IEEE 754 Encoding

The figure below shows the encodings of IEEE 754 floating-point numbers. Other features of IEEE 754 are special symbols to represent unusual events:

S E F = f1 f2 f3 f4 ...
Zero
Zero is represented with exponent E=0 and fraction F=0.

Infinity (∞)
Infinity is a special value represented with maximum E and F=0 where
  • For single precision with 8-bit exponent: maximum E=255, and
  • For double precision with 11-bit exponent: maximum E=2047.
Infinity can result from overflow or division by zero. ±0 and ±∞ are possible according to the sign bit S.

NaN (Not a Number)
NaN is a special value represented with maximum E and F≠0. It is the result from exceptional situations, such as 0/0 or sqrt(negative). Operation on a NaN is NaN: Op(X,NaN)=NaN.
Single Precision Exponent (8 bits) Fraction (23 bits) Value
Normalized numbers 1 — 254 anything ±(1.F)2 × 2E-127
Denormalized numbers 0 non-zero ±(0.F)2 × 2-126
Zero 0 0 ±0
Infinity 255 0 ±∞
NaN (not a number) 255 non-zero NaN
 
Double Precision Exponent (11 bits) Fraction (52 bits) Value
Normalized numbers 1 — 2046 anything ±(1.F)2 × 2E-1023
Denormalized numbers 0 non-zero ±(0.F)2 × 2-1022
Zero 0 0 ±0
Infinity 2047 0 ±∞
NaN (not a number) 2047 non-zero NaN




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