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Derivation

1. Split input into 3 regimes

2. if (/ (* M D) (* 2 d)) < -1.0301112861049044e+93

   1. Initial program 48.2

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

   2. Initial simplification48.3

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

   4. Applied div-inv48.3

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

   5. Applied associate-/r*50.1

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

   7. Applied div-inv50.1

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

   8. Applied *-un-lft-identity50.1

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

   9. Applied times-frac50.1

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

   10. Simplified50.1

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

   11. Simplified39.3

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

   if -1.0301112861049044e+93 < (/ (* M D) (* 2 d)) < 1.6479509164934052e+123

   1. Initial program 6.5

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

   2. Initial simplification6.0

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

   4. Applied div-inv6.0

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

   5. Applied associate-/r*2.2

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

   if 1.6479509164934052e+123 < (/ (* M D) (* 2 d))

   1. Initial program 53.8

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

   2. Initial simplification53.8

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

   4. Applied div-inv53.8

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

   5. Applied times-frac41.7

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

   6. Simplified42.2

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

   8. Applied *-un-lft-identity42.2

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

   9. Applied associate-/l*42.2

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

4. Final simplification8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \le -1.0301112861049044 \cdot 10^{+93}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\frac{\ell}{h}}{\frac{M}{2} \cdot \frac{D}{d}}}} \cdot w0\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le 1.6479509164934052 \cdot 10^{+123}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell}}{\frac{1}{h}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d}}} \cdot \left(\frac{h}{2 \cdot d} \cdot \left(M \cdot D\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 
(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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