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

↓

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

Error

The X axis uses an exponential scale — Bits error versus `l`

Derivation

Initial program 1.1
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
Initial simplification1.1
\[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell}{\frac{Om}{2}} \cdot \frac{\ell}{\frac{Om}{2}}\right) + 1}} + \frac{1}{2}}\]
Using strategy rm
Applied associate-*r*0.8
\[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\color{blue}{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \frac{\ell}{\frac{Om}{2}}\right) \cdot \frac{\ell}{\frac{Om}{2}}} + 1}} + \frac{1}{2}}\]
Final simplification0.8
\[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\left(\frac{\ell}{\frac{Om}{2}} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{\ell}{\frac{Om}{2}} + 1}} + \frac{1}{2}}\]

Runtime

	Baseline	Herbie	Oracle	Span	%
Regimes	0.8	0.8	0.7	0.1	0%

herbie shell --seed 2018296 
(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))))))))))

In
Out

Error

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Derivation

Runtime