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Derivation

1. Split input into 3 regimes

2. if D < -2.1251126957103482e-07

   1. Initial program 56.6

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Initial simplification49.2

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]
   3. Using strategy rm

   4. Applied div-inv50.0

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \color{blue}{\left(\frac{c0}{h} \cdot \frac{1}{w}\right)} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]

   5. Applied associate-*l*51.2

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \color{blue}{\frac{c0}{h} \cdot \left(\frac{1}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)}\right)\]

   if -2.1251126957103482e-07 < D < -5.0907627571985e-242

   1. Initial program 58.0

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Initial simplification54.2

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]
   3. Using strategy rm

   4. Applied associate-*r*54.9

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \color{blue}{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}\right)\]

   5. Taylor expanded around 0 57.9

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \color{blue}{\left(2 \cdot \frac{c0 \cdot {d}^{2}}{{D}^{2} \cdot \left(w \cdot h\right)}\right)}\]

   if -5.0907627571985e-242 < D

   1. Initial program 58.4

      \[\frac{c0}{2 \cdot w} \cdot \left(\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} + \sqrt{\frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} \cdot \frac{c0 \cdot \left(d \cdot d\right)}{\left(w \cdot h\right) \cdot \left(D \cdot D\right)} - M \cdot M}\right)\]

   2. Initial simplification53.6

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right)\]
   3. Using strategy rm

   4. Applied associate-*r*54.5

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right) - M\right)} + \color{blue}{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}}\right)\]
   5. Using strategy rm

   6. Applied associate-*r*54.1

      \[\leadsto \frac{\frac{c0}{2}}{w} \cdot \left(\sqrt{\left(M + \frac{\frac{c0}{h}}{w} \cdot \left(\frac{d}{D} \cdot \frac{d}{D}\right)\right) \cdot \left(\color{blue}{\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}} - M\right)} + \left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification54.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;D \le -2.1251126957103482 \cdot 10^{-07}:\\ \;\;\;\;\left(\frac{c0}{h} \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{1}{w}\right) + \sqrt{\left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} - M\right)}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{elif}\;D \le -5.0907627571985 \cdot 10^{-242}:\\ \;\;\;\;\left(\frac{{d}^{2} \cdot c0}{{D}^{2} \cdot \left(h \cdot w\right)} \cdot 2\right) \cdot \frac{\frac{c0}{2}}{w}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(\left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D} - M\right) \cdot \left(\left(\frac{d}{D} \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{h}}{w} + M\right)} + \left(\frac{\frac{c0}{h}}{w} \cdot \frac{d}{D}\right) \cdot \frac{d}{D}\right) \cdot \frac{\frac{c0}{2}}{w}\\ \end{array}\]

Reproduce

herbie shell --seed 2018365 
(FPCore (c0 w h D d M)
  :name "Henrywood and Agarwal, Equation (13)"
  (* (/ c0 (* 2 w)) (+ (/ (* c0 (* d d)) (* (* w h) (* D D))) (sqrt (- (* (/ (* c0 (* d d)) (* (* w h) (* D D))) (/ (* c0 (* d d)) (* (* w h) (* D D)))) (* M M))))))

Details

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