Rationalise recurring binary
+0
−0
Given a potentially recurring binary string, output the number it represents, as a fraction in lowest terms.
The notation used in this challenge for recurring digits is nonstandard. An r
is used to indicate that the remaining digits recur. For example, 0.000r10
means the last 2 digits recur, like this: 0.000101010...
. This is the binary representation of $\frac{1}{12}$, which cannot be expressed in binary without recurring digits.
Input
 A binary string made up of
 Zero or more of
0
 Zero or more of
1
 Zero or one of
.
 Zero or one of
r
 Zero or more of
 If
r
is present then there will always be a.
somewhere to the left of ther
 There may be zero or more leading zeroes
 There may be zero or more trailing zeroes (in both recurring and nonrecurring inputs)
 The last character will never be an
r
 There will always be at least 1 character (the string will not be empty)
 The maximum length of input required to be handled is 15 characters
Interpretation
 All digits to the right of an
r
are recurring (repeating forever)  Anything to the left of a
.
represents the integer part of the number (in binary)  Anything to the right of a
.
represents the fractional part of the number (in binary)  If the first character is a
.
then the integer part is zero  If the last character is a
.
then the fractional part is zero  If there is no
.
then the fractional part is zero
Output
 A numerator and denominator, in that order, or a rational type if your language supports one
 These can be in a multivalue data type such as a list/array/vector, or as a single string containing a nonnumeric separating character (such as a space or a
/
)  The numerator and denominator must be in lowest terms (that is, their greatest common factor is 1)
 An integer output N is represented as the fraction N/1, a numerator of N and a denominator of 1
 Zero is represented as the fraction 0/1, a numerator of zero and a denominator of 1
 The numerator and denominator are output as base 10 (decimal)  only the input is in binary
Examples
Throughout these examples binary strings are represented as fixed width
and all other numbers are base 10 (decimal).
Nonrecurring examples

10.00
is the integer 2, represented as 2/1 
01.00
is the integer 1, represented as 1/1 
00.10
is the fraction 1/2 
00.01
is the fraction 1/4
Recurring examples

10.r0
is the same as10.000...
, the same as10
, the integer 2, represented as 2/1 
0.r1
is the same as0.111...
, the same as1
, the integer 1, represented as 1/1 
0.r01
is the same as0.010101...
, the fraction 1/3 
0.r10
is the same as0.101010...
, the fraction 2/3 
1.r01
is the same as1.010101...
, the fraction 4/3 
0.r001
is the same as0.001001001...
, the fraction 1/7 
0.r010
is the same as0.010010010...
, the fraction 2/7 
0.r100
is the same as0.100100100...
, the fraction 4/7 
1.r001
is the same as1.001001001...
, the fraction 8/7
Test cases
Test cases are in the format input : output
Outputs are in the format numerator/denominator
. : 0/1
.0 : 0/1
0. : 0/1
.1 : 1/2
1. : 1/1
0.0 : 0/1
00.00 : 0/1
.r0 : 0/1
.r1 : 1/1
00.r00 : 0/1
11.r11 : 4/1
00.0r0 : 0/1
11.1r1 : 4/1
.0101r01 : 1/3
.1010r10 : 2/3
.r01 : 1/3
.r0101 : 1/3
.r010101 : 1/3
.r0101010 : 42/127
.r0011 : 1/5
.r0110 : 2/5
.r1001 : 3/5
.r1100 : 4/5
.0r01 : 1/6
.00r10 : 1/6
.1r10 : 5/6
.r001 : 1/7
.r010 : 2/7
.r011 : 3/7
.r100 : 4/7
.r101 : 5/7
.r110 : 6/7
.r111 : 1/1
0001000.r01001 : 257/31
.r000111 : 1/9
.0r001110 : 1/9
.00r011100 : 1/9
.000r111000 : 1/9
.0001r110001 : 1/9
.00011r100011 : 1/9
.000111r000111 : 1/9
.r001110 : 2/9
.r011100 : 4/9
.r100011 : 5/9
.r110001 : 7/9
.r111000 : 8/9
1 : 1/1
10 : 2/1
11 : 3/1
100 : 4/1
101 : 5/1
110 : 6/1
111 : 7/1
111. : 7/1
111.0 : 7/1
111.r0 : 7/1
111.0r0 : 7/1
.00000000000r01 : 1/6144
.00000000000r10 : 1/3072
.01010101010r10 : 1/3
.10101010101r01 : 2/3
.01010101010r01 : 2047/6144
.10101010101r10 : 4097/6144
111111111111111 : 32767/1
111111111111.r1 : 4096/1
111111111111.r0 : 4095/1
.00000000000001 : 1/16384
1111111.1111111 : 16383/128
Explanations in answers are optional, but I'm more likely to upvote answers that have one.
1 comment thread