I have a function f[b,d] that defines an implicit curve f[b,d]=0. I want to plot that implicit curve but it takes ages when using ContourPlot

I have a messy function f[b,d]. I need to plot the implicit curve f[b,d]=0, and I have tried with the classical function


but even after waiting for 6 hours, I have not been able to get an output.

I was thinking of using the method of Pseudo Arc-Length continuation to plot the function, but I have discovered that there is no general function for that. I have tried to implement the procedure written in the first answer of this page:

Is there any predictor-corrector method in Mathematica for solving nonlinear system of algebraic equations?

but I have not been able to adapt that code to my case.

In particular, after defining the function TrackRootPAL, I have run the following line:

tr = TrackRootPAL[{f[b,d]}, {d}, {b, -0.001, 0.001}, 1.0011*10^(-5), {0.04}]; tempplot = Plot[Evaluate[d[b] /. tr]], {b, 0, 4}] 

However, I get the following output:


The function I am currently trying to plot is quite large (it has more than 100000 characters) so it cannot be copied over here but the code seems easy to adapt so if you need an example, I think you could try with f[b,d]=b+d. In fact, I get a similar problem with that function.

EDIT: I know that one zero of my complicated function f[b,d] is approximately located at b=1.0011*10^(-5) and d=0.4.