Average Error: 15.4 → 0.2
Time: 17.6s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right)\right) \cdot x - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right)\right) \cdot x - z
double f(double x, double y, double z) {
        double r18806827 = x;
        double r18806828 = y;
        double r18806829 = r18806827 / r18806828;
        double r18806830 = log(r18806829);
        double r18806831 = r18806827 * r18806830;
        double r18806832 = z;
        double r18806833 = r18806831 - r18806832;
        return r18806833;
}
↓
double f(double x, double y, double z) {
        double r18806834 = x;
        double r18806835 = cbrt(r18806834);
        double r18806836 = y;
        double r18806837 = cbrt(r18806836);
        double r18806838 = r18806835 / r18806837;
        double r18806839 = log(r18806838);
        double r18806840 = r18806839 + r18806839;
        double r18806841 = cbrt(r18806837);
        double r18806842 = r18806841 * r18806841;
        double r18806843 = r18806841 * r18806842;
        double r18806844 = r18806835 / r18806843;
        double r18806845 = log(r18806844);
        double r18806846 = r18806840 + r18806845;
        double r18806847 = r18806846 * r18806834;
        double r18806848 = z;
        double r18806849 = r18806847 - r18806848;
        return r18806849;
}

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

Original15.4
Target7.9
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.4

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

  3. Applied add-cube-cbrt15.4

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

  4. Applied add-cube-cbrt15.4

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

  5. Applied times-frac15.4

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

  6. Applied log-prod3.6

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

  7. Simplified0.2

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

  9. Applied add-cube-cbrt0.2

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

  10. Final simplification0.2

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(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))