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Derivation

1. Initial program 13.6

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

2. Initial simplification13.2

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

4. Applied div-inv13.2

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

5. Applied associate-/r*10.2

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

7. Applied associate-/l*8.9

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

9. Applied add-cube-cbrt8.9

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

10. Applied sqrt-prod8.9

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

11. Simplified9.9

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

12. Final simplification9.9

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

Runtime

Time bar (total: 34.6s)Debug logProfile

herbie shell --seed 2018274 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))