Average Error: 15.0 → 0.2
Time: 15.5s
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 r358999 = x;
        double r359000 = y;
        double r359001 = r358999 / r359000;
        double r359002 = log(r359001);
        double r359003 = r358999 * r359002;
        double r359004 = z;
        double r359005 = r359003 - r359004;
        return r359005;
}
↓
double f(double x, double y, double z) {
        double r359006 = 2.0;
        double r359007 = x;
        double r359008 = cbrt(r359007);
        double r359009 = y;
        double r359010 = cbrt(r359009);
        double r359011 = r359008 / r359010;
        double r359012 = log(r359011);
        double r359013 = r359006 * r359012;
        double r359014 = r359013 + r359012;
        double r359015 = r359014 * r359007;
        double r359016 = z;
        double r359017 = r359015 - r359016;
        return r359017;
}

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))