// 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)
}
}
// Sqrt returns the square root n.
func Sqrt(n *big.Float) *big.Float {
prec := n.Prec()
x := new(big.Float).SetPrec(prec).SetInt64(1)
z := new(big.Float).SetPrec(prec).SetInt64(1)
half := new(big.Float).SetPrec(prec).SetFloat64(0.5)
t := new(big.Float).SetPrec(prec)
for {
z.Copy(x)
t.Mul(x, x)
t.Sub(t, n)
t.Quo(t, x)
t.Mul(t, half)
x.Sub(x, t)
if x.Cmp(z) == 0 {
break
}
}
return x
}
// NearestInt set res to the nearest integer to x
func (e *Pslq) NearestInt(x *big.Float, res *big.Int) {
prec := x.Prec()
var tmp big.Float
tmp.SetPrec(prec)
if x.Sign() >= 0 {
tmp.Add(x, &e.half)
} else {
tmp.Sub(x, &e.half)
}
tmp.Int(res)
}
// 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)
}
// 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())
}
func withinMandLimit(z *BigComplex, limit *big.Float) bool {
// Approximate cmplx.Abs
negLimit := MakeBigFloat(0.0, limit.Prec())
negLimit.Neg(limit)
r := z.Real()
i := z.Imag()
rLimCmp := r.Cmp(limit)
rNegLimCmp := r.Cmp(&negLimit)
iLimCmp := i.Cmp(limit)
iNegLimCmp := i.Cmp(&negLimit)
within := rLimCmp == -1 && rNegLimCmp == 1
within = within && iLimCmp == -1 && iNegLimCmp == 1
return within
}
// fast path for z**w when w is an integer
func powInt(z *big.Float, w int) *big.Float {
// get mantissa and exponent of z
mant := new(big.Float)
exp := z.MantExp(mant)
// result's exponent
exp = exp * w
// result's mantissa
x := big.NewFloat(1).SetPrec(z.Prec())
// Classic right-to-left binary exponentiation
for w > 0 {
if w%2 == 1 {
x.Mul(x, mant)
}
w >>= 1
mant.Mul(mant, mant)
}
return new(big.Float).SetMantExp(x, exp)
}
// 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())
}
func ExampleFloat_Add() {
// Operating on numbers of different precision.
var x, y, z big.Float
x.SetInt64(1000) // x is automatically set to 64bit precision
y.SetFloat64(2.718281828) // y is automatically set to 53bit precision
z.SetPrec(32)
z.Add(&x, &y)
fmt.Printf("x = %s (%s, prec = %d, acc = %s)\n", &x, x.Format('p', 0), x.Prec(), x.Acc())
fmt.Printf("y = %s (%s, prec = %d, acc = %s)\n", &y, y.Format('p', 0), y.Prec(), y.Acc())
fmt.Printf("z = %s (%s, prec = %d, acc = %s)\n", &z, z.Format('p', 0), z.Prec(), z.Acc())
// Output:
// x = 1000 (0x.fap10, prec = 64, acc = Exact)
// y = 2.718281828 (0x.adf85458248cd8p2, prec = 53, acc = Exact)
// z = 1002.718282 (0x.faadf854p10, prec = 32, acc = Below)
}
// 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())
}
func runTest(encoderDecoder EncoderDecoder, n *big.Float) (nBytes uint64) {
y := newBigFloat(0)
buf := new(bytes.Buffer)
err := encoderDecoder.Encode(buf, n)
if err != nil {
panic(err)
}
nBytes += uint64(buf.Len())
buf = bytes.NewBuffer(buf.Bytes())
err = encoderDecoder.Decode(buf, y)
nBytes += uint64(buf.Len())
if n.Cmp(y) != 0 {
panic(fmt.Sprintf("write and read are not the same: %v, %v - %d - %d, %d", stringOfBigFloat(n), stringOfBigFloat(y), nBytes, n.Prec(), y.Prec()))
}
return
}
// Pow returns a big.Float representation of z**w. Precision is the same as the one
// of the first argument. The function panics when z is negative.
func Pow(z *big.Float, w *big.Float) *big.Float {
if z.Sign() < 0 {
panic("Pow: negative base")
}
// Pow(z, 0) = 1.0
if w.Sign() == 0 {
return big.NewFloat(1).SetPrec(z.Prec())
}
// Pow(z, 1) = z
// Pow(+Inf, n) = +Inf
if w.Cmp(big.NewFloat(1)) == 0 || z.IsInf() {
return new(big.Float).Copy(z)
}
// Pow(z, -w) = 1 / Pow(z, w)
if w.Sign() < 0 {
x := new(big.Float)
zExt := new(big.Float).Copy(z).SetPrec(z.Prec() + 64)
wNeg := new(big.Float).Neg(w)
return x.Quo(big.NewFloat(1), Pow(zExt, wNeg)).SetPrec(z.Prec())
}
// w integer fast path
if w.IsInt() {
wi, _ := w.Int64()
return powInt(z, int(wi))
}
// compute w**z as exp(z log(w))
x := new(big.Float).SetPrec(z.Prec() + 64)
logZ := Log(new(big.Float).Copy(z).SetPrec(z.Prec() + 64))
x.Mul(w, logZ)
x = Exp(x)
return x.SetPrec(z.Prec())
}
// 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
}
