Average Error: 7.4 → 0.4
Time: 17.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\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 r67251 = x;
        double r67252 = 1.0;
        double r67253 = r67251 - r67252;
        double r67254 = y;
        double r67255 = log(r67254);
        double r67256 = r67253 * r67255;
        double r67257 = z;
        double r67258 = r67257 - r67252;
        double r67259 = r67252 - r67254;
        double r67260 = log(r67259);
        double r67261 = r67258 * r67260;
        double r67262 = r67256 + r67261;
        double r67263 = t;
        double r67264 = r67262 - r67263;
        return r67264;
}
↓
double f(double x, double y, double z, double t) {
        double r67265 = x;
        double r67266 = 1.0;
        double r67267 = r67265 - r67266;
        double r67268 = y;
        double r67269 = log(r67268);
        double r67270 = z;
        double r67271 = r67270 - r67266;
        double r67272 = log(r67266);
        double r67273 = 0.5;
        double r67274 = 2.0;
        double r67275 = pow(r67268, r67274);
        double r67276 = pow(r67266, r67274);
        double r67277 = r67275 / r67276;
        double r67278 = r67273 * r67277;
        double r67279 = fma(r67266, r67268, r67278);
        double r67280 = r67272 - r67279;
        double r67281 = r67271 * r67280;
        double r67282 = fma(r67267, r67269, r67281);
        double r67283 = t;
        double r67284 = r67282 - r67283;
        return r67284;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.4

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

  2. Simplified7.4

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2020047 +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))