Jmat.Real.lambertw, estimator

Average Error: 0.3 → 0.0

Time: 2.1s

Precision: binary64

Math

\[\log x - \log \left(\log x\right)\]

↓

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

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-log 0.0

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

5. Applied frac-2neg 0.0

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

6. Simplified 0.0

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

7. Final simplification 0.0

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

Reproduce

herbie shell --seed 2020153 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))