Jmat.Real.lambertw, estimator

Average Accuracy: 99.6% → 100.0%

Time: 3.9s

Precision: binary64

Cost: 13184

Math

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

↓

\[\log \left(\frac{-x}{\log \left(\frac{1}{x}\right)}\right) \]

FPCore

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

↓

(FPCore (x) :precision binary64 (log (/ (- x) (log (/ 1.0 x)))))

C

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

↓

double code(double x) {
	return log((-x / log((1.0 / 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((-x / log((1.0d0 / 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((-x / Math.log((1.0 / x))));
}

Python

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

↓

def code(x):
	return math.log((-x / math.log((1.0 / x))))

Julia

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

↓

function code(x)
	return log(Float64(Float64(-x) / log(Float64(1.0 / x))))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = log((-x / log((1.0 / x))));
end

Wolfram

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

↓

code[x_] := N[Log[N[((-x) / N[Log[N[(1.0 / x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in x around inf 99.6%

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

3. Simplified 100.0%

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

   [Start] 99.6	\[ -1 \cdot \log \left(\frac{1}{x}\right) - \log \left(-1 \cdot \log \left(\frac{1}{x}\right)\right) \]
   mul-1-neg [=>] 99.6	\[ -1 \cdot \log \left(\frac{1}{x}\right) - \log \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} \]
   log-rec [=>] 99.6	\[ -1 \cdot \log \left(\frac{1}{x}\right) - \log \left(-\color{blue}{\left(-\log x\right)}\right) \]
   remove-double-neg [=>] 99.6	\[ -1 \cdot \log \left(\frac{1}{x}\right) - \log \color{blue}{\log x} \]
   mul-1-neg [=>] 99.6	\[ \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} - \log \log x \]
   log-rec [=>] 99.6	\[ \left(-\color{blue}{\left(-\log x\right)}\right) - \log \log x \]
   remove-double-neg [=>] 99.6	\[ \color{blue}{\log x} - \log \log x \]
   log-div [<=] 100.0	\[ \color{blue}{\log \left(\frac{x}{\log x}\right)} \]

4. Taylor expanded in x around inf 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \log \left(\frac{-x}{\log \left(\frac{1}{x}\right)}\right) \]

Alternatives

Alternative 1
Accuracy	100.0%
Cost	12992

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

Reproduce

herbie shell --seed 2023146 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))