Henrywood and Agarwal, Equation (12)

Average Error: 26.1 → 10.9

Time: 1.6m

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r5271501 = d;
        double r5271502 = h;
        double r5271503 = r5271501 / r5271502;
        double r5271504 = 1.0;
        double r5271505 = 2.0;
        double r5271506 = r5271504 / r5271505;
        double r5271507 = pow(r5271503, r5271506);
        double r5271508 = l;
        double r5271509 = r5271501 / r5271508;
        double r5271510 = pow(r5271509, r5271506);
        double r5271511 = r5271507 * r5271510;
        double r5271512 = M;
        double r5271513 = D;
        double r5271514 = r5271512 * r5271513;
        double r5271515 = r5271505 * r5271501;
        double r5271516 = r5271514 / r5271515;
        double r5271517 = pow(r5271516, r5271505);
        double r5271518 = r5271506 * r5271517;
        double r5271519 = r5271502 / r5271508;
        double r5271520 = r5271518 * r5271519;
        double r5271521 = r5271504 - r5271520;
        double r5271522 = r5271511 * r5271521;
        return r5271522;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r5271523 = d;
        double r5271524 = cbrt(r5271523);
        double r5271525 = l;
        double r5271526 = r5271524 / r5271525;
        double r5271527 = sqrt(r5271526);
        double r5271528 = fabs(r5271524);
        double r5271529 = r5271527 * r5271528;
        double r5271530 = 1.0;
        double r5271531 = D;
        double r5271532 = M;
        double r5271533 = r5271531 * r5271532;
        double r5271534 = 2.0;
        double r5271535 = r5271523 * r5271534;
        double r5271536 = r5271533 / r5271535;
        double r5271537 = h;
        double r5271538 = cbrt(r5271537);
        double r5271539 = r5271536 * r5271538;
        double r5271540 = r5271538 / r5271525;
        double r5271541 = r5271539 * r5271540;
        double r5271542 = r5271541 * r5271539;
        double r5271543 = 0.5;
        double r5271544 = r5271542 * r5271543;
        double r5271545 = r5271530 - r5271544;
        double r5271546 = r5271524 / r5271538;
        double r5271547 = sqrt(r5271546);
        double r5271548 = fabs(r5271546);
        double r5271549 = r5271547 * r5271548;
        double r5271550 = r5271545 * r5271549;
        double r5271551 = r5271529 * r5271550;
        return r5271551;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Initial program 26.1

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

3. Applied add-cube-cbrt 26.3

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

4. Applied add-cube-cbrt 26.4

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

5. Applied times-frac 26.4

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

6. Applied unpow-prod-down 21.2

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

7. Simplified 21.2

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

8. Simplified 21.2

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

10. Applied *-un-lft-identity 21.2

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

11. Applied add-cube-cbrt 21.3

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

12. Applied times-frac 21.3

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

13. Applied associate-*r* 20.0

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

14. Simplified 19.0

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

16. Applied *-un-lft-identity 19.0

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

17. Applied add-cube-cbrt 19.2

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

18. Applied times-frac 19.2

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

19. Applied unpow-prod-down 14.1

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

20. Simplified 14.1

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

21. Simplified 14.1

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

23. Applied pow1 14.1

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

24. Applied pow1 14.1

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

25. Applied pow1 14.1

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

26. Applied pow1 14.1

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

27. Applied pow-prod-down 14.1

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

28. Applied pow-prod-down 14.1

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

29. Applied pow-prod-down 14.1

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

30. Simplified 10.9

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

31. Final simplification 10.9

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

Reproduce

herbie shell --seed 2019149 +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)))))