Average Error: 15.8 → 0.2
Time: 18.7s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x\right) - z
double f(double x, double y, double z) {
        double r32686252 = x;
        double r32686253 = y;
        double r32686254 = r32686252 / r32686253;
        double r32686255 = log(r32686254);
        double r32686256 = r32686252 * r32686255;
        double r32686257 = z;
        double r32686258 = r32686256 - r32686257;
        return r32686258;
}
↓
double f(double x, double y, double z) {
        double r32686259 = x;
        double r32686260 = cbrt(r32686259);
        double r32686261 = y;
        double r32686262 = cbrt(r32686261);
        double r32686263 = r32686260 / r32686262;
        double r32686264 = log(r32686263);
        double r32686265 = r32686264 + r32686264;
        double r32686266 = r32686259 * r32686265;
        double r32686267 = r32686264 * r32686259;
        double r32686268 = r32686266 + r32686267;
        double r32686269 = z;
        double r32686270 = r32686268 - r32686269;
        return r32686270;
}

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

Original15.8
Target7.9
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 15.8

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

  3. Applied add-cube-cbrt15.8

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

  4. Applied add-cube-cbrt15.8

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

  5. Applied times-frac15.8

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

  6. Applied log-prod3.9

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

  7. Applied distribute-rgt-in3.9

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

  8. Simplified3.9

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

  10. Applied log-prod0.2

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

  11. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 +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))