Henrywood and Agarwal, Equation (12)

Average Error: 25.5 → 11.3

Time: 1.3m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;h \le 1.2317179435178053 \cdot 10^{-303}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{h}} \cdot \left|\sqrt[3]{d}\right|\right)\right)\right) \cdot \left(1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{D}{\frac{d \cdot 2}{M}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{D}{\frac{d \cdot 2}{M}}\right)}{2}\right)\\ \mathbf{elif}\;h \le 8.093973936929225 \cdot 10^{+277}:\\ \;\;\;\;\frac{\left(\left(\sqrt{\sqrt[3]{d}} \cdot \left|\sqrt[3]{d}\right|\right) \cdot \left(\sqrt{\sqrt[3]{d}} \cdot \left|\sqrt[3]{d}\right|\right)\right) \cdot \left(1 - \frac{\frac{\sqrt[3]{h} \cdot \frac{D}{\frac{d \cdot 2}{M}}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{h} \cdot \frac{D}{\frac{d \cdot 2}{M}}}{\sqrt[3]{\ell}}}{2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}{\left(\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\frac{D}{\frac{d \cdot 2}{M}} \cdot \left(\frac{D}{\frac{d \cdot 2}{M}} \cdot \sqrt{h}\right)}{2} \cdot \frac{\sqrt{h}}{\ell}\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r5305480 = d;
        double r5305481 = h;
        double r5305482 = r5305480 / r5305481;
        double r5305483 = 1.0;
        double r5305484 = 2.0;
        double r5305485 = r5305483 / r5305484;
        double r5305486 = pow(r5305482, r5305485);
        double r5305487 = l;
        double r5305488 = r5305480 / r5305487;
        double r5305489 = pow(r5305488, r5305485);
        double r5305490 = r5305486 * r5305489;
        double r5305491 = M;
        double r5305492 = D;
        double r5305493 = r5305491 * r5305492;
        double r5305494 = r5305484 * r5305480;
        double r5305495 = r5305493 / r5305494;
        double r5305496 = pow(r5305495, r5305484);
        double r5305497 = r5305485 * r5305496;
        double r5305498 = r5305481 / r5305487;
        double r5305499 = r5305497 * r5305498;
        double r5305500 = r5305483 - r5305499;
        double r5305501 = r5305490 * r5305500;
        return r5305501;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r5305502 = h;
        double r5305503 = 1.2317179435178053e-303;
        bool r5305504 = r5305502 <= r5305503;
        double r5305505 = d;
        double r5305506 = cbrt(r5305505);
        double r5305507 = l;
        double r5305508 = cbrt(r5305507);
        double r5305509 = r5305506 / r5305508;
        double r5305510 = sqrt(r5305509);
        double r5305511 = fabs(r5305509);
        double r5305512 = r5305506 / r5305502;
        double r5305513 = sqrt(r5305512);
        double r5305514 = fabs(r5305506);
        double r5305515 = r5305513 * r5305514;
        double r5305516 = r5305511 * r5305515;
        double r5305517 = r5305510 * r5305516;
        double r5305518 = 1.0;
        double r5305519 = cbrt(r5305502);
        double r5305520 = r5305519 / r5305508;
        double r5305521 = D;
        double r5305522 = 2.0;
        double r5305523 = r5305505 * r5305522;
        double r5305524 = M;
        double r5305525 = r5305523 / r5305524;
        double r5305526 = r5305521 / r5305525;
        double r5305527 = r5305520 * r5305526;
        double r5305528 = r5305527 * r5305527;
        double r5305529 = r5305528 / r5305522;
        double r5305530 = r5305520 * r5305529;
        double r5305531 = r5305518 - r5305530;
        double r5305532 = r5305517 * r5305531;
        double r5305533 = 8.093973936929225e+277;
        bool r5305534 = r5305502 <= r5305533;
        double r5305535 = sqrt(r5305506);
        double r5305536 = r5305535 * r5305514;
        double r5305537 = r5305536 * r5305536;
        double r5305538 = r5305519 * r5305526;
        double r5305539 = r5305538 / r5305508;
        double r5305540 = r5305539 * r5305539;
        double r5305541 = r5305540 / r5305522;
        double r5305542 = r5305541 * r5305520;
        double r5305543 = r5305518 - r5305542;
        double r5305544 = r5305537 * r5305543;
        double r5305545 = sqrt(r5305508);
        double r5305546 = r5305508 * r5305508;
        double r5305547 = sqrt(r5305546);
        double r5305548 = r5305545 * r5305547;
        double r5305549 = sqrt(r5305502);
        double r5305550 = r5305548 * r5305549;
        double r5305551 = r5305544 / r5305550;
        double r5305552 = r5305505 / r5305507;
        double r5305553 = 0.5;
        double r5305554 = pow(r5305552, r5305553);
        double r5305555 = r5305505 / r5305502;
        double r5305556 = pow(r5305555, r5305553);
        double r5305557 = r5305554 * r5305556;
        double r5305558 = r5305526 * r5305549;
        double r5305559 = r5305526 * r5305558;
        double r5305560 = r5305559 / r5305522;
        double r5305561 = r5305549 / r5305507;
        double r5305562 = r5305560 * r5305561;
        double r5305563 = r5305518 - r5305562;
        double r5305564 = r5305557 * r5305563;
        double r5305565 = r5305534 ? r5305551 : r5305564;
        double r5305566 = r5305504 ? r5305532 : r5305565;
        return r5305566;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 3 regimes

2. if h < 1.2317179435178053e-303

   1. Initial program 24.9

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

   3. Applied add-cube-cbrt 25.0

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

   4. Applied add-cube-cbrt 25.0

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

   5. Applied times-frac 25.0

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

   6. Applied associate-*r* 23.3

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

   7. Simplified 21.9

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

   9. Applied add-cube-cbrt 22.1

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

   10. Applied add-cube-cbrt 22.2

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

   11. Applied times-frac 22.2

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

   12. Applied unpow-prod-down 16.9

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

   13. Simplified 16.9

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

   14. Simplified 16.9

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

   16. Applied *-un-lft-identity 16.9

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

   17. Applied add-cube-cbrt 17.1

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

   18. Applied times-frac 17.1

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

   19. Applied unpow-prod-down 12.4

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

   20. Simplified 12.4

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

   21. Simplified 12.4

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

   23. Applied *-un-lft-identity 12.4

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

   24. Applied associate-*r* 12.4

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

   25. Simplified 11.7

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

   if 1.2317179435178053e-303 < h < 8.093973936929225e+277

   1. Initial program 25.9

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

   3. Applied add-cube-cbrt 26.0

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

   4. Applied add-cube-cbrt 26.0

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

   5. Applied times-frac 26.0

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

   6. Applied associate-*r* 24.6

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

   7. Simplified 22.9

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

   9. Applied add-cube-cbrt 23.2

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

   10. Applied add-cube-cbrt 23.3

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

   11. Applied times-frac 23.3

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

   12. Applied unpow-prod-down 17.6

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

   13. Simplified 17.6

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

   14. Simplified 17.6

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

   16. Applied *-un-lft-identity 17.6

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

   17. Applied add-cube-cbrt 17.8

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

   18. Applied times-frac 17.8

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

   19. Applied unpow-prod-down 12.4

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

   20. Simplified 12.4

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

   21. Simplified 12.4

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

   23. Applied sqrt-div 12.4

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

   24. Applied frac-times 12.4

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

   25. Applied sqrt-div 11.8

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

   26. Applied frac-times 11.8

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

   27. Applied sqrt-div 10.5

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

   28. Applied associate-*r/ 10.5

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

   29. Applied frac-times 10.5

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

   30. Applied associate-*l/ 10.3

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

   31. Simplified 10.3

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

   if 8.093973936929225e+277 < h

   1. Initial program 31.1

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

   3. Applied *-un-lft-identity 31.1

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

   4. Applied add-sqr-sqrt 31.1

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

   5. Applied times-frac 31.1

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

   6. Applied associate-*r* 27.2

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

   7. Simplified 27.2

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

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \le 1.2317179435178053 \cdot 10^{-303}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{h}} \cdot \left|\sqrt[3]{d}\right|\right)\right)\right) \cdot \left(1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{D}{\frac{d \cdot 2}{M}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{D}{\frac{d \cdot 2}{M}}\right)}{2}\right)\\ \mathbf{elif}\;h \le 8.093973936929225 \cdot 10^{+277}:\\ \;\;\;\;\frac{\left(\left(\sqrt{\sqrt[3]{d}} \cdot \left|\sqrt[3]{d}\right|\right) \cdot \left(\sqrt{\sqrt[3]{d}} \cdot \left|\sqrt[3]{d}\right|\right)\right) \cdot \left(1 - \frac{\frac{\sqrt[3]{h} \cdot \frac{D}{\frac{d \cdot 2}{M}}}{\sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{h} \cdot \frac{D}{\frac{d \cdot 2}{M}}}{\sqrt[3]{\ell}}}{2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}{\left(\sqrt{\sqrt[3]{\ell}} \cdot \sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \sqrt{h}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\frac{D}{\frac{d \cdot 2}{M}} \cdot \left(\frac{D}{\frac{d \cdot 2}{M}} \cdot \sqrt{h}\right)}{2} \cdot \frac{\sqrt{h}}{\ell}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(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)))))