Average Error: 7.1 → 0.4
Time: 9.6s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(2 \cdot \log \left({y}^{\frac{1}{3}}\right)\right) \cdot \left(x - 1\right) + \left(x - 1\right) \cdot \log \left(\sqrt[3]{y}\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(2 \cdot \log \left({y}^{\frac{1}{3}}\right)\right) \cdot \left(x - 1\right) + \left(x - 1\right) \cdot \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
double f(double x, double y, double z, double t) {
        double r58351 = x;
        double r58352 = 1.0;
        double r58353 = r58351 - r58352;
        double r58354 = y;
        double r58355 = log(r58354);
        double r58356 = r58353 * r58355;
        double r58357 = z;
        double r58358 = r58357 - r58352;
        double r58359 = r58352 - r58354;
        double r58360 = log(r58359);
        double r58361 = r58358 * r58360;
        double r58362 = r58356 + r58361;
        double r58363 = t;
        double r58364 = r58362 - r58363;
        return r58364;
}
↓
double f(double x, double y, double z, double t) {
        double r58365 = 2.0;
        double r58366 = y;
        double r58367 = 0.3333333333333333;
        double r58368 = pow(r58366, r58367);
        double r58369 = log(r58368);
        double r58370 = r58365 * r58369;
        double r58371 = x;
        double r58372 = 1.0;
        double r58373 = r58371 - r58372;
        double r58374 = r58370 * r58373;
        double r58375 = cbrt(r58366);
        double r58376 = log(r58375);
        double r58377 = r58373 * r58376;
        double r58378 = r58374 + r58377;
        double r58379 = z;
        double r58380 = r58379 - r58372;
        double r58381 = log(r58372);
        double r58382 = r58372 * r58366;
        double r58383 = 0.5;
        double r58384 = pow(r58366, r58365);
        double r58385 = pow(r58372, r58365);
        double r58386 = r58384 / r58385;
        double r58387 = r58383 * r58386;
        double r58388 = r58382 + r58387;
        double r58389 = r58381 - r58388;
        double r58390 = r58380 * r58389;
        double r58391 = r58378 + r58390;
        double r58392 = t;
        double r58393 = r58391 - r58392;
        return r58393;
}

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 0.3

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

  4. Applied add-cube-cbrt0.3

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

  5. Applied log-prod0.4

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

  6. Applied distribute-lft-in0.4

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

  7. Simplified0.4

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

  9. Applied pow1/30.4

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

  10. Final simplification0.4

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

Reproduce

herbie shell --seed 2019354 
(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))