Average Error: 6.8 → 0.5
Time: 29.0s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(\left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right) - t\right) + \left(z - 1\right) \cdot \left(\left(\log 1 - y \cdot 1\right) - \frac{\frac{1}{2} \cdot \frac{y \cdot y}{1}}{1}\right)\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(\left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right) - t\right) + \left(z - 1\right) \cdot \left(\left(\log 1 - y \cdot 1\right) - \frac{\frac{1}{2} \cdot \frac{y \cdot y}{1}}{1}\right)\right)
double f(double x, double y, double z, double t) {
        double r64540 = x;
        double r64541 = 1.0;
        double r64542 = r64540 - r64541;
        double r64543 = y;
        double r64544 = log(r64543);
        double r64545 = r64542 * r64544;
        double r64546 = z;
        double r64547 = r64546 - r64541;
        double r64548 = r64541 - r64543;
        double r64549 = log(r64548);
        double r64550 = r64547 * r64549;
        double r64551 = r64545 + r64550;
        double r64552 = t;
        double r64553 = r64551 - r64552;
        return r64553;
}
↓
double f(double x, double y, double z, double t) {
        double r64554 = y;
        double r64555 = cbrt(r64554);
        double r64556 = r64555 * r64555;
        double r64557 = log(r64556);
        double r64558 = x;
        double r64559 = 1.0;
        double r64560 = r64558 - r64559;
        double r64561 = r64557 * r64560;
        double r64562 = log(r64555);
        double r64563 = r64560 * r64562;
        double r64564 = t;
        double r64565 = r64563 - r64564;
        double r64566 = z;
        double r64567 = r64566 - r64559;
        double r64568 = log(r64559);
        double r64569 = r64554 * r64559;
        double r64570 = r64568 - r64569;
        double r64571 = 0.5;
        double r64572 = r64554 * r64554;
        double r64573 = r64572 / r64559;
        double r64574 = r64571 * r64573;
        double r64575 = r64574 / r64559;
        double r64576 = r64570 - r64575;
        double r64577 = r64567 * r64576;
        double r64578 = r64565 + r64577;
        double r64579 = r64561 + r64578;
        return r64579;
}

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 6.8

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

  2. Simplified6.8

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  6. Applied add-cube-cbrt0.4

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

  7. Applied log-prod0.5

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

  8. Applied distribute-lft-in0.5

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

  9. Applied associate--l+0.5

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

  10. Applied associate-+l+0.5

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

  11. Simplified0.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2019194 
(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))