Average Error: 15.0 → 0.2
Time: 8.1s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot x + \log \left(\frac{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right) \cdot x\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot x + \log \left(\frac{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right) \cdot x\right) - z
double f(double x, double y, double z) {
        double r538837 = x;
        double r538838 = y;
        double r538839 = r538837 / r538838;
        double r538840 = log(r538839);
        double r538841 = r538837 * r538840;
        double r538842 = z;
        double r538843 = r538841 - r538842;
        return r538843;
}
↓
double f(double x, double y, double z) {
        double r538844 = 2.0;
        double r538845 = x;
        double r538846 = cbrt(r538845);
        double r538847 = y;
        double r538848 = cbrt(r538847);
        double r538849 = r538846 / r538848;
        double r538850 = log(r538849);
        double r538851 = r538844 * r538850;
        double r538852 = r538851 * r538845;
        double r538853 = cbrt(r538846);
        double r538854 = r538853 * r538853;
        double r538855 = r538854 * r538853;
        double r538856 = r538855 / r538848;
        double r538857 = log(r538856);
        double r538858 = r538857 * r538845;
        double r538859 = r538852 + r538858;
        double r538860 = z;
        double r538861 = r538859 - r538860;
        return r538861;
}

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
Target8.0
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.59507779908377277 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

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-prod3.5

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

  7. Applied distribute-lft-in3.5

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

  8. Simplified0.2

    \[\leadsto \left(\color{blue}{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot x} + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\]

  9. Simplified0.2

    \[\leadsto \left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot x + \color{blue}{\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x}\right) - z\]
  10. Using strategy rm

  11. Applied add-cube-cbrt0.2

    \[\leadsto \left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot x + \log \left(\frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}}{\sqrt[3]{y}}\right) \cdot x\right) - z\]

  12. Final simplification0.2

    \[\leadsto \left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot x + \log \left(\frac{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right) \cdot x\right) - z\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))