-2), maxExp64)},
+ u32 := math.Float32frombits(0x7f7fffff) - math.Float32frombits(0x7f7ffffe)
+ if ulp32 != u32 {
+ bug()
+ fmt.Printf("ulp32=%g, want %g", ulp32, u32)
+ }
- {float32(-maxFloat32_0), fmt.Sprintf("-%dp+%d", int32(maxMant32-0), maxExp32)},
- {float32(-maxFloat32_1), fmt.Sprintf("-%dp+%d", int32(maxMant32-1), maxExp32)},
- {float32(-maxFloat32_2), fmt.Sprintf("-%dp+%d", int32(maxMant32-2), maxExp32)},
+ for _, c := range cvt {
+ if bits(c.exact) != c.bits {
+ bug()
+ fmt.Printf("%s: inconsistent table: bits=%#x (%g) but exact=%g (%#x)\n", c.text, c.bits, fromBits(c.bits, c.exact), c.exact, bits(c.exact))
+ }
+ if c.approx != c.exact || bits(c.approx) != c.bits {
+ bug()
+ fmt.Printf("%s: have %g (%#x) want %g (%#x)\n", c.text, c.approx, bits(c.approx), c.exact, c.bits)
+ }
+ }
+}
- {float64(-maxFloat64_0), fmt.Sprintf("-%dp+%d", int64(maxMant64-0), maxExp64)},
- {float64(-maxFloat64_1), fmt.Sprintf("-%dp+%d", int64(maxMant64-1), maxExp64)},
- {float64(-maxFloat64_2), fmt.Sprintf("-%dp+%d", int64(maxMant64-2), maxExp64)},
+func bits(x interface{}) interface{} {
+ switch x := x.(type) {
+ case float32:
+ return uint64(math.Float32bits(x))
+ case float64:
+ return math.Float64bits(x)
+ }
+ return 0
}
-func main() {
- bug := false
- for i, c := range cvt {
- s := fmt.Sprintf("%b", c.val)
- if s != c.binary {
- if !bug {
- bug = true
- fmt.Println("BUG")
- }
- fmt.Printf("#%d: have %s, want %s\n", i, s, c.binary)
- }
+func fromBits(b uint64, x interface{}) interface{} {
+ switch x.(type) {
+ case float32:
+ return math.Float32frombits(uint32(b))
+ case float64:
+ return math.Float64frombits(b)
}
+ return "?"
}
package main
+// See float_lit2.go for motivation for these values.
+const (
+ two24 = 1.0 * (1 << 24)
+ two53 = 1.0 * (1 << 53)
+ two64 = 1.0 * (1 << 64)
+ two128 = two64 * two64
+ two256 = two128 * two128
+ two512 = two256 * two256
+ two768 = two512 * two256
+ two1024 = two512 * two512
+
+ ulp32 = two128 / two24
+ max32 = two128 - ulp32
+
+ ulp64 = two1024 / two53
+ max64 = two1024 - ulp64
+)
+
var x = []interface{}{
- float32(-340282356779733661637539395458142568448), // ERROR "constant -3\.40282e\+38 overflows float32"
- float32(-340282356779733661637539395458142568447),
- float32(-340282326356119256160033759537265639424),
- float32(340282326356119256160033759537265639424),
- float32(340282356779733661637539395458142568447),
- float32(340282356779733661637539395458142568448), // ERROR "constant 3\.40282e\+38 overflows float32"
- -1e1000, // ERROR "constant -1\.00000e\+1000 overflows float64"
- float64(-1.797693134862315907937289714053e+308), // ERROR "constant -1\.79769e\+308 overflows float64"
- float64(-1.797693134862315807937289714053e+308),
- float64(-1.797693134862315708145274237317e+308),
- float64(-1.797693134862315608353258760581e+308),
- float64(1.797693134862315608353258760581e+308),
- float64(1.797693134862315708145274237317e+308),
- float64(1.797693134862315807937289714053e+308),
- float64(1.797693134862315907937289714053e+308), // ERROR "constant 1\.79769e\+308 overflows float64"
- 1e1000, // ERROR "constant 1\.00000e\+1000 overflows float64"
+ float32(max32 + ulp32/2 - 1), // ok
+ float32(max32 + ulp32/2 - two128/two256), // ok
+ float32(max32 + ulp32/2), // ERROR "constant 3\.40282e\+38 overflows float32"
+
+ float32(-max32 - ulp32/2 + 1), // ok
+ float32(-max32 - ulp32/2 + two128/two256), // ok
+ float32(-max32 - ulp32/2), // ERROR "constant -3\.40282e\+38 overflows float32"
+
+ // If the compiler's internal floating point representation
+ // is shorter than 1024 bits, it cannot distinguish max64+ulp64/2-1 and max64+ulp64/2.
+ // gc uses fewer than 1024 bits, so allow it to print the overflow error for the -1 case.
+ float64(max64 + ulp64/2 - two1024/two256), // ok
+ float64(max64 + ulp64/2 - 1), // GC_ERROR "constant 1\.79769e\+308 overflows float64"
+ float64(max64 + ulp64/2), // ERROR "constant 1\.79769e\+308 overflows float64"
+
+ float64(-max64 - ulp64/2 + two1024/two256), // ok
+ float64(-max64 - ulp64/2 + 1), // GC_ERROR "constant -1\.79769e\+308 overflows float64"
+ float64(-max64 - ulp64/2), // ERROR "constant -1\.79769e\+308 overflows float64"
}