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

↓

\[\sqrt{\frac{\frac{1}{2}}{\sqrt{4 \cdot \frac{\frac{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} + 1}} + \frac{1}{2}}\]

Error

The X axis uses an exponential scale — Bits error versus `l`

Derivation

Initial program 1.6
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
Simplified1.6
\[\leadsto \color{blue}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 + \frac{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}{\frac{\frac{Om}{\ell}}{\frac{4}{\frac{Om}{\ell}}}}}}}}\]
Taylor expanded around -inf 9.5
\[\leadsto \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 + \color{blue}{4 \cdot \frac{\left({\left(\sin ky\right)}^{2} + {\left(\sin kx\right)}^{2}\right) \cdot {\ell}^{2}}{{Om}^{2}}}}}}\]
Simplified1.4
\[\leadsto \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{1 + \color{blue}{4 \cdot \frac{\frac{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}{\frac{Om}{\ell}}}{\frac{Om}{\ell}}}}}}\]
Final simplification1.4
\[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{4 \cdot \frac{\frac{\sin kx \cdot \sin kx + \sin ky \cdot \sin ky}{\frac{Om}{\ell}}}{\frac{Om}{\ell}} + 1}} + \frac{1}{2}}\]

Reproduce

herbie shell --seed 2019068 
(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))))))))))