Average Error: 7.0 → 0.3
Time: 37.7s
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 r63651 = x;
        double r63652 = 1.0;
        double r63653 = r63651 - r63652;
        double r63654 = y;
        double r63655 = log(r63654);
        double r63656 = r63653 * r63655;
        double r63657 = z;
        double r63658 = r63657 - r63652;
        double r63659 = r63652 - r63654;
        double r63660 = log(r63659);
        double r63661 = r63658 * r63660;
        double r63662 = r63656 + r63661;
        double r63663 = t;
        double r63664 = r63662 - r63663;
        return r63664;
}
↓
double f(double x, double y, double z, double t) {
        double r63665 = x;
        double r63666 = 1.0;
        double r63667 = r63665 - r63666;
        double r63668 = y;
        double r63669 = log(r63668);
        double r63670 = z;
        double r63671 = r63670 - r63666;
        double r63672 = log(r63666);
        double r63673 = 0.5;
        double r63674 = 2.0;
        double r63675 = pow(r63668, r63674);
        double r63676 = pow(r63666, r63674);
        double r63677 = r63675 / r63676;
        double r63678 = r63673 * r63677;
        double r63679 = fma(r63666, r63668, r63678);
        double r63680 = r63672 - r63679;
        double r63681 = r63671 * r63680;
        double r63682 = fma(r63667, r63669, r63681);
        double r63683 = t;
        double r63684 = r63682 - r63683;
        return r63684;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.0

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

  2. Simplified7.0

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

  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 2019306 +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))