IEEE 754 Floating-Point Standard

Updated 15 Sep 2026

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What it is
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  the standard binary representation for real numbers in computers: $x = (-1)^s \times (1+m) \times 2^{e-\text{bias}}$
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Format
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  single precision (float32) — 1 sign bit, 8 exponent bits (bias = 127), 23 mantissa bits
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  double precision (float64) — 1 sign bit, 11 exponent bits (bias = 1023), 52 mantissa bits
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Worked example: decimal -> IEEE-754 (encoding $-26.625$ , single precision)
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  sign: negative -> $s=1$
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  integer part $26_{10} = 11010_2$ ; fractional part $0.625_{10} = 0.101_2$
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  normalize: $11010.101_2 = 1.1010101_2 \times 2^4$
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  biased exponent: $4 + 127 = 131_{10} = 10000011_2$
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  mantissa (23 bits, padded): $10101010000000000000000$
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  result: $1\ 10000011\ 10101010000000000000000_2 = \texttt{0xC1D54000}$
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Worked example: IEEE-754 -> decimal (decoding 0x41440000)
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  binary: $0100\ 0001\ 0100\ 0100\ 0000\ 0000\ 0000\ 0000$
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  $s=0$ ; exponent bits $= 10000010_2 = 130_{10}$ , true exponent $= 130-127=3$
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  mantissa $= 1 + 2^{-2} + 2^{-6} = 1.265625$
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  result: $(+1) \times 1.265625 \times 2^3 = 10.125_{10}$
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Relatd
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  Machine Epsilon
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  Subnormal Numbers and Underflow

Linked references 3