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Derivation

1. Split input into 2 regimes

2. if V < 7.91776305489967e-278

   1. Initial program 19.5

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

   2. Initial simplification19.2

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

   3. Taylor expanded around -inf 19.5

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

   if 7.91776305489967e-278 < V

   1. Initial program 18.0

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

   2. Initial simplification18.0

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

   4. Applied *-un-lft-identity18.0

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

   5. Applied associate-/l*18.3

      \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{1}{\frac{\ell}{\frac{A}{V}}}}}\]
   6. Using strategy rm

   7. Applied sqrt-div18.1

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

   8. Applied associate-*r/18.0

      \[\leadsto \color{blue}{\frac{c0 \cdot \sqrt{1}}{\sqrt{\frac{\ell}{\frac{A}{V}}}}}\]
   9. Using strategy rm

   10. Applied associate-/r/18.6

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

   11. Applied sqrt-prod11.5

       \[\leadsto \frac{c0 \cdot \sqrt{1}}{\color{blue}{\sqrt{\frac{\ell}{A}} \cdot \sqrt{V}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification15.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \le 7.91776305489967 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\frac{A}{\ell \cdot V}} \cdot c0\\ \mathbf{else}:\\ \;\;\;\;\frac{c0}{\sqrt{V} \cdot \sqrt{\frac{\ell}{A}}}\\ \end{array}\]

Runtime

Time bar (total: 12.3s)Debug logProfile

herbie shell --seed 2018290 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))