Average Error: 6.8 → 0.4
Time: 25.9s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right) \cdot \left(x - 1\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(\left(\log 1 - 1 \cdot y\right) - \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right) \cdot \left(z - 1\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(\log \left(\sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right) \cdot \left(x - 1\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(\left(\log 1 - 1 \cdot y\right) - \frac{\frac{1}{2}}{\frac{1}{y} \cdot \frac{1}{y}}\right) \cdot \left(z - 1\right)\right) - t
double f(double x, double y, double z, double t) {
        double r2407900 = x;
        double r2407901 = 1.0;
        double r2407902 = r2407900 - r2407901;
        double r2407903 = y;
        double r2407904 = log(r2407903);
        double r2407905 = r2407902 * r2407904;
        double r2407906 = z;
        double r2407907 = r2407906 - r2407901;
        double r2407908 = r2407901 - r2407903;
        double r2407909 = log(r2407908);
        double r2407910 = r2407907 * r2407909;
        double r2407911 = r2407905 + r2407910;
        double r2407912 = t;
        double r2407913 = r2407911 - r2407912;
        return r2407913;
}
↓
double f(double x, double y, double z, double t) {
        double r2407914 = y;
        double r2407915 = cbrt(r2407914);
        double r2407916 = log(r2407915);
        double r2407917 = r2407916 + r2407916;
        double r2407918 = x;
        double r2407919 = 1.0;
        double r2407920 = r2407918 - r2407919;
        double r2407921 = r2407917 * r2407920;
        double r2407922 = r2407920 * r2407916;
        double r2407923 = r2407921 + r2407922;
        double r2407924 = log(r2407919);
        double r2407925 = r2407919 * r2407914;
        double r2407926 = r2407924 - r2407925;
        double r2407927 = 0.5;
        double r2407928 = r2407919 / r2407914;
        double r2407929 = r2407928 * r2407928;
        double r2407930 = r2407927 / r2407929;
        double r2407931 = r2407926 - r2407930;
        double r2407932 = z;
        double r2407933 = r2407932 - r2407919;
        double r2407934 = r2407931 * r2407933;
        double r2407935 = r2407923 + r2407934;
        double r2407936 = t;
        double r2407937 = r2407935 - r2407936;
        return r2407937;
}

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 6.8

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

  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

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-rgt-in0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019192 
(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))