Average Error: 7.3 → 0.4
Time: 12.7s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\log \left(\sqrt[3]{1} \cdot {y}^{\frac{2}{3}}\right) \cdot x + \left(\left(-1\right) \cdot \log \left(\sqrt[3]{1} \cdot {y}^{\frac{2}{3}}\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\log \left(\sqrt[3]{1} \cdot {y}^{\frac{2}{3}}\right) \cdot x + \left(\left(-1\right) \cdot \log \left(\sqrt[3]{1} \cdot {y}^{\frac{2}{3}}\right) + \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)\right) - t
double f(double x, double y, double z, double t) {
        double r64895 = x;
        double r64896 = 1.0;
        double r64897 = r64895 - r64896;
        double r64898 = y;
        double r64899 = log(r64898);
        double r64900 = r64897 * r64899;
        double r64901 = z;
        double r64902 = r64901 - r64896;
        double r64903 = r64896 - r64898;
        double r64904 = log(r64903);
        double r64905 = r64902 * r64904;
        double r64906 = r64900 + r64905;
        double r64907 = t;
        double r64908 = r64906 - r64907;
        return r64908;
}
↓
double f(double x, double y, double z, double t) {
        double r64909 = 1.0;
        double r64910 = cbrt(r64909);
        double r64911 = y;
        double r64912 = 0.6666666666666666;
        double r64913 = pow(r64911, r64912);
        double r64914 = r64910 * r64913;
        double r64915 = log(r64914);
        double r64916 = x;
        double r64917 = r64915 * r64916;
        double r64918 = 1.0;
        double r64919 = -r64918;
        double r64920 = r64919 * r64915;
        double r64921 = cbrt(r64911);
        double r64922 = log(r64921);
        double r64923 = r64916 - r64918;
        double r64924 = r64922 * r64923;
        double r64925 = z;
        double r64926 = r64925 - r64918;
        double r64927 = log(r64918);
        double r64928 = r64918 * r64911;
        double r64929 = 0.5;
        double r64930 = 2.0;
        double r64931 = pow(r64911, r64930);
        double r64932 = pow(r64918, r64930);
        double r64933 = r64931 / r64932;
        double r64934 = r64929 * r64933;
        double r64935 = r64928 + r64934;
        double r64936 = r64927 - r64935;
        double r64937 = r64926 * r64936;
        double r64938 = r64924 + r64937;
        double r64939 = r64920 + r64938;
        double r64940 = r64917 + r64939;
        double r64941 = t;
        double r64942 = r64940 - r64941;
        return r64942;
}

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 7.3

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

  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-rgt-in0.4

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

  7. Applied associate-+l+0.4

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

  9. Applied *-un-lft-identity0.4

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

  10. Applied cbrt-prod0.4

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

  11. Applied associate-*l*0.4

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

  12. Simplified0.4

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

  14. Applied sub-neg0.4

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

  15. Applied distribute-lft-in0.4

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

  16. Applied associate-+l+0.4

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

  17. Simplified0.4

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

  18. Final simplification0.4

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

Reproduce

herbie shell --seed 2019322 
(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))