Average Error: 7.3 → 0.4
Time: 11.0s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y} \cdot {y}^{\frac{1}{3}}\right) + \left(\log \left(1 \cdot {y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\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(\sqrt[3]{y} \cdot {y}^{\frac{1}{3}}\right) + \left(\log \left(1 \cdot {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 r63917 = x;
        double r63918 = 1.0;
        double r63919 = r63917 - r63918;
        double r63920 = y;
        double r63921 = log(r63920);
        double r63922 = r63919 * r63921;
        double r63923 = z;
        double r63924 = r63923 - r63918;
        double r63925 = r63918 - r63920;
        double r63926 = log(r63925);
        double r63927 = r63924 * r63926;
        double r63928 = r63922 + r63927;
        double r63929 = t;
        double r63930 = r63928 - r63929;
        return r63930;
}
↓
double f(double x, double y, double z, double t) {
        double r63931 = x;
        double r63932 = 1.0;
        double r63933 = r63931 - r63932;
        double r63934 = y;
        double r63935 = cbrt(r63934);
        double r63936 = 0.3333333333333333;
        double r63937 = pow(r63934, r63936);
        double r63938 = r63935 * r63937;
        double r63939 = log(r63938);
        double r63940 = r63933 * r63939;
        double r63941 = 1.0;
        double r63942 = r63941 * r63937;
        double r63943 = log(r63942);
        double r63944 = r63943 * r63933;
        double r63945 = z;
        double r63946 = r63945 - r63932;
        double r63947 = log(r63932);
        double r63948 = r63932 * r63934;
        double r63949 = 0.5;
        double r63950 = 2.0;
        double r63951 = pow(r63934, r63950);
        double r63952 = pow(r63932, r63950);
        double r63953 = r63951 / r63952;
        double r63954 = r63949 * r63953;
        double r63955 = r63948 + r63954;
        double r63956 = r63947 - r63955;
        double r63957 = r63946 * r63956;
        double r63958 = r63944 + r63957;
        double r63959 = r63940 + r63958;
        double r63960 = t;
        double r63961 = r63959 - r63960;
        return r63961;
}

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 0.3

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \color{blue}{\left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]
  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-prod0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Applied associate-+l+0.4

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

  8. Simplified0.4

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

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

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

  11. Applied cbrt-prod0.4

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

  12. Simplified0.4

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

  13. Simplified0.4

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

  15. Applied pow1/30.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2020034 
(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))