Golang Float.Float64方法代码示例

// 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())
}