Average Error: 0.3 → 0.0
Time: 9.9s
Precision: 64
\[\log x - \log \left(\log x\right)\]
↓
\[\log \left(\frac{\sqrt{x}}{\log x} \cdot \sqrt{x}\right)\]
double f(double x) {
        double r7067559 = x;
        double r7067560 = log(r7067559);
        double r7067561 = log(r7067560);
        double r7067562 = r7067560 - r7067561;
        return r7067562;
}
↓
double f(double x) {
        double r7067563 = x;
        double r7067564 = sqrt(r7067563);
        double r7067565 = log(r7067563);
        double r7067566 = r7067564 / r7067565;
        double r7067567 = r7067566 * r7067564;
        double r7067568 = log(r7067567);
        return r7067568;
}


\log x - \log \left(\log x\right)
↓
\log \left(\frac{\sqrt{x}}{\log x} \cdot \sqrt{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 pow10.0

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

  6. Applied log-pow0.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied times-frac0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019101 
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  (- (log x) (log (log x))))