Average Error: 15.7 → 0.3
Time: 17.5s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) - \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)\right) \cdot x - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) - \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right)\right) \cdot x - z
double f(double x, double y, double z) {
        double r25103526 = x;
        double r25103527 = y;
        double r25103528 = r25103526 / r25103527;
        double r25103529 = log(r25103528);
        double r25103530 = r25103526 * r25103529;
        double r25103531 = z;
        double r25103532 = r25103530 - r25103531;
        return r25103532;
}
↓
double f(double x, double y, double z) {
        double r25103533 = x;
        double r25103534 = cbrt(r25103533);
        double r25103535 = y;
        double r25103536 = cbrt(r25103535);
        double r25103537 = r25103534 / r25103536;
        double r25103538 = log(r25103537);
        double r25103539 = r25103534 * r25103534;
        double r25103540 = log(r25103539);
        double r25103541 = r25103536 * r25103536;
        double r25103542 = log(r25103541);
        double r25103543 = r25103540 - r25103542;
        double r25103544 = r25103538 + r25103543;
        double r25103545 = r25103544 * r25103533;
        double r25103546 = z;
        double r25103547 = r25103545 - r25103546;
        return r25103547;
}

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

Original15.7
Target7.6
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;y \lt 7.595077799083772773657101400994168792118 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.7

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

  3. Applied add-cube-cbrt15.7

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

  4. Applied add-cube-cbrt15.7

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

  5. Applied times-frac15.7

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

  6. Applied log-prod3.8

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

  7. Simplified3.8

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

  9. Applied frac-times3.8

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

  10. Applied log-div0.3

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

  11. Final simplification0.3

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

Reproduce

herbie shell --seed 2019192 
(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))