#!/usr/bin/sed -f
:loop
  # Remove leading non-math stuff
  s/^[^$]*//

  # If we didn't hit a math start yet, get next line
  /^\$\$/! d

  # We now have a string starting with $$
  # Until we have a starting and an ending point, get more lines
  :get
  /\$\$.*\$\$/ bgot
  N
  bget
  :got

  # We now have a string containing at least one string we want
  # Preserve in hold space
  h
  # Remove everything not inside the pattern and print it
  s/^\(\$\$[^$]*\$\$\).*/\1/p
  # get text back from hold space
  g
  # Remove just-printed string
  s/^\$\$[^$]*\$\$//
bloop