The problem is catastrophic cancellation due to limited accuracy of double precision floating point arithmetic:
julia> 1 - (1e-7) * exp(-1e-7)
0.99999990000001
julia> exp(-1e-7)
0.999999900000005
julia> 1 - (1e-7) * exp(-1e-7) - exp(-1e-7)
4.9960036108132044e-15
julia> 1 - (1e-8) * exp(-1e-8)
0.9999999900000001
julia> exp(-1e-8)
0.9999999900000001
julia> 1 - (1e-8) * exp(-1e-8) - exp(-1e-8)
0.0