Average Error: 15.7 → 0.3
Time: 12.2s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\sqrt[3]{{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)}^{3}} \cdot x + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x - z\right)\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\sqrt[3]{{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)}^{3}} \cdot x + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x - z\right)
double f(double x, double y, double z) {
        double r312396 = x;
        double r312397 = y;
        double r312398 = r312396 / r312397;
        double r312399 = log(r312398);
        double r312400 = r312396 * r312399;
        double r312401 = z;
        double r312402 = r312400 - r312401;
        return r312402;
}
↓
double f(double x, double y, double z) {
        double r312403 = 2.0;
        double r312404 = x;
        double r312405 = cbrt(r312404);
        double r312406 = y;
        double r312407 = cbrt(r312406);
        double r312408 = r312405 / r312407;
        double r312409 = log(r312408);
        double r312410 = r312403 * r312409;
        double r312411 = 3.0;
        double r312412 = pow(r312410, r312411);
        double r312413 = cbrt(r312412);
        double r312414 = r312413 * r312404;
        double r312415 = r312409 * r312404;
        double r312416 = z;
        double r312417 = r312415 - r312416;
        double r312418 = r312414 + r312417;
        return r312418;
}

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
Target8.1
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-prod3.5

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

  7. Applied distribute-rgt-in3.5

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

  8. Applied associate--l+3.5

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

  10. Applied add-cbrt-cube3.5

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

  11. Simplified0.3

    \[\leadsto \sqrt[3]{\color{blue}{{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)}^{3}}} \cdot x + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x - z\right)\]

  12. Final simplification0.3

    \[\leadsto \sqrt[3]{{\left(2 \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)}^{3}} \cdot x + \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot x - z\right)\]

Reproduce

herbie shell --seed 2019305 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.59507779908377277e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))