Average Error: 7.2 → 0.3
Time: 48.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(\frac{1}{2}, \frac{y}{1} \cdot \frac{y}{1}, 1 \cdot y\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(\frac{1}{2}, \frac{y}{1} \cdot \frac{y}{1}, 1 \cdot y\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r4995383 = x;
        double r4995384 = 1.0;
        double r4995385 = r4995383 - r4995384;
        double r4995386 = y;
        double r4995387 = log(r4995386);
        double r4995388 = r4995385 * r4995387;
        double r4995389 = z;
        double r4995390 = r4995389 - r4995384;
        double r4995391 = r4995384 - r4995386;
        double r4995392 = log(r4995391);
        double r4995393 = r4995390 * r4995392;
        double r4995394 = r4995388 + r4995393;
        double r4995395 = t;
        double r4995396 = r4995394 - r4995395;
        return r4995396;
}
↓
double f(double x, double y, double z, double t) {
        double r4995397 = x;
        double r4995398 = 1.0;
        double r4995399 = r4995397 - r4995398;
        double r4995400 = y;
        double r4995401 = log(r4995400);
        double r4995402 = z;
        double r4995403 = r4995402 - r4995398;
        double r4995404 = log(r4995398);
        double r4995405 = 0.5;
        double r4995406 = r4995400 / r4995398;
        double r4995407 = r4995406 * r4995406;
        double r4995408 = r4995398 * r4995400;
        double r4995409 = fma(r4995405, r4995407, r4995408);
        double r4995410 = r4995404 - r4995409;
        double r4995411 = r4995403 * r4995410;
        double r4995412 = fma(r4995399, r4995401, r4995411);
        double r4995413 = t;
        double r4995414 = r4995412 - r4995413;
        return r4995414;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.2

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

  2. Simplified7.2

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019174 +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))