Jmat.Real.lambertw, estimator

Average Error: 0.3 → 0.0

Time: 3.8s

Precision: binary64

Math

\[\log x - \log \log x \]

↓

\[\log \left(\sqrt{\frac{\log x}{x}}\right) \cdot -2 \]

FPCore

(FPCore (x) :precision binary64 (- (log x) (log (log x))))

↓

(FPCore (x) :precision binary64 (* (log (sqrt (/ (log x) x))) -2.0))

C

double code(double x) {
	return log(x) - log(log(x));
}

↓

double code(double x) {
	return log(sqrt((log(x) / x))) * -2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(x) - log(log(x))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = log(sqrt((log(x) / x))) * (-2.0d0)
end function

Java

public static double code(double x) {
	return Math.log(x) - Math.log(Math.log(x));
}

↓

public static double code(double x) {
	return Math.log(Math.sqrt((Math.log(x) / x))) * -2.0;
}

Python

def code(x):
	return math.log(x) - math.log(math.log(x))

↓

def code(x):
	return math.log(math.sqrt((math.log(x) / x))) * -2.0

Julia

function code(x)
	return Float64(log(x) - log(log(x)))
end

↓

function code(x)
	return Float64(log(sqrt(Float64(log(x) / x))) * -2.0)
end

MATLAB

function tmp = code(x)
	tmp = log(x) - log(log(x));
end

↓

function tmp = code(x)
	tmp = log(sqrt((log(x) / x))) * -2.0;
end

Wolfram

code[x_] := N[(N[Log[x], $MachinePrecision] - N[Log[N[Log[x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[Log[N[Sqrt[N[(N[Log[x], $MachinePrecision] / x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision]

Error

Bits error versus x

Derivation

1. Initial program 0.3

   \[\log x - \log \log x \]

2. Applied egg-rr 0.0

   \[\leadsto \color{blue}{-\log \left(\frac{\log x}{x}\right)} \]

3. Applied egg-rr 0.0

   \[\leadsto -\color{blue}{2 \cdot \log \left(\sqrt{\frac{\log x}{x}}\right)} \]

4. Final simplification 0.0

   \[\leadsto \log \left(\sqrt{\frac{\log x}{x}}\right) \cdot -2 \]

Reproduce

herbie shell --seed 2022151 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))