Average Error: 7.2 → 0.4
Time: 1.7m
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(z - 1\right) \cdot \left(\log 1 - \left(\left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2} + y \cdot 1\right)\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right) + \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(z - 1\right) \cdot \left(\log 1 - \left(\left(\frac{y}{1} \cdot \frac{y}{1}\right) \cdot \frac{1}{2} + y \cdot 1\right)\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right) + \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1\right)\right) - t
double f(double x, double y, double z, double t) {
        double r1992576 = x;
        double r1992577 = 1.0;
        double r1992578 = r1992576 - r1992577;
        double r1992579 = y;
        double r1992580 = log(r1992579);
        double r1992581 = r1992578 * r1992580;
        double r1992582 = z;
        double r1992583 = r1992582 - r1992577;
        double r1992584 = r1992577 - r1992579;
        double r1992585 = log(r1992584);
        double r1992586 = r1992583 * r1992585;
        double r1992587 = r1992581 + r1992586;
        double r1992588 = t;
        double r1992589 = r1992587 - r1992588;
        return r1992589;
}
↓
double f(double x, double y, double z, double t) {
        double r1992590 = z;
        double r1992591 = 1.0;
        double r1992592 = r1992590 - r1992591;
        double r1992593 = log(r1992591);
        double r1992594 = y;
        double r1992595 = r1992594 / r1992591;
        double r1992596 = r1992595 * r1992595;
        double r1992597 = 0.5;
        double r1992598 = r1992596 * r1992597;
        double r1992599 = r1992594 * r1992591;
        double r1992600 = r1992598 + r1992599;
        double r1992601 = r1992593 - r1992600;
        double r1992602 = r1992592 * r1992601;
        double r1992603 = x;
        double r1992604 = r1992603 - r1992591;
        double r1992605 = cbrt(r1992594);
        double r1992606 = log(r1992605);
        double r1992607 = r1992604 * r1992606;
        double r1992608 = r1992602 + r1992607;
        double r1992609 = r1992605 * r1992605;
        double r1992610 = log(r1992609);
        double r1992611 = r1992610 * r1992604;
        double r1992612 = r1992608 + r1992611;
        double r1992613 = t;
        double r1992614 = r1992612 - r1992613;
        return r1992614;
}

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 7.2

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

  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. Simplified0.3

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

  5. Applied add-cube-cbrt0.3

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

  6. Applied log-prod0.4

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

  7. Applied distribute-lft-in0.4

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

  8. Applied associate-+l+0.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2019200 
(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))