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Derivation

1. Initial program 13.5

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

2. Initial simplification13.1

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

4. Applied associate-/r/10.5

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

6. Applied times-frac11.0

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

8. Applied times-frac10.4

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

10. Applied associate-/l*8.9

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

11. Final simplification8.9

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

Runtime

Time bar (total: 38.6s)Debug logProfile

herbie shell --seed 2018351 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))