Average Error: 25.5 → 11.1
Time: 5.9m
Precision: 64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}} - \frac{\frac{\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\frac{\sqrt[3]{2}}{\frac{M}{2}} \cdot \frac{\sqrt[3]{2}}{\frac{D}{d}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{\sqrt{\frac{\sqrt[3]{\sqrt[3]{d}}}{\ell}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{d}} \cdot \sqrt[3]{\sqrt[3]{d}}}}{\frac{\sqrt[3]{2}}{\frac{D}{d} \cdot \frac{M}{2}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}\right)\]
\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
↓
\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}} - \frac{\frac{\sqrt{\sqrt[3]{d} \cdot \sqrt[3]{d}}}{\frac{\sqrt[3]{2}}{\frac{M}{2}} \cdot \frac{\sqrt[3]{2}}{\frac{D}{d}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}} \cdot \frac{\frac{\sqrt{\frac{\sqrt[3]{\sqrt[3]{d}}}{\ell}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{d}} \cdot \sqrt[3]{\sqrt[3]{d}}}}{\frac{\sqrt[3]{2}}{\frac{D}{d} \cdot \frac{M}{2}}}}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{h}}}\right)
double f(double d, double h, double l, double M, double D) {
        double r15709411 = d;
        double r15709412 = h;
        double r15709413 = r15709411 / r15709412;
        double r15709414 = 1.0;
        double r15709415 = 2.0;
        double r15709416 = r15709414 / r15709415;
        double r15709417 = pow(r15709413, r15709416);
        double r15709418 = l;
        double r15709419 = r15709411 / r15709418;
        double r15709420 = pow(r15709419, r15709416);
        double r15709421 = r15709417 * r15709420;
        double r15709422 = M;
        double r15709423 = D;
        double r15709424 = r15709422 * r15709423;
        double r15709425 = r15709415 * r15709411;
        double r15709426 = r15709424 / r15709425;
        double r15709427 = pow(r15709426, r15709415);
        double r15709428 = r15709416 * r15709427;
        double r15709429 = r15709412 / r15709418;
        double r15709430 = r15709428 * r15709429;
        double r15709431 = r15709414 - r15709430;
        double r15709432 = r15709421 * r15709431;
        return r15709432;
}
↓
double f(double d, double h, double l, double M, double D) {
        double r15709433 = d;
        double r15709434 = cbrt(r15709433);
        double r15709435 = h;
        double r15709436 = cbrt(r15709435);
        double r15709437 = r15709434 / r15709436;
        double r15709438 = r15709437 * r15709437;
        double r15709439 = sqrt(r15709438);
        double r15709440 = sqrt(r15709437);
        double r15709441 = r15709439 * r15709440;
        double r15709442 = fabs(r15709434);
        double r15709443 = l;
        double r15709444 = r15709434 / r15709443;
        double r15709445 = sqrt(r15709444);
        double r15709446 = r15709442 * r15709445;
        double r15709447 = r15709434 * r15709434;
        double r15709448 = sqrt(r15709447);
        double r15709449 = 2.0;
        double r15709450 = cbrt(r15709449);
        double r15709451 = M;
        double r15709452 = r15709451 / r15709449;
        double r15709453 = r15709450 / r15709452;
        double r15709454 = D;
        double r15709455 = r15709454 / r15709433;
        double r15709456 = r15709450 / r15709455;
        double r15709457 = r15709453 * r15709456;
        double r15709458 = r15709448 / r15709457;
        double r15709459 = cbrt(r15709443);
        double r15709460 = r15709459 / r15709436;
        double r15709461 = r15709460 * r15709460;
        double r15709462 = r15709458 / r15709461;
        double r15709463 = cbrt(r15709434);
        double r15709464 = r15709463 / r15709443;
        double r15709465 = sqrt(r15709464);
        double r15709466 = r15709463 * r15709463;
        double r15709467 = sqrt(r15709466);
        double r15709468 = r15709465 * r15709467;
        double r15709469 = r15709455 * r15709452;
        double r15709470 = r15709450 / r15709469;
        double r15709471 = r15709468 / r15709470;
        double r15709472 = r15709471 / r15709460;
        double r15709473 = r15709462 * r15709472;
        double r15709474 = r15709446 - r15709473;
        double r15709475 = r15709441 * r15709474;
        return r15709475;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 25.5

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

  2. Simplified25.5

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

  4. Applied add-cube-cbrt25.8

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

  5. Applied add-cube-cbrt25.9

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

  6. Applied times-frac25.9

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

  7. Applied sqrt-prod20.4

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

  8. Simplified20.4

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

  10. Applied add-cube-cbrt20.4

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

  11. Applied add-cube-cbrt20.5

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

  12. Applied times-frac20.5

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

  13. Applied add-cube-cbrt20.5

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

  14. Applied times-frac20.4

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

  15. Applied *-un-lft-identity20.4

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

  16. Applied add-cube-cbrt20.4

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

  17. Applied times-frac20.4

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

  18. Applied sqrt-prod19.9

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

  19. Applied times-frac18.6

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

  20. Applied times-frac15.0

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

  21. Simplified15.0

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

  23. Applied *-un-lft-identity15.0

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

  24. Applied add-cube-cbrt15.2

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

  25. Applied times-frac15.2

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

  26. Applied sqrt-prod11.3

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

  27. Simplified11.3

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

  29. Applied *-un-lft-identity11.3

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

  30. Applied add-cube-cbrt11.4

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

  31. Applied times-frac11.4

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

  32. Applied sqrt-prod11.1

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

  33. Simplified11.1

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

  34. Final simplification11.1

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

Reproduce

herbie shell --seed 2019151 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))