Average Error: 6.9 → 0.4
Time: 10.2s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(x - 1\right) \cdot \log \left(\sqrt{y}\right) + \left(\log \left(\sqrt{y}\right) \cdot \left(x - 1\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right)\right) - t\]
\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(\sqrt{y}\right) + \left(\log \left(\sqrt{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 r68545 = x;
        double r68546 = 1.0;
        double r68547 = r68545 - r68546;
        double r68548 = y;
        double r68549 = log(r68548);
        double r68550 = r68547 * r68549;
        double r68551 = z;
        double r68552 = r68551 - r68546;
        double r68553 = r68546 - r68548;
        double r68554 = log(r68553);
        double r68555 = r68552 * r68554;
        double r68556 = r68550 + r68555;
        double r68557 = t;
        double r68558 = r68556 - r68557;
        return r68558;
}
↓
double f(double x, double y, double z, double t) {
        double r68559 = x;
        double r68560 = 1.0;
        double r68561 = r68559 - r68560;
        double r68562 = y;
        double r68563 = sqrt(r68562);
        double r68564 = log(r68563);
        double r68565 = r68561 * r68564;
        double r68566 = r68564 * r68561;
        double r68567 = z;
        double r68568 = r68567 - r68560;
        double r68569 = log(r68560);
        double r68570 = r68560 * r68562;
        double r68571 = 0.5;
        double r68572 = 2.0;
        double r68573 = pow(r68562, r68572);
        double r68574 = pow(r68560, r68572);
        double r68575 = r68573 / r68574;
        double r68576 = r68571 * r68575;
        double r68577 = r68570 + r68576;
        double r68578 = r68569 - r68577;
        double r68579 = r68568 * r68578;
        double r68580 = r68566 + r68579;
        double r68581 = r68565 + r68580;
        double r68582 = t;
        double r68583 = r68581 - r68582;
        return r68583;
}

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 6.9

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

  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-sqr-sqrt0.4

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

  5. Applied log-prod0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Applied associate-+l+0.4

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

  8. Simplified0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2020083 
(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))