Average Error: 6.5 → 0.4
Time: 29.9s
Precision: 64
\[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1.0\right) \cdot \log \left(\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)\right) + \left(x - 1.0\right) \cdot \log \left({y}^{\frac{1}{3}}\right)\right) + \left(\left(\log 1.0 - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) - y \cdot 1.0\right) \cdot \left(z - 1.0\right)\right) - t\]
\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(\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) + \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 r2749968 = x;
        double r2749969 = 1.0;
        double r2749970 = r2749968 - r2749969;
        double r2749971 = y;
        double r2749972 = log(r2749971);
        double r2749973 = r2749970 * r2749972;
        double r2749974 = z;
        double r2749975 = r2749974 - r2749969;
        double r2749976 = r2749969 - r2749971;
        double r2749977 = log(r2749976);
        double r2749978 = r2749975 * r2749977;
        double r2749979 = r2749973 + r2749978;
        double r2749980 = t;
        double r2749981 = r2749979 - r2749980;
        return r2749981;
}
↓
double f(double x, double y, double z, double t) {
        double r2749982 = x;
        double r2749983 = 1.0;
        double r2749984 = r2749982 - r2749983;
        double r2749985 = y;
        double r2749986 = cbrt(r2749985);
        double r2749987 = cbrt(r2749986);
        double r2749988 = r2749987 * r2749987;
        double r2749989 = r2749988 * r2749988;
        double r2749990 = r2749989 * r2749988;
        double r2749991 = log(r2749990);
        double r2749992 = r2749984 * r2749991;
        double r2749993 = 0.3333333333333333;
        double r2749994 = pow(r2749985, r2749993);
        double r2749995 = log(r2749994);
        double r2749996 = r2749984 * r2749995;
        double r2749997 = r2749992 + r2749996;
        double r2749998 = log(r2749983);
        double r2749999 = r2749985 / r2749983;
        double r2750000 = r2749999 * r2749999;
        double r2750001 = 0.5;
        double r2750002 = r2750000 * r2750001;
        double r2750003 = r2749998 - r2750002;
        double r2750004 = r2749985 * r2749983;
        double r2750005 = r2750003 - r2750004;
        double r2750006 = z;
        double r2750007 = r2750006 - r2749983;
        double r2750008 = r2750005 * r2750007;
        double r2750009 = r2749997 + r2750008;
        double r2750010 = t;
        double r2750011 = r2750009 - r2750010;
        return r2750011;
}

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-lft-in0.4

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

  9. Applied pow1/30.4

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

  11. Applied add-cube-cbrt0.4

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

  12. Applied add-cube-cbrt0.4

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

  13. Applied swap-sqr0.4

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

  14. Final simplification0.4

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

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))