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Derivation

1. Split input into 3 regimes

2. if (/ A (* V l)) < 8.6955553668059e-322

   1. Initial program 40.8

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

   3. Applied associate-/r*30.0

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

   5. Applied div-inv30.0

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

   if 8.6955553668059e-322 < (/ A (* V l)) < 3.528095059673412e+291

   1. Initial program 0.7

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

   3. Applied associate-/r*8.2

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

   5. Applied div-inv8.2

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

   6. Applied associate-/l*0.8

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

   if 3.528095059673412e+291 < (/ A (* V l))

   1. Initial program 57.3

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

   3. Applied associate-/r*46.0

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

   5. Applied div-inv46.0

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

   6. Applied sqrt-prod42.6

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

   7. Applied associate-*r*42.8

      \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\frac{A}{V}}\right) \cdot \sqrt{\frac{1}{\ell}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification13.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{A}{\ell \cdot V} \le 8.6955553668059 \cdot 10^{-322}:\\ \;\;\;\;\sqrt{\frac{A}{V} \cdot \frac{1}{\ell}} \cdot c0\\ \mathbf{elif}\;\frac{A}{\ell \cdot V} \le 3.528095059673412 \cdot 10^{+291}:\\ \;\;\;\;\sqrt{\frac{A}{\frac{\ell}{\frac{1}{V}}}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\ell}} \cdot \left(\sqrt{\frac{A}{V}} \cdot c0\right)\\ \end{array}\]

Reproduce

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

Details

Time bar (total: 8.1s)Debug log