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Root finding needs higher accuracy


I would like to numerically find the root of the following function
with up to 20 digits.

    f12[x_] := Log[x] + x^2 / (2 Exp[1]) - 2 x / Sqrt[Exp[1]] + 1

This problem is known to be difficult for solvers in double precision
arithmetics. I thought it should be easy with Mathematica, but the
following attempts were not successful.

    SetPrecision[
        x /. FindRoot[f12[x], {x, 1.0, 3.4}, Method -> "Brent",
                 AccuracyGoal -> Infinity, PrecisionGoal -> 20], 16]
    # 1.648732212532746
    SetPrecision[
        x /. FindRoot[f12[x], {x, 1.0, 3.4}, Method -> "Secant",
                 AccuracyGoal -> Infinity, PrecisionGoal -> 20], 16]
    # 1.648710202030051

The true root obviously is Sqrt[Exp[1]]//N = 1.648721270700128...

The symbolic solver explicitely says it cannot solve this expression.
What do I need to do to get a much more exact result out of
Mathematica?

Many thanks, Hans Werner



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