Average Error: 14.8 → 0.2
Time: 18.6s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x - z\right)\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x - z\right)
double f(double x, double y, double z) {
        double r17999474 = x;
        double r17999475 = y;
        double r17999476 = r17999474 / r17999475;
        double r17999477 = log(r17999476);
        double r17999478 = r17999474 * r17999477;
        double r17999479 = z;
        double r17999480 = r17999478 - r17999479;
        return r17999480;
}
↓
double f(double x, double y, double z) {
        double r17999481 = x;
        double r17999482 = cbrt(r17999481);
        double r17999483 = y;
        double r17999484 = cbrt(r17999483);
        double r17999485 = r17999482 / r17999484;
        double r17999486 = log(r17999485);
        double r17999487 = r17999486 + r17999486;
        double r17999488 = r17999481 * r17999487;
        double r17999489 = r17999486 * r17999481;
        double r17999490 = z;
        double r17999491 = r17999489 - r17999490;
        double r17999492 = r17999488 + r17999491;
        return r17999492;
}

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

Original14.8
Target7.7
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 14.8

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

  3. Applied add-cube-cbrt14.8

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

  4. Applied add-cube-cbrt14.8

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

  5. Applied times-frac14.8

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

  6. Applied log-prod3.6

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

  7. Applied distribute-lft-in3.6

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

  8. Applied associate--l+3.6

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

  10. Applied times-frac3.6

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

  11. Applied log-prod0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019163 +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))