Average Error: 6.7 → 0.4
Time: 22.9s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log y + \left(\left(z - 1\right) \cdot \left(\sqrt{\log 1} + \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) - t\]
\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 y + \left(\left(z - 1\right) \cdot \left(\sqrt{\log 1} + \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) - t
double f(double x, double y, double z, double t) {
        double r63631 = x;
        double r63632 = 1.0;
        double r63633 = r63631 - r63632;
        double r63634 = y;
        double r63635 = log(r63634);
        double r63636 = r63633 * r63635;
        double r63637 = z;
        double r63638 = r63637 - r63632;
        double r63639 = r63632 - r63634;
        double r63640 = log(r63639);
        double r63641 = r63638 * r63640;
        double r63642 = r63636 + r63641;
        double r63643 = t;
        double r63644 = r63642 - r63643;
        return r63644;
}
↓
double f(double x, double y, double z, double t) {
        double r63645 = x;
        double r63646 = 1.0;
        double r63647 = r63645 - r63646;
        double r63648 = y;
        double r63649 = log(r63648);
        double r63650 = r63647 * r63649;
        double r63651 = z;
        double r63652 = r63651 - r63646;
        double r63653 = log(r63646);
        double r63654 = sqrt(r63653);
        double r63655 = r63646 * r63648;
        double r63656 = 0.5;
        double r63657 = 2.0;
        double r63658 = pow(r63648, r63657);
        double r63659 = pow(r63646, r63657);
        double r63660 = r63658 / r63659;
        double r63661 = r63656 * r63660;
        double r63662 = r63655 + r63661;
        double r63663 = sqrt(r63662);
        double r63664 = r63654 + r63663;
        double r63665 = r63652 * r63664;
        double r63666 = r63654 - r63663;
        double r63667 = r63665 * r63666;
        double r63668 = r63650 + r63667;
        double r63669 = t;
        double r63670 = r63668 - r63669;
        return r63670;
}

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-sqr-sqrt0.4

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

  5. Applied add-sqr-sqrt0.4

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

  6. Applied difference-of-squares0.4

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

  7. Applied associate-*r*0.4

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

  8. Final simplification0.4

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

Reproduce

herbie shell --seed 2019298 
(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))