Average Error: 15.3 → 0.3
Time: 6.0s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)\right) - z
double f(double x, double y, double z) {
        double r559434 = x;
        double r559435 = y;
        double r559436 = r559434 / r559435;
        double r559437 = log(r559436);
        double r559438 = r559434 * r559437;
        double r559439 = z;
        double r559440 = r559438 - r559439;
        return r559440;
}
↓
double f(double x, double y, double z) {
        double r559441 = x;
        double r559442 = 1.0;
        double r559443 = y;
        double r559444 = cbrt(r559443);
        double r559445 = r559444 * r559444;
        double r559446 = r559442 / r559445;
        double r559447 = log(r559446);
        double r559448 = 2.0;
        double r559449 = cbrt(r559441);
        double r559450 = cbrt(r559444);
        double r559451 = r559449 / r559450;
        double r559452 = log(r559451);
        double r559453 = r559448 * r559452;
        double r559454 = r559453 + r559452;
        double r559455 = r559447 + r559454;
        double r559456 = r559441 * r559455;
        double r559457 = z;
        double r559458 = r559456 - r559457;
        return r559458;
}

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

Original15.3
Target8.0
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt 7.59507779908377277 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.3

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

  3. Applied add-cube-cbrt15.3

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

  4. Applied *-un-lft-identity15.3

    \[\leadsto x \cdot \log \left(\frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right) - z\]

  5. Applied times-frac15.3

    \[\leadsto x \cdot \log \color{blue}{\left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\sqrt[3]{y}}\right)} - z\]

  6. Applied log-prod4.5

    \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\frac{x}{\sqrt[3]{y}}\right)\right)} - z\]
  7. Using strategy rm

  8. Applied add-cube-cbrt4.5

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\frac{x}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}}\right)\right) - z\]

  9. Applied add-cube-cbrt4.5

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}\right)\right) - z\]

  10. Applied times-frac4.5

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)}\right) - z\]

  11. Applied log-prod0.3

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \color{blue}{\left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)}\right) - z\]

  12. Simplified0.3

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(\color{blue}{2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)\right) - z\]

  13. Final simplification0.3

    \[\leadsto x \cdot \left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)\right) - z\]

Reproduce

herbie shell --seed 2020089 
(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))