Average Error: 6.7 → 0.4
Time: 26.5s
Precision: 64
\[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]
↓
\[\left(\left(\left(\log 1.0 - \frac{\frac{1}{2}}{\frac{1.0}{y} \cdot \frac{1.0}{y}}\right) - 1.0 \cdot y\right) \cdot \left(z - 1.0\right) + \left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \log \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x - 1.0\right)\right)\right) - t\]
\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t
↓
\left(\left(\left(\log 1.0 - \frac{\frac{1}{2}}{\frac{1.0}{y} \cdot \frac{1.0}{y}}\right) - 1.0 \cdot y\right) \cdot \left(z - 1.0\right) + \left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \log \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x - 1.0\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r2324433 = x;
        double r2324434 = 1.0;
        double r2324435 = r2324433 - r2324434;
        double r2324436 = y;
        double r2324437 = log(r2324436);
        double r2324438 = r2324435 * r2324437;
        double r2324439 = z;
        double r2324440 = r2324439 - r2324434;
        double r2324441 = r2324434 - r2324436;
        double r2324442 = log(r2324441);
        double r2324443 = r2324440 * r2324442;
        double r2324444 = r2324438 + r2324443;
        double r2324445 = t;
        double r2324446 = r2324444 - r2324445;
        return r2324446;
}
↓
double f(double x, double y, double z, double t) {
        double r2324447 = 1.0;
        double r2324448 = log(r2324447);
        double r2324449 = 0.5;
        double r2324450 = y;
        double r2324451 = r2324447 / r2324450;
        double r2324452 = r2324451 * r2324451;
        double r2324453 = r2324449 / r2324452;
        double r2324454 = r2324448 - r2324453;
        double r2324455 = r2324447 * r2324450;
        double r2324456 = r2324454 - r2324455;
        double r2324457 = z;
        double r2324458 = r2324457 - r2324447;
        double r2324459 = r2324456 * r2324458;
        double r2324460 = cbrt(r2324450);
        double r2324461 = r2324460 * r2324460;
        double r2324462 = log(r2324461);
        double r2324463 = x;
        double r2324464 = r2324463 - r2324447;
        double r2324465 = r2324462 * r2324464;
        double r2324466 = cbrt(r2324460);
        double r2324467 = r2324466 * r2324466;
        double r2324468 = r2324467 * r2324466;
        double r2324469 = log(r2324468);
        double r2324470 = r2324469 * r2324464;
        double r2324471 = r2324465 + r2324470;
        double r2324472 = r2324459 + r2324471;
        double r2324473 = t;
        double r2324474 = r2324472 - r2324473;
        return r2324474;
}

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 6.7

    \[\left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \log \left(1.0 - y\right)\right) - t\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \color{blue}{\left(\log 1.0 - \left(1.0 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1.0}^{2}}\right)\right)}\right) - t\]

  3. Simplified0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log y + \left(z - 1.0\right) \cdot \color{blue}{\left(\left(\log 1.0 - \frac{\frac{1}{2}}{\frac{1.0}{y} \cdot \frac{1.0}{y}}\right) - y \cdot 1.0\right)}\right) - t\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - \frac{\frac{1}{2}}{\frac{1.0}{y} \cdot \frac{1.0}{y}}\right) - y \cdot 1.0\right)\right) - t\]

  6. Applied log-prod0.4

    \[\leadsto \left(\left(x - 1.0\right) \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - \frac{\frac{1}{2}}{\frac{1.0}{y} \cdot \frac{1.0}{y}}\right) - y \cdot 1.0\right)\right) - t\]

  7. Applied distribute-rgt-in0.4

    \[\leadsto \left(\color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \log \left(\sqrt[3]{y}\right) \cdot \left(x - 1.0\right)\right)} + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - \frac{\frac{1}{2}}{\frac{1.0}{y} \cdot \frac{1.0}{y}}\right) - y \cdot 1.0\right)\right) - t\]
  8. Using strategy rm

  9. Applied add-cube-cbrt0.4

    \[\leadsto \left(\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \log \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right)} \cdot \left(x - 1.0\right)\right) + \left(z - 1.0\right) \cdot \left(\left(\log 1.0 - \frac{\frac{1}{2}}{\frac{1.0}{y} \cdot \frac{1.0}{y}}\right) - y \cdot 1.0\right)\right) - t\]

  10. Final simplification0.4

    \[\leadsto \left(\left(\left(\log 1.0 - \frac{\frac{1}{2}}{\frac{1.0}{y} \cdot \frac{1.0}{y}}\right) - 1.0 \cdot y\right) \cdot \left(z - 1.0\right) + \left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \left(x - 1.0\right) + \log \left(\left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x - 1.0\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019165 
(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))