Average Error: 7.0 → 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 r82157 = x;
        double r82158 = 1.0;
        double r82159 = r82157 - r82158;
        double r82160 = y;
        double r82161 = log(r82160);
        double r82162 = r82159 * r82161;
        double r82163 = z;
        double r82164 = r82163 - r82158;
        double r82165 = r82158 - r82160;
        double r82166 = log(r82165);
        double r82167 = r82164 * r82166;
        double r82168 = r82162 + r82167;
        double r82169 = t;
        double r82170 = r82168 - r82169;
        return r82170;
}
↓
double f(double x, double y, double z, double t) {
        double r82171 = x;
        double r82172 = 1.0;
        double r82173 = r82171 - r82172;
        double r82174 = y;
        double r82175 = log(r82174);
        double r82176 = r82173 * r82175;
        double r82177 = z;
        double r82178 = r82177 - r82172;
        double r82179 = log(r82172);
        double r82180 = r82172 * r82174;
        double r82181 = 0.5;
        double r82182 = 2.0;
        double r82183 = pow(r82174, r82182);
        double r82184 = pow(r82172, r82182);
        double r82185 = r82183 / r82184;
        double r82186 = r82181 * r82185;
        double r82187 = r82180 + r82186;
        double r82188 = r82179 - r82187;
        double r82189 = r82178 * r82188;
        double r82190 = r82176 + r82189;
        double r82191 = t;
        double r82192 = r82190 - r82191;
        return r82192;
}

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 0.3

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

  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 2020001 
(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))