Jmat.Real.lambertw, estimator

Average Accuracy: 99.6% → 100.0%

Time: 4.6s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

code[x_] := N[Log[N[(1.0 / N[(N[Log[x], $MachinePrecision] / x), $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	\[ -1 \cdot \log \left(\frac{1}{x}\right) - \log \left(-1 \cdot \log \left(\frac{1}{x}\right)\right) \]
   log-prod [=>] 0.0	\[ -1 \cdot \log \left(\frac{1}{x}\right) - \color{blue}{\left(\log -1 + \log \log \left(\frac{1}{x}\right)\right)} \]
   mul-1-neg [=>] 0.0	\[ \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} - \left(\log -1 + \log \log \left(\frac{1}{x}\right)\right) \]
   log-rec [=>] 0.0	\[ \left(-\color{blue}{\left(-\log x\right)}\right) - \left(\log -1 + \log \log \left(\frac{1}{x}\right)\right) \]
   remove-double-neg [=>] 0.0	\[ \color{blue}{\log x} - \left(\log -1 + \log \log \left(\frac{1}{x}\right)\right) \]
   log-prod [<=] 99.6	\[ \log x - \color{blue}{\log \left(-1 \cdot \log \left(\frac{1}{x}\right)\right)} \]
   mul-1-neg [=>] 99.6	\[ \log x - \log \color{blue}{\left(-\log \left(\frac{1}{x}\right)\right)} \]
   log-rec [=>] 99.6	\[ \log x - \log \left(-\color{blue}{\left(-\log x\right)}\right) \]
   remove-double-neg [=>] 99.6	\[ \log x - \log \color{blue}{\log x} \]
   log-div [<=] 100.0	\[ \color{blue}{\log \left(\frac{x}{\log x}\right)} \]

4. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \log \left(\frac{x}{\log x}\right) \]
   clear-num [=>] 100.0	\[ \log \color{blue}{\left(\frac{1}{\frac{\log x}{x}}\right)} \]
   inv-pow [=>] 100.0	\[ \log \color{blue}{\left({\left(\frac{\log x}{x}\right)}^{-1}\right)} \]

5. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \log \left({\left(\frac{\log x}{x}\right)}^{-1}\right) \]
   unpow-1 [=>] 100.0	\[ \log \color{blue}{\left(\frac{1}{\frac{\log x}{x}}\right)} \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	100.0%
Cost	12992

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

Alternative 2
Accuracy	26.9%
Cost	6848

\[\frac{1}{\frac{1}{\log x + -0.5}} \]

Alternative 3
Accuracy	26.9%
Cost	6592

\[\log x + -0.5 \]

Reproduce

herbie shell --seed 2023130 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))