Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Average Error: 15.3 → 0.3

Time: 18.8s

Precision: 64

Math

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

↓

\[x \cdot \log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(\left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right) \cdot x - z\right)\]

C

double f(double x, double y, double z) {
        double r17941883 = x;
        double r17941884 = y;
        double r17941885 = r17941883 / r17941884;
        double r17941886 = log(r17941885);
        double r17941887 = r17941883 * r17941886;
        double r17941888 = z;
        double r17941889 = r17941887 - r17941888;
        return r17941889;
}

↓

double f(double x, double y, double z) {
        double r17941890 = x;
        double r17941891 = 1.0;
        double r17941892 = y;
        double r17941893 = cbrt(r17941892);
        double r17941894 = r17941893 * r17941893;
        double r17941895 = r17941891 / r17941894;
        double r17941896 = log(r17941895);
        double r17941897 = r17941890 * r17941896;
        double r17941898 = cbrt(r17941890);
        double r17941899 = cbrt(r17941893);
        double r17941900 = r17941898 / r17941899;
        double r17941901 = log(r17941900);
        double r17941902 = r17941901 + r17941901;
        double r17941903 = r17941902 + r17941901;
        double r17941904 = r17941903 * r17941890;
        double r17941905 = z;
        double r17941906 = r17941904 - r17941905;
        double r17941907 = r17941897 + r17941906;
        return r17941907;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.3
Target	7.8
Herbie	0.3

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

Derivation

1. Initial program 15.3

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

3. Applied add-cube-cbrt 15.3

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

4. Applied *-un-lft-identity 15.3

   \[\leadsto x \cdot \log \left(\frac{\color{blue}{1 \cdot x}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right) - z\]

5. Applied times-frac 15.3

   \[\leadsto x \cdot \log \color{blue}{\left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{x}{\sqrt[3]{y}}\right)} - z\]

6. Applied log-prod 4.7

   \[\leadsto x \cdot \color{blue}{\left(\log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \log \left(\frac{x}{\sqrt[3]{y}}\right)\right)} - z\]

7. Applied distribute-lft-in 4.8

   \[\leadsto \color{blue}{\left(x \cdot \log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + x \cdot \log \left(\frac{x}{\sqrt[3]{y}}\right)\right)} - z\]

8. Applied associate--l+ 4.8

   \[\leadsto \color{blue}{x \cdot \log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(x \cdot \log \left(\frac{x}{\sqrt[3]{y}}\right) - z\right)}\]
9. Using strategy rm

10. Applied add-cube-cbrt 4.8

    \[\leadsto x \cdot \log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(x \cdot \log \left(\frac{x}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}}\right) - z\right)\]

11. Applied add-cube-cbrt 4.8

    \[\leadsto x \cdot \log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(x \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}}\right) - z\right)\]

12. Applied times-frac 4.8

    \[\leadsto x \cdot \log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(x \cdot \log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)} - z\right)\]

13. Applied log-prod 0.3

    \[\leadsto x \cdot \log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(x \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)} - z\right)\]

14. Simplified 0.3

    \[\leadsto x \cdot \log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(x \cdot \left(\color{blue}{\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right)} + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right) - z\right)\]

15. Final simplification 0.3

    \[\leadsto x \cdot \log \left(\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\right) + \left(\left(\left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right) + \log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}}}\right)\right) \cdot x - z\right)\]

Reproduce

herbie shell --seed 2019172 +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))