Average Error: 15.0 → 0.2
Time: 15.2s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot x - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) \cdot x - z
double f(double x, double y, double z) {
        double r462545 = x;
        double r462546 = y;
        double r462547 = r462545 / r462546;
        double r462548 = log(r462547);
        double r462549 = r462545 * r462548;
        double r462550 = z;
        double r462551 = r462549 - r462550;
        return r462551;
}
↓
double f(double x, double y, double z) {
        double r462552 = 2.0;
        double r462553 = x;
        double r462554 = cbrt(r462553);
        double r462555 = y;
        double r462556 = cbrt(r462555);
        double r462557 = r462554 / r462556;
        double r462558 = log(r462557);
        double r462559 = r462552 * r462558;
        double r462560 = r462559 + r462558;
        double r462561 = r462560 * r462553;
        double r462562 = z;
        double r462563 = r462561 - r462562;
        return r462563;
}

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
Target7.6
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.0

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

  2. Simplified15.0

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

  4. Applied add-cube-cbrt15.0

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

  5. Applied add-cube-cbrt15.0

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

  6. Applied times-frac15.0

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

  7. Applied log-prod3.4

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

  8. Simplified0.2

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

  9. Final simplification0.2

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

Reproduce

herbie shell --seed 2019194 
(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))