Average Error: 7.3 → 0.4
Time: 32.2s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(\sqrt{\log 1} + \sqrt{\frac{1}{2} \cdot \left(\frac{y}{1} \cdot \frac{y}{1}\right) + y \cdot 1}\right) \cdot \left(z - 1\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{\frac{1}{2} \cdot \left(\frac{y}{1} \cdot \frac{y}{1}\right) + y \cdot 1}\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt{y}\right) + \left(x - 1\right) \cdot \log \left(\sqrt{y}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(\sqrt{\log 1} + \sqrt{\frac{1}{2} \cdot \left(\frac{y}{1} \cdot \frac{y}{1}\right) + y \cdot 1}\right) \cdot \left(z - 1\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{\frac{1}{2} \cdot \left(\frac{y}{1} \cdot \frac{y}{1}\right) + y \cdot 1}\right) + \left(\left(x - 1\right) \cdot \log \left(\sqrt{y}\right) + \left(x - 1\right) \cdot \log \left(\sqrt{y}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r2315587 = x;
        double r2315588 = 1.0;
        double r2315589 = r2315587 - r2315588;
        double r2315590 = y;
        double r2315591 = log(r2315590);
        double r2315592 = r2315589 * r2315591;
        double r2315593 = z;
        double r2315594 = r2315593 - r2315588;
        double r2315595 = r2315588 - r2315590;
        double r2315596 = log(r2315595);
        double r2315597 = r2315594 * r2315596;
        double r2315598 = r2315592 + r2315597;
        double r2315599 = t;
        double r2315600 = r2315598 - r2315599;
        return r2315600;
}
↓
double f(double x, double y, double z, double t) {
        double r2315601 = 1.0;
        double r2315602 = log(r2315601);
        double r2315603 = sqrt(r2315602);
        double r2315604 = 0.5;
        double r2315605 = y;
        double r2315606 = r2315605 / r2315601;
        double r2315607 = r2315606 * r2315606;
        double r2315608 = r2315604 * r2315607;
        double r2315609 = r2315605 * r2315601;
        double r2315610 = r2315608 + r2315609;
        double r2315611 = sqrt(r2315610);
        double r2315612 = r2315603 + r2315611;
        double r2315613 = z;
        double r2315614 = r2315613 - r2315601;
        double r2315615 = r2315612 * r2315614;
        double r2315616 = r2315603 - r2315611;
        double r2315617 = r2315615 * r2315616;
        double r2315618 = x;
        double r2315619 = r2315618 - r2315601;
        double r2315620 = sqrt(r2315605);
        double r2315621 = log(r2315620);
        double r2315622 = r2315619 * r2315621;
        double r2315623 = r2315622 + r2315622;
        double r2315624 = r2315617 + r2315623;
        double r2315625 = t;
        double r2315626 = r2315624 - r2315625;
        return r2315626;
}

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 7.3

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

  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

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

  5. Applied add-sqr-sqrt0.3

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

  6. Applied log-prod0.3

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

  7. Applied distribute-lft-in0.3

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

  9. Applied add-sqr-sqrt0.4

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied difference-of-squares0.4

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

  12. Applied associate-*r*0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2019172 
(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))