Average Error: 6.9 → 0.3
Time: 14.9s
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 r44793 = x;
        double r44794 = 1.0;
        double r44795 = r44793 - r44794;
        double r44796 = y;
        double r44797 = log(r44796);
        double r44798 = r44795 * r44797;
        double r44799 = z;
        double r44800 = r44799 - r44794;
        double r44801 = r44794 - r44796;
        double r44802 = log(r44801);
        double r44803 = r44800 * r44802;
        double r44804 = r44798 + r44803;
        double r44805 = t;
        double r44806 = r44804 - r44805;
        return r44806;
}
↓
double f(double x, double y, double z, double t) {
        double r44807 = x;
        double r44808 = 1.0;
        double r44809 = r44807 - r44808;
        double r44810 = y;
        double r44811 = log(r44810);
        double r44812 = r44809 * r44811;
        double r44813 = z;
        double r44814 = r44813 - r44808;
        double r44815 = log(r44808);
        double r44816 = r44808 * r44810;
        double r44817 = 0.5;
        double r44818 = 2.0;
        double r44819 = pow(r44810, r44818);
        double r44820 = pow(r44808, r44818);
        double r44821 = r44819 / r44820;
        double r44822 = r44817 * r44821;
        double r44823 = r44816 + r44822;
        double r44824 = r44815 - r44823;
        double r44825 = r44814 * r44824;
        double r44826 = r44812 + r44825;
        double r44827 = t;
        double r44828 = r44826 - r44827;
        return r44828;
}

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 6.9

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 1978988140 
(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))