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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. Using strategy rm

3. Applied unpow213.4

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

4. Applied associate-*l*12.0

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

6. Applied associate-*r/8.1

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

8. Applied clear-num8.1

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

9. Final simplification8.1

   \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\ell} \cdot \frac{1}{\frac{2 \cdot d}{M \cdot D}}}\]

Reproduce

herbie shell --seed 2019008 +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))))))

Details

Time bar (total: 26.2s)Debug log