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Derivation

1. Split input into 4 regimes

2. if (* V l) < -6.242604426383306e+129 or -5.160736323445315e-135 < (* V l) < 2.4962980487814e-316

   1. Initial program 31.7

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

   2. Taylor expanded around -inf 31.7

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

   4. Applied associate-/r*23.6

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

   6. Applied *-commutative23.6

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

   if -6.242604426383306e+129 < (* V l) < -5.160736323445315e-135

   1. Initial program 4.7

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

   3. Applied clear-num5.3

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

   if 2.4962980487814e-316 < (* V l) < 5.957024405526943e+287

   1. Initial program 9.6

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

   3. Applied sqrt-div0.5

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

   if 5.957024405526943e+287 < (* V l)

   1. Initial program 39.3

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

   2. Taylor expanded around -inf 39.3

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

   4. Applied associate-/r*22.0

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

   6. Applied *-commutative22.0

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

   8. Applied div-inv22.0

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

   9. Applied sqrt-prod37.1

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

   10. Applied associate-*l*37.4

       \[\leadsto \color{blue}{\sqrt{\frac{A}{\ell}} \cdot \left(\sqrt{\frac{1}{V}} \cdot c0\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification12.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;V \cdot \ell \le -6.242604426383306 \cdot 10^{+129}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \le -5.160736323445315 \cdot 10^{-135}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{1}{\frac{V \cdot \ell}{A}}}\\ \mathbf{elif}\;V \cdot \ell \le 2.4962980487814 \cdot 10^{-316}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{\frac{A}{\ell}}{V}}\\ \mathbf{elif}\;V \cdot \ell \le 5.957024405526943 \cdot 10^{+287}:\\ \;\;\;\;c0 \cdot \frac{\sqrt{A}}{\sqrt{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{1}{V}} \cdot c0\right) \cdot \sqrt{\frac{A}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))

Details

Time bar (total: 8.0s)Debug log