Average Error: 7.1 → 0.3
Time: 38.5s
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 r53160 = x;
        double r53161 = 1.0;
        double r53162 = r53160 - r53161;
        double r53163 = y;
        double r53164 = log(r53163);
        double r53165 = r53162 * r53164;
        double r53166 = z;
        double r53167 = r53166 - r53161;
        double r53168 = r53161 - r53163;
        double r53169 = log(r53168);
        double r53170 = r53167 * r53169;
        double r53171 = r53165 + r53170;
        double r53172 = t;
        double r53173 = r53171 - r53172;
        return r53173;
}
↓
double f(double x, double y, double z, double t) {
        double r53174 = x;
        double r53175 = 1.0;
        double r53176 = r53174 - r53175;
        double r53177 = y;
        double r53178 = log(r53177);
        double r53179 = z;
        double r53180 = r53179 - r53175;
        double r53181 = log(r53175);
        double r53182 = 0.5;
        double r53183 = 2.0;
        double r53184 = pow(r53177, r53183);
        double r53185 = pow(r53175, r53183);
        double r53186 = r53184 / r53185;
        double r53187 = r53182 * r53186;
        double r53188 = fma(r53175, r53177, r53187);
        double r53189 = r53181 - r53188;
        double r53190 = r53180 * r53189;
        double r53191 = fma(r53176, r53178, r53190);
        double r53192 = t;
        double r53193 = r53191 - r53192;
        return r53193;
}

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

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

  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 2019303 +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))