Average Error: 7.1 → 0.3
Time: 7.5s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log y, x - 1, \mathsf{fma}\left(z - 1, \log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), -t\right)\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log y, x - 1, \mathsf{fma}\left(z - 1, \log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right), -t\right)\right)
double f(double x, double y, double z, double t) {
        double r51963 = x;
        double r51964 = 1.0;
        double r51965 = r51963 - r51964;
        double r51966 = y;
        double r51967 = log(r51966);
        double r51968 = r51965 * r51967;
        double r51969 = z;
        double r51970 = r51969 - r51964;
        double r51971 = r51964 - r51966;
        double r51972 = log(r51971);
        double r51973 = r51970 * r51972;
        double r51974 = r51968 + r51973;
        double r51975 = t;
        double r51976 = r51974 - r51975;
        return r51976;
}
↓
double f(double x, double y, double z, double t) {
        double r51977 = y;
        double r51978 = log(r51977);
        double r51979 = x;
        double r51980 = 1.0;
        double r51981 = r51979 - r51980;
        double r51982 = z;
        double r51983 = r51982 - r51980;
        double r51984 = log(r51980);
        double r51985 = r51980 * r51977;
        double r51986 = 0.5;
        double r51987 = 2.0;
        double r51988 = pow(r51977, r51987);
        double r51989 = pow(r51980, r51987);
        double r51990 = r51988 / r51989;
        double r51991 = r51986 * r51990;
        double r51992 = r51985 + r51991;
        double r51993 = r51984 - r51992;
        double r51994 = t;
        double r51995 = -r51994;
        double r51996 = fma(r51983, r51993, r51995);
        double r51997 = fma(r51978, r51981, r51996);
        return r51997;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.1

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

  2. Simplified7.0

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

  3. Taylor expanded around 0 0.3

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

  5. Applied fma-neg0.3

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

  6. Final simplification0.3

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

Reproduce

herbie shell --seed 2020036 +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))