Average Error: 15.0 → 0.3
Time: 19.1s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) \cdot x + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot x\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(\left(\left(\log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) \cdot x + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot x\right) - z
double f(double x, double y, double z) {
        double r19443393 = x;
        double r19443394 = y;
        double r19443395 = r19443393 / r19443394;
        double r19443396 = log(r19443395);
        double r19443397 = r19443393 * r19443396;
        double r19443398 = z;
        double r19443399 = r19443397 - r19443398;
        return r19443399;
}
↓
double f(double x, double y, double z) {
        double r19443400 = x;
        double r19443401 = cbrt(r19443400);
        double r19443402 = cbrt(r19443401);
        double r19443403 = y;
        double r19443404 = cbrt(r19443403);
        double r19443405 = r19443402 / r19443404;
        double r19443406 = log(r19443405);
        double r19443407 = r19443406 + r19443406;
        double r19443408 = r19443407 + r19443406;
        double r19443409 = r19443408 * r19443400;
        double r19443410 = r19443401 * r19443401;
        double r19443411 = log(r19443410);
        double r19443412 = r19443411 * r19443400;
        double r19443413 = r19443409 + r19443412;
        double r19443414 = z;
        double r19443415 = r19443413 - r19443414;
        return r19443415;
}

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
Target8.0
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.0

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

  3. Applied *-un-lft-identity15.0

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

  4. Applied add-cube-cbrt15.0

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

  5. Applied times-frac15.0

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

  6. Applied log-prod4.7

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

  7. Applied distribute-lft-in4.7

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

  8. Simplified4.7

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

  10. Applied add-cube-cbrt4.7

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

  11. Applied add-cube-cbrt4.7

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

  12. Applied times-frac4.7

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

  13. Applied log-prod0.3

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

  14. Simplified0.3

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

  15. Final simplification0.3

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

Reproduce

herbie shell --seed 2019168 
(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))