I need to implement gif from bmp to animate Abelian sandpile model using only c++ standard library.
Ideally, your starting points would be specifications for GIF and BMP.
The GIF Specification, is a pretty easy thing to find.
Unfortunately, (at least to the best of my knowledge) Microsoft has never brought all the information about BMP format into a single document to act as a specification. There's a lot of documentation in various places, but no one place that has all of it together and completely organized (at least of which I'm aware).
That means you're kind of stuck with a piecemeal approach. Fortunately, you probably don't need to read every possible legitimate BMP file--it's been around a long time, so there are lots of variations, many of which are rarely used any more (e.g., 16-color bitmaps).
At a guess, you probably only need to deal with one or two specific variants (e.g., 24 or 32-bits per pixel), which makes life a great deal easier. Here's a page that gives at least a starting point for documentation on how BMP files are formatted.
You'll probably need to consider at least a few ancillary problems though. Unless your input BMP files use 8 bits per pixel with a palette to define the color associated with each of those 255 values, you're probably going to have at least one other problem: you'll most likely be starting with a file that has lots of colors (e.g., as noted above, 24 or 32 bits per pixel), but for a GIF file you need to reduce that to only 8 bits per pixel, so you'll need to choose 255 colors that best represent those in the pictures you care about, then for each input pixel, choose one of those 255 colors to represent that pixel as well as possible.
Depending on how much you care about color fidelity vs. spatial resolution, there are multitudes of ways of doing this job, varying from fairly simple (but with results that may be rather mediocre) to extremely complex (with results that may be somewhat better, but will probably still be fairly mediocre).