Average Error: 6.6 → 0.4
Time: 30.5s
Precision: 64
\[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \left(\left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y}\right) + \left(\left(\log 1.0 - y \cdot 1.0\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) \cdot \left(z - 1.0\right)\right)\right) - t\]
\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t
↓
\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \left(\left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y}\right) + \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) \cdot \left(z - 1.0\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r3145892 = x;
        double r3145893 = 1.0;
        double r3145894 = r3145892 - r3145893;
        double r3145895 = y;
        double r3145896 = log(r3145895);
        double r3145897 = r3145894 * r3145896;
        double r3145898 = z;
        double r3145899 = r3145898 - r3145893;
        double r3145900 = r3145893 - r3145895;
        double r3145901 = log(r3145900);
        double r3145902 = r3145899 * r3145901;
        double r3145903 = r3145897 + r3145902;
        double r3145904 = t;
        double r3145905 = r3145903 - r3145904;
        return r3145905;
}
↓
double f(double x, double y, double z, double t) {
        double r3145906 = y;
        double r3145907 = cbrt(r3145906);
        double r3145908 = r3145907 * r3145907;
        double r3145909 = log(r3145908);
        double r3145910 = x;
        double r3145911 = 1.0;
        double r3145912 = r3145910 - r3145911;
        double r3145913 = r3145909 * r3145912;
        double r3145914 = log(r3145907);
        double r3145915 = r3145912 * r3145914;
        double r3145916 = log(r3145911);
        double r3145917 = r3145906 * r3145911;
        double r3145918 = r3145916 - r3145917;
        double r3145919 = r3145906 / r3145911;
        double r3145920 = r3145919 * r3145919;
        double r3145921 = 0.5;
        double r3145922 = r3145920 * r3145921;
        double r3145923 = r3145918 - r3145922;
        double r3145924 = z;
        double r3145925 = r3145924 - r3145911;
        double r3145926 = r3145923 * r3145925;
        double r3145927 = r3145915 + r3145926;
        double r3145928 = r3145913 + r3145927;
        double r3145929 = t;
        double r3145930 = r3145928 - r3145929;
        return r3145930;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 6.6

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

  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. Simplified0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Applied associate-+l+0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019164 
(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))