Average Error: 6.7 → 0.5
Time: 25.0s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\log \left(\sqrt[3]{y}\right) \cdot \left(3 \cdot x - 3\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\log \left(\sqrt[3]{y}\right) \cdot \left(3 \cdot x - 3\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r49358 = x;
        double r49359 = 1.0;
        double r49360 = r49358 - r49359;
        double r49361 = y;
        double r49362 = log(r49361);
        double r49363 = r49360 * r49362;
        double r49364 = z;
        double r49365 = r49364 - r49359;
        double r49366 = r49359 - r49361;
        double r49367 = log(r49366);
        double r49368 = r49365 * r49367;
        double r49369 = r49363 + r49368;
        double r49370 = t;
        double r49371 = r49369 - r49370;
        return r49371;
}
↓
double f(double x, double y, double z, double t) {
        double r49372 = y;
        double r49373 = cbrt(r49372);
        double r49374 = log(r49373);
        double r49375 = 3.0;
        double r49376 = x;
        double r49377 = r49375 * r49376;
        double r49378 = 3.0;
        double r49379 = r49377 - r49378;
        double r49380 = r49374 * r49379;
        double r49381 = z;
        double r49382 = 1.0;
        double r49383 = r49381 - r49382;
        double r49384 = log(r49382);
        double r49385 = r49382 * r49372;
        double r49386 = 0.5;
        double r49387 = 2.0;
        double r49388 = pow(r49372, r49387);
        double r49389 = pow(r49382, r49387);
        double r49390 = r49388 / r49389;
        double r49391 = r49386 * r49390;
        double r49392 = r49385 + r49391;
        double r49393 = r49384 - r49392;
        double r49394 = r49383 * r49393;
        double r49395 = r49380 + r49394;
        double r49396 = t;
        double r49397 = r49395 - r49396;
        return r49397;
}

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-cube-cbrt0.4

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

  5. Applied log-prod0.5

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

  6. Applied distribute-lft-in0.5

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

  7. Simplified0.5

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

  8. Taylor expanded around 0 0.5

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

  9. Simplified0.5

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

  10. Final simplification0.5

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

Reproduce

herbie shell --seed 2019198 
(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))