Henrywood and Agarwal, Equation (9a)

Average Error: 13.8 → 7.7

Time: 33.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -3.7880103230179996 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\frac{D \cdot M}{2 \cdot d} \cdot \left(\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)\right)} \cdot w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -7.22297199381291 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{D \cdot M}{2 \cdot d}\right) \cdot \frac{D \cdot M}{2 \cdot d}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt[3]{\sqrt{1 - \left(\frac{M}{2 \cdot d} \cdot D\right) \cdot \left(\left(\left(\frac{\sqrt[3]{\sqrt[3]{h}} \cdot \left(\frac{M}{2 \cdot d} \cdot D\right)}{\frac{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\sqrt[3]{h}}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)} \cdot \left(1 - \left(\frac{M}{2 \cdot d} \cdot D\right) \cdot \left(\left(\left(\frac{\sqrt[3]{\sqrt[3]{h}} \cdot \left(\frac{M}{2 \cdot d} \cdot D\right)}{\frac{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\sqrt[3]{h}}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)}\\ \end{array}\]

C

double f(double w0, double M, double D, double h, double l, double d) {
        double r4726392 = w0;
        double r4726393 = 1.0;
        double r4726394 = M;
        double r4726395 = D;
        double r4726396 = r4726394 * r4726395;
        double r4726397 = 2.0;
        double r4726398 = d;
        double r4726399 = r4726397 * r4726398;
        double r4726400 = r4726396 / r4726399;
        double r4726401 = pow(r4726400, r4726397);
        double r4726402 = h;
        double r4726403 = l;
        double r4726404 = r4726402 / r4726403;
        double r4726405 = r4726401 * r4726404;
        double r4726406 = r4726393 - r4726405;
        double r4726407 = sqrt(r4726406);
        double r4726408 = r4726392 * r4726407;
        return r4726408;
}

↓

double f(double w0, double M, double D, double h, double l, double d) {
        double r4726409 = h;
        double r4726410 = l;
        double r4726411 = r4726409 / r4726410;
        double r4726412 = -3.7880103230179996e+184;
        bool r4726413 = r4726411 <= r4726412;
        double r4726414 = 1.0;
        double r4726415 = cbrt(r4726409);
        double r4726416 = cbrt(r4726410);
        double r4726417 = r4726415 / r4726416;
        double r4726418 = D;
        double r4726419 = M;
        double r4726420 = r4726418 * r4726419;
        double r4726421 = 2.0;
        double r4726422 = d;
        double r4726423 = r4726421 * r4726422;
        double r4726424 = r4726420 / r4726423;
        double r4726425 = cbrt(r4726415);
        double r4726426 = r4726425 * r4726425;
        double r4726427 = r4726424 * r4726426;
        double r4726428 = r4726425 / r4726416;
        double r4726429 = r4726427 * r4726428;
        double r4726430 = r4726416 * r4726416;
        double r4726431 = cbrt(r4726430);
        double r4726432 = r4726426 / r4726431;
        double r4726433 = r4726424 * r4726432;
        double r4726434 = cbrt(r4726416);
        double r4726435 = r4726425 / r4726434;
        double r4726436 = r4726433 * r4726435;
        double r4726437 = r4726429 * r4726436;
        double r4726438 = r4726417 * r4726437;
        double r4726439 = r4726414 - r4726438;
        double r4726440 = sqrt(r4726439);
        double r4726441 = w0;
        double r4726442 = r4726440 * r4726441;
        double r4726443 = -7.22297199381291e-296;
        bool r4726444 = r4726411 <= r4726443;
        double r4726445 = r4726411 * r4726424;
        double r4726446 = r4726445 * r4726424;
        double r4726447 = r4726414 - r4726446;
        double r4726448 = sqrt(r4726447);
        double r4726449 = r4726448 * r4726441;
        double r4726450 = r4726419 / r4726423;
        double r4726451 = r4726450 * r4726418;
        double r4726452 = r4726425 * r4726451;
        double r4726453 = r4726431 / r4726425;
        double r4726454 = r4726452 / r4726453;
        double r4726455 = r4726454 * r4726417;
        double r4726456 = r4726455 * r4726435;
        double r4726457 = r4726456 * r4726417;
        double r4726458 = r4726451 * r4726457;
        double r4726459 = r4726414 - r4726458;
        double r4726460 = sqrt(r4726459);
        double r4726461 = r4726460 * r4726459;
        double r4726462 = cbrt(r4726461);
        double r4726463 = r4726441 * r4726462;
        double r4726464 = r4726444 ? r4726449 : r4726463;
        double r4726465 = r4726413 ? r4726442 : r4726464;
        return r4726465;
}

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

Derivation

1. Split input into 3 regimes

2. if (/ h l) < -3.7880103230179996e+184

   1. Initial program 38.0

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

   2. Simplified 38.0

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

   4. Applied add-cube-cbrt 38.0

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

   5. Applied add-cube-cbrt 38.1

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

   6. Applied times-frac 38.0

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

   7. Applied associate-*r* 23.1

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

   8. Simplified 18.2

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

   10. Applied add-cube-cbrt 18.2

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

   11. Applied cbrt-prod 18.2

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

   12. Applied add-cube-cbrt 18.3

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

   13. Applied times-frac 18.3

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

   14. Applied associate-*r* 18.3

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

   16. Applied *-un-lft-identity 18.3

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

   17. Applied add-cube-cbrt 18.3

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

   18. Applied times-frac 18.3

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

   19. Applied associate-*r* 18.3

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

   if -3.7880103230179996e+184 < (/ h l) < -7.22297199381291e-296

   1. Initial program 12.9

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

   2. Simplified 12.9

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

   4. Applied associate-*l* 10.9

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

   if -7.22297199381291e-296 < (/ h l)

   1. Initial program 8.2

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

   2. Simplified 8.2

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

   4. Applied add-cube-cbrt 8.2

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

   5. Applied add-cube-cbrt 8.2

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

   6. Applied times-frac 8.2

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

   7. Applied associate-*r* 5.5

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

   8. Simplified 2.3

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

   10. Applied add-cube-cbrt 2.3

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

   11. Applied cbrt-prod 2.3

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

   12. Applied add-cube-cbrt 2.3

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

   13. Applied times-frac 2.3

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

   14. Applied associate-*r* 2.3

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

   16. Applied add-cbrt-cube 2.3

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

   17. Simplified 2.4

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

4. Final simplification 7.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -3.7880103230179996 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\frac{D \cdot M}{2 \cdot d} \cdot \left(\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}\right)\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{\sqrt[3]{h}} \cdot \sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)\right)} \cdot w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -7.22297199381291 \cdot 10^{-296}:\\ \;\;\;\;\sqrt{1 - \left(\frac{h}{\ell} \cdot \frac{D \cdot M}{2 \cdot d}\right) \cdot \frac{D \cdot M}{2 \cdot d}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt[3]{\sqrt{1 - \left(\frac{M}{2 \cdot d} \cdot D\right) \cdot \left(\left(\left(\frac{\sqrt[3]{\sqrt[3]{h}} \cdot \left(\frac{M}{2 \cdot d} \cdot D\right)}{\frac{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\sqrt[3]{h}}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)} \cdot \left(1 - \left(\frac{M}{2 \cdot d} \cdot D\right) \cdot \left(\left(\left(\frac{\sqrt[3]{\sqrt[3]{h}} \cdot \left(\frac{M}{2 \cdot d} \cdot D\right)}{\frac{\sqrt[3]{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{\sqrt[3]{h}}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{\sqrt[3]{h}}}{\sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019144 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))