Average Error: 7.0 → 0.4
Time: 9.9s
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 r56312 = x;
        double r56313 = 1.0;
        double r56314 = r56312 - r56313;
        double r56315 = y;
        double r56316 = log(r56315);
        double r56317 = r56314 * r56316;
        double r56318 = z;
        double r56319 = r56318 - r56313;
        double r56320 = r56313 - r56315;
        double r56321 = log(r56320);
        double r56322 = r56319 * r56321;
        double r56323 = r56317 + r56322;
        double r56324 = t;
        double r56325 = r56323 - r56324;
        return r56325;
}
↓
double f(double x, double y, double z, double t) {
        double r56326 = x;
        double r56327 = 1.0;
        double r56328 = r56326 - r56327;
        double r56329 = y;
        double r56330 = log(r56329);
        double r56331 = r56328 * r56330;
        double r56332 = z;
        double r56333 = r56332 - r56327;
        double r56334 = log(r56327);
        double r56335 = r56327 * r56329;
        double r56336 = 0.5;
        double r56337 = 2.0;
        double r56338 = pow(r56329, r56337);
        double r56339 = pow(r56327, r56337);
        double r56340 = r56338 / r56339;
        double r56341 = r56336 * r56340;
        double r56342 = r56335 + r56341;
        double r56343 = r56334 - r56342;
        double r56344 = r56333 * r56343;
        double r56345 = r56331 + r56344;
        double r56346 = t;
        double r56347 = r56345 - r56346;
        return r56347;
}

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 0.4

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

  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 2019347 +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))