Average Error: 7.1 → 0.3
Time: 10.2s
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 r51941 = x;
        double r51942 = 1.0;
        double r51943 = r51941 - r51942;
        double r51944 = y;
        double r51945 = log(r51944);
        double r51946 = r51943 * r51945;
        double r51947 = z;
        double r51948 = r51947 - r51942;
        double r51949 = r51942 - r51944;
        double r51950 = log(r51949);
        double r51951 = r51948 * r51950;
        double r51952 = r51946 + r51951;
        double r51953 = t;
        double r51954 = r51952 - r51953;
        return r51954;
}
↓
double f(double x, double y, double z, double t) {
        double r51955 = y;
        double r51956 = log(r51955);
        double r51957 = x;
        double r51958 = 1.0;
        double r51959 = r51957 - r51958;
        double r51960 = z;
        double r51961 = r51960 - r51958;
        double r51962 = log(r51958);
        double r51963 = r51958 * r51955;
        double r51964 = 0.5;
        double r51965 = 2.0;
        double r51966 = pow(r51955, r51965);
        double r51967 = pow(r51958, r51965);
        double r51968 = r51966 / r51967;
        double r51969 = r51964 * r51968;
        double r51970 = r51963 + r51969;
        double r51971 = r51962 - r51970;
        double r51972 = t;
        double r51973 = -r51972;
        double r51974 = fma(r51961, r51971, r51973);
        double r51975 = fma(r51956, r51959, r51974);
        return r51975;
}

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))