Average Error: 1.6 → 1.6
Time: 2.0m
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{\frac{\frac{1}{2}}{\sqrt[3]{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\left(\frac{2}{Om} \cdot \ell\right) \cdot \left(\frac{2}{Om} \cdot \ell\right)\right) + 1}} \cdot \sqrt[3]{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\left(\frac{2}{Om} \cdot \ell\right) \cdot \left(\frac{2}{Om} \cdot \ell\right)\right) + 1}}}}{\sqrt[3]{\sqrt[3]{\sqrt{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\left(\frac{2}{Om} \cdot \ell\right) \cdot \left(\frac{2}{Om} \cdot \ell\right)\right) + 1}} \cdot \sqrt{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\left(\frac{2}{Om} \cdot \ell\right) \cdot \left(\frac{2}{Om} \cdot \ell\right)\right) + 1}}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\left(\frac{2}{Om} \cdot \ell\right) \cdot \left(\frac{2}{Om} \cdot \ell\right)\right) + 1}} \cdot \sqrt[3]{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\left(\frac{2}{Om} \cdot \ell\right) \cdot \left(\frac{2}{Om} \cdot \ell\right)\right) + 1}}}} + \frac{1}{2}}\]

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

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. Simplified1.6

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

  4. Applied add-cube-cbrt1.6

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

  5. Applied associate-/r*1.6

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

  7. Applied add-cube-cbrt1.6

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

  8. Applied cbrt-prod1.6

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

  10. Applied add-sqr-sqrt1.6

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

  11. Applied sqrt-prod1.6

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

  12. Final simplification1.6

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

Reproduce

herbie shell --seed 2019094 
(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))))))))))