Average Error: 6.6 → 0.4
Time: 25.8s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{\left(y \cdot y\right) \cdot \frac{1}{2}}{{1}^{2}}\right)\right) + \left(\left(\left(x - 1\right) \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2\right) + \left(x - 1\right) \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)\right) - t\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{\left(y \cdot y\right) \cdot \frac{1}{2}}{{1}^{2}}\right)\right) + \left(\left(\left(x - 1\right) \cdot \left(\log \left(\sqrt[3]{y}\right) \cdot 2\right) + \left(x - 1\right) \cdot \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r44477 = x;
        double r44478 = 1.0;
        double r44479 = r44477 - r44478;
        double r44480 = y;
        double r44481 = log(r44480);
        double r44482 = r44479 * r44481;
        double r44483 = z;
        double r44484 = r44483 - r44478;
        double r44485 = r44478 - r44480;
        double r44486 = log(r44485);
        double r44487 = r44484 * r44486;
        double r44488 = r44482 + r44487;
        double r44489 = t;
        double r44490 = r44488 - r44489;
        return r44490;
}
↓
double f(double x, double y, double z, double t) {
        double r44491 = z;
        double r44492 = 1.0;
        double r44493 = r44491 - r44492;
        double r44494 = log(r44492);
        double r44495 = y;
        double r44496 = r44492 * r44495;
        double r44497 = r44495 * r44495;
        double r44498 = 0.5;
        double r44499 = r44497 * r44498;
        double r44500 = 2.0;
        double r44501 = pow(r44492, r44500);
        double r44502 = r44499 / r44501;
        double r44503 = r44496 + r44502;
        double r44504 = r44494 - r44503;
        double r44505 = r44493 * r44504;
        double r44506 = x;
        double r44507 = r44506 - r44492;
        double r44508 = cbrt(r44495);
        double r44509 = log(r44508);
        double r44510 = r44509 * r44500;
        double r44511 = r44507 * r44510;
        double r44512 = 1.0;
        double r44513 = r44512 / r44495;
        double r44514 = -0.3333333333333333;
        double r44515 = pow(r44513, r44514);
        double r44516 = log(r44515);
        double r44517 = r44507 * r44516;
        double r44518 = r44511 + r44517;
        double r44519 = t;
        double r44520 = r44518 - r44519;
        double r44521 = r44505 + r44520;
        return r44521;
}

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 6.6

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

  2. Simplified6.6

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

  3. Taylor expanded around 0 0.4

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

  4. Simplified0.4

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

  6. Applied add-cube-cbrt0.4

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

  7. Applied log-prod0.5

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

  8. Applied distribute-lft-in0.5

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

  9. Simplified0.5

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

  10. Simplified0.5

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

  11. Taylor expanded around inf 0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2019196 
(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))