Henrywood and Agarwal, Equation (12)

Average Error: 25.7 → 13.9

Time: 1.4m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le -1.5951901487005102 \cdot 10^{-112}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\left(\sqrt[3]{\frac{h}{\ell}} \cdot \frac{\frac{M}{2} \cdot D}{d}\right) \cdot \left(\sqrt[3]{\frac{h}{\ell}} \cdot \frac{\frac{M}{2} \cdot D}{d}\right)}{2} \cdot \sqrt[3]{\frac{h}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right) \cdot \left(1 - \frac{h}{2} \cdot \frac{\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}}{\ell}\right)\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r5249384 = d;
        double r5249385 = h;
        double r5249386 = r5249384 / r5249385;
        double r5249387 = 1.0;
        double r5249388 = 2.0;
        double r5249389 = r5249387 / r5249388;
        double r5249390 = pow(r5249386, r5249389);
        double r5249391 = l;
        double r5249392 = r5249384 / r5249391;
        double r5249393 = pow(r5249392, r5249389);
        double r5249394 = r5249390 * r5249393;
        double r5249395 = M;
        double r5249396 = D;
        double r5249397 = r5249395 * r5249396;
        double r5249398 = r5249388 * r5249384;
        double r5249399 = r5249397 / r5249398;
        double r5249400 = pow(r5249399, r5249388);
        double r5249401 = r5249389 * r5249400;
        double r5249402 = r5249385 / r5249391;
        double r5249403 = r5249401 * r5249402;
        double r5249404 = r5249387 - r5249403;
        double r5249405 = r5249394 * r5249404;
        return r5249405;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r5249406 = 1.0;
        double r5249407 = h;
        double r5249408 = l;
        double r5249409 = r5249407 / r5249408;
        double r5249410 = M;
        double r5249411 = D;
        double r5249412 = r5249410 * r5249411;
        double r5249413 = 2.0;
        double r5249414 = d;
        double r5249415 = r5249413 * r5249414;
        double r5249416 = r5249412 / r5249415;
        double r5249417 = pow(r5249416, r5249413);
        double r5249418 = 0.5;
        double r5249419 = r5249417 * r5249418;
        double r5249420 = r5249409 * r5249419;
        double r5249421 = r5249406 - r5249420;
        double r5249422 = r5249414 / r5249408;
        double r5249423 = pow(r5249422, r5249418);
        double r5249424 = r5249414 / r5249407;
        double r5249425 = pow(r5249424, r5249418);
        double r5249426 = r5249423 * r5249425;
        double r5249427 = r5249421 * r5249426;
        double r5249428 = -1.5951901487005102e-112;
        bool r5249429 = r5249427 <= r5249428;
        double r5249430 = cbrt(r5249409);
        double r5249431 = r5249410 / r5249413;
        double r5249432 = r5249431 * r5249411;
        double r5249433 = r5249432 / r5249414;
        double r5249434 = r5249430 * r5249433;
        double r5249435 = r5249434 * r5249434;
        double r5249436 = r5249435 / r5249413;
        double r5249437 = r5249436 * r5249430;
        double r5249438 = r5249406 - r5249437;
        double r5249439 = r5249426 * r5249438;
        double r5249440 = cbrt(r5249414);
        double r5249441 = cbrt(r5249407);
        double r5249442 = r5249440 / r5249441;
        double r5249443 = pow(r5249442, r5249418);
        double r5249444 = r5249442 * r5249442;
        double r5249445 = sqrt(r5249444);
        double r5249446 = r5249443 * r5249445;
        double r5249447 = cbrt(r5249408);
        double r5249448 = r5249414 / r5249447;
        double r5249449 = sqrt(r5249448);
        double r5249450 = r5249406 / r5249447;
        double r5249451 = r5249450 / r5249447;
        double r5249452 = sqrt(r5249451);
        double r5249453 = r5249449 * r5249452;
        double r5249454 = r5249407 / r5249413;
        double r5249455 = r5249415 / r5249411;
        double r5249456 = r5249410 / r5249455;
        double r5249457 = r5249456 * r5249456;
        double r5249458 = r5249457 / r5249408;
        double r5249459 = r5249454 * r5249458;
        double r5249460 = r5249406 - r5249459;
        double r5249461 = r5249453 * r5249460;
        double r5249462 = r5249446 * r5249461;
        double r5249463 = r5249429 ? r5249439 : r5249462;
        return r5249463;
}

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)))) < -1.5951901487005102e-112

   1. Initial program 28.4

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

   3. Applied add-cube-cbrt 28.7

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

   4. Applied associate-*r* 28.7

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

   5. Simplified 23.4

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

   if -1.5951901487005102e-112 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

   1. Initial program 25.2

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

   3. Applied add-cube-cbrt 25.5

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

   4. Applied *-un-lft-identity 25.5

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

   5. Applied times-frac 25.5

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

   6. Applied unpow-prod-down 21.1

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

   7. Simplified 21.1

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

   8. Simplified 21.1

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

   10. Applied add-cube-cbrt 21.3

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

   11. Applied add-cube-cbrt 21.3

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

   12. Applied times-frac 21.3

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

   13. Applied unpow-prod-down 15.4

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

   14. Simplified 15.4

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

   16. Applied associate-*l/ 15.4

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

   17. Applied frac-times 12.5

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

   18. Simplified 12.5

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

   20. Applied pow1 12.5

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

   21. Applied pow1 12.5

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

   22. Applied pow1 12.5

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

   23. Applied pow-prod-down 12.5

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

   24. Applied pow-prod-down 12.5

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

   25. Simplified 12.0

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

4. Final simplification 13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le -1.5951901487005102 \cdot 10^{-112}:\\ \;\;\;\;\left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \cdot \left(1 - \frac{\left(\sqrt[3]{\frac{h}{\ell}} \cdot \frac{\frac{M}{2} \cdot D}{d}\right) \cdot \left(\sqrt[3]{\frac{h}{\ell}} \cdot \frac{\frac{M}{2} \cdot D}{d}\right)}{2} \cdot \sqrt[3]{\frac{h}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right) \cdot \left(1 - \frac{h}{2} \cdot \frac{\frac{M}{\frac{2 \cdot d}{D}} \cdot \frac{M}{\frac{2 \cdot d}{D}}}{\ell}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +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)))))