Average Error: 6.4 → 0.4
Time: 58.9s
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 r155824 = x;
        double r155825 = 1.0;
        double r155826 = r155824 - r155825;
        double r155827 = y;
        double r155828 = log(r155827);
        double r155829 = r155826 * r155828;
        double r155830 = z;
        double r155831 = r155830 - r155825;
        double r155832 = r155825 - r155827;
        double r155833 = log(r155832);
        double r155834 = r155831 * r155833;
        double r155835 = r155829 + r155834;
        double r155836 = t;
        double r155837 = r155835 - r155836;
        return r155837;
}
↓
double f(double x, double y, double z, double t) {
        double r155838 = x;
        double r155839 = 1.0;
        double r155840 = r155838 - r155839;
        double r155841 = y;
        double r155842 = log(r155841);
        double r155843 = z;
        double r155844 = r155843 - r155839;
        double r155845 = log(r155839);
        double r155846 = 0.5;
        double r155847 = 2.0;
        double r155848 = pow(r155841, r155847);
        double r155849 = pow(r155839, r155847);
        double r155850 = r155848 / r155849;
        double r155851 = r155846 * r155850;
        double r155852 = fma(r155839, r155841, r155851);
        double r155853 = r155845 - r155852;
        double r155854 = r155844 * r155853;
        double r155855 = fma(r155840, r155842, r155854);
        double r155856 = t;
        double r155857 = r155855 - r155856;
        return r155857;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 6.4

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

  2. Simplified6.4

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

  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 2019326 +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))