Average Error: 7.1 → 0.6
Time: 25.4s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \mathsf{fma}\left(x - 1, \log \left(\sqrt[3]{y}\right), \mathsf{fma}\left(0.5 \cdot y, y, -\mathsf{fma}\left(y, 0.5 \cdot \left(z \cdot y\right), \left(1 \cdot y\right) \cdot z\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \mathsf{fma}\left(x - 1, \log \left(\sqrt[3]{y}\right), \mathsf{fma}\left(0.5 \cdot y, y, -\mathsf{fma}\left(y, 0.5 \cdot \left(z \cdot y\right), \left(1 \cdot y\right) \cdot z\right)\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r42919 = x;
        double r42920 = 1.0;
        double r42921 = r42919 - r42920;
        double r42922 = y;
        double r42923 = log(r42922);
        double r42924 = r42921 * r42923;
        double r42925 = z;
        double r42926 = r42925 - r42920;
        double r42927 = r42920 - r42922;
        double r42928 = log(r42927);
        double r42929 = r42926 * r42928;
        double r42930 = r42924 + r42929;
        double r42931 = t;
        double r42932 = r42930 - r42931;
        return r42932;
}
↓
double f(double x, double y, double z, double t) {
        double r42933 = y;
        double r42934 = cbrt(r42933);
        double r42935 = r42934 * r42934;
        double r42936 = log(r42935);
        double r42937 = x;
        double r42938 = 1.0;
        double r42939 = r42937 - r42938;
        double r42940 = r42936 * r42939;
        double r42941 = log(r42934);
        double r42942 = 0.5;
        double r42943 = r42942 * r42933;
        double r42944 = z;
        double r42945 = r42944 * r42933;
        double r42946 = r42942 * r42945;
        double r42947 = r42938 * r42933;
        double r42948 = r42947 * r42944;
        double r42949 = fma(r42933, r42946, r42948);
        double r42950 = -r42949;
        double r42951 = fma(r42943, r42933, r42950);
        double r42952 = fma(r42939, r42941, r42951);
        double r42953 = r42940 + r42952;
        double r42954 = t;
        double r42955 = r42953 - r42954;
        return r42955;
}

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 0.4

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

  3. Simplified0.4

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

  4. Taylor expanded around inf 0.5

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(0.5 \cdot {y}^{2} - \left(1 \cdot \left(z \cdot y\right) + 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right)}\right) - t\]

  5. Simplified0.5

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(\left(y \cdot y\right) \cdot 0.5 - \mathsf{fma}\left(z \cdot y, 1, \left(\left(y \cdot y\right) \cdot z\right) \cdot 0.5\right)\right)}\right) - t\]
  6. Using strategy rm

  7. Applied add-cube-cbrt0.5

    \[\leadsto \left(\left(x - 1\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + \left(\left(y \cdot y\right) \cdot 0.5 - \mathsf{fma}\left(z \cdot y, 1, \left(\left(y \cdot y\right) \cdot z\right) \cdot 0.5\right)\right)\right) - t\]

  8. Applied log-prod0.6

    \[\leadsto \left(\left(x - 1\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + \left(\left(y \cdot y\right) \cdot 0.5 - \mathsf{fma}\left(z \cdot y, 1, \left(\left(y \cdot y\right) \cdot z\right) \cdot 0.5\right)\right)\right) - t\]

  9. Applied distribute-rgt-in0.6

    \[\leadsto \left(\color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right)} + \left(\left(y \cdot y\right) \cdot 0.5 - \mathsf{fma}\left(z \cdot y, 1, \left(\left(y \cdot y\right) \cdot z\right) \cdot 0.5\right)\right)\right) - t\]

  10. Applied associate-+l+0.6

    \[\leadsto \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(\left(y \cdot y\right) \cdot 0.5 - \mathsf{fma}\left(z \cdot y, 1, \left(\left(y \cdot y\right) \cdot z\right) \cdot 0.5\right)\right)\right)\right)} - t\]

  11. Simplified0.6

    \[\leadsto \left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \color{blue}{\mathsf{fma}\left(x - 1, \log \left(\sqrt[3]{y}\right), \mathsf{fma}\left(0.5 \cdot y, y, -\mathsf{fma}\left(y, \left(y \cdot z\right) \cdot 0.5, z \cdot \left(y \cdot 1\right)\right)\right)\right)}\right) - t\]

  12. Final simplification0.6

    \[\leadsto \left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \mathsf{fma}\left(x - 1, \log \left(\sqrt[3]{y}\right), \mathsf{fma}\left(0.5 \cdot y, y, -\mathsf{fma}\left(y, 0.5 \cdot \left(z \cdot y\right), \left(1 \cdot y\right) \cdot z\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019179 +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))