Average Error: 0.3 → 0.0
Time: 2.8s
Precision: 64
\[\log x - \log \left(\log x\right)\]
↓
\[\log \left(\frac{1}{\frac{\log x}{x}}\right)\]
\log x - \log \left(\log x\right)
↓
\log \left(\frac{1}{\frac{\log x}{x}}\right)
double f(double x) {
        double r107446 = x;
        double r107447 = log(r107446);
        double r107448 = log(r107447);
        double r107449 = r107447 - r107448;
        return r107449;
}
↓
double f(double x) {
        double r107450 = 1.0;
        double r107451 = x;
        double r107452 = log(r107451);
        double r107453 = r107452 / r107451;
        double r107454 = r107450 / r107453;
        double r107455 = log(r107454);
        return r107455;
}

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

    \[\log x - \log \left(\log x\right)\]
  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 clear-num0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  :precision binary64
  (- (log x) (log (log x))))