Henrywood and Agarwal, Equation (12)

Average Error: 25.8 → 12.4

Time: 1.7m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) = -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right) - \frac{\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right) \cdot \left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot h\right)}{\ell \cdot 2}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le -1.7428153322691488 \cdot 10^{-208}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) - \frac{\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot h\right)\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right)}{\ell \cdot 2}\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r7669508 = d;
        double r7669509 = h;
        double r7669510 = r7669508 / r7669509;
        double r7669511 = 1.0;
        double r7669512 = 2.0;
        double r7669513 = r7669511 / r7669512;
        double r7669514 = pow(r7669510, r7669513);
        double r7669515 = l;
        double r7669516 = r7669508 / r7669515;
        double r7669517 = pow(r7669516, r7669513);
        double r7669518 = r7669514 * r7669517;
        double r7669519 = M;
        double r7669520 = D;
        double r7669521 = r7669519 * r7669520;
        double r7669522 = r7669512 * r7669508;
        double r7669523 = r7669521 / r7669522;
        double r7669524 = pow(r7669523, r7669512);
        double r7669525 = r7669513 * r7669524;
        double r7669526 = r7669509 / r7669515;
        double r7669527 = r7669525 * r7669526;
        double r7669528 = r7669511 - r7669527;
        double r7669529 = r7669518 * r7669528;
        return r7669529;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r7669530 = 1.0;
        double r7669531 = h;
        double r7669532 = l;
        double r7669533 = r7669531 / r7669532;
        double r7669534 = M;
        double r7669535 = D;
        double r7669536 = r7669534 * r7669535;
        double r7669537 = 2.0;
        double r7669538 = d;
        double r7669539 = r7669537 * r7669538;
        double r7669540 = r7669536 / r7669539;
        double r7669541 = pow(r7669540, r7669537);
        double r7669542 = 0.5;
        double r7669543 = r7669541 * r7669542;
        double r7669544 = r7669533 * r7669543;
        double r7669545 = r7669530 - r7669544;
        double r7669546 = r7669538 / r7669532;
        double r7669547 = pow(r7669546, r7669542);
        double r7669548 = r7669538 / r7669531;
        double r7669549 = pow(r7669548, r7669542);
        double r7669550 = r7669547 * r7669549;
        double r7669551 = r7669545 * r7669550;
        double r7669552 = -inf.0;
        bool r7669553 = r7669551 <= r7669552;
        double r7669554 = sqrt(r7669548);
        double r7669555 = cbrt(r7669538);
        double r7669556 = r7669555 / r7669532;
        double r7669557 = sqrt(r7669556);
        double r7669558 = fabs(r7669555);
        double r7669559 = r7669557 * r7669558;
        double r7669560 = r7669554 * r7669559;
        double r7669561 = r7669535 / r7669538;
        double r7669562 = r7669537 / r7669561;
        double r7669563 = r7669534 / r7669562;
        double r7669564 = sqrt(r7669546);
        double r7669565 = r7669554 * r7669564;
        double r7669566 = r7669563 * r7669565;
        double r7669567 = r7669563 * r7669531;
        double r7669568 = r7669566 * r7669567;
        double r7669569 = r7669532 * r7669537;
        double r7669570 = r7669568 / r7669569;
        double r7669571 = r7669560 - r7669570;
        double r7669572 = -1.7428153322691488e-208;
        bool r7669573 = r7669551 <= r7669572;
        double r7669574 = cbrt(r7669531);
        double r7669575 = r7669538 / r7669574;
        double r7669576 = sqrt(r7669575);
        double r7669577 = r7669574 * r7669574;
        double r7669578 = r7669530 / r7669577;
        double r7669579 = sqrt(r7669578);
        double r7669580 = r7669576 * r7669579;
        double r7669581 = r7669559 * r7669580;
        double r7669582 = r7669563 * r7669567;
        double r7669583 = cbrt(r7669532);
        double r7669584 = r7669555 / r7669583;
        double r7669585 = fabs(r7669584);
        double r7669586 = sqrt(r7669584);
        double r7669587 = r7669585 * r7669586;
        double r7669588 = r7669580 * r7669587;
        double r7669589 = r7669582 * r7669588;
        double r7669590 = r7669589 / r7669569;
        double r7669591 = r7669581 - r7669590;
        double r7669592 = r7669573 ? r7669551 : r7669591;
        double r7669593 = r7669553 ? r7669571 : r7669592;
        return r7669593;
}

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 (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < -inf.0

   1. Initial program 60.1

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

   2. Simplified 41.1

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

   4. Applied *-un-lft-identity 41.1

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

   5. Applied add-cube-cbrt 41.1

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

   6. Applied times-frac 41.1

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

   7. Applied sqrt-prod 40.9

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

   8. Simplified 40.9

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

   10. Applied associate-*r* 32.5

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

   if -inf.0 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < -1.7428153322691488e-208

   1. Initial program 1.3

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

   if -1.7428153322691488e-208 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

   1. Initial program 25.7

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

   2. Simplified 24.3

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

   4. Applied *-un-lft-identity 24.3

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

   5. Applied add-cube-cbrt 24.5

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

   6. Applied times-frac 24.5

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

   7. Applied sqrt-prod 22.3

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

   8. Simplified 22.3

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

   10. Applied add-cube-cbrt 22.3

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

   11. Applied add-cube-cbrt 22.3

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

   12. Applied times-frac 22.3

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

   13. Applied sqrt-prod 19.0

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

   14. Simplified 18.9

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

   16. Applied add-cube-cbrt 19.0

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

   17. Applied *-un-lft-identity 19.0

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

   18. Applied times-frac 19.0

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

   19. Applied sqrt-prod 18.5

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

   21. Applied add-cube-cbrt 18.5

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

   22. Applied *-un-lft-identity 18.5

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

   23. Applied times-frac 18.5

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

   24. Applied sqrt-prod 12.0

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

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) = -\infty:\\ \;\;\;\;\sqrt{\frac{d}{h}} \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right) - \frac{\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \left(\sqrt{\frac{d}{h}} \cdot \sqrt{\frac{d}{\ell}}\right)\right) \cdot \left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot h\right)}{\ell \cdot 2}\\ \mathbf{elif}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le -1.7428153322691488 \cdot 10^{-208}:\\ \;\;\;\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) - \frac{\left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot \left(\frac{M}{\frac{2}{\frac{D}{d}}} \cdot h\right)\right) \cdot \left(\left(\sqrt{\frac{d}{\sqrt[3]{h}}} \cdot \sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right)\right)}{\ell \cdot 2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(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)))))