Average Error: 15.3 → 0.2
Time: 15.6s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x - z\right)\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x - z\right)
double f(double x, double y, double z) {
        double r9671166 = x;
        double r9671167 = y;
        double r9671168 = r9671166 / r9671167;
        double r9671169 = log(r9671168);
        double r9671170 = r9671166 * r9671169;
        double r9671171 = z;
        double r9671172 = r9671170 - r9671171;
        return r9671172;
}
↓
double f(double x, double y, double z) {
        double r9671173 = x;
        double r9671174 = cbrt(r9671173);
        double r9671175 = y;
        double r9671176 = cbrt(r9671175);
        double r9671177 = r9671174 / r9671176;
        double r9671178 = log(r9671177);
        double r9671179 = r9671178 + r9671178;
        double r9671180 = r9671173 * r9671179;
        double r9671181 = r9671178 * r9671173;
        double r9671182 = z;
        double r9671183 = r9671181 - r9671182;
        double r9671184 = r9671180 + r9671183;
        return r9671184;
}

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
Target7.8
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.3

    \[x \cdot \log \left(\frac{x}{y}\right) - z\]
  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 add-cube-cbrt15.3

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

  5. Applied times-frac15.3

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

  6. Applied log-prod3.6

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

  7. Applied distribute-rgt-in3.6

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

  8. Applied associate--l+3.6

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

  10. Applied times-frac3.6

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

  11. Applied log-prod0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019179 
(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))