Average Error: 6.5 → 0.4
Time: 28.2s
Precision: 64
\[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1.0\right) \cdot \log \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + \left(x - 1.0\right) \cdot \log \left({y}^{\frac{1}{3}}\right)\right) + \left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - y \cdot 1.0\right) \cdot \left(z - 1.0\right)\right) - t\]
\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t
↓
\left(\left(\left(x - 1.0\right) \cdot \log \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + \left(x - 1.0\right) \cdot \log \left({y}^{\frac{1}{3}}\right)\right) + \left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - y \cdot 1.0\right) \cdot \left(z - 1.0\right)\right) - t
double f(double x, double y, double z, double t) {
        double r2749948 = x;
        double r2749949 = 1.0;
        double r2749950 = r2749948 - r2749949;
        double r2749951 = y;
        double r2749952 = log(r2749951);
        double r2749953 = r2749950 * r2749952;
        double r2749954 = z;
        double r2749955 = r2749954 - r2749949;
        double r2749956 = r2749949 - r2749951;
        double r2749957 = log(r2749956);
        double r2749958 = r2749955 * r2749957;
        double r2749959 = r2749953 + r2749958;
        double r2749960 = t;
        double r2749961 = r2749959 - r2749960;
        return r2749961;
}
↓
double f(double x, double y, double z, double t) {
        double r2749962 = x;
        double r2749963 = 1.0;
        double r2749964 = r2749962 - r2749963;
        double r2749965 = y;
        double r2749966 = cbrt(r2749965);
        double r2749967 = cbrt(r2749966);
        double r2749968 = r2749967 * r2749967;
        double r2749969 = r2749968 * r2749968;
        double r2749970 = r2749968 * r2749969;
        double r2749971 = log(r2749970);
        double r2749972 = r2749964 * r2749971;
        double r2749973 = 0.3333333333333333;
        double r2749974 = pow(r2749965, r2749973);
        double r2749975 = log(r2749974);
        double r2749976 = r2749964 * r2749975;
        double r2749977 = r2749972 + r2749976;
        double r2749978 = log(r2749963);
        double r2749979 = r2749965 / r2749963;
        double r2749980 = r2749979 * r2749979;
        double r2749981 = 0.5;
        double r2749982 = r2749980 * r2749981;
        double r2749983 = r2749978 - r2749982;
        double r2749984 = r2749965 * r2749963;
        double r2749985 = r2749983 - r2749984;
        double r2749986 = z;
        double r2749987 = r2749986 - r2749963;
        double r2749988 = r2749985 * r2749987;
        double r2749989 = r2749977 + r2749988;
        double r2749990 = t;
        double r2749991 = r2749989 - r2749990;
        return r2749991;
}

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 6.5

    \[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)\right)}\right) - t\]

  3. Simplified0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \color{blue}{\left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - 1.0 \cdot y\right)}\right) - t\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - 1.0 \cdot y\right)\right) - t\]

  6. Applied log-prod0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - 1.0 \cdot y\right)\right) - t\]

  7. Applied distribute-rgt-in0.4

    \[\leadsto \left(\color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1.0\right)\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - 1.0 \cdot y\right)\right) - t\]
  8. Using strategy rm

  9. Applied pow1/30.4

    \[\leadsto \left(\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \log \color{blue}{\left({y}^{\frac{1}{3}}\right)} \cdot \left(x - 1.0\right)\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - 1.0 \cdot y\right)\right) - t\]
  10. Using strategy rm

  11. Applied add-cube-cbrt0.4

    \[\leadsto \left(\left(\log \left(\sqrt[3]{y} \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right) \cdot \left(x - 1.0\right) + \log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1.0\right)\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - 1.0 \cdot y\right)\right) - t\]

  12. Applied add-cube-cbrt0.4

    \[\leadsto \left(\left(\log \left(\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)} \cdot \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot \left(x - 1.0\right) + \log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1.0\right)\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - 1.0 \cdot y\right)\right) - t\]

  13. Applied swap-sqr0.4

    \[\leadsto \left(\left(\log \color{blue}{\left(\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right)} \cdot \left(x - 1.0\right) + \log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1.0\right)\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - 1.0 \cdot y\right)\right) - t\]

  14. Final simplification0.4

    \[\leadsto \left(\left(\left(x - 1.0\right) \cdot \log \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right)\right) + \left(x - 1.0\right) \cdot \log \left({y}^{\frac{1}{3}}\right)\right) + \left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - y \cdot 1.0\right) \cdot \left(z - 1.0\right)\right) - t\]

Reproduce

herbie shell --seed 2019163 
(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))