Average Error: 0.3 → 0.0
Time: 12.1s
Precision: 64
\[\log x - \log \left(\log x\right)\]
↓
\[\log \left(\frac{1}{\sqrt{\log x}} \cdot \frac{x}{\sqrt{\log x}}\right)\]
\log x - \log \left(\log x\right)
↓
\log \left(\frac{1}{\sqrt{\log x}} \cdot \frac{x}{\sqrt{\log x}}\right)
double f(double x) {
        double r88002 = x;
        double r88003 = log(r88002);
        double r88004 = log(r88003);
        double r88005 = r88003 - r88004;
        return r88005;
}
↓
double f(double x) {
        double r88006 = 1.0;
        double r88007 = x;
        double r88008 = log(r88007);
        double r88009 = sqrt(r88008);
        double r88010 = r88006 / r88009;
        double r88011 = r88007 / r88009;
        double r88012 = r88010 * r88011;
        double r88013 = log(r88012);
        return r88013;
}

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 add-sqr-sqrt0.0

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

  6. Applied *-un-lft-identity0.0

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

  7. Applied times-frac0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.lambertw, estimator"
  (- (log x) (log (log x))))