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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-/l*11.6

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

6. Applied add-cube-cbrt11.6

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

7. Applied div-inv11.6

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

8. Applied times-frac7.8

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

9. Final simplification7.8

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

Runtime

Time bar (total: 44.5s)Debug logProfile

herbie shell --seed 2018286 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))