Average Error: 15.3 → 0.3
Time: 22.3s
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 r25755509 = x;
        double r25755510 = y;
        double r25755511 = r25755509 / r25755510;
        double r25755512 = log(r25755511);
        double r25755513 = r25755509 * r25755512;
        double r25755514 = z;
        double r25755515 = r25755513 - r25755514;
        return r25755515;
}
↓
double f(double x, double y, double z) {
        double r25755516 = y;
        double r25755517 = cbrt(r25755516);
        double r25755518 = r25755517 * r25755517;
        double r25755519 = log(r25755518);
        double r25755520 = x;
        double r25755521 = -r25755520;
        double r25755522 = r25755519 * r25755521;
        double r25755523 = cbrt(r25755520);
        double r25755524 = r25755523 / r25755517;
        double r25755525 = log(r25755524);
        double r25755526 = r25755523 * r25755523;
        double r25755527 = log(r25755526);
        double r25755528 = r25755525 + r25755527;
        double r25755529 = r25755520 * r25755528;
        double r25755530 = r25755522 + r25755529;
        double r25755531 = z;
        double r25755532 = r25755530 - r25755531;
        return r25755532;
}

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))