Average Error: 15.8 → 0.2
Time: 17.5s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \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
double f(double x, double y, double z) {
        double r331241 = x;
        double r331242 = y;
        double r331243 = r331241 / r331242;
        double r331244 = log(r331243);
        double r331245 = r331241 * r331244;
        double r331246 = z;
        double r331247 = r331245 - r331246;
        return r331247;
}
↓
double f(double x, double y, double z) {
        double r331248 = x;
        double r331249 = cbrt(r331248);
        double r331250 = y;
        double r331251 = cbrt(r331250);
        double r331252 = r331249 / r331251;
        double r331253 = log(r331252);
        double r331254 = r331253 + r331253;
        double r331255 = r331248 * r331254;
        double r331256 = r331253 * r331248;
        double r331257 = r331255 + r331256;
        double r331258 = z;
        double r331259 = r331257 - r331258;
        return r331259;
}

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

Original15.8
Target7.9
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 15.8

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

  3. Applied add-cube-cbrt15.8

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

  4. Applied add-cube-cbrt15.8

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

  5. Applied times-frac15.8

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

  6. Applied log-prod3.9

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

  7. Applied distribute-lft-in3.9

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

  8. Simplified3.9

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

  10. Applied add-sqr-sqrt33.4

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

  11. Applied add-sqr-sqrt33.4

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

  12. Applied times-frac33.4

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

  13. Applied times-frac33.4

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

  14. Applied log-prod31.5

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

  15. Simplified31.5

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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))