Average Error: 7.3 → 0.3
Time: 25.2s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(y, 1, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), z - 1, \log y \cdot \left(x - 1\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\mathsf{fma}\left(\log 1 - \mathsf{fma}\left(y, 1, \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right), z - 1, \log y \cdot \left(x - 1\right)\right) - t
double f(double x, double y, double z, double t) {
        double r1805937 = x;
        double r1805938 = 1.0;
        double r1805939 = r1805937 - r1805938;
        double r1805940 = y;
        double r1805941 = log(r1805940);
        double r1805942 = r1805939 * r1805941;
        double r1805943 = z;
        double r1805944 = r1805943 - r1805938;
        double r1805945 = r1805938 - r1805940;
        double r1805946 = log(r1805945);
        double r1805947 = r1805944 * r1805946;
        double r1805948 = r1805942 + r1805947;
        double r1805949 = t;
        double r1805950 = r1805948 - r1805949;
        return r1805950;
}
↓
double f(double x, double y, double z, double t) {
        double r1805951 = 1.0;
        double r1805952 = log(r1805951);
        double r1805953 = y;
        double r1805954 = 0.5;
        double r1805955 = r1805951 / r1805953;
        double r1805956 = r1805955 * r1805955;
        double r1805957 = r1805954 / r1805956;
        double r1805958 = fma(r1805953, r1805951, r1805957);
        double r1805959 = r1805952 - r1805958;
        double r1805960 = z;
        double r1805961 = r1805960 - r1805951;
        double r1805962 = log(r1805953);
        double r1805963 = x;
        double r1805964 = r1805963 - r1805951;
        double r1805965 = r1805962 * r1805964;
        double r1805966 = fma(r1805959, r1805961, r1805965);
        double r1805967 = t;
        double r1805968 = r1805966 - r1805967;
        return r1805968;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 7.3

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

  2. Simplified7.3

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

  3. Taylor expanded around 0 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))