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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 associate-*r/10.0

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

5. Applied unpow210.0

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

6. Applied associate-*l*8.5

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

8. Applied associate-/l*7.8

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

10. Applied add-cube-cbrt7.9

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

11. Final simplification7.9

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

Reproduce

herbie shell --seed 2018360 +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: 26.7s)Debug log