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Derivation

1. Split input into 3 regimes

2. if (* V l) < -6.279111945542786e-272

   1. Initial program 13.9

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

   3. Applied clear-num14.2

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

   5. Applied div-inv14.3

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

   6. Applied associate-/r*13.9

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

   8. Applied add-cube-cbrt14.3

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

   9. Applied add-cube-cbrt14.4

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

   10. Applied times-frac14.4

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

   11. Applied sqrt-prod8.2

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

   12. Applied associate-*r*8.2

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

   if -6.279111945542786e-272 < (* V l) < 0.0

   1. Initial program 54.6

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

   3. Applied clear-num54.6

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

   5. Applied div-inv54.6

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

   6. Applied associate-/r*55.6

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

   8. Applied *-un-lft-identity55.6

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

   9. Applied div-inv55.6

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

   10. Applied times-frac55.6

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

   11. Simplified55.6

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

   12. Simplified33.1

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

   if 0.0 < (* V l)

   1. Initial program 15.7

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

   3. Applied clear-num16.0

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

   5. Applied div-inv16.0

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

   6. Applied associate-/r*16.0

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

   8. Applied associate-/r/16.0

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

   9. Applied sqrt-prod7.4

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

   10. Simplified7.4

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

4. Final simplification10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -6.279111945542786 \cdot 10^{-272}:\\ \;\;\;\;\left(\sqrt{\frac{\sqrt[3]{\frac{1}{V \cdot \ell}} \cdot \sqrt[3]{\frac{1}{V \cdot \ell}}}{\sqrt[3]{\frac{1}{A}} \cdot \sqrt[3]{\frac{1}{A}}}} \cdot c0\right) \cdot \sqrt{\frac{\sqrt[3]{\frac{1}{V \cdot \ell}}}{\sqrt[3]{\frac{1}{A}}}}\\ \mathbf{elif}\;V \cdot \ell \le 0.0:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A} \cdot \sqrt{\frac{1}{V \cdot \ell}}\right) \cdot c0\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))

Details

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