Average Error: 7.2 → 0.4
Time: 28.8s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2}\right)\right)\right) + \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\log \left({y}^{\frac{1}{3}}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2}\right)\right)\right) + \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) - t
double f(double x, double y, double z, double t) {
        double r2145347 = x;
        double r2145348 = 1.0;
        double r2145349 = r2145347 - r2145348;
        double r2145350 = y;
        double r2145351 = log(r2145350);
        double r2145352 = r2145349 * r2145351;
        double r2145353 = z;
        double r2145354 = r2145353 - r2145348;
        double r2145355 = r2145348 - r2145350;
        double r2145356 = log(r2145355);
        double r2145357 = r2145354 * r2145356;
        double r2145358 = r2145352 + r2145357;
        double r2145359 = t;
        double r2145360 = r2145358 - r2145359;
        return r2145360;
}
↓
double f(double x, double y, double z, double t) {
        double r2145361 = y;
        double r2145362 = 0.3333333333333333;
        double r2145363 = pow(r2145361, r2145362);
        double r2145364 = log(r2145363);
        double r2145365 = x;
        double r2145366 = 1.0;
        double r2145367 = r2145365 - r2145366;
        double r2145368 = r2145364 * r2145367;
        double r2145369 = z;
        double r2145370 = r2145369 - r2145366;
        double r2145371 = log(r2145366);
        double r2145372 = r2145366 * r2145361;
        double r2145373 = r2145361 / r2145366;
        double r2145374 = r2145373 * r2145373;
        double r2145375 = 0.5;
        double r2145376 = r2145374 * r2145375;
        double r2145377 = r2145372 + r2145376;
        double r2145378 = r2145371 - r2145377;
        double r2145379 = r2145370 * r2145378;
        double r2145380 = r2145368 + r2145379;
        double r2145381 = cbrt(r2145361);
        double r2145382 = r2145381 * r2145381;
        double r2145383 = log(r2145382);
        double r2145384 = r2145383 * r2145367;
        double r2145385 = r2145380 + r2145384;
        double r2145386 = t;
        double r2145387 = r2145385 - r2145386;
        return r2145387;
}

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

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-rgt-in0.4

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

  8. Applied associate-+l+0.4

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

  10. Applied pow1/30.4

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

  11. Final simplification0.4

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

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))