Average Error: 0.0 → 0.1
Time: 13.8s
Precision: 64
\[-\log \left(\frac{1}{x} - 1\right)\]
↓
\[-\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\sqrt{\frac{1}{x}} + \sqrt{1}} \cdot \sqrt{\sqrt{\frac{1}{x}} - \sqrt{1}}\right)\right)\]
-\log \left(\frac{1}{x} - 1\right)
↓
-\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\sqrt{\frac{1}{x}} + \sqrt{1}} \cdot \sqrt{\sqrt{\frac{1}{x}} - \sqrt{1}}\right)\right)
double f(double x) {
        double r22598 = 1.0;
        double r22599 = x;
        double r22600 = r22598 / r22599;
        double r22601 = r22600 - r22598;
        double r22602 = log(r22601);
        double r22603 = -r22602;
        return r22603;
}
↓
double f(double x) {
        double r22604 = 1.0;
        double r22605 = x;
        double r22606 = r22604 / r22605;
        double r22607 = r22606 - r22604;
        double r22608 = sqrt(r22607);
        double r22609 = log(r22608);
        double r22610 = sqrt(r22606);
        double r22611 = sqrt(r22604);
        double r22612 = r22610 + r22611;
        double r22613 = sqrt(r22612);
        double r22614 = r22610 - r22611;
        double r22615 = sqrt(r22614);
        double r22616 = r22613 * r22615;
        double r22617 = log(r22616);
        double r22618 = r22609 + r22617;
        double r22619 = -r22618;
        return r22619;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.0

    \[-\log \left(\frac{1}{x} - 1\right)\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.0

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

  4. Applied log-prod0.0

    \[\leadsto -\color{blue}{\left(\log \left(\sqrt{\frac{1}{x} - 1}\right) + \log \left(\sqrt{\frac{1}{x} - 1}\right)\right)}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied difference-of-squares0.0

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

  9. Applied sqrt-prod0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1 x) 1))))