Average Error: 6.6 → 0.4
Time: 9.3s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\log \left(\sqrt{y}\right) \cdot \left(x - 1\right) + \left(\log \left(\sqrt{y}\right) \cdot \left(x - 1\right) + \left(\left(z - 1\right) \cdot \left(\sqrt{\log 1} + \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\log \left(\sqrt{y}\right) \cdot \left(x - 1\right) + \left(\log \left(\sqrt{y}\right) \cdot \left(x - 1\right) + \left(\left(z - 1\right) \cdot \left(\sqrt{\log 1} + \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right) \cdot \left(\sqrt{\log 1} - \sqrt{1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r65392 = x;
        double r65393 = 1.0;
        double r65394 = r65392 - r65393;
        double r65395 = y;
        double r65396 = log(r65395);
        double r65397 = r65394 * r65396;
        double r65398 = z;
        double r65399 = r65398 - r65393;
        double r65400 = r65393 - r65395;
        double r65401 = log(r65400);
        double r65402 = r65399 * r65401;
        double r65403 = r65397 + r65402;
        double r65404 = t;
        double r65405 = r65403 - r65404;
        return r65405;
}
↓
double f(double x, double y, double z, double t) {
        double r65406 = y;
        double r65407 = sqrt(r65406);
        double r65408 = log(r65407);
        double r65409 = x;
        double r65410 = 1.0;
        double r65411 = r65409 - r65410;
        double r65412 = r65408 * r65411;
        double r65413 = z;
        double r65414 = r65413 - r65410;
        double r65415 = log(r65410);
        double r65416 = sqrt(r65415);
        double r65417 = r65410 * r65406;
        double r65418 = 0.5;
        double r65419 = 2.0;
        double r65420 = pow(r65406, r65419);
        double r65421 = pow(r65410, r65419);
        double r65422 = r65420 / r65421;
        double r65423 = r65418 * r65422;
        double r65424 = r65417 + r65423;
        double r65425 = sqrt(r65424);
        double r65426 = r65416 + r65425;
        double r65427 = r65414 * r65426;
        double r65428 = r65416 - r65425;
        double r65429 = r65427 * r65428;
        double r65430 = r65412 + r65429;
        double r65431 = r65412 + r65430;
        double r65432 = t;
        double r65433 = r65431 - r65432;
        return r65433;
}

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 6.6

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

  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-rgt-in0.4

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

  7. Applied associate-+l+0.4

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

  9. Applied add-sqr-sqrt0.4

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

  10. Applied add-sqr-sqrt0.4

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

  11. Applied difference-of-squares0.4

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

  12. Applied associate-*r*0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2020021 
(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))