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Derivation

1. Initial program 1.7

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

3. Applied unpow21.7

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

4. Applied associate-*l*1.5

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

6. Applied add-cube-cbrt1.5

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

7. Final simplification1.5

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

Reproduce

herbie shell --seed 2019002 +o rules:numerics
(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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