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Derivation

1. Split input into 3 regimes

2. if (* V l) < -3.334876225266701e+183 or -1.175718995950725e-219 < (* V l) < 1.98613222286689e-309 or 4.4098284317428e+306 < (* V l)

   1. Initial program 37.3

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

   3. Applied associate-/r*24.9

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

   5. Applied add-cube-cbrt25.2

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

   6. Applied sqrt-prod25.2

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

   7. Applied associate-*r*25.2

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

   9. Applied div-inv25.1

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

   10. Applied cbrt-prod25.1

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

   12. Applied cbrt-div25.1

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

   13. Applied cbrt-div15.3

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

   14. Applied frac-times15.3

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

   15. Applied sqrt-div12.0

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

   16. Simplified12.0

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

   if -3.334876225266701e+183 < (* V l) < -1.175718995950725e-219

   1. Initial program 6.8

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

   3. Applied clear-num7.1

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

   if 1.98613222286689e-309 < (* V l) < 4.4098284317428e+306

   1. Initial program 9.7

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

   3. Applied div-inv9.7

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

   4. Applied sqrt-prod0.4

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

4. Final simplification6.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -3.334876225266701 \cdot 10^{+183}:\\ \;\;\;\;\left(c0 \cdot \frac{\left|\sqrt[3]{\frac{A}{V}}\right|}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \sqrt{\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{1}{\ell}}}\\ \mathbf{elif}\;V \cdot \ell \le -1.175718995950725 \cdot 10^{-219}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \le 1.98613222286689 \cdot 10^{-309} \lor \neg \left(V \cdot \ell \le 4.4098284317428 \cdot 10^{+306}\right):\\ \;\;\;\;\left(c0 \cdot \frac{\left|\sqrt[3]{\frac{A}{V}}\right|}{\sqrt{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \sqrt{\sqrt[3]{\frac{A}{V}} \cdot \sqrt[3]{\frac{1}{\ell}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right) \cdot c0\\ \end{array}\]

Reproduce

herbie shell --seed 2019016 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))

Details

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