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

Average Error: 6.7 → 0.4

Time: 33.8s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r3119106 = x;
        double r3119107 = 1.0;
        double r3119108 = r3119106 - r3119107;
        double r3119109 = y;
        double r3119110 = log(r3119109);
        double r3119111 = r3119108 * r3119110;
        double r3119112 = z;
        double r3119113 = r3119112 - r3119107;
        double r3119114 = r3119107 - r3119109;
        double r3119115 = log(r3119114);
        double r3119116 = r3119113 * r3119115;
        double r3119117 = r3119111 + r3119116;
        double r3119118 = t;
        double r3119119 = r3119117 - r3119118;
        return r3119119;
}

↓

double f(double x, double y, double z, double t) {
        double r3119120 = -0.5;
        double r3119121 = y;
        double r3119122 = 1.0;
        double r3119123 = r3119121 / r3119122;
        double r3119124 = r3119123 * r3119123;
        double r3119125 = r3119120 * r3119124;
        double r3119126 = log(r3119122);
        double r3119127 = r3119121 * r3119122;
        double r3119128 = r3119126 - r3119127;
        double r3119129 = r3119125 + r3119128;
        double r3119130 = z;
        double r3119131 = r3119130 - r3119122;
        double r3119132 = r3119129 * r3119131;
        double r3119133 = x;
        double r3119134 = r3119133 - r3119122;
        double r3119135 = cbrt(r3119121);
        double r3119136 = log(r3119135);
        double r3119137 = r3119134 * r3119136;
        double r3119138 = r3119132 + r3119137;
        double r3119139 = 0.6666666666666666;
        double r3119140 = pow(r3119121, r3119139);
        double r3119141 = log(r3119140);
        double r3119142 = r3119141 * r3119134;
        double r3119143 = r3119138 + r3119142;
        double r3119144 = t;
        double r3119145 = r3119143 - r3119144;
        return r3119145;
}

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 sub-neg 0.3

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

6. Applied distribute-lft-in 0.3

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

7. Simplified 0.3

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

9. Applied add-cube-cbrt 0.3

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

10. Applied log-prod 0.4

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

11. Applied distribute-lft-in 0.4

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

12. Applied associate-+l+ 0.4

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

13. Simplified 0.4

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

15. Applied pow1/3 0.4

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

16. Applied pow1/3 0.4

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

17. Applied pow-prod-up 0.4

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

18. Simplified 0.4

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

19. Final simplification 0.4

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

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))