Average Error: 1.0 → 0.7
Time: 2.0m
Precision: 64
Internal Precision: 320
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\sqrt{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\left(\frac{\ell}{\frac{Om}{2}} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{\ell}{\frac{Om}{2}} + 1}}}\right)} \cdot \sqrt{\log \left(e^{\frac{\frac{1}{2}}{\sqrt{\left(\frac{\ell}{\frac{Om}{2}} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{\ell}{\frac{Om}{2}} + 1}}}\right)} + \frac{1}{2}}\]

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 1.0

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

  2. Initial simplification1.0

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

  4. Applied associate-*r*0.7

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

  6. Applied add-log-exp0.7

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

  8. Applied add-sqr-sqrt0.7

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

  9. Final simplification0.7

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018248 
(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))))))))))