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Derivation

1. Split input into 3 regimes

2. if (* c0 (sqrt (/ A (* V l)))) < -inf.0 or -7.78990101570949e-271 < (* c0 (sqrt (/ A (* V l)))) < 5.625650864506304e-184

   1. Initial program 34.4

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

   2. Initial simplification27.0

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

   4. Applied add-sqr-sqrt27.0

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

   5. Applied sqrt-prod27.1

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

   6. Applied associate-*r*27.1

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

   8. Applied add-cube-cbrt27.1

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

   9. Applied sqrt-prod27.1

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

   10. Simplified27.1

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

   if -inf.0 < (* c0 (sqrt (/ A (* V l)))) < -7.78990101570949e-271 or 5.625650864506304e-184 < (* c0 (sqrt (/ A (* V l)))) < 1.1213961670905229e+300

   1. Initial program 0.9

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

   if 1.1213961670905229e+300 < (* c0 (sqrt (/ A (* V l))))

   1. Initial program 59.7

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

   2. Initial simplification46.1

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

   4. Applied sqrt-div40.0

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

   5. Applied associate-*r/40.1

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

4. Final simplification14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} = -\infty:\\ \;\;\;\;\left(\sqrt{\sqrt{\frac{\frac{A}{V}}{\ell}}} \cdot c0\right) \cdot \sqrt{\left|\sqrt[3]{\frac{\frac{A}{V}}{\ell}}\right| \cdot \sqrt{\sqrt[3]{\frac{\frac{A}{V}}{\ell}}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \le -7.78990101570949 \cdot 10^{-271}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \le 5.625650864506304 \cdot 10^{-184}:\\ \;\;\;\;\left(\sqrt{\sqrt{\frac{\frac{A}{V}}{\ell}}} \cdot c0\right) \cdot \sqrt{\left|\sqrt[3]{\frac{\frac{A}{V}}{\ell}}\right| \cdot \sqrt{\sqrt[3]{\frac{\frac{A}{V}}{\ell}}}}\\ \mathbf{elif}\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}} \le 1.1213961670905229 \cdot 10^{+300}:\\ \;\;\;\;c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{A}{V}} \cdot c0}{\sqrt{\ell}}\\ \end{array}\]

Runtime

Time bar (total: 17.3s)Debug logProfile

herbie shell --seed 2018353 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))