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Derivation

1. Initial program 13.4

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

2. Initial simplification12.9

   \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
3. Using strategy rm

4. Applied associate-/r/10.0

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

6. Applied associate-*l/9.9

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

7. Simplified8.3

   \[\leadsto \sqrt{1 - \frac{\color{blue}{\left(h \cdot \frac{M}{\frac{d}{D}}\right) \cdot \frac{\frac{M}{\frac{d}{D}}}{4}}}{\ell}} \cdot w0\]
8. Using strategy rm

9. Applied associate-/l*7.6

   \[\leadsto \sqrt{1 - \color{blue}{\frac{h \cdot \frac{M}{\frac{d}{D}}}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{4}}}}} \cdot w0\]

10. Final simplification7.6

    \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{d}{D}} \cdot h}{\frac{\ell}{\frac{\frac{M}{\frac{d}{D}}}{4}}}}\]

Reproduce

herbie shell --seed 2018362 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))

Details

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