Average Error: 7.0 → 0.4
Time: 9.6s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(x - 1\right) \cdot \log y + \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
double f(double x, double y, double z, double t) {
        double r61891 = x;
        double r61892 = 1.0;
        double r61893 = r61891 - r61892;
        double r61894 = y;
        double r61895 = log(r61894);
        double r61896 = r61893 * r61895;
        double r61897 = z;
        double r61898 = r61897 - r61892;
        double r61899 = r61892 - r61894;
        double r61900 = log(r61899);
        double r61901 = r61898 * r61900;
        double r61902 = r61896 + r61901;
        double r61903 = t;
        double r61904 = r61902 - r61903;
        return r61904;
}
↓
double f(double x, double y, double z, double t) {
        double r61905 = x;
        double r61906 = 1.0;
        double r61907 = r61905 - r61906;
        double r61908 = y;
        double r61909 = log(r61908);
        double r61910 = r61907 * r61909;
        double r61911 = z;
        double r61912 = r61911 - r61906;
        double r61913 = log(r61906);
        double r61914 = r61906 * r61908;
        double r61915 = 0.5;
        double r61916 = 2.0;
        double r61917 = pow(r61908, r61916);
        double r61918 = pow(r61906, r61916);
        double r61919 = r61917 / r61918;
        double r61920 = r61915 * r61919;
        double r61921 = r61914 + r61920;
        double r61922 = r61913 - r61921;
        double r61923 = r61912 * r61922;
        double r61924 = r61910 + r61923;
        double r61925 = t;
        double r61926 = r61924 - r61925;
        return r61926;
}

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 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 0.4

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

  3. Final simplification0.4

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

Reproduce

herbie shell --seed 2019347 
(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))