Average Error: 14.6 → 0.2
Time: 20.2s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(x \cdot \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) \cdot x\right) - z
double f(double x, double y, double z) {
        double r10882845 = x;
        double r10882846 = y;
        double r10882847 = r10882845 / r10882846;
        double r10882848 = log(r10882847);
        double r10882849 = r10882845 * r10882848;
        double r10882850 = z;
        double r10882851 = r10882849 - r10882850;
        return r10882851;
}
↓
double f(double x, double y, double z) {
        double r10882852 = x;
        double r10882853 = cbrt(r10882852);
        double r10882854 = y;
        double r10882855 = cbrt(r10882854);
        double r10882856 = r10882853 / r10882855;
        double r10882857 = log(r10882856);
        double r10882858 = r10882857 + r10882857;
        double r10882859 = r10882852 * r10882858;
        double r10882860 = r10882857 * r10882852;
        double r10882861 = r10882859 + r10882860;
        double r10882862 = z;
        double r10882863 = r10882861 - r10882862;
        return r10882863;
}

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
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

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-lft-in3.3

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 +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))