Error

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. Initial simplification1.7

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

3. Taylor expanded around -inf 16.5

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

4. Simplified0.7

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

6. Applied add-cube-cbrt0.7

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

7. Applied sqrt-prod0.7

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

8. Simplified0.7

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

10. Applied add-cube-cbrt0.7

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

11. Final simplification0.7

    \[\leadsto \sqrt{\frac{\frac{1}{2}}{\left|\sqrt[3]{\sqrt[3]{(\left((\left(\frac{\sin kx}{\frac{Om}{\ell}}\right) \cdot \left(\frac{\sin kx}{\frac{Om}{\ell}}\right) + \left(\frac{\sin ky}{\frac{Om}{\ell}} \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right))_*\right) \cdot 4 + 1)_*}} \cdot \left(\sqrt[3]{\sqrt[3]{(\left((\left(\frac{\sin kx}{\frac{Om}{\ell}}\right) \cdot \left(\frac{\sin kx}{\frac{Om}{\ell}}\right) + \left(\frac{\sin ky}{\frac{Om}{\ell}} \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right))_*\right) \cdot 4 + 1)_*}} \cdot \sqrt[3]{\sqrt[3]{(\left((\left(\frac{\sin kx}{\frac{Om}{\ell}}\right) \cdot \left(\frac{\sin kx}{\frac{Om}{\ell}}\right) + \left(\frac{\sin ky}{\frac{Om}{\ell}} \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right))_*\right) \cdot 4 + 1)_*}}\right)\right| \cdot \sqrt{\sqrt[3]{(4 \cdot \left((\left(\frac{\sin kx}{\frac{Om}{\ell}}\right) \cdot \left(\frac{\sin kx}{\frac{Om}{\ell}}\right) + \left(\frac{\sin ky}{\frac{Om}{\ell}} \cdot \frac{\sin ky}{\frac{Om}{\ell}}\right))_*\right) + 1)_*}}} + \frac{1}{2}}\]

Runtime

Time bar (total: 34.2s)Debug logProfile

herbie shell --seed 2018353 +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))))))))))