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Derivation

1. Split input into 4 regimes

2. if (* V l) < -1.432883117701101e+185

   1. Initial program 29.0

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

   3. Applied clear-num29.4

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

   5. Applied sqrt-div29.4

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

   6. Applied associate-*r/29.4

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

   8. Applied *-un-lft-identity29.4

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

   9. Applied times-frac29.4

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

   10. Simplified29.4

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

   11. Simplified21.6

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

   13. Applied pow1/221.6

       \[\leadsto c0 \cdot \frac{1}{\color{blue}{{\left(\frac{V}{A} \cdot \ell\right)}^{\frac{1}{2}}}}\]

   14. Applied pow-flip21.6

       \[\leadsto c0 \cdot \color{blue}{{\left(\frac{V}{A} \cdot \ell\right)}^{\left(-\frac{1}{2}\right)}}\]

   15. Simplified21.6

       \[\leadsto c0 \cdot {\left(\frac{V}{A} \cdot \ell\right)}^{\color{blue}{\frac{-1}{2}}}\]

   if -1.432883117701101e+185 < (* V l) < -2.5461010569718647e-302

   1. Initial program 8.8

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

   3. Applied clear-num9.2

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

   5. Applied sqrt-div8.8

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

   6. Applied associate-*r/8.8

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

   if -2.5461010569718647e-302 < (* V l) < 7.3133984123809e-315

   1. Initial program 58.0

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

   3. Applied clear-num58.0

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

   5. Applied associate-/l*32.5

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

   if 7.3133984123809e-315 < (* V l)

   1. Initial program 15.3

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

   3. Applied sqrt-div6.7

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

4. Final simplification11.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.432883117701101 \cdot 10^{+185}:\\ \;\;\;\;c0 \cdot {\left(\ell \cdot \frac{V}{A}\right)}^{\frac{-1}{2}}\\ \mathbf{elif}\;V \cdot \ell \le -2.5461010569718647 \cdot 10^{-302}:\\ \;\;\;\;\frac{c0}{\sqrt{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \le 7.3133984123809 \cdot 10^{-315}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V}{\frac{A}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019005 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))

Details

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