Average Error: 7.2 → 0.4
Time: 8.8s
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 r64803 = x;
        double r64804 = 1.0;
        double r64805 = r64803 - r64804;
        double r64806 = y;
        double r64807 = log(r64806);
        double r64808 = r64805 * r64807;
        double r64809 = z;
        double r64810 = r64809 - r64804;
        double r64811 = r64804 - r64806;
        double r64812 = log(r64811);
        double r64813 = r64810 * r64812;
        double r64814 = r64808 + r64813;
        double r64815 = t;
        double r64816 = r64814 - r64815;
        return r64816;
}
↓
double f(double x, double y, double z, double t) {
        double r64817 = x;
        double r64818 = 1.0;
        double r64819 = r64817 - r64818;
        double r64820 = y;
        double r64821 = log(r64820);
        double r64822 = r64819 * r64821;
        double r64823 = z;
        double r64824 = r64823 - r64818;
        double r64825 = log(r64818);
        double r64826 = r64818 * r64820;
        double r64827 = 0.5;
        double r64828 = 2.0;
        double r64829 = pow(r64820, r64828);
        double r64830 = pow(r64818, r64828);
        double r64831 = r64829 / r64830;
        double r64832 = r64827 * r64831;
        double r64833 = r64826 + r64832;
        double r64834 = r64825 - r64833;
        double r64835 = r64824 * r64834;
        double r64836 = r64822 + r64835;
        double r64837 = t;
        double r64838 = r64836 - r64837;
        return r64838;
}

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 7.2

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

  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 2020025 
(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))