Average Error: 15.7 → 0.3
Time: 42.8s
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 r23845364 = x;
        double r23845365 = y;
        double r23845366 = r23845364 / r23845365;
        double r23845367 = log(r23845366);
        double r23845368 = r23845364 * r23845367;
        double r23845369 = z;
        double r23845370 = r23845368 - r23845369;
        return r23845370;
}
↓
double f(double x, double y, double z) {
        double r23845371 = x;
        double r23845372 = cbrt(r23845371);
        double r23845373 = cbrt(r23845372);
        double r23845374 = y;
        double r23845375 = cbrt(r23845374);
        double r23845376 = r23845373 / r23845375;
        double r23845377 = log(r23845376);
        double r23845378 = r23845377 + r23845377;
        double r23845379 = r23845378 + r23845377;
        double r23845380 = r23845379 * r23845371;
        double r23845381 = r23845372 * r23845372;
        double r23845382 = log(r23845381);
        double r23845383 = r23845382 * r23845371;
        double r23845384 = r23845380 + r23845383;
        double r23845385 = z;
        double r23845386 = r23845384 - r23845385;
        return r23845386;
}

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

Original15.7
Target8.3
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 15.7

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

  3. Applied *-un-lft-identity15.7

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

  4. Applied add-cube-cbrt15.7

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

  5. Applied times-frac15.7

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

  6. Applied log-prod4.9

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

  7. Applied distribute-lft-in4.9

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

  8. Simplified4.9

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

  10. Applied add-cube-cbrt4.9

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

  11. Applied add-cube-cbrt4.9

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

  12. Applied times-frac4.9

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

  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))