Average Error: 6.4 → 0.4
Time: 1.0m
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 r155821 = x;
        double r155822 = 1.0;
        double r155823 = r155821 - r155822;
        double r155824 = y;
        double r155825 = log(r155824);
        double r155826 = r155823 * r155825;
        double r155827 = z;
        double r155828 = r155827 - r155822;
        double r155829 = r155822 - r155824;
        double r155830 = log(r155829);
        double r155831 = r155828 * r155830;
        double r155832 = r155826 + r155831;
        double r155833 = t;
        double r155834 = r155832 - r155833;
        return r155834;
}
↓
double f(double x, double y, double z, double t) {
        double r155835 = x;
        double r155836 = 1.0;
        double r155837 = r155835 - r155836;
        double r155838 = y;
        double r155839 = log(r155838);
        double r155840 = z;
        double r155841 = r155840 - r155836;
        double r155842 = log(r155836);
        double r155843 = 0.5;
        double r155844 = 2.0;
        double r155845 = pow(r155838, r155844);
        double r155846 = pow(r155836, r155844);
        double r155847 = r155845 / r155846;
        double r155848 = r155843 * r155847;
        double r155849 = fma(r155836, r155838, r155848);
        double r155850 = r155842 - r155849;
        double r155851 = r155841 * r155850;
        double r155852 = fma(r155837, r155839, r155851);
        double r155853 = t;
        double r155854 = r155852 - r155853;
        return r155854;
}

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))