Average Error: 7.4 → 0.4
Time: 24.6s
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 r64703 = x;
        double r64704 = 1.0;
        double r64705 = r64703 - r64704;
        double r64706 = y;
        double r64707 = log(r64706);
        double r64708 = r64705 * r64707;
        double r64709 = z;
        double r64710 = r64709 - r64704;
        double r64711 = r64704 - r64706;
        double r64712 = log(r64711);
        double r64713 = r64710 * r64712;
        double r64714 = r64708 + r64713;
        double r64715 = t;
        double r64716 = r64714 - r64715;
        return r64716;
}
↓
double f(double x, double y, double z, double t) {
        double r64717 = x;
        double r64718 = 1.0;
        double r64719 = r64717 - r64718;
        double r64720 = y;
        double r64721 = log(r64720);
        double r64722 = z;
        double r64723 = r64722 - r64718;
        double r64724 = log(r64718);
        double r64725 = 0.5;
        double r64726 = 2.0;
        double r64727 = pow(r64720, r64726);
        double r64728 = pow(r64718, r64726);
        double r64729 = r64727 / r64728;
        double r64730 = r64725 * r64729;
        double r64731 = fma(r64718, r64720, r64730);
        double r64732 = r64724 - r64731;
        double r64733 = r64723 * r64732;
        double r64734 = fma(r64719, r64721, r64733);
        double r64735 = t;
        double r64736 = r64734 - r64735;
        return r64736;
}

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))