Average Error: 6.8 → 0.4
Time: 40.9s
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 r118859 = x;
        double r118860 = 1.0;
        double r118861 = r118859 - r118860;
        double r118862 = y;
        double r118863 = log(r118862);
        double r118864 = r118861 * r118863;
        double r118865 = z;
        double r118866 = r118865 - r118860;
        double r118867 = r118860 - r118862;
        double r118868 = log(r118867);
        double r118869 = r118866 * r118868;
        double r118870 = r118864 + r118869;
        double r118871 = t;
        double r118872 = r118870 - r118871;
        return r118872;
}
↓
double f(double x, double y, double z, double t) {
        double r118873 = x;
        double r118874 = 1.0;
        double r118875 = r118873 - r118874;
        double r118876 = y;
        double r118877 = log(r118876);
        double r118878 = z;
        double r118879 = r118878 - r118874;
        double r118880 = log(r118874);
        double r118881 = 0.5;
        double r118882 = 2.0;
        double r118883 = pow(r118876, r118882);
        double r118884 = pow(r118874, r118882);
        double r118885 = r118883 / r118884;
        double r118886 = r118881 * r118885;
        double r118887 = fma(r118874, r118876, r118886);
        double r118888 = r118880 - r118887;
        double r118889 = r118879 * r118888;
        double r118890 = fma(r118875, r118877, r118889);
        double r118891 = t;
        double r118892 = r118890 - r118891;
        return r118892;
}

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 0.4

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

  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 2019305 +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))