Jmat.Real.lambertw, estimator

Average Error: 0.3 → 0.0

Time: 2.5s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

(FPCore (x) :precision binary64 (log (/ (sqrt x) (/ (log x) (sqrt x)))))

C

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

↓

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

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(x) / (log(x) / sqrt(x))))
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(x) / (Math.log(x) / Math.sqrt(x))));
}

Python

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

↓

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

Julia

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

↓

function code(x)
	return log(Float64(sqrt(x) / Float64(log(x) / sqrt(x))))
end

MATLAB

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

↓

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

Wolfram

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 0.3

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

2. Applied egg-rr 0.0

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

3. Applied egg-rr 0.0

   \[\leadsto \log \color{blue}{\left(\frac{1}{\sqrt{\log x}} \cdot \frac{x}{\sqrt{\log x}}\right)} \]

4. Applied egg-rr 0.0

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

5. Final simplification 0.0

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

Reproduce

herbie shell --seed 2022153 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))