Average Error: 6.5 → 0.5
Time: 23.1s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(x - 1\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right) + \frac{2}{3} \cdot \log y\right) + \log \left(\sqrt[3]{\sqrt[3]{y}}\right) \cdot \left(x - 1\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(x - 1\right) \cdot \left(2 \cdot \log \left(\sqrt[3]{\sqrt[3]{y}}\right) + \frac{2}{3} \cdot \log y\right) + \log \left(\sqrt[3]{\sqrt[3]{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
double f(double x, double y, double z, double t) {
        double r72251 = x;
        double r72252 = 1.0;
        double r72253 = r72251 - r72252;
        double r72254 = y;
        double r72255 = log(r72254);
        double r72256 = r72253 * r72255;
        double r72257 = z;
        double r72258 = r72257 - r72252;
        double r72259 = r72252 - r72254;
        double r72260 = log(r72259);
        double r72261 = r72258 * r72260;
        double r72262 = r72256 + r72261;
        double r72263 = t;
        double r72264 = r72262 - r72263;
        return r72264;
}
↓
double f(double x, double y, double z, double t) {
        double r72265 = x;
        double r72266 = 1.0;
        double r72267 = r72265 - r72266;
        double r72268 = 2.0;
        double r72269 = y;
        double r72270 = cbrt(r72269);
        double r72271 = cbrt(r72270);
        double r72272 = log(r72271);
        double r72273 = r72268 * r72272;
        double r72274 = 0.6666666666666666;
        double r72275 = log(r72269);
        double r72276 = r72274 * r72275;
        double r72277 = r72273 + r72276;
        double r72278 = r72267 * r72277;
        double r72279 = r72272 * r72267;
        double r72280 = r72278 + r72279;
        double r72281 = z;
        double r72282 = r72281 - r72266;
        double r72283 = log(r72266);
        double r72284 = r72266 * r72269;
        double r72285 = 0.5;
        double r72286 = pow(r72269, r72268);
        double r72287 = pow(r72266, r72268);
        double r72288 = r72286 / r72287;
        double r72289 = r72285 * r72288;
        double r72290 = r72284 + r72289;
        double r72291 = r72283 - r72290;
        double r72292 = r72282 * r72291;
        double r72293 = r72280 + r72292;
        double r72294 = t;
        double r72295 = r72293 - r72294;
        return r72295;
}

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 6.5

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

  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-cube-cbrt0.4

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

  5. Applied log-prod0.5

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

  6. Applied distribute-lft-in0.5

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

  7. Simplified0.5

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

  9. Applied add-cube-cbrt0.5

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

  10. Applied log-prod0.4

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

  11. Applied distribute-rgt-in0.4

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

  12. Applied associate-+r+0.4

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

  13. Simplified0.5

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

  14. Final simplification0.5

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

Reproduce

herbie shell --seed 2020043 
(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))