Average Error: 6.8 → 0.4
Time: 21.8s
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 r68756 = x;
        double r68757 = 1.0;
        double r68758 = r68756 - r68757;
        double r68759 = y;
        double r68760 = log(r68759);
        double r68761 = r68758 * r68760;
        double r68762 = z;
        double r68763 = r68762 - r68757;
        double r68764 = r68757 - r68759;
        double r68765 = log(r68764);
        double r68766 = r68763 * r68765;
        double r68767 = r68761 + r68766;
        double r68768 = t;
        double r68769 = r68767 - r68768;
        return r68769;
}
↓
double f(double x, double y, double z, double t) {
        double r68770 = x;
        double r68771 = 1.0;
        double r68772 = r68770 - r68771;
        double r68773 = y;
        double r68774 = log(r68773);
        double r68775 = r68772 * r68774;
        double r68776 = z;
        double r68777 = r68776 - r68771;
        double r68778 = log(r68771);
        double r68779 = sqrt(r68778);
        double r68780 = r68771 * r68773;
        double r68781 = 0.5;
        double r68782 = 2.0;
        double r68783 = pow(r68773, r68782);
        double r68784 = pow(r68771, r68782);
        double r68785 = r68783 / r68784;
        double r68786 = r68781 * r68785;
        double r68787 = r68780 + r68786;
        double r68788 = sqrt(r68787);
        double r68789 = r68779 + r68788;
        double r68790 = r68777 * r68789;
        double r68791 = r68779 - r68788;
        double r68792 = r68790 * r68791;
        double r68793 = r68775 + r68792;
        double r68794 = t;
        double r68795 = r68793 - r68794;
        return r68795;
}

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 6.8

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

  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 2019308 
(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))