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Derivation

1. Split input into 2 regimes

2. if (pow (/ (* 2 l) Om) 2) < 7.382935019665496e+203

   1. Initial program 1.1

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt1.1

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\color{blue}{\sqrt{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}} \cdot \sqrt{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\]

   4. Applied associate-/r*1.1

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \color{blue}{\frac{\frac{1}{\sqrt{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}{\sqrt{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}}}\right)}\]

   if 7.382935019665496e+203 < (pow (/ (* 2 l) Om) 2)

   1. Initial program 2.7

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]

   2. Taylor expanded around 0 23.2

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{\left(4 \cdot \frac{{kx}^{2} \cdot {\ell}^{2}}{{Om}^{2}} + 4 \cdot \frac{{ky}^{2} \cdot {\ell}^{2}}{{Om}^{2}}\right)}}}\right)}\]

   3. Simplified1.9

      \[\leadsto \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + \color{blue}{\left(\left(\ell \cdot \ell\right) \cdot 4\right) \cdot \left(\frac{ky}{Om} \cdot \frac{ky}{Om} + \frac{kx}{Om} \cdot \frac{kx}{Om}\right)}}}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \le 7.382935019665496 \cdot 10^{+203}:\\ \;\;\;\;\sqrt{\left(1 + \frac{\frac{1}{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}}}{\sqrt{\sqrt{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} + 1}}}\right) \cdot \frac{1}{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{\left(4 \cdot \left(\ell \cdot \ell\right)\right) \cdot \left(\frac{kx}{Om} \cdot \frac{kx}{Om} + \frac{ky}{Om} \cdot \frac{ky}{Om}\right) + 1}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019004 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))

Details

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