Average Error: 15.7 → 0.2
Time: 18.9s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\right) \cdot x - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)\right) \cdot x - z
double f(double x, double y, double z) {
        double r19039535 = x;
        double r19039536 = y;
        double r19039537 = r19039535 / r19039536;
        double r19039538 = log(r19039537);
        double r19039539 = r19039535 * r19039538;
        double r19039540 = z;
        double r19039541 = r19039539 - r19039540;
        return r19039541;
}
↓
double f(double x, double y, double z) {
        double r19039542 = x;
        double r19039543 = cbrt(r19039542);
        double r19039544 = y;
        double r19039545 = cbrt(r19039544);
        double r19039546 = r19039543 / r19039545;
        double r19039547 = log(r19039546);
        double r19039548 = r19039547 + r19039547;
        double r19039549 = r19039547 + r19039548;
        double r19039550 = r19039549 * r19039542;
        double r19039551 = z;
        double r19039552 = r19039550 - r19039551;
        return r19039552;
}

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

Derivation

  1. Initial program 15.7

    \[x \cdot \log \left(\frac{x}{y}\right) - z\]
  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. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2019192 +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))