Average Error: 7.2 → 0.4
Time: 23.5s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(z - 1\right) \cdot \left(\left(\log 1 - y \cdot 1\right) - \frac{\left(y \cdot y\right) \cdot \frac{1}{2}}{1 \cdot 1}\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(x - 1\right) \cdot \log \left({y}^{\frac{2}{3}}\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(z - 1\right) \cdot \left(\left(\log 1 - y \cdot 1\right) - \frac{\left(y \cdot y\right) \cdot \frac{1}{2}}{1 \cdot 1}\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(x - 1\right) \cdot \log \left({y}^{\frac{2}{3}}\right)\right) - t
double f(double x, double y, double z, double t) {
        double r48643 = x;
        double r48644 = 1.0;
        double r48645 = r48643 - r48644;
        double r48646 = y;
        double r48647 = log(r48646);
        double r48648 = r48645 * r48647;
        double r48649 = z;
        double r48650 = r48649 - r48644;
        double r48651 = r48644 - r48646;
        double r48652 = log(r48651);
        double r48653 = r48650 * r48652;
        double r48654 = r48648 + r48653;
        double r48655 = t;
        double r48656 = r48654 - r48655;
        return r48656;
}
↓
double f(double x, double y, double z, double t) {
        double r48657 = z;
        double r48658 = 1.0;
        double r48659 = r48657 - r48658;
        double r48660 = log(r48658);
        double r48661 = y;
        double r48662 = r48661 * r48658;
        double r48663 = r48660 - r48662;
        double r48664 = r48661 * r48661;
        double r48665 = 0.5;
        double r48666 = r48664 * r48665;
        double r48667 = r48658 * r48658;
        double r48668 = r48666 / r48667;
        double r48669 = r48663 - r48668;
        double r48670 = r48659 * r48669;
        double r48671 = x;
        double r48672 = r48671 - r48658;
        double r48673 = cbrt(r48661);
        double r48674 = log(r48673);
        double r48675 = r48672 * r48674;
        double r48676 = r48670 + r48675;
        double r48677 = 0.6666666666666666;
        double r48678 = pow(r48661, r48677);
        double r48679 = log(r48678);
        double r48680 = r48672 * r48679;
        double r48681 = r48676 + r48680;
        double r48682 = t;
        double r48683 = r48681 - r48682;
        return r48683;
}

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 7.2

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

  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. Simplified0.3

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-rgt-in0.4

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

  8. Applied associate-+l+0.4

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

  9. Simplified0.4

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

  11. Applied *-un-lft-identity0.4

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

  12. Applied cbrt-prod0.4

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

  13. Applied associate-*l*0.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(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))