Average Error: 6.6 → 0.3
Time: 11.0s
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 r52696 = x;
        double r52697 = 1.0;
        double r52698 = r52696 - r52697;
        double r52699 = y;
        double r52700 = log(r52699);
        double r52701 = r52698 * r52700;
        double r52702 = z;
        double r52703 = r52702 - r52697;
        double r52704 = r52697 - r52699;
        double r52705 = log(r52704);
        double r52706 = r52703 * r52705;
        double r52707 = r52701 + r52706;
        double r52708 = t;
        double r52709 = r52707 - r52708;
        return r52709;
}
↓
double f(double x, double y, double z, double t) {
        double r52710 = x;
        double r52711 = 1.0;
        double r52712 = r52710 - r52711;
        double r52713 = y;
        double r52714 = log(r52713);
        double r52715 = r52712 * r52714;
        double r52716 = z;
        double r52717 = r52716 - r52711;
        double r52718 = log(r52711);
        double r52719 = r52711 * r52713;
        double r52720 = 0.5;
        double r52721 = 2.0;
        double r52722 = pow(r52713, r52721);
        double r52723 = pow(r52711, r52721);
        double r52724 = r52722 / r52723;
        double r52725 = r52720 * r52724;
        double r52726 = r52719 + r52725;
        double r52727 = r52718 - r52726;
        double r52728 = r52717 * r52727;
        double r52729 = r52715 + r52728;
        double r52730 = t;
        double r52731 = r52729 - r52730;
        return r52731;
}

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

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

  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 2020033 +o rules:numerics
(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))