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Derivation

1. Split input into 3 regimes

2. if (* V l) < 2.598372258245e-317

   1. Initial program 22.1

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

   3. Applied associate-/r*20.5

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\]
   4. Using strategy rm

   5. Applied div-inv20.5

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{V} \cdot \frac{1}{\ell}}}\]

   if 2.598372258245e-317 < (* V l) < 1.1237130726494876e+286

   1. Initial program 9.6

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

   2. Taylor expanded around inf 9.6

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{A}{\ell \cdot V}}}\]
   3. Using strategy rm

   4. Applied div-inv9.8

      \[\leadsto c0 \cdot \sqrt{\color{blue}{A \cdot \frac{1}{\ell \cdot V}}}\]

   5. Applied sqrt-prod0.9

      \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{A} \cdot \sqrt{\frac{1}{\ell \cdot V}}\right)}\]

   if 1.1237130726494876e+286 < (* V l)

   1. Initial program 37.8

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

   3. Applied associate-/r*21.8

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\]
   4. Using strategy rm

   5. Applied sqrt-div36.1

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

4. Final simplification14.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le 2.598372258245 \cdot 10^{-317}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\ell} \cdot \frac{A}{V}}\\ \mathbf{elif}\;V \cdot \ell \le 1.1237130726494876 \cdot 10^{+286}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{1}{V \cdot \ell}} \cdot \sqrt{A}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}}}{\sqrt{\ell}} \cdot c0\\ \end{array}\]

Reproduce

herbie shell --seed 2019002 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))

Details

Time bar (total: 7.7s)Debug log