Average Error: 7.0 → 0.4
Time: 27.5s
Precision: 64
\[\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(x - 1\right) + \log \left({\left(\frac{1}{{y}^{\frac{2}{3}}}\right)}^{\frac{-1}{3}} \cdot {\left(\frac{1}{\sqrt[3]{y}}\right)}^{\frac{-1}{3}}\right) \cdot \left(x - 1\right)\right) + \left(z - 1\right) \cdot \left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)\right) - t\]
\left(\left(x - 1\right) \cdot \log y + \left(z - 1\right) \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot \left(x - 1\right) + \log \left({\left(\frac{1}{{y}^{\frac{2}{3}}}\right)}^{\frac{-1}{3}} \cdot {\left(\frac{1}{\sqrt[3]{y}}\right)}^{\frac{-1}{3}}\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 r53201 = x;
        double r53202 = 1.0;
        double r53203 = r53201 - r53202;
        double r53204 = y;
        double r53205 = log(r53204);
        double r53206 = r53203 * r53205;
        double r53207 = z;
        double r53208 = r53207 - r53202;
        double r53209 = r53202 - r53204;
        double r53210 = log(r53209);
        double r53211 = r53208 * r53210;
        double r53212 = r53206 + r53211;
        double r53213 = t;
        double r53214 = r53212 - r53213;
        return r53214;
}
↓
double f(double x, double y, double z, double t) {
        double r53215 = 2.0;
        double r53216 = y;
        double r53217 = cbrt(r53216);
        double r53218 = log(r53217);
        double r53219 = r53215 * r53218;
        double r53220 = x;
        double r53221 = 1.0;
        double r53222 = r53220 - r53221;
        double r53223 = r53219 * r53222;
        double r53224 = 1.0;
        double r53225 = 0.6666666666666666;
        double r53226 = pow(r53216, r53225);
        double r53227 = r53224 / r53226;
        double r53228 = -0.3333333333333333;
        double r53229 = pow(r53227, r53228);
        double r53230 = r53224 / r53217;
        double r53231 = pow(r53230, r53228);
        double r53232 = r53229 * r53231;
        double r53233 = log(r53232);
        double r53234 = r53233 * r53222;
        double r53235 = r53223 + r53234;
        double r53236 = z;
        double r53237 = r53236 - r53221;
        double r53238 = log(r53221);
        double r53239 = r53221 * r53216;
        double r53240 = 0.5;
        double r53241 = pow(r53216, r53215);
        double r53242 = pow(r53221, r53215);
        double r53243 = r53241 / r53242;
        double r53244 = r53240 * r53243;
        double r53245 = r53239 + r53244;
        double r53246 = r53238 - r53245;
        double r53247 = r53237 * r53246;
        double r53248 = r53235 + r53247;
        double r53249 = t;
        double r53250 = r53248 - r53249;
        return r53250;
}

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 7.0

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

  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. Simplified0.4

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

  9. Taylor expanded around inf 0.4

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

  11. Applied add-cube-cbrt0.4

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

  12. Applied add-cube-cbrt0.4

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

  13. Applied times-frac0.4

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

  14. Applied unpow-prod-down0.4

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

  15. Simplified0.4

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

  16. Simplified0.4

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

  17. Final simplification0.4

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

Reproduce

herbie shell --seed 2019304 
(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))