// Compute the square root of n using Newton's Method. We start with
// an initial estimate for sqrt(n), and then iterate
// x_{i+1} = 1/2 * ( x_i + (n / x_i) )
// Result is returned in x
func (e *Pslq) Sqrt(n, x *big.Float) {
if n == x {
panic("need distinct input and output")
}
if n.Sign() == 0 {
x.Set(n)
return
} else if n.Sign() < 0 {
panic("Sqrt of negative number")
}
prec := n.Prec()
// Use the floating point square root as initial estimate
nFloat64, _ := n.Float64()
x.SetPrec(prec).SetFloat64(math.Sqrt(nFloat64))
// We use t as a temporary variable. There's no need to set its precision
// since big.Float values with unset (== 0) precision automatically assume
// the largest precision of the arguments when used as the result (receiver)
// of a big.Float operation.
var t big.Float
// Iterate.
for {
t.Quo(n, x) // t = n / x_i
t.Add(x, &t) // t = x_i + (n / x_i)
t.Mul(&e.half, &t) // x_{i+1} = 0.5 * t
if x.Cmp(&t) == 0 {
// Exit loop if no change to result
break
}
x.Set(&t)
}
}
func fconv(fvp *Mpflt, flag FmtFlag) string {
if flag&FmtSharp == 0 {
return fvp.Val.Text('b', 0)
}
// use decimal format for error messages
// determine sign
f := &fvp.Val
var sign string
if f.Sign() < 0 {
sign = "-"
f = new(big.Float).Abs(f)
} else if flag&FmtSign != 0 {
sign = "+"
}
// Don't try to convert infinities (will not terminate).
if f.IsInf() {
return sign + "Inf"
}
// Use exact fmt formatting if in float64 range (common case):
// proceed if f doesn't underflow to 0 or overflow to inf.
if x, _ := f.Float64(); f.Sign() == 0 == (x == 0) && !math.IsInf(x, 0) {
return fmt.Sprintf("%s%.6g", sign, x)
}
// Out of float64 range. Do approximate manual to decimal
// conversion to avoid precise but possibly slow Float
// formatting.
// f = mant * 2**exp
var mant big.Float
exp := f.MantExp(&mant) // 0.5 <= mant < 1.0
// approximate float64 mantissa m and decimal exponent d
// f ~ m * 10**d
m, _ := mant.Float64() // 0.5 <= m < 1.0
d := float64(exp) * (math.Ln2 / math.Ln10) // log_10(2)
// adjust m for truncated (integer) decimal exponent e
e := int64(d)
m *= math.Pow(10, d-float64(e))
// ensure 1 <= m < 10
switch {
case m < 1-0.5e-6:
// The %.6g format below rounds m to 5 digits after the
// decimal point. Make sure that m*10 < 10 even after
// rounding up: m*10 + 0.5e-5 < 10 => m < 1 - 0.5e6.
m *= 10
e--
case m >= 10:
m /= 10
e++
}
return fmt.Sprintf("%s%.6ge%+d", sign, m, e)
}
// compute √z using newton to solve
// t² - z = 0 for t
func sqrtDirect(z *big.Float) *big.Float {
// f(t)/f'(t) = 0.5(t² - z)/t
half := big.NewFloat(0.5)
f := func(t *big.Float) *big.Float {
x := new(big.Float).Mul(t, t) // x = t²
x.Sub(x, z) // x = t² - z
x.Mul(half, x) // x = 0.5(t² - z)
return x.Quo(x, t) // return x = 0.5(t² - z)/t
}
// initial guess
zf, _ := z.Float64()
guess := big.NewFloat(math.Sqrt(zf))
return newton(f, guess, z.Prec())
}
// String returns returns a decimal approximation of the Float value.
func (x floatVal) String() string {
f := x.val
// Don't try to convert infinities (will not terminate).
if f.IsInf() {
return f.String()
}
// Use exact fmt formatting if in float64 range (common case):
// proceed if f doesn't underflow to 0 or overflow to inf.
if x, _ := f.Float64(); f.Sign() == 0 == (x == 0) && !math.IsInf(x, 0) {
return fmt.Sprintf("%.6g", x)
}
// Out of float64 range. Do approximate manual to decimal
// conversion to avoid precise but possibly slow Float
// formatting.
// f = mant * 2**exp
var mant big.Float
exp := f.MantExp(&mant) // 0.5 <= |mant| < 1.0
// approximate float64 mantissa m and decimal exponent d
// f ~ m * 10**d
m, _ := mant.Float64() // 0.5 <= |m| < 1.0
d := float64(exp) * (math.Ln2 / math.Ln10) // log_10(2)
// adjust m for truncated (integer) decimal exponent e
e := int64(d)
m *= math.Pow(10, d-float64(e))
// ensure 1 <= |m| < 10
switch am := math.Abs(m); {
case am < 1-0.5e-6:
// The %.6g format below rounds m to 5 digits after the
// decimal point. Make sure that m*10 < 10 even after
// rounding up: m*10 + 0.5e-5 < 10 => m < 1 - 0.5e6.
m *= 10
e--
case am >= 10:
m /= 10
e++
}
return fmt.Sprintf("%.6ge%+d", m, e)
}
// Exp returns a big.Float representation of exp(z). Precision is
// the same as the one of the argument. The function returns +Inf
// when z = +Inf, and 0 when z = -Inf.
func Exp(z *big.Float) *big.Float {
// exp(0) == 1
if z.Sign() == 0 {
return big.NewFloat(1).SetPrec(z.Prec())
}
// Exp(+Inf) = +Inf
if z.IsInf() && z.Sign() > 0 {
return big.NewFloat(math.Inf(+1)).SetPrec(z.Prec())
}
// Exp(-Inf) = 0
if z.IsInf() && z.Sign() < 0 {
return big.NewFloat(0).SetPrec(z.Prec())
}
guess := new(big.Float)
// try to get initial estimate using IEEE-754 math
zf, _ := z.Float64()
if zfs := math.Exp(zf); zfs == math.Inf(+1) || zfs == 0 {
// too big or too small for IEEE-754 math,
// perform argument reduction using
// e^{2z} = (e^z)²
halfZ := new(big.Float).Mul(z, big.NewFloat(0.5))
halfExp := Exp(halfZ.SetPrec(z.Prec() + 64))
return new(big.Float).Mul(halfExp, halfExp).SetPrec(z.Prec())
} else {
// we got a nice IEEE-754 estimate
guess.SetFloat64(zfs)
}
// f(t)/f'(t) = t*(log(t) - z)
f := func(t *big.Float) *big.Float {
x := new(big.Float)
x.Sub(Log(t), z)
return x.Mul(x, t)
}
x := newton(f, guess, z.Prec())
return x
}
// compute √z using newton to solve
// 1/t² - z = 0 for x and then inverting.
func sqrtInverse(z *big.Float) *big.Float {
// f(t)/f'(t) = -0.5t(1 - zt²)
nhalf := big.NewFloat(-0.5)
one := big.NewFloat(1)
f := func(t *big.Float) *big.Float {
u := new(big.Float)
u.Mul(t, t) // u = t²
u.Mul(u, z) // u = zt²
u.Sub(one, u) // u = 1 - zt²
u.Mul(u, nhalf) // u = -0.5(1 - zt²)
return new(big.Float).Mul(t, u) // x = -0.5t(1 - zt²)
}
// initial guess
zf, _ := z.Float64()
guess := big.NewFloat(1 / math.Sqrt(zf))
// There's another operation after newton,
// so we need to force it to return at least
// a few guard digits. Use 32.
x := newton(f, guess, z.Prec()+32)
return x.Mul(z, x).SetPrec(z.Prec())
}
