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

Average Error: 7.0 → 0.4

Time: 37.6s

Precision: 64

Math

\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]

↓

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

C

double f(double x, double y, double z, double t) {
        double r34692 = x;
        double r34693 = 1.0;
        double r34694 = r34692 - r34693;
        double r34695 = y;
        double r34696 = log(r34695);
        double r34697 = r34694 * r34696;
        double r34698 = z;
        double r34699 = r34698 - r34693;
        double r34700 = r34693 - r34695;
        double r34701 = log(r34700);
        double r34702 = r34699 * r34701;
        double r34703 = r34697 + r34702;
        double r34704 = t;
        double r34705 = r34703 - r34704;
        return r34705;
}

↓

double f(double x, double y, double z, double t) {
        double r34706 = 2.0;
        double r34707 = y;
        double r34708 = cbrt(r34707);
        double r34709 = r34708 * r34708;
        double r34710 = cbrt(r34709);
        double r34711 = log(r34710);
        double r34712 = r34706 * r34711;
        double r34713 = x;
        double r34714 = 1.0;
        double r34715 = r34713 - r34714;
        double r34716 = r34712 * r34715;
        double r34717 = cbrt(r34708);
        double r34718 = log(r34717);
        double r34719 = log(r34708);
        double r34720 = fma(r34718, r34706, r34719);
        double r34721 = r34715 * r34720;
        double r34722 = r34716 + r34721;
        double r34723 = z;
        double r34724 = r34723 - r34714;
        double r34725 = log(r34714);
        double r34726 = 0.5;
        double r34727 = pow(r34707, r34706);
        double r34728 = pow(r34714, r34706);
        double r34729 = r34727 / r34728;
        double r34730 = r34726 * r34729;
        double r34731 = fma(r34714, r34707, r34730);
        double r34732 = r34725 - r34731;
        double r34733 = r34724 * r34732;
        double r34734 = r34722 + r34733;
        double r34735 = t;
        double r34736 = r34734 - r34735;
        return r34736;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Initial program 7.0

   \[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]

2. Simplified 7.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t}\]

3. Taylor expanded around 0 0.3

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

4. Simplified 0.3

   \[\leadsto \mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \color{blue}{\left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]
5. Using strategy rm

6. Applied fma-udef 0.3

   \[\leadsto \color{blue}{\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)} - t\]
7. Using strategy rm

8. Applied add-cube-cbrt 0.3

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

9. Applied log-prod 0.4

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

10. Applied distribute-lft-in 0.4

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

11. Simplified 0.4

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

12. Simplified 0.4

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

14. Applied add-cube-cbrt 0.4

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

15. Applied cbrt-prod 0.4

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

16. Applied log-prod 0.4

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

17. Applied distribute-lft-in 0.4

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

18. Applied distribute-rgt-in 0.4

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

19. Applied associate-+l+ 0.4

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

20. Simplified 0.4

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

21. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1) (log y)) (* (- z 1) (log (- 1 y)))) t))