Average Error: 6.7 → 0.4
Time: 24.0s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\left(x - 1\right) \cdot \log \left(e^{\log y \cdot \frac{1}{3}}\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\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 \sqrt[3]{y}\right) + \left(\left(x - 1\right) \cdot \log \left(e^{\log y \cdot \frac{1}{3}}\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 r53597 = x;
        double r53598 = 1.0;
        double r53599 = r53597 - r53598;
        double r53600 = y;
        double r53601 = log(r53600);
        double r53602 = r53599 * r53601;
        double r53603 = z;
        double r53604 = r53603 - r53598;
        double r53605 = r53598 - r53600;
        double r53606 = log(r53605);
        double r53607 = r53604 * r53606;
        double r53608 = r53602 + r53607;
        double r53609 = t;
        double r53610 = r53608 - r53609;
        return r53610;
}
↓
double f(double x, double y, double z, double t) {
        double r53611 = x;
        double r53612 = 1.0;
        double r53613 = r53611 - r53612;
        double r53614 = y;
        double r53615 = cbrt(r53614);
        double r53616 = r53615 * r53615;
        double r53617 = log(r53616);
        double r53618 = r53613 * r53617;
        double r53619 = log(r53614);
        double r53620 = 0.3333333333333333;
        double r53621 = r53619 * r53620;
        double r53622 = exp(r53621);
        double r53623 = log(r53622);
        double r53624 = r53613 * r53623;
        double r53625 = z;
        double r53626 = r53625 - r53612;
        double r53627 = log(r53612);
        double r53628 = r53612 * r53614;
        double r53629 = 0.5;
        double r53630 = 2.0;
        double r53631 = pow(r53614, r53630);
        double r53632 = pow(r53612, r53630);
        double r53633 = r53631 / r53632;
        double r53634 = r53629 * r53633;
        double r53635 = r53628 + r53634;
        double r53636 = r53627 - r53635;
        double r53637 = r53626 * r53636;
        double r53638 = r53624 + r53637;
        double r53639 = r53618 + r53638;
        double r53640 = t;
        double r53641 = r53639 - r53640;
        return r53641;
}

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

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

  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-prod0.4

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

  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. Taylor expanded around inf 0.4

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

  10. Applied add-exp-log0.4

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

  11. Applied rec-exp0.4

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

  12. Applied pow-exp0.4

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

  13. Simplified0.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2019212 
(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))