Question: LambertW Simplify Bug

I found the following inconsitencies when studying the 2nd branch of the LambertW function, and i was hoping someone can help me understand what i am doing wrong here:



#Forming an equation based on what the simplify command produces as output with the lhs on the rhs:

exp(-(1/2)*LambertW(1, 2*exp(2))+1) = simplify(exp(-(1/2)*LambertW(1, 2*exp(2))+1));

#A float approximation made directly for the first branch of LambertW directly:
exp(-(1/2)*LambertW(1, 2*exp(2))+1)=evalf[10](convert(evalf[10](LambertW(1, 2*exp(2))), 'rational'));
# A float approximation made for the value implied by equation stated in the first line gives a slightly differing value:
evalf[10](rhs(isolate((1/2)*sqrt(2)/sqrt(1/x) = convert(evalf[10](exp(-(1/2)*LambertW(1, 2*exp(2))+1)), 'rational'), x)));
#The symbolic simplify command gives the following output, which is contradictory to the evalf command's output for that argument:
exp(-(1/2)*LambertW(1, 2*exp(2))+1)=simplify(exp(-(1/2)*LambertW(1, 2*exp(2))+1),'symbolic');
#So i investigated this a little further in observing that this seemed like a few of the functions i have found very interesting, have a Number theoretic proportionality to the number of significant digits we have restricted the evalf command to:
F:=Re(exp(-(1/2)*LambertW(1, 2*exp(2))+1))+Re((1/2)*sqrt(2)/sqrt(1/LambertW(1, 2*exp(2)))):
fLOATY:=proc (k) options operator, arrow; evalf[k](F) end proc;
DATA := [seq([k, convert(fLOATY(k)*10^(k-1), 'rational')], k = 1 .. 40)];
plots[:-pointplot](DATA, color = "Black");

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