Average Error: 15.3 → 0.3
Time: 20.1s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(-x\right) + x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(-x\right) + x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right) - z
double f(double x, double y, double z) {
        double r17853635 = x;
        double r17853636 = y;
        double r17853637 = r17853635 / r17853636;
        double r17853638 = log(r17853637);
        double r17853639 = r17853635 * r17853638;
        double r17853640 = z;
        double r17853641 = r17853639 - r17853640;
        return r17853641;
}
↓
double f(double x, double y, double z) {
        double r17853642 = y;
        double r17853643 = cbrt(r17853642);
        double r17853644 = r17853643 * r17853643;
        double r17853645 = log(r17853644);
        double r17853646 = x;
        double r17853647 = -r17853646;
        double r17853648 = r17853645 * r17853647;
        double r17853649 = cbrt(r17853646);
        double r17853650 = r17853649 / r17853643;
        double r17853651 = log(r17853650);
        double r17853652 = r17853649 * r17853649;
        double r17853653 = log(r17853652);
        double r17853654 = r17853651 + r17853653;
        double r17853655 = r17853646 * r17853654;
        double r17853656 = r17853648 + r17853655;
        double r17853657 = z;
        double r17853658 = r17853656 - r17853657;
        return r17853658;
}

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
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.3

    \[x \cdot \log \left(\frac{x}{y}\right) - z\]
  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-prod4.7

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

  7. Applied distribute-lft-in4.8

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

  8. Simplified4.8

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

  10. Applied *-un-lft-identity4.8

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

  11. Applied cbrt-prod4.8

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

  12. Applied add-cube-cbrt4.8

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

  13. Applied times-frac4.8

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

  14. Applied log-prod0.3

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

  15. Simplified0.3

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

  16. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +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))