Average Error: 15.3 → 0.3
Time: 5.7s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)\right) - z
double f(double x, double y, double z) {
        double r436516 = x;
        double r436517 = y;
        double r436518 = r436516 / r436517;
        double r436519 = log(r436518);
        double r436520 = r436516 * r436519;
        double r436521 = z;
        double r436522 = r436520 - r436521;
        return r436522;
}
↓
double f(double x, double y, double z) {
        double r436523 = x;
        double r436524 = 1.0;
        double r436525 = y;
        double r436526 = cbrt(r436525);
        double r436527 = r436526 * r436526;
        double r436528 = r436524 / r436527;
        double r436529 = log(r436528);
        double r436530 = cbrt(r436523);
        double r436531 = r436530 * r436530;
        double r436532 = cbrt(r436527);
        double r436533 = r436531 / r436532;
        double r436534 = log(r436533);
        double r436535 = cbrt(r436526);
        double r436536 = r436530 / r436535;
        double r436537 = log(r436536);
        double r436538 = r436534 + r436537;
        double r436539 = r436529 + r436538;
        double r436540 = r436523 * r436539;
        double r436541 = z;
        double r436542 = r436540 - r436541;
        return r436542;
}

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

Original15.3
Target8.0
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt 7.59507779908377277 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.3

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

  3. Applied add-cube-cbrt15.3

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

  4. Applied *-un-lft-identity15.3

    \[\leadsto x \cdot \log \left(\frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right) - z\]

  5. Applied times-frac15.3

    \[\leadsto x \cdot \log \color{blue}{\left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\sqrt[3]{y}}\right)} - z\]

  6. Applied log-prod4.5

    \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\frac{x}{\sqrt[3]{y}}\right)\right)} - z\]
  7. Using strategy rm

  8. Applied add-cube-cbrt4.5

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\frac{x}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}\right)\right) - z\]

  9. Applied cbrt-prod4.5

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\frac{x}{\color{blue}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}}\right)\right) - z\]

  10. Applied add-cube-cbrt4.5

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}\right)\right) - z\]

  11. Applied times-frac4.5

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)}\right) - z\]

  12. Applied log-prod0.3

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \color{blue}{\left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)}\right) - z\]

  13. Final simplification0.3

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)\right) - z\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

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

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