func (p *exporter) float(x *Mpflt) {
// extract sign (there is no -0)
f := &x.Val
sign := f.Sign()
if sign == 0 {
// x == 0
p.int(0)
return
}
// x != 0
// extract exponent such that 0.5 <= m < 1.0
var m big.Float
exp := f.MantExp(&m)
// extract mantissa as *big.Int
// - set exponent large enough so mant satisfies mant.IsInt()
// - get *big.Int from mant
m.SetMantExp(&m, int(m.MinPrec()))
mant, acc := m.Int(nil)
if acc != big.Exact {
Fatalf("exporter: internal error")
}
p.int(sign)
p.int(exp)
p.string(string(mant.Bytes()))
}
func (bed BinaryVarintEncoderDecoder) Decode(r io.Reader, n *big.Float) error {
var isInteger int8
var f float64
var exponent int32
n.SetUint64(0)
if err := binary.Read(r, binary.BigEndian, &isInteger); err != nil {
return err
}
if isInteger <= 0 {
var x int64
var err error
if x, err = binary.ReadVarint(miniByteReader{r}); err != nil {
return err
}
n.SetInt64(x)
n.SetPrec(ENCODER_DECODER_PREC)
return nil
} else {
if err := binary.Read(r, binary.BigEndian, &f); err != nil {
return err
}
if err := binary.Read(r, binary.BigEndian, &exponent); err != nil {
return err
}
bed.tmp.SetFloat64(f)
bed.tmp.SetPrec(ENCODER_DECODER_PREC)
n.SetMantExp(bed.tmp, int(exponent))
return nil
}
}
// Sqrt returns a big.Float representation of the square root of
// z. Precision is the same as the one of the argument. The function
// panics if z is negative, returns ±0 when z = ±0, and +Inf when z =
// +Inf.
func Sqrt(z *big.Float) *big.Float {
// panic on negative z
if z.Sign() == -1 {
panic("Sqrt: argument is negative")
}
// √±0 = ±0
if z.Sign() == 0 {
return big.NewFloat(float64(z.Sign()))
}
// √+Inf = +Inf
if z.IsInf() {
return big.NewFloat(math.Inf(+1))
}
// Compute √(a·2**b) as
// √(a)·2**b/2 if b is even
// √(2a)·2**b/2 if b > 0 is odd
// √(0.5a)·2**b/2 if b < 0 is odd
//
// The difference in the odd exponent case is due to the fact that
// exp/2 is rounded in different directions when exp is negative.
mant := new(big.Float)
exp := z.MantExp(mant)
switch exp % 2 {
case 1:
mant.Mul(big.NewFloat(2), mant)
case -1:
mant.Mul(big.NewFloat(0.5), mant)
}
// Solving x² - z = 0 directly requires a Quo call, but it's
// faster for small precisions.
//
// Solving 1/x² - z = 0 avoids the Quo call and is much faster for
// high precisions.
//
// Use sqrtDirect for prec <= 128 and sqrtInverse for prec > 128.
var x *big.Float
if z.Prec() <= 128 {
x = sqrtDirect(mant)
} else {
x = sqrtInverse(mant)
}
// re-attach the exponent and return
return x.SetMantExp(x, exp/2)
}
func (bed BinaryEncoderDecoder) Decode(r io.Reader, n *big.Float) error {
var f float64
var exponent int32
n.SetUint64(0)
if err := binary.Read(r, binary.BigEndian, &f); err != nil {
return err
}
if err := binary.Read(r, binary.BigEndian, &exponent); err != nil {
return err
}
bed.tmp.SetFloat64(f)
bed.tmp.SetPrec(ENCODER_DECODER_PREC)
n.SetMantExp(bed.tmp, int(exponent))
return nil
}
func (p *exporter) float(x constant.Value) {
if x.Kind() != constant.Float {
log.Fatalf("gcimporter: unexpected constant %v, want float", x)
}
// extract sign (there is no -0)
sign := constant.Sign(x)
if sign == 0 {
// x == 0
p.int(0)
return
}
// x != 0
var f big.Float
if v, exact := constant.Float64Val(x); exact {
// float64
f.SetFloat64(v)
} else if num, denom := constant.Num(x), constant.Denom(x); num.Kind() == constant.Int {
// TODO(gri): add big.Rat accessor to constant.Value.
r := valueToRat(num)
f.SetRat(r.Quo(r, valueToRat(denom)))
} else {
// Value too large to represent as a fraction => inaccessible.
// TODO(gri): add big.Float accessor to constant.Value.
f.SetFloat64(math.MaxFloat64) // FIXME
}
// extract exponent such that 0.5 <= m < 1.0
var m big.Float
exp := f.MantExp(&m)
// extract mantissa as *big.Int
// - set exponent large enough so mant satisfies mant.IsInt()
// - get *big.Int from mant
m.SetMantExp(&m, int(m.MinPrec()))
mant, acc := m.Int(nil)
if acc != big.Exact {
log.Fatalf("gcimporter: internal error")
}
p.int(sign)
p.int(exp)
p.string(string(mant.Bytes()))
}
// Log returns a big.Float representation of the natural logarithm of
// z. Precision is the same as the one of the argument. The function
// panics if z is negative, returns -Inf when z = 0, and +Inf when z =
// +Inf
func Log(z *big.Float) *big.Float {
// panic on negative z
if z.Sign() == -1 {
panic("Log: argument is negative")
}
// Log(0) = -Inf
if z.Sign() == 0 {
return big.NewFloat(math.Inf(-1)).SetPrec(z.Prec())
}
prec := z.Prec() + 64 // guard digits
one := big.NewFloat(1).SetPrec(prec)
two := big.NewFloat(2).SetPrec(prec)
four := big.NewFloat(4).SetPrec(prec)
// Log(1) = 0
if z.Cmp(one) == 0 {
return big.NewFloat(0).SetPrec(z.Prec())
}
// Log(+Inf) = +Inf
if z.IsInf() {
return big.NewFloat(math.Inf(+1)).SetPrec(z.Prec())
}
x := new(big.Float).SetPrec(prec)
// if 0 < z < 1 we compute log(z) as -log(1/z)
var neg bool
if z.Cmp(one) < 0 {
x.Quo(one, z)
neg = true
} else {
x.Set(z)
}
// We scale up x until x >= 2**(prec/2), and then we'll be allowed
// to use the AGM formula for Log(x).
//
// Double x until the condition is met, and keep track of the
// number of doubling we did (needed to scale back later).
lim := new(big.Float)
lim.SetMantExp(two, int(prec/2))
k := 0
for x.Cmp(lim) < 0 {
x.Mul(x, x)
k++
}
// Compute the natural log of x using the fact that
// log(x) = π / (2 * AGM(1, 4/x))
// if
// x >= 2**(prec/2),
// where prec is the desired precision (in bits)
pi := pi(prec)
agm := agm(one, x.Quo(four, x)) // agm = AGM(1, 4/x)
x.Quo(pi, x.Mul(two, agm)) // reuse x, we don't need it
if neg {
x.Neg(x)
}
// scale the result back multiplying by 2**-k
// reuse lim to reduce allocations.
x.Mul(x, lim.SetMantExp(one, -k))
return x.SetPrec(z.Prec())
}
