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Derivation

1. Split input into 3 regimes

2. if (* V l) < -1.125220159166195e-80

   1. Initial program 15.1

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

   3. Applied div-inv15.1

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

   5. Applied add-sqr-sqrt15.3

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

   6. Applied associate-*r*15.2

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

   7. Simplified15.2

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

   if -1.125220159166195e-80 < (* V l) < 2.2677613144113e-321

   1. Initial program 33.8

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

   3. Applied div-inv34.1

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

   5. Applied *-un-lft-identity34.1

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

   6. Applied associate-*r*34.1

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

   7. Simplified25.7

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

   if 2.2677613144113e-321 < (* V l)

   1. Initial program 14.3

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

   3. Applied div-inv14.6

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

   5. Applied un-div-inv14.3

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

   6. Applied sqrt-div6.4

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

4. Final simplification13.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -1.125220159166195 \cdot 10^{-80}:\\ \;\;\;\;\sqrt{\sqrt{\frac{A}{V \cdot \ell}}} \cdot \left(c0 \cdot \sqrt{\sqrt{A \cdot \frac{1}{V \cdot \ell}}}\right)\\ \mathbf{elif}\;V \cdot \ell \le 2.2677613144113 \cdot 10^{-321}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{A}}{\sqrt{V \cdot \ell}} \cdot c0\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))

Details

Time bar (total: 9.5s)Debug log