Average Error: 7.0 → 0.4
Time: 9.9s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log \left({y}^{\frac{1}{3}} \cdot \sqrt[3]{y}\right) + \left(\log \left(\sqrt[3]{y}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(x - 1\right) \cdot \log \left({y}^{\frac{1}{3}} \cdot \sqrt[3]{y}\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
double f(double x, double y, double z, double t) {
        double r68529 = x;
        double r68530 = 1.0;
        double r68531 = r68529 - r68530;
        double r68532 = y;
        double r68533 = log(r68532);
        double r68534 = r68531 * r68533;
        double r68535 = z;
        double r68536 = r68535 - r68530;
        double r68537 = r68530 - r68532;
        double r68538 = log(r68537);
        double r68539 = r68536 * r68538;
        double r68540 = r68534 + r68539;
        double r68541 = t;
        double r68542 = r68540 - r68541;
        return r68542;
}
↓
double f(double x, double y, double z, double t) {
        double r68543 = x;
        double r68544 = 1.0;
        double r68545 = r68543 - r68544;
        double r68546 = y;
        double r68547 = 0.3333333333333333;
        double r68548 = pow(r68546, r68547);
        double r68549 = cbrt(r68546);
        double r68550 = r68548 * r68549;
        double r68551 = log(r68550);
        double r68552 = r68545 * r68551;
        double r68553 = log(r68549);
        double r68554 = r68553 * r68545;
        double r68555 = z;
        double r68556 = r68555 - r68544;
        double r68557 = log(r68544);
        double r68558 = r68544 * r68546;
        double r68559 = 0.5;
        double r68560 = 2.0;
        double r68561 = pow(r68546, r68560);
        double r68562 = pow(r68544, r68560);
        double r68563 = r68561 / r68562;
        double r68564 = r68559 * r68563;
        double r68565 = r68558 + r68564;
        double r68566 = r68557 - r68565;
        double r68567 = r68556 * r68566;
        double r68568 = r68554 + r68567;
        double r68569 = r68552 + r68568;
        double r68570 = t;
        double r68571 = r68569 - r68570;
        return r68571;
}

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 7.0

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

  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. Using strategy rm

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Applied associate-+l+0.4

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

  8. Simplified0.4

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

  10. Applied pow1/30.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2020081 
(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))