Average Error: 1.7 → 1.7
Time: 34.8s
Precision: 64
Internal Precision: 128
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\left(1 + \frac{1}{\sqrt{\left(\sqrt[3]{\sqrt[3]{1 + \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}}} \cdot \sqrt[3]{\sqrt[3]{1 + \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}}}\right) \cdot \sqrt[3]{\sqrt[3]{1 + \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2}}}} \cdot \left|\sqrt[3]{\sqrt[3]{\sqrt[3]{\left(\frac{2}{\frac{Om}{\ell}} \cdot \frac{2}{\frac{Om}{\ell}}\right) \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) + 1}} \cdot \left(\sqrt[3]{\sqrt[3]{\left(\frac{2}{\frac{Om}{\ell}} \cdot \frac{2}{\frac{Om}{\ell}}\right) \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) + 1}} \cdot \sqrt[3]{\sqrt[3]{\left(\frac{2}{\frac{Om}{\ell}} \cdot \frac{2}{\frac{Om}{\ell}}\right) \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) + 1}}\right)} \cdot \left(\sqrt[3]{\sqrt[3]{\left(\frac{2}{\frac{Om}{\ell}} \cdot \frac{2}{\frac{Om}{\ell}}\right) \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) + 1}} \cdot \sqrt[3]{\sqrt[3]{\left(\frac{2}{\frac{Om}{\ell}} \cdot \frac{2}{\frac{Om}{\ell}}\right) \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) + 1}}\right)\right|}\right) \cdot \frac{1}{2}}\]

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied add-cube-cbrt1.7

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

  4. Applied sqrt-prod1.7

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

  5. Simplified1.7

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

  7. Applied add-cube-cbrt1.7

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

  9. Applied add-cube-cbrt1.7

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

  11. Applied add-cube-cbrt1.7

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

  12. Final simplification1.7

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

Reproduce

herbie shell --seed 2019026 
(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))))))))))