Average Error: 7.2 → 0.4
Time: 31.9s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right) + \left(z - 1\right) \cdot \left(\left(\log 1 - \left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2}\right) - 1 \cdot y\right)\right)\right) - t\]
\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(\left(\log 1 - \left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2}\right) - 1 \cdot y\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r4077417 = x;
        double r4077418 = 1.0;
        double r4077419 = r4077417 - r4077418;
        double r4077420 = y;
        double r4077421 = log(r4077420);
        double r4077422 = r4077419 * r4077421;
        double r4077423 = z;
        double r4077424 = r4077423 - r4077418;
        double r4077425 = r4077418 - r4077420;
        double r4077426 = log(r4077425);
        double r4077427 = r4077424 * r4077426;
        double r4077428 = r4077422 + r4077427;
        double r4077429 = t;
        double r4077430 = r4077428 - r4077429;
        return r4077430;
}
↓
double f(double x, double y, double z, double t) {
        double r4077431 = x;
        double r4077432 = 1.0;
        double r4077433 = r4077431 - r4077432;
        double r4077434 = y;
        double r4077435 = cbrt(r4077434);
        double r4077436 = r4077435 * r4077435;
        double r4077437 = log(r4077436);
        double r4077438 = r4077433 * r4077437;
        double r4077439 = log(r4077435);
        double r4077440 = r4077433 * r4077439;
        double r4077441 = z;
        double r4077442 = r4077441 - r4077432;
        double r4077443 = log(r4077432);
        double r4077444 = r4077434 / r4077432;
        double r4077445 = r4077444 * r4077444;
        double r4077446 = 0.5;
        double r4077447 = r4077445 * r4077446;
        double r4077448 = r4077443 - r4077447;
        double r4077449 = r4077432 * r4077434;
        double r4077450 = r4077448 - r4077449;
        double r4077451 = r4077442 * r4077450;
        double r4077452 = r4077440 + r4077451;
        double r4077453 = r4077438 + r4077452;
        double r4077454 = t;
        double r4077455 = r4077453 - r4077454;
        return r4077455;
}

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 7.2

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

  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. Simplified0.3

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Applied associate-+l+0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019174 
(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))