Shorten an absolute path

JavaScript (ES6), 107 106 bytes

Takes the absolute path a and the current path c in currying syntax (a)(c).

a=>c=>(A=a.split`/`,s='',c.split`/`.map(d=>!s&A[0]==d?A.shift():s+='../'),s+=A.join`/`)[a.length]?a:s||'.'

Test cases

let f =

a=>c=>(A=a.split`/`,s='',c.split`/`.map(d=>!s&A[0]==d?A.shift():s+='../'),s+=A.join`/`)[a.length]?a:s||'.'

console.log(f
  ('/home/user/mydir/myfile')
  ('/home/user')
);

console.log(f
  ('/var/users/admin/secret/passwd')
  ('/var/users/joe/hack')
);

console.log(f
  ('/home/user/myfile')
  ('/tmp/someplace')
);

console.log(f
  ('/dir1/dir2')
  ('/dir1/dir2/dir3/dir4')
);

console.log(f
  ('/dir1/dir2')
  ('/dir1/dir2')
);

Retina, 85 83 82 bytes

1 byte saved thanks to @MartinEnder

^(..+)(.*;)\1
%$2
(%?)(.*);(.*)
$1$3;$2
\w+(?=.*;)
..
%;/

;
/
.*//
/
%/?|/$

^$
.

Try it online!

Julia 0.5, 32 bytes

!,~=relpath,endof
t->~t<~!t?t:!t

This uses the current working directory as base and cannot be tested on TIO at the moment.

Example run

Warning: This will alter your file system.

$ sudo julia --quiet
julia> function test(target,base)
       mkpath(base)
       cd(base)
       shorten(target)
       end
test (generic function with 1 method)
julia> !,~=relpath,endof
(relpath,endof)

julia> shorten = t->~t<~!t?t:!t
(::#1) (generic function with 1 method)

julia> test("/home/user/mydir/myfile","/home/user")
"mydir/myfile"

julia> test("/var/users/admin/secret/passwd","/var/users/joe/hack")
"../../admin/secret/passwd"

julia> test("/home/user/myfile","/tmp/someplace")
"/home/user/myfile"

julia> test("/dir1/dir2","/dir1/dir2/dir3/dir4")
"../.."

julia> test("/dir1/dir2","/dir1/dir2")
"."

Alternate version, 35 bytes (dyadic)

^,~=relpath,endof
t-b=~t<~t^b?t:t^b

This takes the base directory as input, so it can be tested without modifying the file system.

Try it online!