Average Error: 7.3 → 0.3
Time: 31.8s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(y, 1, \left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2}\right), z - 1, \left(\left(x - 1\right) \cdot \log \left(\sqrt{y}\right) + \left(x - 1\right) \cdot \log \left(\sqrt{y}\right)\right) - t\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(y, 1, \left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2}\right), z - 1, \left(\left(x - 1\right) \cdot \log \left(\sqrt{y}\right) + \left(x - 1\right) \cdot \log \left(\sqrt{y}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r2101642 = x;
        double r2101643 = 1.0;
        double r2101644 = r2101642 - r2101643;
        double r2101645 = y;
        double r2101646 = log(r2101645);
        double r2101647 = r2101644 * r2101646;
        double r2101648 = z;
        double r2101649 = r2101648 - r2101643;
        double r2101650 = r2101643 - r2101645;
        double r2101651 = log(r2101650);
        double r2101652 = r2101649 * r2101651;
        double r2101653 = r2101647 + r2101652;
        double r2101654 = t;
        double r2101655 = r2101653 - r2101654;
        return r2101655;
}
↓
double f(double x, double y, double z, double t) {
        double r2101656 = 1.0;
        double r2101657 = log(r2101656);
        double r2101658 = y;
        double r2101659 = r2101658 / r2101656;
        double r2101660 = r2101659 * r2101659;
        double r2101661 = 0.5;
        double r2101662 = r2101660 * r2101661;
        double r2101663 = fma(r2101658, r2101656, r2101662);
        double r2101664 = r2101657 - r2101663;
        double r2101665 = z;
        double r2101666 = r2101665 - r2101656;
        double r2101667 = x;
        double r2101668 = r2101667 - r2101656;
        double r2101669 = sqrt(r2101658);
        double r2101670 = log(r2101669);
        double r2101671 = r2101668 * r2101670;
        double r2101672 = r2101671 + r2101671;
        double r2101673 = t;
        double r2101674 = r2101672 - r2101673;
        double r2101675 = fma(r2101664, r2101666, r2101674);
        return r2101675;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.3

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

  2. Simplified7.3

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

    \[\leadsto \mathsf{fma}\left(\color{blue}{\log 1 - \mathsf{fma}\left(y, 1, \left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2}\right)}, z - 1, \left(x - 1\right) \cdot \log y - t\right)\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.3

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

  7. Applied log-prod0.3

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

  8. Applied distribute-lft-in0.3

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

  9. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))