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Derivation

1. Initial program 13.3

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
2. Using strategy rm

3. Applied div-inv13.3

   \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(h \cdot \frac{1}{\ell}\right)}}\]

4. Applied associate-*r*10.1

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
5. Using strategy rm

6. Applied unpow210.1

   \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right)} \cdot h\right) \cdot \frac{1}{\ell}}\]

7. Applied associate-*l*8.8

   \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)\right)} \cdot \frac{1}{\ell}}\]
8. Using strategy rm

9. Applied add-cube-cbrt8.9

   \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\]

10. Applied associate-*l*8.9

    \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot h\right)\right)}\right) \cdot \frac{1}{\ell}}\]
11. Using strategy rm

12. Applied pow18.9

    \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot h\right)\right)\right) \cdot \color{blue}{{\left(\frac{1}{\ell}\right)}^{1}}}\]

13. Applied pow18.9

    \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \color{blue}{{\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot h\right)}^{1}}\right)\right) \cdot {\left(\frac{1}{\ell}\right)}^{1}}\]

14. Applied pow18.9

    \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\color{blue}{{\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{1}} \cdot {\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot h\right)}^{1}\right)\right) \cdot {\left(\frac{1}{\ell}\right)}^{1}}\]

15. Applied pow-prod-down8.9

    \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{{\left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot h\right)\right)}^{1}}\right) \cdot {\left(\frac{1}{\ell}\right)}^{1}}\]

16. Applied pow18.9

    \[\leadsto w0 \cdot \sqrt{1 - \left(\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{1}} \cdot {\left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot h\right)\right)}^{1}\right) \cdot {\left(\frac{1}{\ell}\right)}^{1}}\]

17. Applied pow-prod-down8.9

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot h\right)\right)\right)}^{1}} \cdot {\left(\frac{1}{\ell}\right)}^{1}}\]

18. Applied pow-prod-down8.9

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot h\right)\right)\right) \cdot \frac{1}{\ell}\right)}^{1}}}\]

19. Simplified8.1

    \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{3} \cdot \left(h \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right)\right)}}^{1}}\]

20. Final simplification8.1

    \[\leadsto w0 \cdot \sqrt{1 - {\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{3} \cdot \left(h \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\ell}\right)}\]

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))