Average Error: 0.0 → 0.1
Time: 9.3s
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 r21392 = 1.0;
        double r21393 = x;
        double r21394 = r21392 / r21393;
        double r21395 = r21394 - r21392;
        double r21396 = log(r21395);
        double r21397 = -r21396;
        return r21397;
}
↓
double f(double x) {
        double r21398 = 1.0;
        double r21399 = x;
        double r21400 = r21398 / r21399;
        double r21401 = r21400 - r21398;
        double r21402 = sqrt(r21401);
        double r21403 = log(r21402);
        double r21404 = sqrt(r21400);
        double r21405 = sqrt(r21398);
        double r21406 = r21404 + r21405;
        double r21407 = sqrt(r21406);
        double r21408 = r21404 - r21405;
        double r21409 = sqrt(r21408);
        double r21410 = r21407 * r21409;
        double r21411 = log(r21410);
        double r21412 = r21403 + r21411;
        double r21413 = -r21412;
        return r21413;
}

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 
(FPCore (x)
  :name "neg log"
  :precision binary64
  (- (log (- (/ 1 x) 1))))