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

Average Error: 6.9 → 0.4

Time: 30.0s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r2223739 = x;
        double r2223740 = 1.0;
        double r2223741 = r2223739 - r2223740;
        double r2223742 = y;
        double r2223743 = log(r2223742);
        double r2223744 = r2223741 * r2223743;
        double r2223745 = z;
        double r2223746 = r2223745 - r2223740;
        double r2223747 = r2223740 - r2223742;
        double r2223748 = log(r2223747);
        double r2223749 = r2223746 * r2223748;
        double r2223750 = r2223744 + r2223749;
        double r2223751 = t;
        double r2223752 = r2223750 - r2223751;
        return r2223752;
}

↓

double f(double x, double y, double z, double t) {
        double r2223753 = y;
        double r2223754 = 0.6666666666666666;
        double r2223755 = pow(r2223753, r2223754);
        double r2223756 = log(r2223755);
        double r2223757 = x;
        double r2223758 = 1.0;
        double r2223759 = r2223757 - r2223758;
        double r2223760 = r2223756 * r2223759;
        double r2223761 = cbrt(r2223753);
        double r2223762 = log(r2223761);
        double r2223763 = r2223762 * r2223759;
        double r2223764 = r2223760 + r2223763;
        double r2223765 = z;
        double r2223766 = r2223765 - r2223758;
        double r2223767 = log(r2223758);
        double r2223768 = r2223753 / r2223758;
        double r2223769 = r2223768 * r2223768;
        double r2223770 = 0.5;
        double r2223771 = r2223769 * r2223770;
        double r2223772 = r2223767 - r2223771;
        double r2223773 = r2223758 * r2223753;
        double r2223774 = r2223772 - r2223773;
        double r2223775 = r2223766 * r2223774;
        double r2223776 = r2223764 + r2223775;
        double r2223777 = t;
        double r2223778 = r2223776 - r2223777;
        return r2223778;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

1. Initial program 6.9

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

2. Taylor expanded around 0 0.4

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

3. Simplified 0.4

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

5. Applied add-cube-cbrt 0.4

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

6. Applied log-prod 0.4

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

7. Applied distribute-rgt-in 0.4

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

9. Applied pow1/3 0.4

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

10. Applied pow1/3 0.4

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

11. Applied pow-prod-up 0.4

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

12. Simplified 0.4

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

13. Final simplification 0.4

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

Reproduce

herbie shell --seed 2019158 
(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))