Average Error: 0.3 → 0.0
Time: 2.7s
Precision: 64
\[\log x - \log \left(\log x\right)\]
↓
\[\log \left(\frac{-1 \cdot x}{\log \left(\frac{1}{x}\right)}\right)\]
\log x - \log \left(\log x\right)
↓
\log \left(\frac{-1 \cdot x}{\log \left(\frac{1}{x}\right)}\right)
double f(double x) {
        double r107718 = x;
        double r107719 = log(r107718);
        double r107720 = log(r107719);
        double r107721 = r107719 - r107720;
        return r107721;
}
↓
double f(double x) {
        double r107722 = -1.0;
        double r107723 = x;
        double r107724 = r107722 * r107723;
        double r107725 = 1.0;
        double r107726 = r107725 / r107723;
        double r107727 = log(r107726);
        double r107728 = r107724 / r107727;
        double r107729 = log(r107728);
        return r107729;
}

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 frac-2neg0.0

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

  6. Simplified0.0

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

  7. Simplified0.0

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

  8. Final simplification0.0

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

Reproduce

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