Average Error: 13.6 → 7.3
Time: 53.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{h}}{2} \cdot \left(\left(\left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \sqrt[3]{D}\right) \cdot \left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \sqrt[3]{D}\right)\right) \cdot \left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \sqrt[3]{D}\right)\right)\right)\right)\right) \cdot \left(\left(\left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right) \cdot \frac{\sqrt[3]{M}}{\frac{\sqrt[3]{d}}{D}}\right) \cdot \frac{\sqrt[3]{h}}{2}\right)\right)} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{h}}{2} \cdot \left(\left(\left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \sqrt[3]{D}\right) \cdot \left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \sqrt[3]{D}\right)\right) \cdot \left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \sqrt[3]{D}\right)\right)\right)\right)\right) \cdot \left(\left(\left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right) \cdot \frac{\sqrt[3]{M}}{\frac{\sqrt[3]{d}}{D}}\right) \cdot \frac{\sqrt[3]{h}}{2}\right)\right)} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r6032294 = w0;
        double r6032295 = 1.0;
        double r6032296 = M;
        double r6032297 = D;
        double r6032298 = r6032296 * r6032297;
        double r6032299 = 2.0;
        double r6032300 = d;
        double r6032301 = r6032299 * r6032300;
        double r6032302 = r6032298 / r6032301;
        double r6032303 = pow(r6032302, r6032299);
        double r6032304 = h;
        double r6032305 = l;
        double r6032306 = r6032304 / r6032305;
        double r6032307 = r6032303 * r6032306;
        double r6032308 = r6032295 - r6032307;
        double r6032309 = sqrt(r6032308);
        double r6032310 = r6032294 * r6032309;
        return r6032310;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r6032311 = 1.0;
        double r6032312 = h;
        double r6032313 = cbrt(r6032312);
        double r6032314 = cbrt(r6032313);
        double r6032315 = l;
        double r6032316 = cbrt(r6032315);
        double r6032317 = r6032314 / r6032316;
        double r6032318 = 2.0;
        double r6032319 = r6032313 / r6032318;
        double r6032320 = M;
        double r6032321 = cbrt(r6032320);
        double r6032322 = d;
        double r6032323 = cbrt(r6032322);
        double r6032324 = r6032321 / r6032323;
        double r6032325 = D;
        double r6032326 = cbrt(r6032325);
        double r6032327 = r6032324 * r6032326;
        double r6032328 = r6032327 * r6032327;
        double r6032329 = r6032328 * r6032327;
        double r6032330 = r6032319 * r6032329;
        double r6032331 = r6032317 * r6032330;
        double r6032332 = r6032317 * r6032331;
        double r6032333 = r6032324 * r6032324;
        double r6032334 = r6032323 / r6032325;
        double r6032335 = r6032321 / r6032334;
        double r6032336 = r6032333 * r6032335;
        double r6032337 = r6032336 * r6032319;
        double r6032338 = r6032332 * r6032337;
        double r6032339 = r6032317 * r6032338;
        double r6032340 = r6032311 - r6032339;
        double r6032341 = sqrt(r6032340);
        double r6032342 = w0;
        double r6032343 = r6032341 * r6032342;
        return r6032343;
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.6

    \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

  2. Simplified13.6

    \[\leadsto \color{blue}{\sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \cdot w0}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity13.6

    \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\color{blue}{1 \cdot \ell}}} \cdot w0\]

  5. Applied add-cube-cbrt13.6

    \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}{1 \cdot \ell}} \cdot w0\]

  6. Applied times-frac13.6

    \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1} \cdot \frac{\sqrt[3]{h}}{\ell}\right)}} \cdot w0\]

  7. Applied associate-*r*11.3

    \[\leadsto \sqrt{1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}} \cdot w0\]

  8. Simplified10.2

    \[\leadsto \sqrt{1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right)\right)} \cdot \frac{\sqrt[3]{h}}{\ell}} \cdot w0\]
  9. Using strategy rm

  10. Applied add-cube-cbrt10.2

    \[\leadsto \sqrt{1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right)\right) \cdot \frac{\sqrt[3]{h}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \cdot w0\]

  11. Applied add-cube-cbrt10.2

    \[\leadsto \sqrt{1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right)\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}\right) \cdot \sqrt[3]{\sqrt[3]{h}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot w0\]

  12. Applied times-frac10.2

    \[\leadsto \sqrt{1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right)\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)}} \cdot w0\]

  13. Applied associate-*r*9.1

    \[\leadsto \sqrt{1 - \color{blue}{\left(\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right) \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}}} \cdot w0\]

  14. Simplified7.9

    \[\leadsto \sqrt{1 - \color{blue}{\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{d}{D}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{d}{D}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right)} \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]
  15. Using strategy rm

  16. Applied add-cube-cbrt7.9

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{d}{D}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{d}{\color{blue}{\left(\sqrt[3]{D} \cdot \sqrt[3]{D}\right) \cdot \sqrt[3]{D}}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  17. Applied add-cube-cbrt7.9

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{d}{D}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{\left(\sqrt[3]{D} \cdot \sqrt[3]{D}\right) \cdot \sqrt[3]{D}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  18. Applied times-frac7.9

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{d}{D}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{D} \cdot \sqrt[3]{D}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{D}}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  19. Applied add-cube-cbrt7.9

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{d}{D}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{M} \cdot \sqrt[3]{M}\right) \cdot \sqrt[3]{M}}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{D} \cdot \sqrt[3]{D}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{D}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  20. Applied times-frac7.9

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{d}{D}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{D} \cdot \sqrt[3]{D}}} \cdot \frac{\sqrt[3]{M}}{\frac{\sqrt[3]{d}}{\sqrt[3]{D}}}\right)}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  21. Simplified7.9

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{d}{D}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right)} \cdot \frac{\sqrt[3]{M}}{\frac{\sqrt[3]{d}}{\sqrt[3]{D}}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  22. Simplified7.9

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{d}{D}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \left(\left(\left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right) \cdot \color{blue}{\left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]
  23. Using strategy rm

  24. Applied *-un-lft-identity7.9

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{d}{\color{blue}{1 \cdot D}}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \left(\left(\left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  25. Applied add-cube-cbrt7.9

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\frac{\color{blue}{\left(\sqrt[3]{d} \cdot \sqrt[3]{d}\right) \cdot \sqrt[3]{d}}}{1 \cdot D}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \left(\left(\left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  26. Applied times-frac7.9

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{M}{\color{blue}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{1} \cdot \frac{\sqrt[3]{d}}{D}}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \left(\left(\left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  27. Applied add-cube-cbrt8.0

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{M} \cdot \sqrt[3]{M}\right) \cdot \sqrt[3]{M}}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{1} \cdot \frac{\sqrt[3]{d}}{D}}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \left(\left(\left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  28. Applied times-frac7.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{1}} \cdot \frac{\sqrt[3]{M}}{\frac{\sqrt[3]{d}}{D}}\right)}\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \left(\left(\left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  29. Simplified7.3

    \[\leadsto \sqrt{1 - \left(\left(\frac{\sqrt[3]{h}}{2} \cdot \left(\color{blue}{\left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)} \cdot \frac{\sqrt[3]{M}}{\frac{\sqrt[3]{d}}{D}}\right)\right) \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{2} \cdot \left(\left(\left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right) \cdot \left(\sqrt[3]{D} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right)\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}} \cdot w0\]

  30. Final simplification7.3

    \[\leadsto \sqrt{1 - \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}} \cdot \left(\frac{\sqrt[3]{h}}{2} \cdot \left(\left(\left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \sqrt[3]{D}\right) \cdot \left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \sqrt[3]{D}\right)\right) \cdot \left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \sqrt[3]{D}\right)\right)\right)\right)\right) \cdot \left(\left(\left(\frac{\sqrt[3]{M}}{\sqrt[3]{d}} \cdot \frac{\sqrt[3]{M}}{\sqrt[3]{d}}\right) \cdot \frac{\sqrt[3]{M}}{\frac{\sqrt[3]{d}}{D}}\right) \cdot \frac{\sqrt[3]{h}}{2}\right)\right)} \cdot w0\]

Reproduce

herbie shell --seed 2019151 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))