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Challenges

# Rationalise recurring binary

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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 non-standard. 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
• If r is present then there will always be a . somewhere to the left of the r
• There may be zero or more leading zeroes
• There may be zero or more trailing zeroes (in both recurring and non-recurring 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 multi-value data type such as a list/array/vector, or as a single string containing a non-numeric 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).

### Non-recurring 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 as 10.000..., the same as 10, the integer 2, represented as 2/1
• 0.r1 is the same as 0.111..., the same as 1, the integer 1, represented as 1/1
• 0.r01 is the same as 0.010101..., the fraction 1/3
• 0.r10 is the same as 0.101010..., the fraction 2/3
• 1.r01 is the same as 1.010101..., the fraction 4/3
• 0.r001 is the same as 0.001001001..., the fraction 1/7
• 0.r010 is the same as 0.010010010..., the fraction 2/7
• 0.r100 is the same as 0.100100100..., the fraction 4/7
• 1.r001 is the same as 1.001001001..., 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.

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