Floating Point Numbers

Real numbers are represented on computers by floating point numbers.  These contain finite numbers of sign, exponent and fraction bits:

Here is Python code that interprets IEEE 64 bit standard floating point numbers:

#!/usr/bin/env python3

FP_LEN   = 64
SIGN_LEN = 1
EXP_LEN  = 11
EXP_HALF = 2 ** EXP_LEN / 2
FRAC_LEN = FP_LEN - SIGN_LEN - EXP_LEN
BINARY   = 2

def floating_point(bit_string):
        """
        Interprets IEEE 64 bit standard floating point numbers.
        """

        sign     = int(bit_string[0],                BINARY)
        exponent = int(bit_string[1:1 + EXP_LEN],    BINARY) - EXP_HALF + 1
        fraction = int("1" + bit_string[-FRAC_LEN:], BINARY) / 2 ** FRAC_LEN
        if   exponent == -EXP_HALF + 1:
                print("SPECIAL CASE: subnormal")
                exponent += 1
                fraction -= (1 << FRAC_LEN) / 2 ** FRAC_LEN
        elif exponent == EXP_HALF:
                if fraction == (1 << FRAC_LEN) / 2 ** FRAC_LEN:
                        if sign:
                                print("SPECIAL CASE: -infinity")
                        else:
                                print("SPECIAL CASE: infinity")
                else:
                        print("SPECIAL CASE: undefined")
                exponent = fraction = 0

        return (-1) ** sign * (2 ** exponent) * fraction