Average Error: 6.7 → 0.4
Time: 20.9s
Precision: 64
\[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1.0\right) \cdot \log \left({y}^{\frac{1}{3}}\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) + \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right)\right) - t\]
\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t
↓
\left(\left(\left(x - 1.0\right) \cdot \log \left({y}^{\frac{1}{3}}\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) + \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right)\right) - t
double f(double x, double y, double z, double t) {
        double r1022751 = x;
        double r1022752 = 1.0;
        double r1022753 = r1022751 - r1022752;
        double r1022754 = y;
        double r1022755 = log(r1022754);
        double r1022756 = r1022753 * r1022755;
        double r1022757 = z;
        double r1022758 = r1022757 - r1022752;
        double r1022759 = r1022752 - r1022754;
        double r1022760 = log(r1022759);
        double r1022761 = r1022758 * r1022760;
        double r1022762 = r1022756 + r1022761;
        double r1022763 = t;
        double r1022764 = r1022762 - r1022763;
        return r1022764;
}
↓
double f(double x, double y, double z, double t) {
        double r1022765 = x;
        double r1022766 = 1.0;
        double r1022767 = r1022765 - r1022766;
        double r1022768 = y;
        double r1022769 = 0.3333333333333333;
        double r1022770 = pow(r1022768, r1022769);
        double r1022771 = log(r1022770);
        double r1022772 = r1022767 * r1022771;
        double r1022773 = z;
        double r1022774 = r1022773 - r1022766;
        double r1022775 = log(r1022766);
        double r1022776 = r1022768 * r1022766;
        double r1022777 = r1022775 - r1022776;
        double r1022778 = r1022768 / r1022766;
        double r1022779 = r1022778 * r1022778;
        double r1022780 = 0.5;
        double r1022781 = r1022779 * r1022780;
        double r1022782 = r1022777 - r1022781;
        double r1022783 = r1022774 * r1022782;
        double r1022784 = r1022772 + r1022783;
        double r1022785 = cbrt(r1022768);
        double r1022786 = r1022785 * r1022785;
        double r1022787 = log(r1022786);
        double r1022788 = r1022787 * r1022767;
        double r1022789 = r1022784 + r1022788;
        double r1022790 = t;
        double r1022791 = r1022789 - r1022790;
        return r1022791;
}

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 6.7

    \[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)\right)}\right) - t\]

  3. Simplified0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \color{blue}{\left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)}\right) - t\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) - t\]

  6. Applied log-prod0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) - t\]

  7. Applied distribute-lft-in0.4

    \[\leadsto \left(\color{blue}{\left(\left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) - t\]

  8. Applied associate-+l+0.4

    \[\leadsto \color{blue}{\left(\left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y}\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right)} - t\]
  9. Using strategy rm

  10. Applied pow1/30.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\left(x - 1.0\right) \cdot \log \color{blue}{\left({y}^{\frac{1}{3}}\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

  11. Final simplification0.4

    \[\leadsto \left(\left(\left(x - 1.0\right) \cdot \log \left({y}^{\frac{1}{3}}\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) + \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right)\right) - t\]

Reproduce

herbie shell --seed 2019156 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  (- (+ (* (- x 1.0) (log y)) (* (- z 1.0) (log (- 1.0 y)))) t))