Average Error: 6.7 → 0.3
Time: 33.0s
Precision: 64
\[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log 1.0 - \mathsf{fma}\left(\frac{1}{2}, \frac{y}{1.0} \cdot \frac{y}{1.0}, y \cdot 1.0\right), z - 1.0, \log y \cdot \left(x - 1.0\right) - t\right)\]
\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t
↓
\mathsf{fma}\left(\log 1.0 - \mathsf{fma}\left(\frac{1}{2}, \frac{y}{1.0} \cdot \frac{y}{1.0}, y \cdot 1.0\right), z - 1.0, \log y \cdot \left(x - 1.0\right) - t\right)
double f(double x, double y, double z, double t) {
        double r2013436 = x;
        double r2013437 = 1.0;
        double r2013438 = r2013436 - r2013437;
        double r2013439 = y;
        double r2013440 = log(r2013439);
        double r2013441 = r2013438 * r2013440;
        double r2013442 = z;
        double r2013443 = r2013442 - r2013437;
        double r2013444 = r2013437 - r2013439;
        double r2013445 = log(r2013444);
        double r2013446 = r2013443 * r2013445;
        double r2013447 = r2013441 + r2013446;
        double r2013448 = t;
        double r2013449 = r2013447 - r2013448;
        return r2013449;
}
↓
double f(double x, double y, double z, double t) {
        double r2013450 = 1.0;
        double r2013451 = log(r2013450);
        double r2013452 = 0.5;
        double r2013453 = y;
        double r2013454 = r2013453 / r2013450;
        double r2013455 = r2013454 * r2013454;
        double r2013456 = r2013453 * r2013450;
        double r2013457 = fma(r2013452, r2013455, r2013456);
        double r2013458 = r2013451 - r2013457;
        double r2013459 = z;
        double r2013460 = r2013459 - r2013450;
        double r2013461 = log(r2013453);
        double r2013462 = x;
        double r2013463 = r2013462 - r2013450;
        double r2013464 = r2013461 * r2013463;
        double r2013465 = t;
        double r2013466 = r2013464 - r2013465;
        double r2013467 = fma(r2013458, r2013460, r2013466);
        return r2013467;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 6.7

    \[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]

  2. Simplified6.7

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log \left(1.0 - y\right), z - 1.0, \left(x - 1.0\right) \cdot \log y - t\right)}\]

  3. Taylor expanded around 0 0.3

    \[\leadsto \mathsf{fma}\left(\color{blue}{\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)}, z - 1.0, \left(x - 1.0\right) \cdot \log y - t\right)\]

  4. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\color{blue}{\log 1.0 - \mathsf{fma}\left(\frac{1}{2}, \frac{y}{1.0} \cdot \frac{y}{1.0}, 1.0 \cdot y\right)}, z - 1.0, \left(x - 1.0\right) \cdot \log y - t\right)\]

  5. Final simplification0.3

    \[\leadsto \mathsf{fma}\left(\log 1.0 - \mathsf{fma}\left(\frac{1}{2}, \frac{y}{1.0} \cdot \frac{y}{1.0}, y \cdot 1.0\right), z - 1.0, \log y \cdot \left(x - 1.0\right) - t\right)\]

Reproduce

herbie shell --seed 2019162 +o rules:numerics
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))