Average Error: 1.7 → 1.7
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} + \left(\sqrt[3]{\frac{\frac{1}{2}}{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}}\right) \cdot \sqrt[3]{\frac{\frac{1}{2}}{\sqrt{(\left(\frac{2 \cdot \ell}{Om} \cdot \frac{2 \cdot \ell}{Om}\right) \cdot \left((\left(\sin ky\right) \cdot \left(\sin ky\right) + \left(\sin kx \cdot \sin kx\right))_*\right) + 1)_*}}}}\]

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

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. Simplified1.7

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

  4. Applied add-cube-cbrt1.7

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

  5. Final simplification1.7

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

Reproduce

herbie shell --seed 2019072 +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))))))))))