This is a difficult problem at the algorithmic level. Here is a way to do this in Julia (to check your own solution):

function reversebitstring(::Type{T}, str::String) where {T<:Base.IEEEFloat}
    unsignedbits = Meta.parse(string("0b", str))
    thefloat  = reinterpret(T, unsignedbits)
    return thefloat
end

reversebitstring(Float64, "0100000001011110110111010010111100011010100111111011111001110111")

julia> bitstring(3.141592653589793238462643)
"0100000000001001001000011111101101010100010001000010110100011000"

julia> reversebitstring(Float64, ans)
3.141592653589793

julia> pi
π = 3.1415926535897...

julia> 1.0*pi
3.141592653589793