Average Error: 7.0 → 0.3
Time: 43.8s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, 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 r64261 = x;
        double r64262 = 1.0;
        double r64263 = r64261 - r64262;
        double r64264 = y;
        double r64265 = log(r64264);
        double r64266 = r64263 * r64265;
        double r64267 = z;
        double r64268 = r64267 - r64262;
        double r64269 = r64262 - r64264;
        double r64270 = log(r64269);
        double r64271 = r64268 * r64270;
        double r64272 = r64266 + r64271;
        double r64273 = t;
        double r64274 = r64272 - r64273;
        return r64274;
}
↓
double f(double x, double y, double z, double t) {
        double r64275 = x;
        double r64276 = 1.0;
        double r64277 = r64275 - r64276;
        double r64278 = y;
        double r64279 = log(r64278);
        double r64280 = z;
        double r64281 = r64280 - r64276;
        double r64282 = log(r64276);
        double r64283 = 0.5;
        double r64284 = 2.0;
        double r64285 = pow(r64278, r64284);
        double r64286 = pow(r64276, r64284);
        double r64287 = r64285 / r64286;
        double r64288 = r64283 * r64287;
        double r64289 = fma(r64276, r64278, r64288);
        double r64290 = r64282 - r64289;
        double r64291 = r64281 * r64290;
        double r64292 = fma(r64277, r64279, r64291);
        double r64293 = t;
        double r64294 = r64292 - r64293;
        return r64294;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.0

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

  2. Simplified7.0

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

  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. Final simplification0.3

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

Reproduce

herbie shell --seed 2019323 +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))