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Derivation

1. Split input into 2 regimes

2. if (* (pow (/ (* M D) (* 2 d)) 2) (/ h l)) < -1.228549641598271e+240

   1. Initial program 56.7

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

   3. Applied associate-*r/53.6

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

   5. Applied unpow253.6

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

   6. Applied associate-*l*49.0

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

   8. Applied associate-*l/50.4

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

   9. Applied associate-*r/50.6

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

   10. Applied associate-/l/50.6

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

   if -1.228549641598271e+240 < (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))

   1. Initial program 5.7

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

   3. Applied associate-*r/2.9

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

   5. Applied unpow22.9

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

   6. Applied associate-*l*1.9

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

4. Final simplification8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \le -1.228549641598271 \cdot 10^{+240}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \left(\left(M \cdot D\right) \cdot h\right)}{\ell \cdot \left(2 \cdot d\right)}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\frac{M \cdot D}{2 \cdot d} \cdot h\right) \cdot \frac{M \cdot D}{2 \cdot d}}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(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

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