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Derivation

1. Initial program 13.8

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

2. Initial simplification13.4

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

4. Applied div-inv13.4

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

5. Applied times-frac8.4

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

6. Simplified11.6

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

8. Applied associate-/r/11.6

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

9. Applied associate-*r*9.3

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

11. Applied associate-/l*9.0

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

12. Final simplification9.0

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

Runtime

Time bar (total: 36.0s)Debug logProfile

herbie shell --seed 2018290 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))