Average Error: 7.3 → 0.4
Time: 31.4s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\log \left({y}^{\frac{2}{3}}\right) \cdot x + \left(\left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) + \left(-1 \cdot \log \left({y}^{\frac{2}{3}}\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\log \left({y}^{\frac{2}{3}}\right) \cdot x + \left(\left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\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) + \left(-1 \cdot \log \left({y}^{\frac{2}{3}}\right)\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r43460 = x;
        double r43461 = 1.0;
        double r43462 = r43460 - r43461;
        double r43463 = y;
        double r43464 = log(r43463);
        double r43465 = r43462 * r43464;
        double r43466 = z;
        double r43467 = r43466 - r43461;
        double r43468 = r43461 - r43463;
        double r43469 = log(r43468);
        double r43470 = r43467 * r43469;
        double r43471 = r43465 + r43470;
        double r43472 = t;
        double r43473 = r43471 - r43472;
        return r43473;
}
↓
double f(double x, double y, double z, double t) {
        double r43474 = y;
        double r43475 = 0.6666666666666666;
        double r43476 = pow(r43474, r43475);
        double r43477 = log(r43476);
        double r43478 = x;
        double r43479 = r43477 * r43478;
        double r43480 = cbrt(r43474);
        double r43481 = log(r43480);
        double r43482 = 1.0;
        double r43483 = r43478 - r43482;
        double r43484 = r43481 * r43483;
        double r43485 = z;
        double r43486 = r43485 - r43482;
        double r43487 = log(r43482);
        double r43488 = r43482 * r43474;
        double r43489 = 0.5;
        double r43490 = 2.0;
        double r43491 = pow(r43474, r43490);
        double r43492 = pow(r43482, r43490);
        double r43493 = r43491 / r43492;
        double r43494 = r43489 * r43493;
        double r43495 = r43488 + r43494;
        double r43496 = r43487 - r43495;
        double r43497 = r43486 * r43496;
        double r43498 = r43484 + r43497;
        double r43499 = r43482 * r43477;
        double r43500 = -r43499;
        double r43501 = r43498 + r43500;
        double r43502 = r43479 + r43501;
        double r43503 = t;
        double r43504 = r43502 - r43503;
        return r43504;
}

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 7.3

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

  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-rgt-in0.4

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

  7. Applied associate-+l+0.4

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

  9. Applied *-un-lft-identity0.4

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

  10. Applied cbrt-prod0.4

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

  11. Applied associate-*l*0.4

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

  12. Simplified0.4

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

  14. Applied sub-neg0.4

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

  15. Applied distribute-rgt-in0.4

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

  16. Applied associate-+l+0.4

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

  17. Simplified0.4

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

  18. Final simplification0.4

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

Reproduce

herbie shell --seed 2019322 
(FPCore (x y z t)
  :name "Statistics.Distribution.Beta:$cdensity from math-functions-0.1.5.2"
  :precision binary64
  (- (+ (* (- x 1) (log y)) (* (- z 1) (log (- 1 y)))) t))