Average Error: 0.3 → 0.0
Time: 1.7s
Precision: binary64
\[\log x - \log \log x\]
↓
\[\log \left(\sqrt{x} \cdot \frac{\sqrt{x}}{\log x}\right)\]
\log x - \log \log x
↓
\log \left(\sqrt{x} \cdot \frac{\sqrt{x}}{\log x}\right)
(FPCore (x) :precision binary64 (- (log x) (log (log x))))
↓
(FPCore (x) :precision binary64 (log (* (sqrt x) (/ (sqrt x) (log x)))))
double code(double x) {
	return log(x) - log(log(x));
}
↓
double code(double x) {
	return log(sqrt(x) * (sqrt(x) / log(x)));
}

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. Using strategy rm

  3. Applied diff-log_binary640.0

    \[\leadsto \color{blue}{\log \left(\frac{x}{\log x}\right)}\]
  4. Using strategy rm

  5. Applied pow1_binary640.0

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

  6. Applied log-pow_binary640.0

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

  7. Applied add-sqr-sqrt_binary640.0

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

  8. Applied times-frac_binary640.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020219 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))