Error

Derivation

1. Initial program 1.6

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

2. Simplified1.5

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}}}\]
3. Using strategy rm

4. Applied add-cube-cbrt1.5

   \[\leadsto \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\color{blue}{\left(\sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}} \cdot \sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}\right) \cdot \sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}}}}\]
5. Using strategy rm

6. Applied add-cube-cbrt1.5

   \[\leadsto \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\left(\sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}} \cdot \sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}}\right)}}}\]
7. Using strategy rm

8. Applied add-log-exp1.7

   \[\leadsto \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\left(\sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}} \cdot \sqrt[3]{\color{blue}{\log \left(e^{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}\right)}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}}\right)}}\]

9. Final simplification1.7

   \[\leadsto \sqrt{\frac{\frac{1}{2}}{\left(\left(\sqrt[3]{\sqrt[3]{\sqrt{\frac{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}} + 1}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}} + 1}}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt{\frac{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}} + 1}}}\right) \cdot \left(\sqrt[3]{\sqrt{\frac{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}} + 1}} \cdot \sqrt[3]{\log \left(e^{\sqrt{\frac{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}} + 1}}\right)}\right)} + \frac{1}{2}}\]

Reproduce

herbie shell --seed 2019053 
(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))))))))))