Average Error: 7.0 → 0.3
Time: 9.6s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log y, x - 1, \mathsf{fma}\left(z - 1, \log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), -t\right)\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log y, x - 1, \mathsf{fma}\left(z - 1, \log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), -t\right)\right)
double f(double x, double y, double z, double t) {
        double r60260 = x;
        double r60261 = 1.0;
        double r60262 = r60260 - r60261;
        double r60263 = y;
        double r60264 = log(r60263);
        double r60265 = r60262 * r60264;
        double r60266 = z;
        double r60267 = r60266 - r60261;
        double r60268 = r60261 - r60263;
        double r60269 = log(r60268);
        double r60270 = r60267 * r60269;
        double r60271 = r60265 + r60270;
        double r60272 = t;
        double r60273 = r60271 - r60272;
        return r60273;
}
↓
double f(double x, double y, double z, double t) {
        double r60274 = y;
        double r60275 = log(r60274);
        double r60276 = x;
        double r60277 = 1.0;
        double r60278 = r60276 - r60277;
        double r60279 = z;
        double r60280 = r60279 - r60277;
        double r60281 = log(r60277);
        double r60282 = r60277 * r60274;
        double r60283 = 0.5;
        double r60284 = 2.0;
        double r60285 = pow(r60274, r60284);
        double r60286 = pow(r60277, r60284);
        double r60287 = r60285 / r60286;
        double r60288 = r60283 * r60287;
        double r60289 = r60282 + r60288;
        double r60290 = r60281 - r60289;
        double r60291 = t;
        double r60292 = -r60291;
        double r60293 = fma(r60280, r60290, r60292);
        double r60294 = fma(r60275, r60278, r60293);
        return r60294;
}

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

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

  3. Taylor expanded around 0 0.3

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

  5. Applied fma-neg0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020001 +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))