## Floating-point Binary Fractions: Do math in base 2!

An implementation of a floating-point binary fractions class and module in Python. Work with binary fractions and binary floats with ease!

This module allows one to represent integers, floats and fractions as binary strings.

- e.g. the integer 3 will be represented as string '0b11'.
- e.g. the float -3.75 will be represented as string '-0b11.11'.
- e.g. the fraction 1/2 will be represented as string '0b0.1'
- Exponential representation is also possible: '-0b0.01111e3', '-0b11.1e1' or '-0b1110e-2' all represent float -3.75.
- two's complement representation is possible too: '11.11' for -1.25 in decimal, or '-0b1.01' in binary fraction.

Many operations and transformations are offered. You can sum, subtract, multiply, and divide long floating-point binary fractions. You can compute power of them, shift them left, shift them right, etc.

Basic representation of binary fractions and binary floats: A binary fraction is a subset of binary floats. Basically, a binary fraction is a binary float without an exponent (e.g. '-0b101.0101'). Let's have a look at an example binary float value to see how it is represented.

```
prefix '0b' to indicate "binary" or "base 2"
||
|| decimal point
|| |
|| | exponent separator
|| | |
|| | | exponent in base 10 (not in base 2!)
|| | | ||
-0b101.0101e-34 <-- example floating-point binary fraction
| ||| |||| |
sign ||| |||| exponent sign
||| ||||
||| fraction bits in base 2
|||
integer bits in base 2
```

If you are curious about floating point binary fractions, have a look at:

- https://en.wikipedia.org/wiki/Computer_number_format#Representing_fractions_in_binary
- https://www.electronics-tutorials.ws/binary/binary-fractions.html
- https://ryanstutorials.net/binary-tutorial/binary-floating-point.php
- https://planetcalc.com/862/

If you are curious about Two's complement:

- https://janmr.com/blog/2010/07/bitwise-operators-and-negative-numbers/
- https://en.wikipedia.org/wiki/Two%27s_complement

## License:

- GPL v3 or later

## Features:

- Python 3
- constructors for various types: int, float, Fraction, str, TwosComplement, Binary
- supports many operators: +, -, *, /, //, %, **, <<, >>, ~, &, ...
- supports many methods: not, abs, round, floor, ceil, ...
- internally the value is kept as a Fraction and most operations are done in Fractions. This results in better performance and infinite precision. Only a few limited operations such as 'and', 'or', 'xor', and 'invert' are done on strings.
- very high precision
- many operations are lossless, i.e. with no rounding errors or loss of precision
- supports very long binary fractions
- supports exponential representations
- well documented
- well tested
- over 1600 test cases

## Sample usage, Example calls:

Please have a look at the short example program that uses the `Binary`

class and module. See file binary_sample.py.

The sample source code looks like this:

```
#!/usr/bin/python3
# Sample program using the Binary class and module.
# Install with: pip3 install --upgrade binary-fractions
if __name__ == "__main__":
from binary_fractions import TwosComplement, Binary
from math import ceil, floor
bf1str: str = "-1.01" # -1.25
bf2str: str = "10.1" # 2.5
bf3str: str = "10.1e-3" # 2.5/8
tcstr1: str = "10.1" # -1.5 in two's complement, '-0b1.1' as binary fraction
tcstr2: str = "100001001000.1" # -1975.5 in two's complement, '-0b11110111000.1'
fl1: float = 2.3
fl2: float = -1975.5
bf1: Binary = Binary(bf1str)
bf2: Binary = Binary(bf2str)
bf3: Binary = Binary(bf3str)
tc1: TwosComplement = TwosComplement(tcstr1)
tc2: TwosComplement = TwosComplement(tcstr2)
tc3: TwosComplement = TwosComplement(fl2)
print("Sample program demonstrating binary fractions class and module:")
print(f"Binary({fl1}) = {Binary(fl1)}")
print(f"Binary({fl2}) = {Binary(fl2)}")
print(f"Binary({bf3str}) = {Binary(bf3str)}")
print(f"{bf1} = {bf1}")
print(f"{bf1} + {bf2} = {bf1+bf2}")
print(f"{bf1} - {bf2} = {bf1-bf2}")
print(f"{bf1} * {bf2} = {bf1*bf2}")
print(f"{bf1} / {bf2} = {bf1/bf2}")
print(f"{bf1} // {bf2} = {bf1//bf2}")
print(f"{bf1} % {bf2} = {bf1%bf2}")
print(f"{bf2} ** {bf1} = {bf2**bf1}")
print(f"{bf1} >> {1} = {bf1>>1}")
print(f"{bf1} << {1} = {bf1<<1}")
print(f"abs({bf1}) = {abs(bf1)}")
print(f"round({bf1}) = {round(bf1)}")
print(f"ceil({bf1}) = {ceil(bf1)} (int)")
print(f"Binary('{bf1}').ceil() = {bf1.ceil()} (Binary)")
print(f"floor({bf1}) = {floor(bf1)} (int)")
print(f"Binary('{bf1}').floor() = {bf1.floor()} (Binary)")
print(f"int({bf1}) = {int(bf1)}")
print(f"float({bf1}) = {float(bf1)}")
print(f"str({bf1}) = {str(bf1)}")
print(f"str({bf3}) = {str(bf3)}")
print(f"Fraction({bf1}) = {bf1.fraction}")
print(f"Binary({bf1}).fraction = {bf1.fraction}")
print(f"Binary({fl2}).string = {Binary(fl2).string}")
print(f"{bf1} & {bf2} = {bf1&bf2}")
print(f"{bf1} | {bf2} = {bf1|bf2}")
print(f"{bf1} ^ {bf2} = {bf1^bf2}")
print(f"~(floor({bf2})) = {~(floor(bf2))}")
print(f"type({bf1}) = {type(bf1)}")
print(f"type({tc1}) = {type(tc1)}")
print(f"Binary('{bf3}').to_no_exponent() = {bf3.to_no_exponent()}")
print(f"Binary('{bf3}').to_no_mantissa() = {bf3.to_no_mantissa()}")
# scientific notation
print(f"Binary('{bf3}').to_sci_exponent() = {bf3.to_sci_exponent()}")
# engineering notation
print(f"Binary('{bf3}').to_eng_exponent() = {bf3.to_eng_exponent()}")
print(f"Binary('{bf1}').to_twos_complement() = {bf1.to_twoscomplement()}")
print(f"Binary(TwosComplement('{tcstr1}')) = {Binary.from_twoscomplement(tc1)}")
print(f"Binary(TwosComplement('{tcstr2}')) = {Binary.from_twoscomplement(tc2)}")
print(f"Binary(TwosComplement({fl2})) = {Binary.from_twoscomplement(tc3)}")
print(f"TwosComplement({fl2}) = {TwosComplement(fl2)}")
print("And there are more operands, more methods, more functions, ...")
print("For more information read the documentation at:")
print("https://raw.githubusercontent.com/Jonny-exe/binary-fractions")
```

When executed with the command `python3 binary_sample.py`

, it returns these results:

```
Sample program demonstrating binary fractions class and module:
Binary(2.3) = 0b10.01001100110011001100110011001100110011001100110011
Binary(-1975.5) = -0b11110110111.1
Binary(10.1e-3) = 0b10.1e-3
-0b1.01 = -0b1.01
-0b1.01 + 0b10.1 = 0b1.01
-0b1.01 - 0b10.1 = -0b11.11
-0b1.01 * 0b10.1 = -0b11.001
-0b1.01 / 0b10.1 = -0b0.1
-0b1.01 // 0b10.1 = -0b1
-0b1.01 % 0b10.1 = 0b1.01
0b10.1 ** -0b1.01 = 0b0.010100010110111110001011100001001001101110110100110011
-0b1.01 >> 1 = -0b0.101
-0b1.01 << 1 = -0b10.1
abs(-0b1.01) = 0b1.01
round(-0b1.01) = -0b1
ceil(-0b1.01) = -1 (int)
Binary('-0b1.01').ceil() = -0b1 (Binary)
floor(-0b1.01) = -2 (int)
Binary('-0b1.01').floor() = -0b10 (Binary)
int(-0b1.01) = -1
float(-0b1.01) = -1.25
str(-0b1.01) = -0b1.01
str(0b10.1e-3) = 0b10.1e-3
Fraction(-0b1.01) = -5/4
-0b1.01 & 0b10.1 = 0b10.1
-0b1.01 | 0b10.1 = -0b1.01
-0b1.01 ^ 0b10.1 = -0b11.11
~(floor(0b10.1)) = -3
type(-0b1.01) = <class 'binary.Binary'>
type(10.1) = <class 'binary.TwosComplement'>
Binary('0b10.1e-3').to_no_exponent() = 0b0.0101
Binary('0b10.1e-3').to_no_mantissa() = 0b101e-4
Binary('0b10.1e-3').to_sci_exponent() = 0b1.01e-2
Binary('0b10.1e-3').to_eng_exponent() = 0b101000000e-10
Binary('-0b1.01').to_twos_complement() = 10.11
Binary(TwosComplement('10.1')) = -1.1
Binary(TwosComplement('100001001000.1')) = -11110110111.1
Binary(TwosComplement(-1975.5)) = -11110110111.1
TwosComplement(-1975.5) = 100001001000.1
```

## Requirements:

- Python 3
- requires no
`pip`

packages (uses built-in`math`

and`fractions`

modules for math operations, uses`unittest`

for unit testing)

## Installation:

- see https://pypi.org/project/binary-fractions/
`pip install binary-fractions`

## Testing, Maturity

- run
`python3 binary_sample.py`

to execute a simple sample program - run
`python3 binary_test.py`

to execute all unit tests `Binary`

is relatively mature, more than 1600 test cases have been written and all passed.