Average Error: 0.3 → 0.0
Time: 3.3s
Precision: binary64
\[\log x - \log \log x \]
↓
\[2 \cdot \log \left(\sqrt{\frac{x}{\log x}}\right) \]
(FPCore (x) :precision binary64 (- (log x) (log (log x))))
↓
(FPCore (x) :precision binary64 (* 2.0 (log (sqrt (/ x (log x))))))
double code(double x) {
	return log(x) - log(log(x));
}
↓
double code(double x) {
	return 2.0 * log(sqrt((x / log(x))));
}

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 = 2.0d0 * log(sqrt((x / log(x))))
end function

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

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

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

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

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

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

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.3

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

  2. Applied egg-rr0.0

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

  3. Applied egg-rr0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2022150 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))