Average Error: 7.1 → 0.4
Time: 10.1s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log y + \left(\left(z - 1\right) \cdot \left(\sqrt{\log 1} + \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) - t\]
\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 y + \left(\left(z - 1\right) \cdot \left(\sqrt{\log 1} + \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) - t
double f(double x, double y, double z, double t) {
        double r55852 = x;
        double r55853 = 1.0;
        double r55854 = r55852 - r55853;
        double r55855 = y;
        double r55856 = log(r55855);
        double r55857 = r55854 * r55856;
        double r55858 = z;
        double r55859 = r55858 - r55853;
        double r55860 = r55853 - r55855;
        double r55861 = log(r55860);
        double r55862 = r55859 * r55861;
        double r55863 = r55857 + r55862;
        double r55864 = t;
        double r55865 = r55863 - r55864;
        return r55865;
}
↓
double f(double x, double y, double z, double t) {
        double r55866 = x;
        double r55867 = 1.0;
        double r55868 = r55866 - r55867;
        double r55869 = y;
        double r55870 = log(r55869);
        double r55871 = r55868 * r55870;
        double r55872 = z;
        double r55873 = r55872 - r55867;
        double r55874 = log(r55867);
        double r55875 = sqrt(r55874);
        double r55876 = r55867 * r55869;
        double r55877 = 0.5;
        double r55878 = 2.0;
        double r55879 = pow(r55869, r55878);
        double r55880 = pow(r55867, r55878);
        double r55881 = r55879 / r55880;
        double r55882 = r55877 * r55881;
        double r55883 = r55876 + r55882;
        double r55884 = sqrt(r55883);
        double r55885 = r55875 + r55884;
        double r55886 = r55873 * r55885;
        double r55887 = r55875 - r55884;
        double r55888 = r55886 * r55887;
        double r55889 = r55871 + r55888;
        double r55890 = t;
        double r55891 = r55889 - r55890;
        return r55891;
}

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 7.1

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

  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-sqr-sqrt0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied difference-of-squares0.4

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

  7. Applied associate-*r*0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2020024 
(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))