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Derivation

1. Split input into 4 regimes

2. if (* V l) < -1.474781728144545e+163 or 3.4613075593469834e+287 < (* V l)

   1. Initial program 30.7

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

   3. Applied associate-/r*20.8

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

   5. Applied add-cube-cbrt21.0

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\left(\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{A}{V}}\right) \cdot \sqrt[3]{\frac{A}{V}}}}{\ell}}\]

   6. Applied associate-/l*21.0

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

   8. Applied div-inv21.0

      \[\leadsto c0 \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{A}{V}}}{\color{blue}{\ell \cdot \frac{1}{\sqrt[3]{\frac{A}{V}}}}}}\]

   9. Applied times-frac21.0

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\sqrt[3]{\frac{A}{V}}}{\ell} \cdot \frac{\sqrt[3]{\frac{A}{V}}}{\frac{1}{\sqrt[3]{\frac{A}{V}}}}}}\]

   10. Applied sqrt-prod12.9

       \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\ell}} \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\frac{1}{\sqrt[3]{\frac{A}{V}}}}}\right)}\]
   11. Using strategy rm

   12. Applied div-inv12.9

       \[\leadsto c0 \cdot \left(\sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\ell}} \cdot \sqrt{\frac{\sqrt[3]{\color{blue}{A \cdot \frac{1}{V}}}}{\frac{1}{\sqrt[3]{\frac{A}{V}}}}}\right)\]

   13. Applied cbrt-prod12.8

       \[\leadsto c0 \cdot \left(\sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\ell}} \cdot \sqrt{\frac{\color{blue}{\sqrt[3]{A} \cdot \sqrt[3]{\frac{1}{V}}}}{\frac{1}{\sqrt[3]{\frac{A}{V}}}}}\right)\]

   if -1.474781728144545e+163 < (* V l) < -2.8793218738207274e-256

   1. Initial program 7.9

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

   3. Applied add-sqr-sqrt8.2

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

   4. Applied associate-*r*8.1

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

   if -2.8793218738207274e-256 < (* V l) < 0.0

   1. Initial program 53.0

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

   3. Applied associate-/r*32.2

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

   5. Applied add-cube-cbrt32.5

      \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\left(\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{A}{V}}\right) \cdot \sqrt[3]{\frac{A}{V}}}}{\ell}}\]

   6. Applied associate-/l*32.5

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

   8. Applied add-sqr-sqrt32.5

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

   9. Applied times-frac32.5

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

   10. Applied rem-sqrt-square18.9

       \[\leadsto c0 \cdot \color{blue}{\left|\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt{\frac{\ell}{\sqrt[3]{\frac{A}{V}}}}}\right|}\]

   if 0.0 < (* V l) < 3.4613075593469834e+287

   1. Initial program 10.4

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

   3. Applied sqrt-div0.6

      \[\leadsto c0 \cdot \color{blue}{\frac{\sqrt{A}}{\sqrt{V \cdot \ell}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.474781728144545 \cdot 10^{+163}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{A}}{\frac{1}{\sqrt[3]{\frac{A}{V}}}}} \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\ell}}\right)\\ \mathbf{elif}\;V \cdot \ell \le -2.8793218738207274 \cdot 10^{-256}:\\ \;\;\;\;\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \left(c0 \cdot \sqrt{\sqrt{\frac{A}{V \cdot \ell}}}\right)\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;\left|\frac{\sqrt[3]{\frac{A}{V}}}{\sqrt{\frac{\ell}{\sqrt[3]{\frac{A}{V}}}}}\right| \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le 3.4613075593469834 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \left(\sqrt{\frac{\sqrt[3]{\frac{1}{V}} \cdot \sqrt[3]{A}}{\frac{1}{\sqrt[3]{\frac{A}{V}}}}} \cdot \sqrt{\frac{\sqrt[3]{\frac{A}{V}}}{\ell}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019010 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))

Details

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