float_visualizer_textual.py

import struct
import math
import decimal
from fractions import Fraction

def float_to_bits(f):
    bits = [(b>>i)%2 for b in struct.pack(">d", f) for i in range(7, -1, -1)]
    return "".join(str(b) for b in bits)

def bits_to_float(bits):
    result = 0
    for b in bits:
        result <<= 1
        result += int(b)
    return struct.unpack("d", struct.pack("Q", result))[0]

EXPONENT_OFFSET = 1023
BASE = 2
PRECISION = 53

regions = {
    "sign": slice(0, 1),
    "exponent": slice(1, 12),
    "mantissa": slice(12, 64)
}

#normals
test_cases = [math.pi, 1.0, 0.25, 123.456, -777.0]
#denormals
test_cases += [0.0, -0.0, 1.1125369292536007e-308]
#specials
test_cases += [float("nan"), float("inf"), float("-inf")]

for idx, f in enumerate(test_cases,1):
    print(f"Example #{idx}: {f}")
    bits = float_to_bits(f)
    print(f"  Stored in memory as {bits}")
    for name, s in regions.items():
        label = f"  {name} bit{'s' if s.stop-s.start>1 else ''}:"
        padding = " " * (20 + s.start - len(label))
        print(label, padding, bits[s])
    sign = 1 if bits[0] == "0" else -1

    raw_exponent = int(bits[regions["exponent"]], 2)
    if raw_exponent == 0b11111111111:
        print("  an exponent of 0b11111111111 signals that the number is special -- either NAN or an infinity")
        if bits[regions["mantissa"]][0] == "1":
            print("  If the mantissa's leftmost bit is 1, the value is NaN.")
        else:
            print("  If the mantissa's leftmost bit is 0, the value is an infinity.")
            print(f"  Taking the sign bit into account, the value is {'positive' if sign == 1 else 'negative'} infinity.")
        print("\n\n")
        continue

    normal = raw_exponent != 0
    if normal:
        print(f"  mantissa with implied `1` bit: 1{bits[regions['mantissa']]}")
    else:
        print(f"  mantissa with implied `0` bit: 0{bits[regions['mantissa']]}")

    print(f"  raw exponent: {raw_exponent}")

    if normal:
        exponent_bias = EXPONENT_OFFSET
    if not normal:
        exponent_bias = EXPONENT_OFFSET - 1
        print(f"  a raw exponent of 00000000000 signals that the float is DENORMAL.")
        print(f"  exponent bias for denormal numbers is {exponent_bias} instead of the usual {EXPONENT_OFFSET}")
    exponent = raw_exponent - exponent_bias
    print(f"  actual exponent (subtracting exponent bias {exponent_bias} from raw): {exponent}")
    # mantissa = sum([Fraction(1, 2**i) for i, bit in enumerate(raw_mantissa,1) if bit=="1"], Fraction(1))
    # print(f"  raw mantissa: 0b1.{raw_mantissa} \n  calculated mantissa: {mantissa}")
    # value = mantissa * Fraction(BASE)**adjusted_exponent
    # print(f"  final value: {value} ~= {float(value)}")
    raw_mantissa = int(bits[regions["mantissa"]], 2)
    if normal:
        mantissa = raw_mantissa + 2**(PRECISION-1)
    else:
        mantissa = raw_mantissa
    print(f"  mantissa with implied bit: {mantissa}")
    print(f"  final value formula: sign * (mantissa / b^(p-1)) * b^exponent")
    print(f"  b (aka base) = {BASE}")
    print(f"  p (aka precision) = {PRECISION}")
    print(f"  final value  = {sign} * ({mantissa} / {BASE}^{PRECISION-1}) * {BASE}^{exponent}")
    print(f"  final value  = {sign} * ({mantissa} / {BASE**(PRECISION-1)}) * {Fraction(BASE)**exponent}")
    result = sign * Fraction(mantissa, BASE**(PRECISION-1)) * Fraction(BASE)**exponent
    print(f"  final value  = {result}")
    if str(result) != str(decimal.Decimal(f)):
        print(f"  final value  = {decimal.Decimal(f)}")
    if str(float(result)) != str(decimal.Decimal(f)):
        print(f"  final value ~= {float(result)}")
    print("\n\n")

output.txt

Example #1: 3.141592653589793
  Stored in memory as 0100000000001001001000011111101101010100010001000010110100011000
  sign bit:           0
  exponent bits:       10000000000
  mantissa bits:                  1001001000011111101101010100010001000010110100011000
  mantissa with implied `1` bit: 11001001000011111101101010100010001000010110100011000
  raw exponent: 1024
  actual exponent (subtracting exponent bias 1023 from raw): 1
  mantissa with implied bit: 7074237752028440
  final value formula: sign * (mantissa / b^(p-1)) * b^exponent
  b (aka base) = 2
  p (aka precision) = 53
  final value  = 1 * (7074237752028440 / 2^52) * 2^1
  final value  = 1 * (7074237752028440 / 4503599627370496) * 2
  final value  = 884279719003555/281474976710656
  final value  = 3.141592653589793115997963468544185161590576171875
  final value ~= 3.141592653589793



Example #2: 1.0
  Stored in memory as 0011111111110000000000000000000000000000000000000000000000000000
  sign bit:           0
  exponent bits:       01111111111
  mantissa bits:                  0000000000000000000000000000000000000000000000000000
  mantissa with implied `1` bit: 10000000000000000000000000000000000000000000000000000
  raw exponent: 1023
  actual exponent (subtracting exponent bias 1023 from raw): 0
  mantissa with implied bit: 4503599627370496
  final value formula: sign * (mantissa / b^(p-1)) * b^exponent
  b (aka base) = 2
  p (aka precision) = 53
  final value  = 1 * (4503599627370496 / 2^52) * 2^0
  final value  = 1 * (4503599627370496 / 4503599627370496) * 1
  final value  = 1
  final value ~= 1.0



Example #3: 0.25
  Stored in memory as 0011111111010000000000000000000000000000000000000000000000000000
  sign bit:           0
  exponent bits:       01111111101
  mantissa bits:                  0000000000000000000000000000000000000000000000000000
  mantissa with implied `1` bit: 10000000000000000000000000000000000000000000000000000
  raw exponent: 1021
  actual exponent (subtracting exponent bias 1023 from raw): -2
  mantissa with implied bit: 4503599627370496
  final value formula: sign * (mantissa / b^(p-1)) * b^exponent
  b (aka base) = 2
  p (aka precision) = 53
  final value  = 1 * (4503599627370496 / 2^52) * 2^-2
  final value  = 1 * (4503599627370496 / 4503599627370496) * 1/4
  final value  = 1/4
  final value  = 0.25



Example #4: 123.456
  Stored in memory as 0100000001011110110111010010111100011010100111111011111001110111
  sign bit:           0
  exponent bits:       10000000101
  mantissa bits:                  1110110111010010111100011010100111111011111001110111
  mantissa with implied `1` bit: 11110110111010010111100011010100111111011111001110111
  raw exponent: 1029
  actual exponent (subtracting exponent bias 1023 from raw): 6
  mantissa with implied bit: 8687443681197687
  final value formula: sign * (mantissa / b^(p-1)) * b^exponent
  b (aka base) = 2
  p (aka precision) = 53
  final value  = 1 * (8687443681197687 / 2^52) * 2^6
  final value  = 1 * (8687443681197687 / 4503599627370496) * 64
  final value  = 8687443681197687/70368744177664
  final value  = 123.4560000000000030695446184836328029632568359375
  final value ~= 123.456



Example #5: -777.0
  Stored in memory as 1100000010001000010010000000000000000000000000000000000000000000
  sign bit:           1
  exponent bits:       10000001000
  mantissa bits:                  1000010010000000000000000000000000000000000000000000
  mantissa with implied `1` bit: 11000010010000000000000000000000000000000000000000000
  raw exponent: 1032
  actual exponent (subtracting exponent bias 1023 from raw): 9
  mantissa with implied bit: 6834564278255616
  final value formula: sign * (mantissa / b^(p-1)) * b^exponent
  b (aka base) = 2
  p (aka precision) = 53
  final value  = -1 * (6834564278255616 / 2^52) * 2^9
  final value  = -1 * (6834564278255616 / 4503599627370496) * 512
  final value  = -777
  final value ~= -777.0



Example #6: 0.0
  Stored in memory as 0000000000000000000000000000000000000000000000000000000000000000
  sign bit:           0
  exponent bits:       00000000000
  mantissa bits:                  0000000000000000000000000000000000000000000000000000
  mantissa with implied `0` bit: 00000000000000000000000000000000000000000000000000000
  raw exponent: 0
  a raw exponent of 00000000000 signals that the float is DENORMAL.
  exponent bias for denormal numbers is 1022 instead of the usual 1023
  actual exponent (subtracting exponent bias 1022 from raw): -1022
  mantissa with implied bit: 0
  final value formula: sign * (mantissa / b^(p-1)) * b^exponent
  b (aka base) = 2
  p (aka precision) = 53
  final value  = 1 * (0 / 2^52) * 2^-1022
  final value  = 1 * (0 / 4503599627370496) * 1/44942328371557897693232629769725618340449424473557664318357520289433168951375240783177119330601884005280028469967848339414697442203604155623211857659868531094441973356216371319075554900311523529863270738021251442209537670585615720368478277635206809290837627671146574559986811484619929076208839082406056034304
  final value  = 0
  final value ~= 0.0



Example #7: -0.0
  Stored in memory as 1000000000000000000000000000000000000000000000000000000000000000
  sign bit:           1
  exponent bits:       00000000000
  mantissa bits:                  0000000000000000000000000000000000000000000000000000
  mantissa with implied `0` bit: 00000000000000000000000000000000000000000000000000000
  raw exponent: 0
  a raw exponent of 00000000000 signals that the float is DENORMAL.
  exponent bias for denormal numbers is 1022 instead of the usual 1023
  actual exponent (subtracting exponent bias 1022 from raw): -1022
  mantissa with implied bit: 0
  final value formula: sign * (mantissa / b^(p-1)) * b^exponent
  b (aka base) = 2
  p (aka precision) = 53
  final value  = -1 * (0 / 2^52) * 2^-1022
  final value  = -1 * (0 / 4503599627370496) * 1/44942328371557897693232629769725618340449424473557664318357520289433168951375240783177119330601884005280028469967848339414697442203604155623211857659868531094441973356216371319075554900311523529863270738021251442209537670585615720368478277635206809290837627671146574559986811484619929076208839082406056034304
  final value  = 0
  final value  = -0
  final value ~= 0.0



Example #8: 1.1125369292536007e-308
  Stored in memory as 0000000000001000000000000000000000000000000000000000000000000000
  sign bit:           0
  exponent bits:       00000000000
  mantissa bits:                  1000000000000000000000000000000000000000000000000000
  mantissa with implied `0` bit: 01000000000000000000000000000000000000000000000000000
  raw exponent: 0
  a raw exponent of 00000000000 signals that the float is DENORMAL.
  exponent bias for denormal numbers is 1022 instead of the usual 1023
  actual exponent (subtracting exponent bias 1022 from raw): -1022
  mantissa with implied bit: 2251799813685248
  final value formula: sign * (mantissa / b^(p-1)) * b^exponent
  b (aka base) = 2
  p (aka precision) = 53
  final value  = 1 * (2251799813685248 / 2^52) * 2^-1022
  final value  = 1 * (2251799813685248 / 4503599627370496) * 1/44942328371557897693232629769725618340449424473557664318357520289433168951375240783177119330601884005280028469967848339414697442203604155623211857659868531094441973356216371319075554900311523529863270738021251442209537670585615720368478277635206809290837627671146574559986811484619929076208839082406056034304
  final value  = 1/89884656743115795386465259539451236680898848947115328636715040578866337902750481566354238661203768010560056939935696678829394884407208311246423715319737062188883946712432742638151109800623047059726541476042502884419075341171231440736956555270413618581675255342293149119973622969239858152417678164812112068608
  final value  = 1.1125369292536006915451163586662020321096079902311659152766637084436022174069590979271415795062555102820336698655179055025762170807767300544280061926888594105653889967660011652398050737212918180359607825234712518671041876254033253083290794743602455899842958198242503179543850591524373998904438768749747257902258025254576999282912354093225567689679024960579905428830259962166760571761950743978498047956444458014963207555317331566968317387932565146858810236628158907428321754360614143188210224234057038069557385314008449266220550120807237108092835830752700771425423583764509515806613894483648536865616670434944915875339194234630463869889864293298274705456845477030682337843511993391576453404923086054623126983642578125E-308
  final value ~= 1.1125369292536007e-308



Example #9: nan
  Stored in memory as 0111111111111000000000000000000000000000000000000000000000000000
  sign bit:           0
  exponent bits:       11111111111
  mantissa bits:                  1000000000000000000000000000000000000000000000000000
  an exponent of 0b11111111111 signals that the number is special -- either NAN or an infinity
  If the mantissa's leftmost bit is 1, the value is NaN.



Example #10: inf
  Stored in memory as 0111111111110000000000000000000000000000000000000000000000000000
  sign bit:           0
  exponent bits:       11111111111
  mantissa bits:                  0000000000000000000000000000000000000000000000000000
  an exponent of 0b11111111111 signals that the number is special -- either NAN or an infinity
  If the mantissa's leftmost bit is 0, the value is an infinity.
  Taking the sign bit into account, the value is positive infinity.



Example #11: -inf
  Stored in memory as 1111111111110000000000000000000000000000000000000000000000000000
  sign bit:           1
  exponent bits:       11111111111
  mantissa bits:                  0000000000000000000000000000000000000000000000000000
  an exponent of 0b11111111111 signals that the number is special -- either NAN or an infinity
  If the mantissa's leftmost bit is 0, the value is an infinity.
  Taking the sign bit into account, the value is negative infinity.