How can I find the best fit to my data?

Question

I have a table of data given by Table[{x3,v3},{r,SetPrecision[1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000001, 100],SetPrecision[1.00000000000000000000000000000000000000000000000000000000000000000000000000000000001,100],10^-87}] with x=r+Log[-1+r] and v=(1-1/r)/r^3. How can I find the best fit (function) of these data? I will be thankful if someone help.

Answer 1

x[r_] := r + Log[-1 + r];
v[r_] := (1 - 1/r)/r^3;
data = Table[{x[r], v[r]}, {r, 1 + 10^-88, 1 + 10^-83, 10^-87}];
fit = FindFormula[data, x]
0.367879 E^x
Show[ListPlot[data, PlotStyle -> Red], 
 Plot[fit, {x, -194.26, -190}, PlotRange -> All, PlotStyle -> Green]]

Answer 2

Data

First your data

data = Table[
       {
        r + Log[-1 + r],
        (1 - 1/r)/r^3
        }
       , {
        r,SetPrecision[1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000001, 100],
        SetPrecision[1.00000000000000000000000000000000000000000000000000000000000000000000000000000000001, 100],
        10^-87
        }
       ];

Model

You already have the equations, so there is nothing to guess or fit.

model = FullSimplify[
  ReplaceAll[
   (1 - 1/r)/r^3,
   First@Solve[
     x == r + Log[-1 + r]
     , r
     ]
   ]
  ]

(* ProductLog[E^(-1 + x)]/(1 + ProductLog[E^(-1 + x)])^4 *)

Or because

FullSimplify@N@Normal@Series[Log[model], {x, -200, 2}]
(* -1. + (1. - 1.27277*10^-87 x) x*)

To very good approximation

modelaprx = Exp[x - 1] 

Plot

Show[
 ListPlot[
  data
  , PlotRange -> All
  , ScalingFunctions -> "Log"
  , PlotTheme -> "Scientific"
  , PlotMarkers -> {Blue, Medium}
  ]
 , Plot[
  model
  , {x, -201, -180}
  , PlotStyle -> Red
  , ScalingFunctions -> "Log"
  ]
 ]