What is this one parameter subgroup of the Lie group Diff(R)?

I found a useful definition of a one parameter subgroup of a Lie group G. It is a map f from R to G so that f(s)*f(t) = f(s+t) for all s and t. here * is the law in G. We have f(0)*f(t) = f(t) so f(0) is the identity. f(-t)*f(0) = Id. so f(-t) is the inverse of f(t). In Diff(R) the elements are differentiable bijective maps on R. The binary law * is the composition law $circ$ .The inverse with this law is the reciprocal function. I choose an element in Diff(R) here the function sinh, and the aim is to build the one parameter subgroup that maps 0 to Id and 1 to sinh. I will use the one parameter definition to define f(s) first on Z tben on Q and next on R. As $f(1)circ f(1)$ = sinh(sinh) = f(2) we get the value on 2 then on integer values of t. i introduce the notation $sinh^{(n)}$ for sinh(sinh(...))) n times and in the same way the notation with n negative by using arsinh. we can consider the function g so that $g^{(q)} = sinh$ so $sinh^{(1/q)}$ is well defined and $sinh^{(p/q)}$ also. As real numbers are limits of fractions we get a way to define $sinh^{(r)}$ for all real numbers r. for any z in Diff(R) the one parameter mapping 0 to Id and 1 to z may be constructed in the same way. a usual notation is exp(tz)