Average Error: 7.1 → 0.5
Time: 23.5s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(x - 1\right) \cdot \left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right) + \left(\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(x - 1\right) \cdot \left(\log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right) + \left(\left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right) - t\right)
double f(double x, double y, double z, double t) {
        double r68567 = x;
        double r68568 = 1.0;
        double r68569 = r68567 - r68568;
        double r68570 = y;
        double r68571 = log(r68570);
        double r68572 = r68569 * r68571;
        double r68573 = z;
        double r68574 = r68573 - r68568;
        double r68575 = r68568 - r68570;
        double r68576 = log(r68575);
        double r68577 = r68574 * r68576;
        double r68578 = r68572 + r68577;
        double r68579 = t;
        double r68580 = r68578 - r68579;
        return r68580;
}
↓
double f(double x, double y, double z, double t) {
        double r68581 = x;
        double r68582 = 1.0;
        double r68583 = r68581 - r68582;
        double r68584 = 1.0;
        double r68585 = y;
        double r68586 = r68584 / r68585;
        double r68587 = -0.3333333333333333;
        double r68588 = pow(r68586, r68587);
        double r68589 = cbrt(r68585);
        double r68590 = r68588 * r68589;
        double r68591 = log(r68590);
        double r68592 = log(r68589);
        double r68593 = r68591 + r68592;
        double r68594 = r68583 * r68593;
        double r68595 = z;
        double r68596 = r68595 - r68582;
        double r68597 = log(r68582);
        double r68598 = r68582 * r68585;
        double r68599 = 0.5;
        double r68600 = 2.0;
        double r68601 = pow(r68585, r68600);
        double r68602 = pow(r68582, r68600);
        double r68603 = r68601 / r68602;
        double r68604 = r68599 * r68603;
        double r68605 = r68598 + r68604;
        double r68606 = r68597 - r68605;
        double r68607 = r68596 * r68606;
        double r68608 = t;
        double r68609 = r68607 - r68608;
        double r68610 = r68594 + r68609;
        return r68610;
}

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. 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-in0.5

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

  7. Applied associate-+l+0.5

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

  8. Taylor expanded around inf 0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2019291 
(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))