// Subtract
func (a Scalar) Sub_S(b S) S {
var x, y big.Float
x = big.Float(a)
y = big.Float(b.(Scalar))
z := x.Sub(&x, &y)
return (Scalar)(*z)
}
func TestSqrt(t *testing.T) {
tests := []struct {
prec uint
in float64
}{
{16, 0},
{16, 1},
{16, 4},
{16, 10000},
{16, 2},
{64, 2},
{256, 2},
{1024, 1.5},
}
for _, test := range tests {
x := new(big.Float).SetPrec(test.prec)
x.SetFloat64(test.in)
var got, got2, diff big.Float
pslq := New(test.prec)
pslq.Sqrt(x, &got)
got2.SetPrec(test.prec).Mul(&got, &got)
diff.Sub(&got2, x)
if diff.MinPrec() > 1 {
t.Errorf("sqrt(%f) prec %d wrong got %.20f square %.20f expecting %f diff %g minprec %d", test.in, test.prec, &got, &got2, x, &diff, diff.MinPrec())
}
}
}
// 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
}
// sincos iterates a sin or cos Taylor series.
func sincos(name string, index int, x, z, exponent, factorial *big.Float) *big.Float {
plus := false
term := newF().Set(floatOne)
for j := 0; j < index; j++ {
term.Mul(term, x)
}
xN := newF().Set(term)
x2 := newF().Mul(x, x)
loop := newLoop(name, x, 4)
for {
// Invariant: factorial holds exponent!.
term.Quo(term, factorial)
if plus {
z.Add(z, term)
} else {
z.Sub(z, term)
}
plus = !plus
if loop.terminate(z) {
break
}
// Advance x**index (multiply by x²).
term.Mul(xN, x2)
xN.Set(term)
// Advance exponent and factorial.
exponent.Add(exponent, floatOne)
factorial.Mul(factorial, exponent)
exponent.Add(exponent, floatOne)
factorial.Mul(factorial, exponent)
}
return z
}
// twoPiReduce guarantees x < 2𝛑; x is known to be >= 0 coming in.
func twoPiReduce(x *big.Float) {
// TODO: Is there an easy better algorithm?
twoPi := newF().Set(floatTwo)
twoPi.Mul(twoPi, floatPi)
// Do something clever(er) if it's large.
if x.Cmp(newF().SetInt64(1000)) > 0 {
multiples := make([]*big.Float, 0, 100)
sixteen := newF().SetInt64(16)
multiple := newF().Set(twoPi)
for {
multiple.Mul(multiple, sixteen)
if x.Cmp(multiple) < 0 {
break
}
multiples = append(multiples, newF().Set(multiple))
}
// From the right, subtract big multiples.
for i := len(multiples) - 1; i >= 0; i-- {
multiple := multiples[i]
for x.Cmp(multiple) >= 0 {
x.Sub(x, multiple)
}
}
}
for x.Cmp(twoPi) >= 0 {
x.Sub(x, twoPi)
}
}
func mandelbrotFloat(a, b *big.Float) color.Color {
var x, y, nx, ny, x2, y2, f2, f4, r2, tmp big.Float
f2.SetInt64(2)
f4.SetInt64(4)
x.SetInt64(0)
y.SetInt64(0)
defer func() { recover() }()
for n := uint8(0); n < iterations; n++ {
// Not update x2 and y2
// because they are already updated in the previous loop
nx.Sub(&x2, &y2)
nx.Add(&nx, a)
tmp.Mul(&x, &y)
ny.Mul(&f2, &tmp)
ny.Add(&ny, b)
x.Set(&nx)
y.Set(&ny)
x2.Mul(&x, &x)
y2.Mul(&y, &y)
r2.Add(&x2, &y2)
if r2.Cmp(&f4) > 0 {
return color.Gray{255 - contrast*n}
}
}
return color.Black
}
// Returns pi using Machin's formula
func pi(prec uint, result *big.Float) {
var tmp, _4 big.Float
_4.SetPrec(prec).SetInt64(4)
acot(prec, 5, &tmp)
tmp.SetPrec(prec).Mul(&tmp, &_4)
acot(prec, 239, result)
result.Sub(&tmp, result)
result.SetPrec(prec).Mul(result, &_4)
}
// 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)
}
// 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())
}
// 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
}
// This example shows how to use big.Float to compute the square root of 2 with
// a precision of 200 bits, and how to print the result as a decimal number.
func Example_sqrt2() {
// We'll do computations with 200 bits of precision in the mantissa.
const prec = 200
// Compute the square root of 2 using Newton's Method. We start with
// an initial estimate for sqrt(2), and then iterate:
// x_{n+1} = 1/2 * ( x_n + (2.0 / x_n) )
// Since Newton's Method doubles the number of correct digits at each
// iteration, we need at least log_2(prec) steps.
steps := int(math.Log2(prec))
// Initialize values we need for the computation.
two := new(big.Float).SetPrec(prec).SetInt64(2)
half := new(big.Float).SetPrec(prec).SetFloat64(0.5)
// Use 1 as the initial estimate.
x := new(big.Float).SetPrec(prec).SetInt64(1)
// 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.
t := new(big.Float)
// Iterate.
for i := 0; i <= steps; i++ {
t.Quo(two, x) // t = 2.0 / x_n
t.Add(x, t) // t = x_n + (2.0 / x_n)
x.Mul(half, t) // x_{n+1} = 0.5 * t
}
// We can use the usual fmt.Printf verbs since big.Float implements fmt.Formatter
fmt.Printf("sqrt(2) = %.50f\n", x)
// Print the error between 2 and x*x.
t.Mul(x, x) // t = x*x
fmt.Printf("error = %e\n", t.Sub(two, t))
// Output:
// sqrt(2) = 1.41421356237309504880168872420969807856967187537695
// error = 0.000000e+00
}
// Estimates the Variance based on the data set
// If the data set is relatively small (< 1000 examples), then remove 1 from the total
func (ad *AnomalyDetection) estimateVariance() *big.Float {
// this means that the mean was never calculated before, therefore do it now
// the means is needed for the cimputation of the deviation
if ad.mean.Cmp(zero) == 0 {
ad.estimateMean()
}
// initialize the total to zero
totalVariance := big.NewFloat(0)
totalDeviation := big.NewFloat(0)
var deviation big.Float
var deviationCopy big.Float
var singleVariance big.Float
// Loop while a is smaller than 1e100.
for _, element := range ad.dataSet {
// first calculate the deviation for each element, by subtracting the mean, take the absolute value
deviation.Sub(&element, &ad.mean).Abs(&deviation)
// add it to the total
totalDeviation.Add(totalDeviation, &deviation)
// calculate the variance by squaring it
singleVariance = *deviationCopy.Mul(&deviation, &deviation) // ^2
// the calculate the variance
totalVariance.Add(totalVariance, &singleVariance)
}
// calculate the variance
// assign the variance to the anomaly detection object
ad.variance = *totalVariance.Quo(totalVariance, &ad.totalSamples)
// calculate the deviation
ad.deviation = *totalDeviation.Quo(totalDeviation, &ad.totalSamples)
return &ad.variance
}
// 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())
}
