Average Error: 0.3 → 0.0
Time: 32.5s
Precision: 64
Internal Precision: 128
\[\log x - \log \left(\log x\right)\]
↓
\[\log \left(\frac{\sqrt{x}}{\sqrt{\log x}} \cdot \frac{\sqrt{x}}{\sqrt{\log x}}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 0.3

    \[\log x - \log \left(\log x\right)\]
  2. Using strategy rm

  3. Applied diff-log0.0

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

  5. Applied add-sqr-sqrt0.0

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

  6. Applied add-sqr-sqrt0.0

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

  7. Applied times-frac0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019094 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  (- (log x) (log (log x))))