Henrywood and Agarwal, Equation (12)

Average Error: 25.7 → 11.3

Time: 2.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \le -0.0:\\ \;\;\;\;\left(\left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{M \cdot D}{d}}{2}}{\frac{2}{h} \cdot \frac{\ell}{\frac{\frac{M \cdot D}{d}}{2}}}\right)\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{\frac{M \cdot \frac{\frac{D}{d}}{2}}{\frac{\ell}{\frac{M \cdot \frac{\frac{D}{d}}{2}}{\frac{2}{h}}}}}\right) \cdot \left(\left(1 - \sqrt{\frac{M \cdot \frac{\frac{D}{d}}{2}}{\frac{\ell}{\frac{M \cdot \frac{\frac{D}{d}}{2}}{\frac{2}{h}}}}}\right) \cdot \left(\left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)\right)\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r7097410 = d;
        double r7097411 = h;
        double r7097412 = r7097410 / r7097411;
        double r7097413 = 1.0;
        double r7097414 = 2.0;
        double r7097415 = r7097413 / r7097414;
        double r7097416 = pow(r7097412, r7097415);
        double r7097417 = l;
        double r7097418 = r7097410 / r7097417;
        double r7097419 = pow(r7097418, r7097415);
        double r7097420 = r7097416 * r7097419;
        double r7097421 = M;
        double r7097422 = D;
        double r7097423 = r7097421 * r7097422;
        double r7097424 = r7097414 * r7097410;
        double r7097425 = r7097423 / r7097424;
        double r7097426 = pow(r7097425, r7097414);
        double r7097427 = r7097415 * r7097426;
        double r7097428 = r7097411 / r7097417;
        double r7097429 = r7097427 * r7097428;
        double r7097430 = r7097413 - r7097429;
        double r7097431 = r7097420 * r7097430;
        return r7097431;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r7097432 = 1.0;
        double r7097433 = h;
        double r7097434 = l;
        double r7097435 = r7097433 / r7097434;
        double r7097436 = M;
        double r7097437 = D;
        double r7097438 = r7097436 * r7097437;
        double r7097439 = 2.0;
        double r7097440 = d;
        double r7097441 = r7097439 * r7097440;
        double r7097442 = r7097438 / r7097441;
        double r7097443 = pow(r7097442, r7097439);
        double r7097444 = r7097432 / r7097439;
        double r7097445 = r7097443 * r7097444;
        double r7097446 = r7097435 * r7097445;
        double r7097447 = r7097432 - r7097446;
        double r7097448 = r7097440 / r7097434;
        double r7097449 = pow(r7097448, r7097444);
        double r7097450 = r7097440 / r7097433;
        double r7097451 = pow(r7097450, r7097444);
        double r7097452 = r7097449 * r7097451;
        double r7097453 = r7097447 * r7097452;
        double r7097454 = -0.0;
        bool r7097455 = r7097453 <= r7097454;
        double r7097456 = cbrt(r7097434);
        double r7097457 = r7097456 * r7097456;
        double r7097458 = r7097432 / r7097457;
        double r7097459 = pow(r7097458, r7097444);
        double r7097460 = r7097440 / r7097456;
        double r7097461 = pow(r7097460, r7097444);
        double r7097462 = r7097459 * r7097461;
        double r7097463 = r7097438 / r7097440;
        double r7097464 = r7097463 / r7097439;
        double r7097465 = r7097439 / r7097433;
        double r7097466 = r7097434 / r7097464;
        double r7097467 = r7097465 * r7097466;
        double r7097468 = r7097464 / r7097467;
        double r7097469 = r7097432 - r7097468;
        double r7097470 = r7097462 * r7097469;
        double r7097471 = cbrt(r7097440);
        double r7097472 = cbrt(r7097433);
        double r7097473 = r7097471 / r7097472;
        double r7097474 = pow(r7097473, r7097444);
        double r7097475 = r7097473 * r7097473;
        double r7097476 = pow(r7097475, r7097444);
        double r7097477 = r7097474 * r7097476;
        double r7097478 = r7097470 * r7097477;
        double r7097479 = r7097437 / r7097440;
        double r7097480 = r7097479 / r7097439;
        double r7097481 = r7097436 * r7097480;
        double r7097482 = r7097481 / r7097465;
        double r7097483 = r7097434 / r7097482;
        double r7097484 = r7097481 / r7097483;
        double r7097485 = sqrt(r7097484);
        double r7097486 = r7097432 + r7097485;
        double r7097487 = r7097432 - r7097485;
        double r7097488 = r7097462 * r7097477;
        double r7097489 = r7097487 * r7097488;
        double r7097490 = r7097486 * r7097489;
        double r7097491 = r7097455 ? r7097478 : r7097490;
        return r7097491;
}

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 2 regimes

2. if (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < -0.0

   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. Simplified 28.1

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

   4. Applied add-cube-cbrt 28.1

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

   5. Applied add-cube-cbrt 28.2

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

   6. Applied times-frac 28.2

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

   7. Applied unpow-prod-down 25.3

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

   8. Simplified 25.3

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

   10. Applied add-cube-cbrt 25.4

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

   11. Applied *-un-lft-identity 25.4

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

   12. Applied times-frac 25.4

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

   13. Applied unpow-prod-down 17.5

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

   15. Applied *-un-lft-identity 17.5

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

   16. Applied times-frac 17.5

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

   17. Applied associate-/l* 17.3

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

   19. Applied associate-*r* 16.8

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

   20. Simplified 13.6

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

   if -0.0 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

   1. Initial program 23.6

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

   2. Simplified 23.2

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

   4. Applied add-cube-cbrt 23.5

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

   5. Applied add-cube-cbrt 23.7

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

   6. Applied times-frac 23.7

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

   7. Applied unpow-prod-down 17.3

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

   8. Simplified 17.3

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

   10. Applied add-cube-cbrt 17.4

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

   11. Applied *-un-lft-identity 17.4

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

   12. Applied times-frac 17.4

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

   13. Applied unpow-prod-down 14.9

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

   15. Applied *-un-lft-identity 14.9

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

   16. Applied times-frac 14.9

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

   17. Applied associate-/l* 10.5

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

   19. Applied add-sqr-sqrt 10.5

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

   20. Applied *-un-lft-identity 10.5

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

   21. Applied difference-of-squares 10.5

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

   22. Applied associate-*l* 10.5

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

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \le -0.0:\\ \;\;\;\;\left(\left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{M \cdot D}{d}}{2}}{\frac{2}{h} \cdot \frac{\ell}{\frac{\frac{M \cdot D}{d}}{2}}}\right)\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \sqrt{\frac{M \cdot \frac{\frac{D}{d}}{2}}{\frac{\ell}{\frac{M \cdot \frac{\frac{D}{d}}{2}}{\frac{2}{h}}}}}\right) \cdot \left(\left(1 - \sqrt{\frac{M \cdot \frac{\frac{D}{d}}{2}}{\frac{\ell}{\frac{M \cdot \frac{\frac{D}{d}}{2}}{\frac{2}{h}}}}}\right) \cdot \left(\left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +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)))))