Average Error: 6.6 → 0.5
Time: 24.6s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(2 \cdot \log \left(\left(\sqrt[3]{\sqrt[3]{{y}^{\frac{2}{3}}}} \cdot \sqrt[3]{\sqrt[3]{{y}^{\frac{2}{3}}}}\right) \cdot \sqrt[3]{\sqrt[3]{{y}^{\frac{2}{3}}}}\right)\right) \cdot \left(x - 1\right) + \left(x - 1\right) \cdot \mathsf{fma}\left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right), 2, \log \left(\sqrt[3]{y}\right)\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, y, \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\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(\left(\sqrt[3]{\sqrt[3]{{y}^{\frac{2}{3}}}} \cdot \sqrt[3]{\sqrt[3]{{y}^{\frac{2}{3}}}}\right) \cdot \sqrt[3]{\sqrt[3]{{y}^{\frac{2}{3}}}}\right)\right) \cdot \left(x - 1\right) + \left(x - 1\right) \cdot \mathsf{fma}\left(\log \left(\sqrt[3]{\sqrt[3]{y}}\right), 2, \log \left(\sqrt[3]{y}\right)\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \mathsf{fma}\left(1, 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 r80251 = x;
        double r80252 = 1.0;
        double r80253 = r80251 - r80252;
        double r80254 = y;
        double r80255 = log(r80254);
        double r80256 = r80253 * r80255;
        double r80257 = z;
        double r80258 = r80257 - r80252;
        double r80259 = r80252 - r80254;
        double r80260 = log(r80259);
        double r80261 = r80258 * r80260;
        double r80262 = r80256 + r80261;
        double r80263 = t;
        double r80264 = r80262 - r80263;
        return r80264;
}
↓
double f(double x, double y, double z, double t) {
        double r80265 = 2.0;
        double r80266 = y;
        double r80267 = 0.6666666666666666;
        double r80268 = pow(r80266, r80267);
        double r80269 = cbrt(r80268);
        double r80270 = cbrt(r80269);
        double r80271 = r80270 * r80270;
        double r80272 = r80271 * r80270;
        double r80273 = log(r80272);
        double r80274 = r80265 * r80273;
        double r80275 = x;
        double r80276 = 1.0;
        double r80277 = r80275 - r80276;
        double r80278 = r80274 * r80277;
        double r80279 = cbrt(r80266);
        double r80280 = cbrt(r80279);
        double r80281 = log(r80280);
        double r80282 = log(r80279);
        double r80283 = fma(r80281, r80265, r80282);
        double r80284 = r80277 * r80283;
        double r80285 = r80278 + r80284;
        double r80286 = z;
        double r80287 = r80286 - r80276;
        double r80288 = log(r80276);
        double r80289 = 0.5;
        double r80290 = pow(r80266, r80265);
        double r80291 = pow(r80276, r80265);
        double r80292 = r80290 / r80291;
        double r80293 = r80289 * r80292;
        double r80294 = fma(r80276, r80266, r80293);
        double r80295 = r80288 - r80294;
        double r80296 = r80287 * r80295;
        double r80297 = r80285 + r80296;
        double r80298 = t;
        double r80299 = r80297 - r80298;
        return r80299;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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. Simplified0.4

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

  5. Applied add-cube-cbrt0.4

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

  6. Applied log-prod0.5

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

  7. Applied distribute-lft-in0.5

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

  8. Simplified0.5

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

  9. Simplified0.5

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

  11. Applied add-cube-cbrt0.5

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

  12. Applied cbrt-prod0.5

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

  13. Applied log-prod0.5

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

  14. Applied distribute-lft-in0.5

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

  15. Applied distribute-rgt-in0.4

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

  16. Applied associate-+l+0.5

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

  17. Simplified0.4

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

  19. Applied add-cube-cbrt0.5

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

  20. Simplified0.5

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

  21. Simplified0.5

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

  22. Final simplification0.5

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(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))