BigFloats give Float64 accuracy — doesn't use the precision of the arguments

[stevengj]

This is related to #98 — Float64 precision seems to be baked into the package, whereas it would be more flexible (and more Julian) to use the precision of the arguments. For example, using BigFloat in the following example only gives 16 digits of accuracy:

julia> using FiniteDifferences

julia> extrapolate_fdm(central_fdm(2, 1), sin, big"1.0")[1] - cos(big"1.0")
-6.71174699531887290713204926350530055924229547805016487900793424335727248883486e-17

In contrast, "manually" calling Richardson extrapolation with a 2nd-order finite-difference approximation gives 70 digits (in 10 iterations):

julia> using Richardson

julia> dsin,_ = extrapolate(big"0.1", rtol=0, power=2) do h
           @show Float64(h)
           (sin(1+h) - sin(1-h)) / 2h
       end
Float64(h) = 0.1
Float64(h) = 0.0125
Float64(h) = 0.0015625
Float64(h) = 0.0001953125
Float64(h) = 2.44140625e-5
Float64(h) = 3.0517578125e-6
Float64(h) = 3.814697265625e-7
Float64(h) = 4.76837158203125e-8
Float64(h) = 5.960464477539063e-9
Float64(h) = 7.450580596923829e-10
(0.5403023058681397174009366074429766037323104206179222276700972553811007395485809, 3.557003874037244232691736616015521320518299017661431723628543167235436096570656e-70)

julia> dsin - cos(big"1.0")
3.44774105579695699101361233110829910923644073773066771188994182638266121185316e-70

So, someplace in FiniteDifferences is either hard-coding a tolerance (rather than using eps(float(x))), or "contaminating" the calculation with an inexact Float64 literal.

cc @hammy4815

[oxinabox]

This is true.
It is hard coded to use Float64 in a bunch of places, like when storing the precomputed grids.
It would take a fair bit to fix, but would be good if someone was interested in doing so.
I don't have the need or the time right now