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Derivation

1. Split input into 2 regimes

2. if h < -2.2498668262508357e+184 or 1.5451301861912833e+40 < h

   1. Initial program 21.8

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

   2. Initial simplification20.6

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
   3. Using strategy rm

   4. Applied associate-/r/10.3

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell} \cdot h}} \cdot w0\]

   if -2.2498668262508357e+184 < h < 1.5451301861912833e+40

   1. Initial program 10.7

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

   2. Initial simplification10.3

      \[\leadsto \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\frac{\ell}{h}}} \cdot w0\]
   3. Using strategy rm

   4. Applied associate-/l*8.4

      \[\leadsto \sqrt{1 - \color{blue}{\frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h}}{\frac{M \cdot D}{2 \cdot d}}}}} \cdot w0\]
3. Recombined 2 regimes into one program.

4. Final simplification8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \le -2.2498668262508357 \cdot 10^{+184} \lor \neg \left(h \le 1.5451301861912833 \cdot 10^{+40}\right):\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}}{\ell} \cdot h} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h}}{\frac{M \cdot D}{2 \cdot d}}}} \cdot w0\\ \end{array}\]

Runtime

Time bar (total: 39.3s)Debug logProfile

herbie shell --seed 2018353 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))