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

↓

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

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
\[\leadsto \color{blue}{\sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\left(\ell \cdot \frac{2}{Om}\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right) + 1}}}}\]
Using strategy rm
Applied associate-*r*1.3
\[\leadsto \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{\color{blue}{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)} + 1}}}\]
Using strategy rm
Applied add-log-exp1.5
\[\leadsto \sqrt{\frac{1}{2} + \frac{\frac{1}{2}}{\sqrt{\color{blue}{\log \left(e^{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)\right) \cdot \left(\ell \cdot \frac{2}{Om}\right)}\right)} + 1}}}\]
Final simplification1.5
\[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{1 + \log \left(e^{\left(\left(\frac{2}{Om} \cdot \ell\right) \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \left(\frac{2}{Om} \cdot \ell\right)}\right)}} + \frac{1}{2}}\]

Reproduce

herbie shell --seed 2019100 
(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))))))))))

Error

Derivation

Reproduce