Average Error: 15.4 → 0.3
Time: 16.5s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[x \cdot \left(\left(-\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + \left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
x \cdot \left(\left(-\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\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 r312413 = x;
        double r312414 = y;
        double r312415 = r312413 / r312414;
        double r312416 = log(r312415);
        double r312417 = r312413 * r312416;
        double r312418 = z;
        double r312419 = r312417 - r312418;
        return r312419;
}
↓
double f(double x, double y, double z) {
        double r312420 = x;
        double r312421 = y;
        double r312422 = cbrt(r312421);
        double r312423 = r312422 * r312422;
        double r312424 = log(r312423);
        double r312425 = -r312424;
        double r312426 = 2.0;
        double r312427 = cbrt(r312420);
        double r312428 = cbrt(r312422);
        double r312429 = r312427 / r312428;
        double r312430 = log(r312429);
        double r312431 = r312426 * r312430;
        double r312432 = r312431 + r312430;
        double r312433 = r312425 + r312432;
        double r312434 = r312420 * r312433;
        double r312435 = z;
        double r312436 = r312434 - r312435;
        return r312436;
}

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

Original15.4
Target7.8
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 15.4

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

  3. Applied add-cube-cbrt15.4

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

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

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

  5. Applied times-frac15.4

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

  6. Applied log-prod4.7

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

  7. Simplified4.7

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

  9. Applied add-cube-cbrt4.7

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

  10. Applied add-cube-cbrt4.7

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

  11. Applied times-frac4.7

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

  12. Applied log-prod0.3

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

  13. Simplified0.3

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

  14. Final simplification0.3

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.59507779908377277e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

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