machine precision - Mathematica Plot: Inconsistency when plotting large values

I am working with a function in Mathematica and I am getting some inconsistencies when I plot it. As I really need to understand were this comes from I would appreciate any help.

I am working with a function of the following form:

f2b[b_] = Exp[-1000 - 2 Sqrt[1000*b] - b]*(Erfi[Sqrt[1000]] - Erfi[Sqrt[1000] + Sqrt[b]])

From my understanding this function should go to zero for very small b because both parts, the Exp function as well as the difference between the imaginary error functions, should vanish.

Mathematica calculates:

f2b[E^-30] // N = -3.4517046881*10^-7 as expected.

BUT if I plot this function:

Plot[{f2b[E^b], 0}, {b, -50, 0}, PlotStyle -> {{Blue, Thick}, {Red, Thick}}]

Mathematica shows me a plot which is approaching a value of about -0.0175 for small b and has therefore an absolute value much larger the the value at E^-30 calculated above.

If I tabulate, interpolate and plot the function:

t = Table[{b, f2b[E^b] // N}, {b, -50, 0}]; f = Interpolation[t]; Plot[{f[b], 0}, {b, -50, 0}, PlotStyle -> {{Blue, Thick}, {Red, Thick}}]

The plot is looking much different and is approaching zero as expected.

I am wondering whether this has to do something with overflow problems in graphics but I am not sure.

Answer

Your function isn't evaluating correctly when given inexact input:

In[16]:= Table[f2b[N[E^-k]], {k, 0, 50, 10}]


Out[16]= {-0.01730248257001, -0.01784636881283, -0.01785014954397,
          -0.01785017502377, -0.01785017519545, -0.01785017519660}

If we force f2b to be evaluated with exact inputs (delaying the numericization of the result) we get the expected plot:

f2b2[b_?NumericQ] := N[f2b[SetPrecision[b, Infinity]]] 

Plot[{f2b2[E^b], 0}, {b, -50, 0}, PlotStyle -> {{Blue, Thick}, {Red, Thick}}]