It is known that the Hooke's law relating the restoring force of a spring to the distance of retraction from the equilibrium position, is only an approximation.
That is, the equation $F=-kx$ is only the linear term that approximates the relationship, but gets less accurate the more the spring is retracted from the equilibrium position. There is also an elastic limit for the spring, which makes the relationship vertically asymptotic at distances large enough and the law completely breaks down there.
The relationship looks to me more like a tangent function than a line when considering the increasing deviation and asymptotic behavior.
Since $F=-kx$ is only an approximation, then what is the full story? What is the actual relationship? I couldn't find the answer anywere on the internet. Is it a transcendental function, or perhaps some non-elementary function? Or does the function defining the relationship depend on the material used and local physical quantities?