Average Error: 7.2 → 0.3
Time: 34.8s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(x - 1, \log y, \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\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 r124743 = x;
        double r124744 = 1.0;
        double r124745 = r124743 - r124744;
        double r124746 = y;
        double r124747 = log(r124746);
        double r124748 = r124745 * r124747;
        double r124749 = z;
        double r124750 = r124749 - r124744;
        double r124751 = r124744 - r124746;
        double r124752 = log(r124751);
        double r124753 = r124750 * r124752;
        double r124754 = r124748 + r124753;
        double r124755 = t;
        double r124756 = r124754 - r124755;
        return r124756;
}
↓
double f(double x, double y, double z, double t) {
        double r124757 = x;
        double r124758 = 1.0;
        double r124759 = r124757 - r124758;
        double r124760 = y;
        double r124761 = log(r124760);
        double r124762 = z;
        double r124763 = r124762 - r124758;
        double r124764 = log(r124758);
        double r124765 = 0.5;
        double r124766 = 2.0;
        double r124767 = pow(r124760, r124766);
        double r124768 = pow(r124758, r124766);
        double r124769 = r124767 / r124768;
        double r124770 = r124765 * r124769;
        double r124771 = fma(r124758, r124760, r124770);
        double r124772 = r124764 - r124771;
        double r124773 = r124763 * r124772;
        double r124774 = fma(r124759, r124761, r124773);
        double r124775 = t;
        double r124776 = r124774 - r124775;
        return r124776;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.2

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

  2. Simplified7.2

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019350 +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))