Average Error: 15.0 → 0.3
Time: 18.7s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) - z\right) + \sqrt[3]{\left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\right) \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)} \cdot x\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) - z\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)} \cdot x
double f(double x, double y, double z) {
        double r19676866 = x;
        double r19676867 = y;
        double r19676868 = r19676866 / r19676867;
        double r19676869 = log(r19676868);
        double r19676870 = r19676866 * r19676869;
        double r19676871 = z;
        double r19676872 = r19676870 - r19676871;
        return r19676872;
}
↓
double f(double x, double y, double z) {
        double r19676873 = x;
        double r19676874 = cbrt(r19676873);
        double r19676875 = y;
        double r19676876 = cbrt(r19676875);
        double r19676877 = cbrt(r19676876);
        double r19676878 = r19676876 * r19676876;
        double r19676879 = cbrt(r19676878);
        double r19676880 = r19676877 * r19676879;
        double r19676881 = r19676874 / r19676880;
        double r19676882 = log(r19676881);
        double r19676883 = r19676873 * r19676882;
        double r19676884 = z;
        double r19676885 = r19676883 - r19676884;
        double r19676886 = r19676874 / r19676876;
        double r19676887 = log(r19676886);
        double r19676888 = r19676887 + r19676887;
        double r19676889 = r19676888 * r19676888;
        double r19676890 = r19676889 * r19676888;
        double r19676891 = cbrt(r19676890);
        double r19676892 = r19676891 * r19676873;
        double r19676893 = r19676885 + r19676892;
        return r19676893;
}

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

Original15.0
Target7.6
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 15.0

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

  3. Applied add-cube-cbrt15.0

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

  4. Applied add-cube-cbrt15.0

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

  5. Applied times-frac15.0

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

  6. Applied log-prod3.2

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

  7. Applied distribute-rgt-in3.2

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

  8. Applied associate--l+3.2

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

  10. Applied add-cbrt-cube3.3

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

  11. Simplified0.3

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

  13. Applied add-cube-cbrt0.3

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

  14. Applied cbrt-prod0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019165 
(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))