Average Error: 6.7 → 0.4
Time: 1.0m
Precision: 64
\[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \left(\left(z - 1.0\right) \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) + \log \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x - 1.0\right)\right)\right) - t\]
\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t
↓
\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \left(\left(z - 1.0\right) \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) + \log \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x - 1.0\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r1836730 = x;
        double r1836731 = 1.0;
        double r1836732 = r1836730 - r1836731;
        double r1836733 = y;
        double r1836734 = log(r1836733);
        double r1836735 = r1836732 * r1836734;
        double r1836736 = z;
        double r1836737 = r1836736 - r1836731;
        double r1836738 = r1836731 - r1836733;
        double r1836739 = log(r1836738);
        double r1836740 = r1836737 * r1836739;
        double r1836741 = r1836735 + r1836740;
        double r1836742 = t;
        double r1836743 = r1836741 - r1836742;
        return r1836743;
}
↓
double f(double x, double y, double z, double t) {
        double r1836744 = y;
        double r1836745 = cbrt(r1836744);
        double r1836746 = r1836745 * r1836745;
        double r1836747 = log(r1836746);
        double r1836748 = x;
        double r1836749 = 1.0;
        double r1836750 = r1836748 - r1836749;
        double r1836751 = r1836747 * r1836750;
        double r1836752 = z;
        double r1836753 = r1836752 - r1836749;
        double r1836754 = log(r1836749);
        double r1836755 = r1836744 * r1836749;
        double r1836756 = r1836754 - r1836755;
        double r1836757 = r1836744 / r1836749;
        double r1836758 = r1836757 * r1836757;
        double r1836759 = 0.5;
        double r1836760 = r1836758 * r1836759;
        double r1836761 = r1836756 - r1836760;
        double r1836762 = r1836753 * r1836761;
        double r1836763 = cbrt(r1836745);
        double r1836764 = r1836763 * r1836763;
        double r1836765 = r1836764 * r1836763;
        double r1836766 = log(r1836765);
        double r1836767 = r1836766 * r1836750;
        double r1836768 = r1836762 + r1836767;
        double r1836769 = r1836751 + r1836768;
        double r1836770 = t;
        double r1836771 = r1836769 - r1836770;
        return r1836771;
}

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

    \[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)\right)}\right) - t\]

  3. Simplified0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \color{blue}{\left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)}\right) - t\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) - t\]

  6. Applied log-prod0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) - t\]

  7. Applied distribute-lft-in0.4

    \[\leadsto \left(\color{blue}{\left(\left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right) - t\]

  8. Applied associate-+l+0.4

    \[\leadsto \color{blue}{\left(\left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y}\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right)} - t\]
  9. Using strategy rm

  10. Applied add-cube-cbrt0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\left(x - 1.0\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - 1.0 \cdot y\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right)\right)\right) - t\]

  11. Final simplification0.4

    \[\leadsto \left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \left(\left(z - 1.0\right) \cdot \left(\left(\log 1.0 - y \cdot 1.0\right) - \left(\frac{y}{1.0} \cdot \frac{y}{1.0}\right) \cdot \frac{1}{2}\right) + \log \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x - 1.0\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019165 
(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))