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Derivation

1. Initial program 1.5

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

2. Initial simplification1.5

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

4. Applied add-cube-cbrt1.5

   \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}}}\right) \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}}}} + \frac{1}{2}}\]
5. Using strategy rm

6. Applied add-cube-cbrt1.5

   \[\leadsto \sqrt{\left(\sqrt[3]{\frac{\frac{1}{2}}{\color{blue}{\left(\sqrt[3]{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}} \cdot \sqrt[3]{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}}\right) \cdot \sqrt[3]{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}}}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}}}\right) \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}}} + \frac{1}{2}}\]

7. Final simplification1.5

   \[\leadsto \sqrt{\frac{1}{2} + \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}}} \cdot \left(\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt[3]{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}} \cdot \left(\sqrt[3]{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}} \cdot \sqrt[3]{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}}\right)}}\right)}\]

Reproduce

herbie shell --seed 2018362 
(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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