Average Error: 7.1 → 0.4
Time: 23.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(z - 1\right) \cdot \left(\left(\log 1 - \frac{{y}^{2}}{1} \cdot \frac{\frac{1}{2}}{1}\right) - y \cdot 1\right) + \left(\left(\log \left(\left|\sqrt[3]{y}\right|\right) \cdot \left(x - 1\right) + \left(x - 1\right) \cdot \log \left(\sqrt{\sqrt[3]{y}}\right)\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt{y}\right) - t\right)\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(z - 1\right) \cdot \left(\left(\log 1 - \frac{{y}^{2}}{1} \cdot \frac{\frac{1}{2}}{1}\right) - y \cdot 1\right) + \left(\left(\log \left(\left|\sqrt[3]{y}\right|\right) \cdot \left(x - 1\right) + \left(x - 1\right) \cdot \log \left(\sqrt{\sqrt[3]{y}}\right)\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt{y}\right) - t\right)\right)
double f(double x, double y, double z, double t) {
        double r44673 = x;
        double r44674 = 1.0;
        double r44675 = r44673 - r44674;
        double r44676 = y;
        double r44677 = log(r44676);
        double r44678 = r44675 * r44677;
        double r44679 = z;
        double r44680 = r44679 - r44674;
        double r44681 = r44674 - r44676;
        double r44682 = log(r44681);
        double r44683 = r44680 * r44682;
        double r44684 = r44678 + r44683;
        double r44685 = t;
        double r44686 = r44684 - r44685;
        return r44686;
}
↓
double f(double x, double y, double z, double t) {
        double r44687 = z;
        double r44688 = 1.0;
        double r44689 = r44687 - r44688;
        double r44690 = log(r44688);
        double r44691 = y;
        double r44692 = 2.0;
        double r44693 = pow(r44691, r44692);
        double r44694 = r44693 / r44688;
        double r44695 = 0.5;
        double r44696 = r44695 / r44688;
        double r44697 = r44694 * r44696;
        double r44698 = r44690 - r44697;
        double r44699 = r44691 * r44688;
        double r44700 = r44698 - r44699;
        double r44701 = r44689 * r44700;
        double r44702 = cbrt(r44691);
        double r44703 = fabs(r44702);
        double r44704 = log(r44703);
        double r44705 = x;
        double r44706 = r44705 - r44688;
        double r44707 = r44704 * r44706;
        double r44708 = sqrt(r44702);
        double r44709 = log(r44708);
        double r44710 = r44706 * r44709;
        double r44711 = r44707 + r44710;
        double r44712 = sqrt(r44691);
        double r44713 = log(r44712);
        double r44714 = r44706 * r44713;
        double r44715 = t;
        double r44716 = r44714 - r44715;
        double r44717 = r44711 + r44716;
        double r44718 = r44701 + r44717;
        return r44718;
}

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

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. Simplified7.1

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  6. Applied add-sqr-sqrt0.4

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

  7. Applied log-prod0.4

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

  8. Applied distribute-lft-in0.4

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

  9. Applied associate--l+0.4

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

  10. Simplified0.4

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

  12. Applied add-cube-cbrt0.4

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

  13. Applied sqrt-prod0.4

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

  14. Applied log-prod0.4

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

  15. Applied distribute-lft-in0.4

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

  16. Simplified0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019195 
(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))