Average Error: 6.5 → 0.4
Time: 39.6s
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 r172011 = x;
        double r172012 = 1.0;
        double r172013 = r172011 - r172012;
        double r172014 = y;
        double r172015 = log(r172014);
        double r172016 = r172013 * r172015;
        double r172017 = z;
        double r172018 = r172017 - r172012;
        double r172019 = r172012 - r172014;
        double r172020 = log(r172019);
        double r172021 = r172018 * r172020;
        double r172022 = r172016 + r172021;
        double r172023 = t;
        double r172024 = r172022 - r172023;
        return r172024;
}
↓
double f(double x, double y, double z, double t) {
        double r172025 = x;
        double r172026 = 1.0;
        double r172027 = r172025 - r172026;
        double r172028 = y;
        double r172029 = log(r172028);
        double r172030 = z;
        double r172031 = r172030 - r172026;
        double r172032 = log(r172026);
        double r172033 = 0.5;
        double r172034 = 2.0;
        double r172035 = pow(r172028, r172034);
        double r172036 = pow(r172026, r172034);
        double r172037 = r172035 / r172036;
        double r172038 = r172033 * r172037;
        double r172039 = fma(r172026, r172028, r172038);
        double r172040 = r172032 - r172039;
        double r172041 = r172031 * r172040;
        double r172042 = fma(r172027, r172029, r172041);
        double r172043 = t;
        double r172044 = r172042 - r172043;
        return r172044;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 6.5

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

  2. Simplified6.5

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

  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 2020043 +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))