Average Error: 15.6 → 0.3
Time: 23.2s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x + x \cdot \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) - \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x + x \cdot \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) - \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)\right) - z
double f(double x, double y, double z) {
        double r21401938 = x;
        double r21401939 = y;
        double r21401940 = r21401938 / r21401939;
        double r21401941 = log(r21401940);
        double r21401942 = r21401938 * r21401941;
        double r21401943 = z;
        double r21401944 = r21401942 - r21401943;
        return r21401944;
}
↓
double f(double x, double y, double z) {
        double r21401945 = x;
        double r21401946 = cbrt(r21401945);
        double r21401947 = y;
        double r21401948 = cbrt(r21401947);
        double r21401949 = r21401946 / r21401948;
        double r21401950 = log(r21401949);
        double r21401951 = r21401950 * r21401945;
        double r21401952 = r21401946 * r21401946;
        double r21401953 = log(r21401952);
        double r21401954 = r21401948 * r21401948;
        double r21401955 = log(r21401954);
        double r21401956 = r21401953 - r21401955;
        double r21401957 = r21401945 * r21401956;
        double r21401958 = r21401951 + r21401957;
        double r21401959 = z;
        double r21401960 = r21401958 - r21401959;
        return r21401960;
}

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

Original15.6
Target7.9
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.6

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

  3. Applied add-cube-cbrt15.6

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

  4. Applied add-cube-cbrt15.6

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

  5. Applied times-frac15.6

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

  6. Applied log-prod3.7

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

  7. Applied distribute-rgt-in3.7

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

  8. Simplified3.7

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

  10. Applied frac-times3.7

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

  11. Applied log-div0.3

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

  12. Final simplification0.3

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

Reproduce

herbie shell --seed 2019170 
(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))