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Derivation

1. Split input into 2 regimes

2. if (pow (/ (* M D) (* 2 d)) 2) < 2.4802095421231e-321 or 1.7789904050843808e+308 < (pow (/ (* M D) (* 2 d)) 2)

   1. Initial program 18.7

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   2. Initial simplification17.6

      \[\leadsto \sqrt{(\left(\frac{\frac{M}{2}}{\frac{d}{D}} \cdot \frac{\frac{M}{2}}{\frac{d}{D}}\right) \cdot \left(-\frac{h}{\ell}\right) + 1)_*} \cdot w0\]

   3. Taylor expanded around 0 12.0

      \[\leadsto \color{blue}{1} \cdot w0\]

   if 2.4802095421231e-321 < (pow (/ (* M D) (* 2 d)) 2) < 1.7789904050843808e+308

   1. Initial program 5.8

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
3. Recombined 2 regimes into one program.

4. Final simplification9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 2.4802095421231 \cdot 10^{-321} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.7789904050843808 \cdot 10^{+308}\right):\\ \;\;\;\;w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}\\ \end{array}\]

Runtime

Time bar (total: 31.8s)Debug logProfile

herbie shell --seed 2018340 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))