Make an almost quine
CJam, 0.000884

{_,6/[{64md}6*](124+\+'�f++`"1$~"}""1$~


Here, � denotes the unprintable character with code point 128. Try it online.

Idea

This approach appends all UCS characters (original specification) with code points between U+4000000 and U+7FFFFFFF to the initially empty string that follows the code block.

We choose UTF-8, which encodes each of these characters using a 6 byte string as follows:

1111110u₂     10vvvvvv₂     10wwwwww₂     10xxxxxx₂     10yyyyyy₂     10zzzzzz₂

252 + u       128 + vvvvvv  128 + wwwwww  128 + xxxxxx  128 + yyyyyy  128 + zzzzzz


Thus, we can encode the nth character in this range by computing its 6 least significant digits in base 64 and adding 252 to the most significant and 128 to the remaining ones.

Scoring

There are 2 ** 31 = 2,147,483,648 6 byte UTF-8 characters and the length of the original code is 39, so the score is 39 / floor(2 ** 15.5) = 39 / 46340 = 0.0008416055243849806.

How it works

{                                }""    e# Push the code block and an empty string.
                                    1$~ e# Execute a copy of the code block.
 _,                                     e# Push the length of the string.
   6/                                   e# Divide by 6 to get the number of chars.
      {64md}6*                          e# Perform modular division six times.
     [        ]                         e# Collect the results in an array.
               (124+\+                  e# Add 124 to the first element.
                      '�f+              e# Add 128 to all and cast each to Char.
                          +             e# Concatenate the strings.
                           `            e# Push a string representation.
                            "1$~"       e# Push '1$~' to complete the quine.