Average Error: 7.0 → 0.4
Time: 30.4s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right)\right)\right) + \left(\left(\log 1 - \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right) - 1 \cdot y\right) \cdot \left(z - 1\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right)\right)\right) + \left(\left(\log 1 - \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right) - 1 \cdot y\right) \cdot \left(z - 1\right)\right) - t
double f(double x, double y, double z, double t) {
        double r2387878 = x;
        double r2387879 = 1.0;
        double r2387880 = r2387878 - r2387879;
        double r2387881 = y;
        double r2387882 = log(r2387881);
        double r2387883 = r2387880 * r2387882;
        double r2387884 = z;
        double r2387885 = r2387884 - r2387879;
        double r2387886 = r2387879 - r2387881;
        double r2387887 = log(r2387886);
        double r2387888 = r2387885 * r2387887;
        double r2387889 = r2387883 + r2387888;
        double r2387890 = t;
        double r2387891 = r2387889 - r2387890;
        return r2387891;
}
↓
double f(double x, double y, double z, double t) {
        double r2387892 = y;
        double r2387893 = 0.3333333333333333;
        double r2387894 = pow(r2387892, r2387893);
        double r2387895 = log(r2387894);
        double r2387896 = x;
        double r2387897 = 1.0;
        double r2387898 = r2387896 - r2387897;
        double r2387899 = r2387895 * r2387898;
        double r2387900 = cbrt(r2387892);
        double r2387901 = log(r2387900);
        double r2387902 = r2387898 * r2387901;
        double r2387903 = cbrt(r2387900);
        double r2387904 = r2387903 * r2387903;
        double r2387905 = r2387903 * r2387904;
        double r2387906 = log(r2387905);
        double r2387907 = r2387898 * r2387906;
        double r2387908 = r2387902 + r2387907;
        double r2387909 = r2387899 + r2387908;
        double r2387910 = log(r2387897);
        double r2387911 = 0.5;
        double r2387912 = r2387897 / r2387892;
        double r2387913 = r2387912 * r2387912;
        double r2387914 = r2387911 / r2387913;
        double r2387915 = r2387910 - r2387914;
        double r2387916 = r2387897 * r2387892;
        double r2387917 = r2387915 - r2387916;
        double r2387918 = z;
        double r2387919 = r2387918 - r2387897;
        double r2387920 = r2387917 * r2387919;
        double r2387921 = r2387909 + r2387920;
        double r2387922 = t;
        double r2387923 = r2387921 - r2387922;
        return r2387923;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 7.0

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

  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

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Simplified0.4

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

  9. Simplified0.4

    \[\leadsto \left(\left(\left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) + \color{blue}{\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)}\right) + \left(z - 1\right) \cdot \left(\left(\log 1 - \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right) - 1 \cdot y\right)\right) - t\]
  10. Using strategy rm

  11. Applied pow1/30.4

    \[\leadsto \left(\left(\left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) + \log \color{blue}{\left({y}^{\frac{1}{3}}\right)} \cdot \left(x - 1\right)\right) + \left(z - 1\right) \cdot \left(\left(\log 1 - \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right) - 1 \cdot y\right)\right) - t\]
  12. Using strategy rm

  13. Applied add-cube-cbrt0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019170 
(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))