Enable tolerance for ExtrapolationError checks

[mleseach]

Is your feature request related to a problem? Please describe.

When dealing with pre-computed interpolations but with a different time parameterization, mapping from one parameterization to another can cause precision errors, leading to an extrapolation error if extrapolate=false.

using DataInterpolations

t1 = range(0.000000000000001, 10, length=10)
u = rand(10)

itp = LinearInterpolation(u, t1)

f(t) = t * 2 + 1
f⁻¹(t) = (t - 1) / 2 

t2 = f⁻¹.(f.(t1))

itp(t2[1]) # ExtrapolationError
# note that itp(t2) doesn't check properly for extrapolation

Describe the solution you’d like

I think the best solution is to allow some tolerance when checking for extrapolation error (e.g. something like t < itp.t[1] - eps(t) instead of t < itp.t[1] for the lower bound)

Additionally, the tolerance could be parameterized, allowing users to specify a custom tolerance level if the default isn't ideal for their use case. But in this case maybe extrapolate=true could be sufficient.

If you think this feature is within the scope of this package, I can create a draft PR and discuss the details.

[ChrisRackauckas]

To me, extrapolate=true is sufficient and the cost of maintaining some odd boundary behavior likely is not worth it. @sathvikbhagavan can have an opinion

[sathvikbhagavan]

Yes, I don't think it is useful supporting this in this package. extrapolate=true should be sufficient.

[ChrisRackauckas]

Closing as not planned, but thank you for the input @mleseach as it is an interesting suggestion but we don't think it's the right idea at this time. As mentioned in the OP, we currently think extrapolate=true should be sufficient for most users, but if this is a frequently asked for change we'd be willing to reconsider it.