Average Error: 7.1 → 0.6
Time: 25.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left({y}^{2} \cdot 0.5 - z \cdot \left({y}^{2} \cdot 0.5\right)\right) - z \cdot \left(1 \cdot y\right)\right) + \left(\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left({y}^{2} \cdot 0.5 - z \cdot \left({y}^{2} \cdot 0.5\right)\right) - z \cdot \left(1 \cdot y\right)\right) + \left(\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r48349 = x;
        double r48350 = 1.0;
        double r48351 = r48349 - r48350;
        double r48352 = y;
        double r48353 = log(r48352);
        double r48354 = r48351 * r48353;
        double r48355 = z;
        double r48356 = r48355 - r48350;
        double r48357 = r48350 - r48352;
        double r48358 = log(r48357);
        double r48359 = r48356 * r48358;
        double r48360 = r48354 + r48359;
        double r48361 = t;
        double r48362 = r48360 - r48361;
        return r48362;
}
↓
double f(double x, double y, double z, double t) {
        double r48363 = y;
        double r48364 = 2.0;
        double r48365 = pow(r48363, r48364);
        double r48366 = 0.5;
        double r48367 = r48365 * r48366;
        double r48368 = z;
        double r48369 = r48368 * r48367;
        double r48370 = r48367 - r48369;
        double r48371 = 1.0;
        double r48372 = r48371 * r48363;
        double r48373 = r48368 * r48372;
        double r48374 = r48370 - r48373;
        double r48375 = cbrt(r48363);
        double r48376 = log(r48375);
        double r48377 = r48364 * r48376;
        double r48378 = x;
        double r48379 = r48378 - r48371;
        double r48380 = r48377 * r48379;
        double r48381 = r48376 * r48379;
        double r48382 = r48380 + r48381;
        double r48383 = r48374 + r48382;
        double r48384 = t;
        double r48385 = r48383 - r48384;
        return r48385;
}

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 7.1

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

  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. Simplified0.4

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

  4. Taylor expanded around inf 0.5

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(0.5 \cdot {y}^{2} - \left(1 \cdot \left(z \cdot y\right) + 0.5 \cdot \left(z \cdot {y}^{2}\right)\right)\right)}\right) - t\]

  5. Simplified0.5

    \[\leadsto \left(\left(x - 1\right) \cdot \log y + \color{blue}{\left(\left({y}^{2} \cdot 0.5 - \left({y}^{2} \cdot 0.5\right) \cdot z\right) - \left(1 \cdot y\right) \cdot z\right)}\right) - t\]
  6. Using strategy rm

  7. Applied add-cube-cbrt0.5

    \[\leadsto \left(\left(x - 1\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + \left(\left({y}^{2} \cdot 0.5 - \left({y}^{2} \cdot 0.5\right) \cdot z\right) - \left(1 \cdot y\right) \cdot z\right)\right) - t\]

  8. Applied log-prod0.6

    \[\leadsto \left(\left(x - 1\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + \left(\left({y}^{2} \cdot 0.5 - \left({y}^{2} \cdot 0.5\right) \cdot z\right) - \left(1 \cdot y\right) \cdot z\right)\right) - t\]

  9. Applied distribute-lft-in0.6

    \[\leadsto \left(\color{blue}{\left(\left(x - 1\right) \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(\left({y}^{2} \cdot 0.5 - \left({y}^{2} \cdot 0.5\right) \cdot z\right) - \left(1 \cdot y\right) \cdot z\right)\right) - t\]

  10. Simplified0.6

    \[\leadsto \left(\left(\color{blue}{\left(x - 1\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right)} + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(\left({y}^{2} \cdot 0.5 - \left({y}^{2} \cdot 0.5\right) \cdot z\right) - \left(1 \cdot y\right) \cdot z\right)\right) - t\]

  11. Simplified0.6

    \[\leadsto \left(\left(\left(x - 1\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) + \color{blue}{\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)}\right) + \left(\left({y}^{2} \cdot 0.5 - \left({y}^{2} \cdot 0.5\right) \cdot z\right) - \left(1 \cdot y\right) \cdot z\right)\right) - t\]

  12. Final simplification0.6

    \[\leadsto \left(\left(\left({y}^{2} \cdot 0.5 - z \cdot \left({y}^{2} \cdot 0.5\right)\right) - z \cdot \left(1 \cdot y\right)\right) + \left(\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(x - 1\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019179 
(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))