Average Error: 15.1 → 0.2
Time: 5.2s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(\left(\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot 1\right) \cdot x + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z
double f(double x, double y, double z) {
        double r403441 = x;
        double r403442 = y;
        double r403443 = r403441 / r403442;
        double r403444 = log(r403443);
        double r403445 = r403441 * r403444;
        double r403446 = z;
        double r403447 = r403445 - r403446;
        return r403447;
}
↓
double f(double x, double y, double z) {
        double r403448 = 2.0;
        double r403449 = x;
        double r403450 = cbrt(r403449);
        double r403451 = y;
        double r403452 = cbrt(r403451);
        double r403453 = r403450 / r403452;
        double r403454 = log(r403453);
        double r403455 = r403448 * r403454;
        double r403456 = 1.0;
        double r403457 = r403455 * r403456;
        double r403458 = r403457 * r403449;
        double r403459 = r403449 * r403454;
        double r403460 = r403458 + r403459;
        double r403461 = z;
        double r403462 = r403460 - r403461;
        return r403462;
}

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

Original15.1
Target7.7
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.59507779908377277 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.1

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

  3. Applied add-cube-cbrt15.1

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

  4. Applied add-cube-cbrt15.1

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

  5. Applied times-frac15.1

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

  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. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))