Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2

Average Error: 6.7 → 0.4

Time: 32.1s

Precision: 64

Math

\[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]

↓

\[\left(\left(\left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{1}{2} \cdot \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right)\right) \cdot \left(z - 1.0\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1.0\right)\right) + \log \left({y}^{\frac{2}{3}}\right) \cdot \left(x - 1.0\right)\right) - t\]

C

double f(double x, double y, double z, double t) {
        double r2852834 = x;
        double r2852835 = 1.0;
        double r2852836 = r2852834 - r2852835;
        double r2852837 = y;
        double r2852838 = log(r2852837);
        double r2852839 = r2852836 * r2852838;
        double r2852840 = z;
        double r2852841 = r2852840 - r2852835;
        double r2852842 = r2852835 - r2852837;
        double r2852843 = log(r2852842);
        double r2852844 = r2852841 * r2852843;
        double r2852845 = r2852839 + r2852844;
        double r2852846 = t;
        double r2852847 = r2852845 - r2852846;
        return r2852847;
}

↓

double f(double x, double y, double z, double t) {
        double r2852848 = 1.0;
        double r2852849 = log(r2852848);
        double r2852850 = y;
        double r2852851 = r2852848 * r2852850;
        double r2852852 = r2852849 - r2852851;
        double r2852853 = 0.5;
        double r2852854 = r2852850 / r2852848;
        double r2852855 = r2852854 * r2852854;
        double r2852856 = r2852853 * r2852855;
        double r2852857 = r2852852 - r2852856;
        double r2852858 = z;
        double r2852859 = r2852858 - r2852848;
        double r2852860 = r2852857 * r2852859;
        double r2852861 = cbrt(r2852850);
        double r2852862 = log(r2852861);
        double r2852863 = x;
        double r2852864 = r2852863 - r2852848;
        double r2852865 = r2852862 * r2852864;
        double r2852866 = r2852860 + r2852865;
        double r2852867 = 0.6666666666666666;
        double r2852868 = pow(r2852850, r2852867);
        double r2852869 = log(r2852868);
        double r2852870 = r2852869 * r2852864;
        double r2852871 = r2852866 + r2852870;
        double r2852872 = t;
        double r2852873 = r2852871 - r2852872;
        return r2852873;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Initial program 6.7

   \[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]

2. Taylor expanded around 0 0.3

   \[\leadsto \left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)\right)}\right) - t\]

3. Simplified 0.3

   \[\leadsto \left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \color{blue}{\left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)}\right) - t\]
4. Using strategy rm

5. Applied add-cube-cbrt 0.3

   \[\leadsto \left(\left(x - 1.0\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) - t\]

6. Applied log-prod 0.4

   \[\leadsto \left(\left(x - 1.0\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) - t\]

7. Applied distribute-rgt-in 0.4

   \[\leadsto \left(\color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1.0\right)\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) - t\]

8. Applied associate-+l+ 0.4

   \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right)} - t\]
9. Using strategy rm

10. Applied pow1/3 0.4

    \[\leadsto \left(\log \left(\sqrt[3]{y} \cdot \color{blue}{{y}^{\frac{1}{3}}}\right) \cdot \left(x - 1.0\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

11. Applied pow1/3 0.4

    \[\leadsto \left(\log \left(\color{blue}{{y}^{\frac{1}{3}}} \cdot {y}^{\frac{1}{3}}\right) \cdot \left(x - 1.0\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

12. Applied pow-prod-up 0.4

    \[\leadsto \left(\log \color{blue}{\left({y}^{\left(\frac{1}{3} + \frac{1}{3}\right)}\right)} \cdot \left(x - 1.0\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

13. Simplified 0.4

    \[\leadsto \left(\log \left({y}^{\color{blue}{\frac{2}{3}}}\right) \cdot \left(x - 1.0\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

14. Final simplification 0.4

    \[\leadsto \left(\left(\left(\left(\log 1.0 - 1.0 \cdot y\right) - \frac{1}{2} \cdot \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right)\right) \cdot \left(z - 1.0\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1.0\right)\right) + \log \left({y}^{\frac{2}{3}}\right) \cdot \left(x - 1.0\right)\right) - t\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))