Average Error: 15.2 → 0.4
Time: 22.4s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) \cdot x - z\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot x\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(\left(2 \cdot \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) \cdot x - z\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot x
double f(double x, double y, double z) {
        double r218513 = x;
        double r218514 = y;
        double r218515 = r218513 / r218514;
        double r218516 = log(r218515);
        double r218517 = r218513 * r218516;
        double r218518 = z;
        double r218519 = r218517 - r218518;
        return r218519;
}
↓
double f(double x, double y, double z) {
        double r218520 = 2.0;
        double r218521 = x;
        double r218522 = cbrt(r218521);
        double r218523 = cbrt(r218522);
        double r218524 = y;
        double r218525 = cbrt(r218524);
        double r218526 = r218523 / r218525;
        double r218527 = log(r218526);
        double r218528 = r218520 * r218527;
        double r218529 = r218528 + r218527;
        double r218530 = r218529 * r218521;
        double r218531 = z;
        double r218532 = r218530 - r218531;
        double r218533 = r218522 * r218522;
        double r218534 = log(r218533);
        double r218535 = r218534 * r218521;
        double r218536 = r218532 + r218535;
        return r218536;
}

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

Original15.2
Target7.5
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;y \lt 7.59507779908377277 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.2

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

  3. Applied *-un-lft-identity15.2

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

  4. Applied add-cube-cbrt15.2

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

  5. Applied times-frac15.2

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

  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-rgt-in5.0

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

  8. Applied associate--l+5.0

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

  10. Applied add-cube-cbrt5.0

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

  11. Applied add-cube-cbrt5.0

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

  12. Applied times-frac5.0

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

  13. Applied log-prod0.4

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

  14. Simplified0.4

    \[\leadsto \log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{1}\right) \cdot x + \left(\left(\color{blue}{2 \cdot \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)} + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) \cdot x - z\right)\]

  15. Final simplification0.4

    \[\leadsto \left(\left(2 \cdot \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{\sqrt[3]{x}}}{\sqrt[3]{y}}\right)\right) \cdot x - z\right) + \log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot x\]

Reproduce

herbie shell --seed 2019199 +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))