Average Error: 6.6 → 0.3
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 r68156 = x;
        double r68157 = 1.0;
        double r68158 = r68156 - r68157;
        double r68159 = y;
        double r68160 = log(r68159);
        double r68161 = r68158 * r68160;
        double r68162 = z;
        double r68163 = r68162 - r68157;
        double r68164 = r68157 - r68159;
        double r68165 = log(r68164);
        double r68166 = r68163 * r68165;
        double r68167 = r68161 + r68166;
        double r68168 = t;
        double r68169 = r68167 - r68168;
        return r68169;
}
↓
double f(double x, double y, double z, double t) {
        double r68170 = x;
        double r68171 = 1.0;
        double r68172 = r68170 - r68171;
        double r68173 = y;
        double r68174 = log(r68173);
        double r68175 = r68172 * r68174;
        double r68176 = z;
        double r68177 = r68176 - r68171;
        double r68178 = log(r68171);
        double r68179 = r68171 * r68173;
        double r68180 = 0.5;
        double r68181 = 2.0;
        double r68182 = pow(r68173, r68181);
        double r68183 = pow(r68171, r68181);
        double r68184 = r68182 / r68183;
        double r68185 = r68180 * r68184;
        double r68186 = r68179 + r68185;
        double r68187 = r68178 - r68186;
        double r68188 = r68177 * r68187;
        double r68189 = r68175 + r68188;
        double r68190 = t;
        double r68191 = r68189 - r68190;
        return r68191;
}

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 2020062 
(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))