Average Error: 6.8 → 0.4
Time: 16.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(x - 1, \log y, 0.5 \cdot {y}^{2} - \mathsf{fma}\left(1, z \cdot y, 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(x - 1, \log y, 0.5 \cdot {y}^{2} - \mathsf{fma}\left(1, z \cdot y, 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r48275 = x;
        double r48276 = 1.0;
        double r48277 = r48275 - r48276;
        double r48278 = y;
        double r48279 = log(r48278);
        double r48280 = r48277 * r48279;
        double r48281 = z;
        double r48282 = r48281 - r48276;
        double r48283 = r48276 - r48278;
        double r48284 = log(r48283);
        double r48285 = r48282 * r48284;
        double r48286 = r48280 + r48285;
        double r48287 = t;
        double r48288 = r48286 - r48287;
        return r48288;
}
↓
double f(double x, double y, double z, double t) {
        double r48289 = x;
        double r48290 = 1.0;
        double r48291 = r48289 - r48290;
        double r48292 = y;
        double r48293 = log(r48292);
        double r48294 = 0.5;
        double r48295 = 2.0;
        double r48296 = pow(r48292, r48295);
        double r48297 = r48294 * r48296;
        double r48298 = z;
        double r48299 = r48298 * r48292;
        double r48300 = r48298 * r48296;
        double r48301 = r48294 * r48300;
        double r48302 = fma(r48290, r48299, r48301);
        double r48303 = r48297 - r48302;
        double r48304 = fma(r48291, r48293, r48303);
        double r48305 = t;
        double r48306 = r48304 - r48305;
        return r48306;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 6.8

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

  2. Simplified6.8

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

  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

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

  5. Taylor expanded around inf 0.4

    \[\leadsto \mathsf{fma}\left(x - 1, \log y, \color{blue}{0.5 \cdot {y}^{2} - \left(1 \cdot \left(z \cdot y\right) + 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)}\right) - t\]

  6. Simplified0.4

    \[\leadsto \mathsf{fma}\left(x - 1, \log y, \color{blue}{0.5 \cdot {y}^{2} - \mathsf{fma}\left(1, z \cdot y, 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)}\right) - t\]

  7. Final simplification0.4

    \[\leadsto \mathsf{fma}\left(x - 1, \log y, 0.5 \cdot {y}^{2} - \mathsf{fma}\left(1, z \cdot y, 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020045 +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))