Average Error: 7.0 → 0.3
Time: 9.3s
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 r63184 = x;
        double r63185 = 1.0;
        double r63186 = r63184 - r63185;
        double r63187 = y;
        double r63188 = log(r63187);
        double r63189 = r63186 * r63188;
        double r63190 = z;
        double r63191 = r63190 - r63185;
        double r63192 = r63185 - r63187;
        double r63193 = log(r63192);
        double r63194 = r63191 * r63193;
        double r63195 = r63189 + r63194;
        double r63196 = t;
        double r63197 = r63195 - r63196;
        return r63197;
}
↓
double f(double x, double y, double z, double t) {
        double r63198 = x;
        double r63199 = 1.0;
        double r63200 = r63198 - r63199;
        double r63201 = y;
        double r63202 = log(r63201);
        double r63203 = r63200 * r63202;
        double r63204 = z;
        double r63205 = r63204 - r63199;
        double r63206 = log(r63199);
        double r63207 = r63199 * r63201;
        double r63208 = 0.5;
        double r63209 = 2.0;
        double r63210 = pow(r63201, r63209);
        double r63211 = pow(r63199, r63209);
        double r63212 = r63210 / r63211;
        double r63213 = r63208 * r63212;
        double r63214 = r63207 + r63213;
        double r63215 = r63206 - r63214;
        double r63216 = r63205 * r63215;
        double r63217 = r63203 + r63216;
        double r63218 = t;
        double r63219 = r63217 - r63218;
        return r63219;
}

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))