Average Error: 15.0 → 0.2
Time: 17.9s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(x \cdot \left(\log \left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right) + \log \left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(x \cdot \left(\log \left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right) + \log \left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z
double f(double x, double y, double z) {
        double r309088 = x;
        double r309089 = y;
        double r309090 = r309088 / r309089;
        double r309091 = log(r309090);
        double r309092 = r309088 * r309091;
        double r309093 = z;
        double r309094 = r309092 - r309093;
        return r309094;
}
↓
double f(double x, double y, double z) {
        double r309095 = x;
        double r309096 = cbrt(r309095);
        double r309097 = y;
        double r309098 = cbrt(r309097);
        double r309099 = r309096 / r309098;
        double r309100 = fabs(r309099);
        double r309101 = log(r309100);
        double r309102 = r309101 + r309101;
        double r309103 = r309095 * r309102;
        double r309104 = log(r309099);
        double r309105 = r309095 * r309104;
        double r309106 = r309103 + r309105;
        double r309107 = z;
        double r309108 = r309106 - r309107;
        return r309108;
}

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
Target7.7
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt 7.59507779908377277 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array}\]

Derivation

  1. Initial program 15.0

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

  3. Applied add-cube-cbrt15.0

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

  4. Applied add-cube-cbrt15.0

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

  5. Applied times-frac15.0

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

  6. Applied log-prod3.4

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

  7. Applied distribute-lft-in3.4

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

  9. Applied add-sqr-sqrt3.4

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

  10. Applied log-prod3.4

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

  11. Simplified3.4

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

  12. Simplified0.2

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

  13. Final simplification0.2

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

Reproduce

herbie shell --seed 2019198 +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))