Average Error: 7.1 → 0.4
Time: 27.6s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(y, 1, \left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2}\right), z - 1, \log y \cdot \left(x - 1\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(y, 1, \left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2}\right), z - 1, \log y \cdot \left(x - 1\right)\right) - t
double f(double x, double y, double z, double t) {
        double r1862391 = x;
        double r1862392 = 1.0;
        double r1862393 = r1862391 - r1862392;
        double r1862394 = y;
        double r1862395 = log(r1862394);
        double r1862396 = r1862393 * r1862395;
        double r1862397 = z;
        double r1862398 = r1862397 - r1862392;
        double r1862399 = r1862392 - r1862394;
        double r1862400 = log(r1862399);
        double r1862401 = r1862398 * r1862400;
        double r1862402 = r1862396 + r1862401;
        double r1862403 = t;
        double r1862404 = r1862402 - r1862403;
        return r1862404;
}
↓
double f(double x, double y, double z, double t) {
        double r1862405 = 1.0;
        double r1862406 = log(r1862405);
        double r1862407 = y;
        double r1862408 = r1862407 / r1862405;
        double r1862409 = r1862408 * r1862408;
        double r1862410 = 0.5;
        double r1862411 = r1862409 * r1862410;
        double r1862412 = fma(r1862407, r1862405, r1862411);
        double r1862413 = r1862406 - r1862412;
        double r1862414 = z;
        double r1862415 = r1862414 - r1862405;
        double r1862416 = log(r1862407);
        double r1862417 = x;
        double r1862418 = r1862417 - r1862405;
        double r1862419 = r1862416 * r1862418;
        double r1862420 = fma(r1862413, r1862415, r1862419);
        double r1862421 = t;
        double r1862422 = r1862420 - r1862421;
        return r1862422;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.1

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

  2. Simplified7.1

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2019179 +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))