Average Error: 7.4 → 0.4
Time: 23.7s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{5}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(x - 1\right) \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{5}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r58880 = x;
        double r58881 = 1.0;
        double r58882 = r58880 - r58881;
        double r58883 = y;
        double r58884 = log(r58883);
        double r58885 = r58882 * r58884;
        double r58886 = z;
        double r58887 = r58886 - r58881;
        double r58888 = r58881 - r58883;
        double r58889 = log(r58888);
        double r58890 = r58887 * r58889;
        double r58891 = r58885 + r58890;
        double r58892 = t;
        double r58893 = r58891 - r58892;
        return r58893;
}
↓
double f(double x, double y, double z, double t) {
        double r58894 = x;
        double r58895 = 1.0;
        double r58896 = r58894 - r58895;
        double r58897 = y;
        double r58898 = cbrt(r58897);
        double r58899 = 1.6666666666666667;
        double r58900 = pow(r58898, r58899);
        double r58901 = cbrt(r58898);
        double r58902 = r58900 * r58901;
        double r58903 = log(r58902);
        double r58904 = r58896 * r58903;
        double r58905 = 0.3333333333333333;
        double r58906 = pow(r58897, r58905);
        double r58907 = log(r58906);
        double r58908 = r58907 * r58896;
        double r58909 = z;
        double r58910 = r58909 - r58895;
        double r58911 = log(r58895);
        double r58912 = r58895 * r58897;
        double r58913 = 0.5;
        double r58914 = 2.0;
        double r58915 = pow(r58897, r58914);
        double r58916 = pow(r58895, r58914);
        double r58917 = r58915 / r58916;
        double r58918 = r58913 * r58917;
        double r58919 = r58912 + r58918;
        double r58920 = r58911 - r58919;
        double r58921 = r58910 * r58920;
        double r58922 = r58908 + r58921;
        double r58923 = r58904 + r58922;
        double r58924 = t;
        double r58925 = r58923 - r58924;
        return r58925;
}

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 7.4

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

  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. Using strategy rm

  4. Applied add-cube-cbrt0.4

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

  5. Applied log-prod0.5

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

  6. Applied distribute-lft-in0.5

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

  7. Applied associate-+l+0.5

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

  8. Simplified0.5

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

  10. Applied pow1/30.5

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

  12. Applied add-cube-cbrt0.5

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

  13. Applied associate-*r*0.5

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

  14. Simplified0.4

    \[\leadsto \left(\left(x - 1\right) \cdot \log \left(\color{blue}{{\left(\sqrt[3]{y}\right)}^{\frac{5}{3}}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]

  15. Final simplification0.4

    \[\leadsto \left(\left(x - 1\right) \cdot \log \left({\left(\sqrt[3]{y}\right)}^{\frac{5}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) + \left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1) (log y)) (* (- z 1) (log (- 1 y)))) t))