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

Average Error: 7.0 → 0.4

Time: 14.3s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r50581 = x;
        double r50582 = 1.0;
        double r50583 = r50581 - r50582;
        double r50584 = y;
        double r50585 = log(r50584);
        double r50586 = r50583 * r50585;
        double r50587 = z;
        double r50588 = r50587 - r50582;
        double r50589 = r50582 - r50584;
        double r50590 = log(r50589);
        double r50591 = r50588 * r50590;
        double r50592 = r50586 + r50591;
        double r50593 = t;
        double r50594 = r50592 - r50593;
        return r50594;
}

↓

double f(double x, double y, double z, double t) {
        double r50595 = x;
        double r50596 = 1.0;
        double r50597 = r50595 - r50596;
        double r50598 = y;
        double r50599 = cbrt(r50598);
        double r50600 = r50599 * r50599;
        double r50601 = cbrt(r50600);
        double r50602 = log(r50601);
        double r50603 = 2.0;
        double r50604 = r50602 * r50603;
        double r50605 = cbrt(r50599);
        double r50606 = log(r50605);
        double r50607 = r50606 * r50603;
        double r50608 = log(r50599);
        double r50609 = r50607 + r50608;
        double r50610 = r50604 + r50609;
        double r50611 = r50597 * r50610;
        double r50612 = z;
        double r50613 = r50612 - r50596;
        double r50614 = log(r50596);
        double r50615 = r50596 * r50598;
        double r50616 = 0.5;
        double r50617 = pow(r50598, r50603);
        double r50618 = pow(r50596, r50603);
        double r50619 = r50617 / r50618;
        double r50620 = r50616 * r50619;
        double r50621 = r50615 + r50620;
        double r50622 = r50614 - r50621;
        double r50623 = r50613 * r50622;
        double r50624 = t;
        double r50625 = r50623 - r50624;
        double r50626 = r50611 + r50625;
        return r50626;
}

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. Taylor expanded around 0 0.3

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

4. Applied add-cube-cbrt 0.3

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

5. Applied log-prod 0.4

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

6. Applied distribute-lft-in 0.4

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

7. Simplified 0.4

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

9. Applied add-cube-cbrt 0.4

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

10. Applied cbrt-prod 0.4

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

11. Applied log-prod 0.4

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

12. Applied distribute-rgt-in 0.4

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

13. Applied distribute-lft-in 0.4

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

14. Applied associate-+l+ 0.4

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

15. Simplified 0.4

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

16. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019304 
(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))