Average Error: 14.6 → 0.3
Time: 21.5s
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 r11516151 = x;
        double r11516152 = y;
        double r11516153 = r11516151 / r11516152;
        double r11516154 = log(r11516153);
        double r11516155 = r11516151 * r11516154;
        double r11516156 = z;
        double r11516157 = r11516155 - r11516156;
        return r11516157;
}
↓
double f(double x, double y, double z) {
        double r11516158 = x;
        double r11516159 = cbrt(r11516158);
        double r11516160 = y;
        double r11516161 = cbrt(r11516160);
        double r11516162 = r11516159 / r11516161;
        double r11516163 = log(r11516162);
        double r11516164 = r11516163 * r11516158;
        double r11516165 = r11516159 * r11516159;
        double r11516166 = log(r11516165);
        double r11516167 = r11516161 * r11516161;
        double r11516168 = log(r11516167);
        double r11516169 = r11516166 - r11516168;
        double r11516170 = r11516158 * r11516169;
        double r11516171 = r11516164 + r11516170;
        double r11516172 = z;
        double r11516173 = r11516171 - r11516172;
        return r11516173;
}

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
Target7.5
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-prod3.3

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

  7. Applied distribute-rgt-in3.3

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

  8. Simplified3.3

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

  10. Applied frac-times3.3

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

  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 2019158 
(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))