Average Error: 6.4 → 0.4
Time: 9.4s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\log \left({\left({y}^{\frac{1}{3}}\right)}^{\frac{5}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x - 1\right) + \mathsf{fma}\left(\log \left({y}^{\frac{1}{3}}\right), x - 1, \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\log \left({\left({y}^{\frac{1}{3}}\right)}^{\frac{5}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x - 1\right) + \mathsf{fma}\left(\log \left({y}^{\frac{1}{3}}\right), x - 1, \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 r63584 = x;
        double r63585 = 1.0;
        double r63586 = r63584 - r63585;
        double r63587 = y;
        double r63588 = log(r63587);
        double r63589 = r63586 * r63588;
        double r63590 = z;
        double r63591 = r63590 - r63585;
        double r63592 = r63585 - r63587;
        double r63593 = log(r63592);
        double r63594 = r63591 * r63593;
        double r63595 = r63589 + r63594;
        double r63596 = t;
        double r63597 = r63595 - r63596;
        return r63597;
}
↓
double f(double x, double y, double z, double t) {
        double r63598 = y;
        double r63599 = 0.3333333333333333;
        double r63600 = pow(r63598, r63599);
        double r63601 = 1.6666666666666667;
        double r63602 = pow(r63600, r63601);
        double r63603 = cbrt(r63598);
        double r63604 = cbrt(r63603);
        double r63605 = r63602 * r63604;
        double r63606 = log(r63605);
        double r63607 = x;
        double r63608 = 1.0;
        double r63609 = r63607 - r63608;
        double r63610 = r63606 * r63609;
        double r63611 = log(r63600);
        double r63612 = z;
        double r63613 = r63612 - r63608;
        double r63614 = log(r63608);
        double r63615 = r63608 * r63598;
        double r63616 = 0.5;
        double r63617 = 2.0;
        double r63618 = pow(r63598, r63617);
        double r63619 = pow(r63608, r63617);
        double r63620 = r63618 / r63619;
        double r63621 = r63616 * r63620;
        double r63622 = r63615 + r63621;
        double r63623 = r63614 - r63622;
        double r63624 = r63613 * r63623;
        double r63625 = fma(r63611, r63609, r63624);
        double r63626 = r63610 + r63625;
        double r63627 = t;
        double r63628 = r63626 - r63627;
        return r63628;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Derivation

  1. Initial program 6.4

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

  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-cbrt0.3

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

  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-rgt-in0.4

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

  7. Applied associate-+l+0.4

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

  8. Simplified0.4

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

  10. Applied pow1/30.4

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

  12. Applied add-cube-cbrt0.4

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

  13. Applied associate-*r*0.4

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

  14. Simplified0.4

    \[\leadsto \left(\log \left(\color{blue}{{\left({y}^{\frac{1}{3}}\right)}^{\frac{5}{3}}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x - 1\right) + \mathsf{fma}\left(\log \left({y}^{\frac{1}{3}}\right), x - 1, \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]

  15. Final simplification0.4

    \[\leadsto \left(\log \left({\left({y}^{\frac{1}{3}}\right)}^{\frac{5}{3}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x - 1\right) + \mathsf{fma}\left(\log \left({y}^{\frac{1}{3}}\right), x - 1, \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019352 +o rules:numerics
(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))