IEEE 754 Floating-Point Standard

Updated 02 Oct 2026

•
What it is
- •
  the standard binary representation for real numbers in computers: $x = (-1)^s \times (1+m) \times 2^{e-\text{bias}}$
•
Format
- •
  single precision (float32) — 1 sign bit, 8 exponent bits (bias = 127), 23 mantissa bits
- •
  double precision (float64) — 1 sign bit, 11 exponent bits (bias = 1023), 52 mantissa bits
•
Worked example: decimal -> IEEE-754 (encoding $-26.625$ , single precision)
- •
  sign: negative -> $s=1$
- •
  integer part $26_{10} = 11010_2$ ; fractional part $0.625_{10} = 0.101_2$
- •
  normalize: $11010.101_2 = 1.1010101_2 \times 2^4$
- •
  biased exponent: $4 + 127 = 131_{10} = 10000011_2$
- •
  mantissa (23 bits, padded): $10101010000000000000000$
- •
  result: $1\ 10000011\ 10101010000000000000000_2 = \texttt{0xC1D50000}$
•
Worked example: IEEE-754 -> decimal (decoding 0x41440000)
- •
  binary: $0100\ 0001\ 0100\ 0100\ 0000\ 0000\ 0000\ 0000$
- •
  $s=0$ ; exponent bits $= 10000010_2 = 130_{10}$ , true exponent $= 130-127=3$
- •
  mantissa $= 1+2^{-1}+2^{-5}=1.53125$
- •
  result: $(+1) \times 1.265625 \times 2^3 = 12.25_{10}$
•
Relatd
- •
  Machine Epsilon
- •
  Subnormal Numbers and Underflow

What it is