ClearAll[CloseEnough]
CloseEnough[a_, b_, tol_] := Chop[a - b, tol] == 0
numbers = {
   {100000000000000.01, 100000000000000.011},
   {100.01, 100.011},
   {10000000000000.001/10000.0, 1000000000.0000001000},
   {0.001, 0.0010000001},
   {0.000000000000000000000101, 0.0},
   {Sqrt[2.0] Sqrt[2.0], 2.0}, {-Sqrt[2.0] Sqrt[2.0], -2.0},
   {3.14159265358979323846, 3.14159265358979324}
   };
(*And@@Flatten[Map[MachineNumberQ,numbers,{2}]]*)
{#1, #2, CloseEnough[#1, #2, 10^-9]} & @@@ numbers // Grid