How to use MMA to solve this numerical analysis problem?


The equidistant node function table of the function $ f(x)=e^{x}$ is given on $-4 leqslant x leqslant 4 quad$.

In order to find the value of $e^{x}$ by linear interpolation and make the truncation error less than $10 ^ {- 6}$, what is the maximum step size of the function table used?

The source of this problem(中国科学院武汉岩石力学研究所2000年数值分析科目入学考试):

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After class exercises of numerical analysis

piecewiseLinear(
  points : {{_, _} ..} /; ¬ OrderedQ(points((All, 1)))) :=
  (*保证输入点按横坐标顺序排列*)
  piecewiseLinear(points // SortBy(First))
piecewiseLinear(points : {{_, _} ..})(x_) :=
  With({px = points((All, 1)), pfx = points((All, 2)), 
    n = Length@points},
     Do(
       If(px((i)) <= x <= px((i + 1)),
         Throw((pfx((i + 1)) - pfx((i)))/(
        px((i + 1)) - px((i))) (x - px((i))) + pfx((i)))
        ),
       {i, n - 1});
     Undefined
    ) // Catch


Block({φ = piecewiseLinear(Table({x, E^x}, {x, -4, 4, 1}))},
 Plot({φ(x), E^x}, {x, -4, 4}, PlotTheme -> "Monochrome", 
  PlotLegends -> {Defer(φ(x)), Defer(f(x))})
 )

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