IEEE 754 Floating Point

Scientific notation, in binary

In science we write 6 020 000 as 6.02 × 10⁶: a sign, a number with one digit before the point, and an exponent. IEEE 754 — the standard used by almost every computer — does the same in base 2:

value = (−1)^sign × 1.mantissa × 2^(exponent − 127)

A 32-bit float has three fields:

Converting −6.75 step by step

1. Sign: negative → 1.
2. Integer part: 6 = 110₂ (divide by 2: remainders 0, 1, 1, read upwards).
3. Fraction part: multiply by 2 repeatedly: 0.75 × 2 = 1.5 → 1, 0.5 × 2 = 1.0 → 1. So 0.75 = .11₂.
4. So 6.75 = 110.11₂. Normalise by moving the point 2 places left: 1.1011 × 2².
5. Exponent: 2 + 127 = 129 = 10000001₂.
6. Mantissa: the bits after the leading 1, padded to 23 bits: 10110000000000000000000.

Result: 1 10000001 10110000000000000000000 = 0xC0D80000.

The hidden bit

After normalising, the first bit is always 1 — so IEEE 754 doesn’t store it. That free “hidden bit” gives 24 bits of precision from 23 stored bits.

Why 0.1 can’t be stored exactly

0.1 × 2 = 0.2 → 0, 0.4 → 0, 0.8 → 0, 1.6 → 1, 1.2 → 1, then 0.4 again… In binary 0.1 = 0.000110011001100… forever. It must be rounded to 23 bits, so the stored float is 0.100000001490116… That’s why:

>>> 0.1 + 0.2
0.30000000000000004
>>> 0.1 + 0.2 == 0.3
False
>>> abs((0.1 + 0.2) - 0.3) < 1e-9     # compare floats with a tolerance
True

Never use floats for money — use integers (paise or cents) or a decimal type.

Special values

So normal exponents run from 1 to 254, meaning 2⁻¹²⁶ to 2¹²⁷.

Code: see the bits yourself

import struct

def float_bits(x):
    [u] = struct.unpack(">I", struct.pack(">f", x))   # the raw 32 bits
    b = f"{u:032b}"
    return b[0], b[1:9], b[9:], hex(u)

print(float_bits(-6.75))
# ('1', '10000001', '10110000000000000000000', '0xc0d80000')
print(float_bits(0.1))
# ('0', '01111011', '10011001100110011001101', '0x3dcccccd')

Common mistakes

• Forgetting the bias: the stored exponent is e + 127, not e.
• Including the leading 1 in the 23 mantissa bits.
• Moving the binary point the wrong way: if the point moves left, the exponent is positive.