Average Error: 1.1 → 0.0
Time: 3.0m
Precision: 64
Internal Precision: 320
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\left(2 \cdot \ell\right) \cdot \frac{\sqrt{\left(\sin ky\right)^2 + \left(\sin kx\right)^2}^*}{Om}\right)}^{2}}}\right)}\]

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 1.1

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

  3. Applied div-inv1.1

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

  4. Applied unpow-prod-down9.1

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

  5. Applied associate-*l*9.2

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

  6. Applied simplify9.3

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

  8. Applied add-sqr-sqrt9.3

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

  9. Applied times-frac8.4

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

  10. Applied simplify8.4

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

  11. Applied simplify7.7

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

  13. Applied pow27.7

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

  14. Applied pow-prod-down0.0

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

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed 2019053 +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))))))))))