Average Error: 6.9 → 0.4
Time: 10.3s
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 r61568 = x;
        double r61569 = 1.0;
        double r61570 = r61568 - r61569;
        double r61571 = y;
        double r61572 = log(r61571);
        double r61573 = r61570 * r61572;
        double r61574 = z;
        double r61575 = r61574 - r61569;
        double r61576 = r61569 - r61571;
        double r61577 = log(r61576);
        double r61578 = r61575 * r61577;
        double r61579 = r61573 + r61578;
        double r61580 = t;
        double r61581 = r61579 - r61580;
        return r61581;
}
↓
double f(double x, double y, double z, double t) {
        double r61582 = x;
        double r61583 = 1.0;
        double r61584 = r61582 - r61583;
        double r61585 = y;
        double r61586 = log(r61585);
        double r61587 = r61584 * r61586;
        double r61588 = z;
        double r61589 = r61588 - r61583;
        double r61590 = log(r61583);
        double r61591 = r61583 * r61585;
        double r61592 = 0.5;
        double r61593 = 2.0;
        double r61594 = pow(r61585, r61593);
        double r61595 = pow(r61583, r61593);
        double r61596 = r61594 / r61595;
        double r61597 = r61592 * r61596;
        double r61598 = r61591 + r61597;
        double r61599 = r61590 - r61598;
        double r61600 = r61589 * r61599;
        double r61601 = r61587 + r61600;
        double r61602 = t;
        double r61603 = r61601 - r61602;
        return r61603;
}

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 6.9

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

  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 2020083 +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))