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. Initial simplification1.6

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

3. Taylor expanded around inf 16.8

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

4. Simplified0.6

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

5. Final simplification0.6

   \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{(4 \cdot \left((\left(\frac{\sin kx}{\frac{Om}{\ell}}\right) \cdot \left(\frac{\sin kx}{\frac{Om}{\ell}}\right) + \left(\frac{\sin ky}{\frac{Om}{\ell}} \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right))_*\right) + 1)_*}} + \frac{1}{2}}\]

Reproduce

herbie shell --seed 2018365 +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

Time bar (total: 25.2s)Debug log