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Derivation

1. Split input into 3 regimes

2. if (* V l) < -2.498603521546299e-308

   1. Initial program 13.8

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

   if -2.498603521546299e-308 < (* V l) < -0.0

   1. Initial program 59.6

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

   3. Applied associate-/r*34.6

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

   5. Applied add-sqr-sqrt34.6

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

   6. Applied sqrt-prod34.7

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

   if -0.0 < (* V l)

   1. Initial program 14.9

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

   3. Applied sqrt-div6.9

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

4. Final simplification12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -2.498603521546299 \cdot 10^{-308}:\\ \;\;\;\;\sqrt{\frac{A}{V \cdot \ell}} \cdot c0\\ \mathbf{elif}\;V \cdot \ell \le -0.0:\\ \;\;\;\;\left(\sqrt{\sqrt{\frac{\frac{A}{V}}{\ell}}} \cdot \sqrt{\sqrt{\frac{\frac{A}{V}}{\ell}}}\right) \cdot c0\\ \mathbf{else}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019007 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))

Details

Time bar (total: 8.6s)Debug log