If you get \d from an external source, it will be represented as "\\d" as a string, and this will work fine with Regex.compile.

Elixir string "\d" does not represent two characters, but a single one. A user entering text into, say, text field, does not use Elixir syntax, though, so when they type \d, this will result in a two-byte string, represented as "\\d" in Elixir. When you use a string in a test, you use Elixir syntax, so additional escaping is necessary there.

That is because just using " as the string delimiters turns on special characters like that.

To turn it off there are a few ways, the usual one is via using the ~S (capital S) sigil, which turns interpolation and special characters off, thus:

iex> "\d"
"\d"
iex> ~s"\d"
"\d"
iex> ~S"\d"
"\\d"

EDIT1: For note, " basically is like having an implied ~s before it. You can change the delimiter to a certain set if you want, ", ', / and a half dozen others are all valid, so ~s/blah/ == "blah" is true.

EDIT2: Also, the r sigil both defines a string and passes it to regex compile all in the same step, so your original example could be done like:

iex> regex_str = "^\\d+,$"
"^\\d+,$"
iex> Regex.compile(regex_str)
{:ok, ~r/^\d+,$/}
iex> ~r"\\d+,$"
~r/\\d+,$/
iex> ~r/\\d+,$/
~r/\\d+,$/
iex> ~r{\\d+,$}
~r/\\d+,$/

The inspection protocol for compiled regex’s just converts the compiles regex into the sigil form for easy copy/pasting into the shell, but that is not how it really is internally. The inspection protocol is for ease of ‘you’ reading it, not how it really is.

EDIT3: There are lots of sigils, you can even make your own, all documented at:

As a side note, be careful running user input as a regex. It’s very easy to create a pathologically expensive regex that will DOS your server.