Average Error: 7.4 → 0.5
Time: 17.6s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-2}{3}}\right) \cdot \left(x - 1\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-2}{3}}\right) \cdot \left(x - 1\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
double f(double x, double y, double z, double t) {
        double r69908 = x;
        double r69909 = 1.0;
        double r69910 = r69908 - r69909;
        double r69911 = y;
        double r69912 = log(r69911);
        double r69913 = r69910 * r69912;
        double r69914 = z;
        double r69915 = r69914 - r69909;
        double r69916 = r69909 - r69911;
        double r69917 = log(r69916);
        double r69918 = r69915 * r69917;
        double r69919 = r69913 + r69918;
        double r69920 = t;
        double r69921 = r69919 - r69920;
        return r69921;
}
↓
double f(double x, double y, double z, double t) {
        double r69922 = 1.0;
        double r69923 = y;
        double r69924 = r69922 / r69923;
        double r69925 = -0.6666666666666666;
        double r69926 = pow(r69924, r69925);
        double r69927 = log(r69926);
        double r69928 = x;
        double r69929 = 1.0;
        double r69930 = r69928 - r69929;
        double r69931 = r69927 * r69930;
        double r69932 = cbrt(r69923);
        double r69933 = log(r69932);
        double r69934 = r69930 * r69933;
        double r69935 = z;
        double r69936 = r69935 - r69929;
        double r69937 = log(r69929);
        double r69938 = r69929 * r69923;
        double r69939 = 0.5;
        double r69940 = 2.0;
        double r69941 = pow(r69923, r69940);
        double r69942 = pow(r69929, r69940);
        double r69943 = r69941 / r69942;
        double r69944 = r69939 * r69943;
        double r69945 = r69938 + r69944;
        double r69946 = r69937 - r69945;
        double r69947 = r69936 * r69946;
        double r69948 = r69934 + r69947;
        double r69949 = r69931 + r69948;
        double r69950 = t;
        double r69951 = r69949 - r69950;
        return r69951;
}

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-rgt-in0.5

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

  7. Applied associate-+l+0.5

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

  8. Simplified0.5

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

  9. Taylor expanded around inf 0.5

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

  10. Final simplification0.5

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

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))