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Derivation

1. Split input into 3 regimes

2. if (* V l) < -2.5999771263643773e-121

   1. Initial program 15.2

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt15.2

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\sqrt{\frac{A}{V \cdot \ell}} \cdot \sqrt{\frac{A}{V \cdot \ell}}}}\]

   4. Applied sqrt-prod15.4

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)}\]

   if -2.5999771263643773e-121 < (* V l) < 6.0308871737048e-314 or 1.0699470049292665e+287 < (* V l)

   1. Initial program 37.7

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
   2. Using strategy rm

   3. Applied associate-/r*25.8

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\]

   if 6.0308871737048e-314 < (* V l) < 1.0699470049292665e+287

   1. Initial program 10.2

      \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
   2. Using strategy rm

   3. Applied sqrt-div0.4

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -2.5999771263643773 \cdot 10^{-121}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)\\ \mathbf{elif}\;V \cdot \ell \le 6.0308871737048 \cdot 10^{-314} \lor \neg \left(V \cdot \ell \le 1.0699470049292665 \cdot 10^{+287}\right):\\ \;\;\;\;\sqrt{\frac{\frac{A}{V}}{\ell}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array}\]

Reproduce

herbie shell --seed 2018360 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))

Details

Time bar (total: 8.4s)Debug log