Average Error: 0.3 → 0.0
Time: 2.8s
Precision: binary64
\[\log x - \log \log x\]
↓
\[\log \left(\frac{x}{\log x}\right)\]
\log x - \log \log x
↓
\log \left(\frac{x}{\log x}\right)
(FPCore (x) :precision binary64 (- (log x) (log (log x))))
↓
(FPCore (x) :precision binary64 (log (/ x (log x))))
double code(double x) {
	return log(x) - log(log(x));
}
↓
double code(double x) {
	return log(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 add-cube-cbrt_binary640.3

    \[\leadsto \log x - \log \color{blue}{\left(\left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) \cdot \sqrt[3]{\log x}\right)}\]

  4. Applied log-prod_binary640.3

    \[\leadsto \log x - \color{blue}{\left(\log \left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right) + \log \left(\sqrt[3]{\log x}\right)\right)}\]

  5. Applied associate--r+_binary640.3

    \[\leadsto \color{blue}{\left(\log x - \log \left(\sqrt[3]{\log x} \cdot \sqrt[3]{\log x}\right)\right) - \log \left(\sqrt[3]{\log x}\right)}\]

  6. Simplified0.3

    \[\leadsto \color{blue}{\left(\log x - \log \left(\sqrt[3]{\log x}\right) \cdot 2\right)} - \log \left(\sqrt[3]{\log x}\right)\]
  7. Using strategy rm

  8. Applied *-un-lft-identity_binary640.3

    \[\leadsto \left(\log \color{blue}{\left(1 \cdot x\right)} - \log \left(\sqrt[3]{\log x}\right) \cdot 2\right) - \log \left(\sqrt[3]{\log x}\right)\]

  9. Applied log-prod_binary640.3

    \[\leadsto \left(\color{blue}{\left(\log 1 + \log x\right)} - \log \left(\sqrt[3]{\log x}\right) \cdot 2\right) - \log \left(\sqrt[3]{\log x}\right)\]

  10. Applied associate--l+_binary640.3

    \[\leadsto \color{blue}{\left(\log 1 + \left(\log x - \log \left(\sqrt[3]{\log x}\right) \cdot 2\right)\right)} - \log \left(\sqrt[3]{\log x}\right)\]

  11. Applied associate--l+_binary640.3

    \[\leadsto \color{blue}{\log 1 + \left(\left(\log x - \log \left(\sqrt[3]{\log x}\right) \cdot 2\right) - \log \left(\sqrt[3]{\log x}\right)\right)}\]

  12. Simplified0.3

    \[\leadsto \log 1 + \color{blue}{\left(\log x - \log \log x\right)}\]
  13. Using strategy rm

  14. Applied diff-log_binary640.0

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

  15. Final simplification0.0

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

Reproduce

herbie shell --seed 2021173 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))