Average Error: 7.1 → 0.4
Time: 34.0s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 1\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(x - 1\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 1\right)\right) + \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 r52927 = x;
        double r52928 = 1.0;
        double r52929 = r52927 - r52928;
        double r52930 = y;
        double r52931 = log(r52930);
        double r52932 = r52929 * r52931;
        double r52933 = z;
        double r52934 = r52933 - r52928;
        double r52935 = r52928 - r52930;
        double r52936 = log(r52935);
        double r52937 = r52934 * r52936;
        double r52938 = r52932 + r52937;
        double r52939 = t;
        double r52940 = r52938 - r52939;
        return r52940;
}
↓
double f(double x, double y, double z, double t) {
        double r52941 = x;
        double r52942 = 1.0;
        double r52943 = r52941 - r52942;
        double r52944 = 2.0;
        double r52945 = y;
        double r52946 = cbrt(r52945);
        double r52947 = log(r52946);
        double r52948 = r52944 * r52947;
        double r52949 = r52943 * r52948;
        double r52950 = 1.0;
        double r52951 = r52950 / r52945;
        double r52952 = -0.3333333333333333;
        double r52953 = pow(r52951, r52952);
        double r52954 = log(r52953);
        double r52955 = r52954 * r52943;
        double r52956 = r52949 + r52955;
        double r52957 = z;
        double r52958 = r52957 - r52942;
        double r52959 = log(r52942);
        double r52960 = 0.5;
        double r52961 = pow(r52945, r52944);
        double r52962 = pow(r52942, r52944);
        double r52963 = r52961 / r52962;
        double r52964 = r52960 * r52963;
        double r52965 = fma(r52942, r52945, r52964);
        double r52966 = r52959 - r52965;
        double r52967 = r52958 * r52966;
        double r52968 = r52956 + r52967;
        double r52969 = t;
        double r52970 = r52968 - r52969;
        return r52970;
}

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. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied add-cube-cbrt0.3

    \[\leadsto \left(\left(x - 1\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

  6. Applied log-prod0.4

    \[\leadsto \left(\left(x - 1\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

  7. Applied distribute-lft-in0.4

    \[\leadsto \left(\color{blue}{\left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

  8. Simplified0.4

    \[\leadsto \left(\left(\color{blue}{\left(x - 1\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

  9. Simplified0.4

    \[\leadsto \left(\left(\left(x - 1\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)}\right) + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

  10. Taylor expanded around inf 0.4

    \[\leadsto \left(\left(\left(x - 1\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \log \color{blue}{\left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)} \cdot \left(x - 1\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

  11. Final simplification0.4

    \[\leadsto \left(\left(\left(x - 1\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 1\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019303 +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))