Average Error: 6.7 → 0.5
Time: 16.6s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\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(\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
double f(double x, double y, double z, double t) {
        double r58639 = x;
        double r58640 = 1.0;
        double r58641 = r58639 - r58640;
        double r58642 = y;
        double r58643 = log(r58642);
        double r58644 = r58641 * r58643;
        double r58645 = z;
        double r58646 = r58645 - r58640;
        double r58647 = r58640 - r58642;
        double r58648 = log(r58647);
        double r58649 = r58646 * r58648;
        double r58650 = r58644 + r58649;
        double r58651 = t;
        double r58652 = r58650 - r58651;
        return r58652;
}
↓
double f(double x, double y, double z, double t) {
        double r58653 = x;
        double r58654 = 1.0;
        double r58655 = r58653 - r58654;
        double r58656 = y;
        double r58657 = cbrt(r58656);
        double r58658 = r58657 * r58657;
        double r58659 = log(r58658);
        double r58660 = r58655 * r58659;
        double r58661 = log(r58657);
        double r58662 = r58655 * r58661;
        double r58663 = z;
        double r58664 = r58663 - r58654;
        double r58665 = log(r58654);
        double r58666 = r58654 * r58656;
        double r58667 = 0.5;
        double r58668 = 2.0;
        double r58669 = pow(r58656, r58668);
        double r58670 = pow(r58654, r58668);
        double r58671 = r58669 / r58670;
        double r58672 = r58667 * r58671;
        double r58673 = r58666 + r58672;
        double r58674 = r58665 - r58673;
        double r58675 = r58664 * r58674;
        double r58676 = r58662 + r58675;
        double r58677 = r58660 + r58676;
        double r58678 = t;
        double r58679 = r58677 - r58678;
        return r58679;
}

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-prod0.5

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

  6. Applied distribute-lft-in0.5

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

  7. Applied associate-+l+0.5

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

  8. Final simplification0.5

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

Reproduce

herbie shell --seed 2020046 
(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))