Average Error: 15.0 → 31.9
Time: 16.1s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(\left(-z\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(3 \cdot x\right)\right) + \left(-\log \left(\sqrt[3]{y}\right)\right) \cdot \left(3 \cdot x\right)\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(\left(-z\right) + \log \left(\sqrt[3]{x}\right) \cdot \left(3 \cdot x\right)\right) + \left(-\log \left(\sqrt[3]{y}\right)\right) \cdot \left(3 \cdot x\right)
double f(double x, double y, double z) {
        double r405893 = x;
        double r405894 = y;
        double r405895 = r405893 / r405894;
        double r405896 = log(r405895);
        double r405897 = r405893 * r405896;
        double r405898 = z;
        double r405899 = r405897 - r405898;
        return r405899;
}
↓
double f(double x, double y, double z) {
        double r405900 = z;
        double r405901 = -r405900;
        double r405902 = x;
        double r405903 = cbrt(r405902);
        double r405904 = log(r405903);
        double r405905 = 3.0;
        double r405906 = r405905 * r405902;
        double r405907 = r405904 * r405906;
        double r405908 = r405901 + r405907;
        double r405909 = y;
        double r405910 = cbrt(r405909);
        double r405911 = log(r405910);
        double r405912 = -r405911;
        double r405913 = r405912 * r405906;
        double r405914 = r405908 + r405913;
        return r405914;
}

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
Target7.2
Herbie31.9
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.0

    \[x \cdot \log \left(\frac{x}{y}\right) - z\]
  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-prod3.7

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

  7. Applied distribute-rgt-in3.7

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

  8. Simplified0.2

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

  10. Applied add-cube-cbrt0.2

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

  11. Final simplification31.9

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

Reproduce

herbie shell --seed 2019291 
(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))