Convert float to/from string to illustrate/understand how a double precision floating-point number is encoded based on IEEE 754

floatify.py

#!/usr/bin/env python3
import struct

def to_float(bins):
    """
    Convert a binary string to float.
    """
    def _expand(s, length):
        ei = s.find('..')
        if ei == -1: return s
        d = length - len(s) + 2
        return s[:ei] + (d * s[ei - 1]) + s[(ei + 2):]
    sign, exp, mantissa = (i.replace('_', '') for i in bins.split())
    exp = _expand(exp, 11)
    mantissa = _expand(mantissa, 52)
    if len(sign) != 1 or len(exp) != 11 or len(mantissa) != 52:
        raise ValueError((sign, exp, mantissa))
    int64 = (int(sign, 2) << 63) | (int(exp, 2) << 52) | int(mantissa, 2)
    return struct.unpack('>d', struct.pack('>Q', int64))[0]

def to_bin(floatv, grouplen=8):
    """
    Convert a float to binary string grouped per `grouplen` digits.
    """
    def _group(s, w):
        rem = len(s) % -w
        return '_'.join(s[max(0, i):i + w] for i in range(rem, len(s), w))
    s = ''.join(format(i, '08b') for i in struct.pack('>d', floatv))
    return f"{s[0]} {_group(s[1:12], grouplen)} {_group(s[12:], grouplen)}"


def to_sn2(floatv):
    """
    Convert a float to string representing the float in scientific notation
    with base 2.
    """
    def _superscript(s):
        m = {'0': '⁰', '1': '¹', '2': '²', '3': '³', '4': '⁴', '5': '⁵',
            '6': '⁶', '7': '⁷', '8': '⁸', '9': '⁹', '-': '⁻'}
        return ''.join(m[i] for i in s)
    int64 = struct.unpack('>Q', struct.pack('>d', floatv))[0]
    sign = '+' if int64 >> 63 == 0 else '-'
    mantissa = (int64 & 4503599627370495) / (1 << 52)
    exp = (int64 >> 52) & 2047
    if exp == 2047: return str(floatv)
    elif exp == 0: exp = -1022
    else: exp -= 1023; mantissa += 1
    return f"{sign}{mantissa} × 2{_superscript(str(exp))}"