Average Error: 14.6 → 0.3
Time: 19.4s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \sqrt[3]{\left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\right) \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)}\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \sqrt[3]{\left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\right) \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)}\right) - z
double f(double x, double y, double z) {
        double r25039606 = x;
        double r25039607 = y;
        double r25039608 = r25039606 / r25039607;
        double r25039609 = log(r25039608);
        double r25039610 = r25039606 * r25039609;
        double r25039611 = z;
        double r25039612 = r25039610 - r25039611;
        return r25039612;
}
↓
double f(double x, double y, double z) {
        double r25039613 = x;
        double r25039614 = cbrt(r25039613);
        double r25039615 = y;
        double r25039616 = cbrt(r25039615);
        double r25039617 = r25039614 / r25039616;
        double r25039618 = log(r25039617);
        double r25039619 = r25039618 + r25039618;
        double r25039620 = r25039619 * r25039619;
        double r25039621 = r25039620 * r25039619;
        double r25039622 = cbrt(r25039621);
        double r25039623 = r25039618 + r25039622;
        double r25039624 = r25039613 * r25039623;
        double r25039625 = z;
        double r25039626 = r25039624 - r25039625;
        return r25039626;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.6
Target7.3
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 14.6

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

  3. Applied add-cube-cbrt14.6

    \[\leadsto x \cdot \log \left(\frac{x}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right) - z\]

  4. Applied add-cube-cbrt14.6

    \[\leadsto x \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right) - z\]

  5. Applied times-frac14.6

    \[\leadsto x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} - z\]

  6. Applied log-prod3.4

    \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)} - z\]

  7. Simplified3.4

    \[\leadsto x \cdot \left(\color{blue}{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\]
  8. Using strategy rm

  9. Applied add-cbrt-cube3.5

    \[\leadsto x \cdot \left(\color{blue}{\sqrt[3]{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)}} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\]

  10. Simplified0.3

    \[\leadsto x \cdot \left(\sqrt[3]{\color{blue}{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot \left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\right)}} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\]

  11. Final simplification0.3

    \[\leadsto x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \sqrt[3]{\left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\right) \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)}\right) - z\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))