Average Error: 7.0 → 0.6
Time: 30.8s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log y + \left({\left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}}}\right)}^{2} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}}}\right)}^{4} \cdot \left(\sqrt[3]{\sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}}\right)\right) \cdot \sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]
\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({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}}}\right)}^{2} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}}}\right)}^{4} \cdot \left(\sqrt[3]{\sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\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) \cdot \sqrt[3]{\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
double f(double x, double y, double z, double t) {
        double r65640 = x;
        double r65641 = 1.0;
        double r65642 = r65640 - r65641;
        double r65643 = y;
        double r65644 = log(r65643);
        double r65645 = r65642 * r65644;
        double r65646 = z;
        double r65647 = r65646 - r65641;
        double r65648 = r65641 - r65643;
        double r65649 = log(r65648);
        double r65650 = r65647 * r65649;
        double r65651 = r65645 + r65650;
        double r65652 = t;
        double r65653 = r65651 - r65652;
        return r65653;
}
↓
double f(double x, double y, double z, double t) {
        double r65654 = x;
        double r65655 = 1.0;
        double r65656 = r65654 - r65655;
        double r65657 = y;
        double r65658 = log(r65657);
        double r65659 = r65656 * r65658;
        double r65660 = z;
        double r65661 = r65660 - r65655;
        double r65662 = log(r65655);
        double r65663 = r65655 * r65657;
        double r65664 = 0.5;
        double r65665 = 2.0;
        double r65666 = pow(r65657, r65665);
        double r65667 = pow(r65655, r65665);
        double r65668 = r65666 / r65667;
        double r65669 = r65664 * r65668;
        double r65670 = r65663 + r65669;
        double r65671 = r65662 - r65670;
        double r65672 = r65661 * r65671;
        double r65673 = cbrt(r65672);
        double r65674 = cbrt(r65673);
        double r65675 = cbrt(r65674);
        double r65676 = pow(r65675, r65665);
        double r65677 = r65676 * r65675;
        double r65678 = 4.0;
        double r65679 = pow(r65677, r65678);
        double r65680 = r65674 * r65674;
        double r65681 = r65679 * r65680;
        double r65682 = r65681 * r65673;
        double r65683 = r65659 + r65682;
        double r65684 = t;
        double r65685 = r65683 - r65684;
        return r65685;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Your Program's Arguments

Results

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Derivation

  1. Initial program 7.0

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

  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

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

  6. Applied add-cube-cbrt0.5

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

  7. Applied associate-*l*0.5

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

  8. Simplified0.5

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

  10. Applied add-cube-cbrt0.6

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

  11. Simplified0.6

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

  12. Final simplification0.6

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \left({\left({\left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}}}\right)}^{2} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}}}\right)}^{4} \cdot \left(\sqrt[3]{\sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}} \cdot \sqrt[3]{\sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}}\right)\right) \cdot \sqrt[3]{\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]

Reproduce

herbie shell --seed 2019306 
(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))