Average Error: 15.4 → 0.3
Time: 15.1s
Precision: 64
\[x \cdot \log \left(\frac{x}{y}\right) - z\]
↓
\[\left(x \cdot \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) - \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right)\right) + x \cdot \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right) - z\]
x \cdot \log \left(\frac{x}{y}\right) - z
↓
\left(x \cdot \left(\log \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) - \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\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 r343971 = x;
        double r343972 = y;
        double r343973 = r343971 / r343972;
        double r343974 = log(r343973);
        double r343975 = r343971 * r343974;
        double r343976 = z;
        double r343977 = r343975 - r343976;
        return r343977;
}
↓
double f(double x, double y, double z) {
        double r343978 = x;
        double r343979 = cbrt(r343978);
        double r343980 = r343979 * r343979;
        double r343981 = log(r343980);
        double r343982 = y;
        double r343983 = cbrt(r343982);
        double r343984 = r343983 * r343983;
        double r343985 = log(r343984);
        double r343986 = r343981 - r343985;
        double r343987 = r343978 * r343986;
        double r343988 = r343979 / r343983;
        double r343989 = log(r343988);
        double r343990 = r343978 * r343989;
        double r343991 = r343987 + r343990;
        double r343992 = z;
        double r343993 = r343991 - r343992;
        return r343993;
}

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

Original15.4
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 15.4

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

  3. Applied add-cube-cbrt15.4

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

  4. Applied add-cube-cbrt15.4

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

  5. Applied times-frac15.4

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

  6. Applied log-prod3.7

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

  7. Applied distribute-lft-in3.7

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

  9. Applied add-exp-log3.7

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

  10. Applied add-exp-log3.8

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

  11. Applied div-exp3.8

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

  12. Applied rem-log-exp0.3

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

  13. Final simplification0.3

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(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))