Infinity and NaNs

The IEEE standard specifies a variety of special exponent and mantissa values in order to support the concepts of plus and minus infinity and “Not-a-Number” (NaN).

Infinity (∞)

Arithmetic involving infinity is treated as the limiting case of real arithmetic, with infinite values defined as those outside the range of representable numbers, or

−∞ < representable numbers < +∞

With the exception of the special cases discussed below (NaNs), any arithmetic operation involving infinity yields infinity. Infinity is represented by the largest biased exponent allowed by the format and a mantissa of zero.

NaNs

A NaN (Not-a-Number) is a symbolic entity encoded in floating-point format. There are two types of NaNs:

• Signalling NaN

  signals an invalid operation exception

• Quiet NaN

  propagates through almost every arithmetic operation without signalling an exception

NaNs are produced by these operations:

∞ − ∞,   −∞ + ∞,   0 × ∞,   0 ÷ 0,   ∞ ÷ ∞

Both types of NaNs are represented by the largest biased exponent allowed by the format (single- or double-precision) and a mantissa that is non-zero.

• The bit pattern of the mantissa for a signalling NaN has the most significant digit set to zero and at least one of the remaining digits set to one.

• The bit pattern of the mantissa for a quiet NaN has the most significant digit set to one

For single-precision values:

• Positive infinity is represented by the bit pattern 7F800000
• Negative infinity is represented by the bit pattern FF800000
• A signalling NaN (NANS) is represented by any bit pattern
  between 7F800001 and 7FBFFFFF or between FF800001 and FFBFFFFF
• A quiet NaN (NANQ) is represented by any bit pattern
  between 7FC00000 and 7FFFFFFF or between FFC00000 and FFFFFFFF

For double-precision values:

• Positive infinity is represented by the bit pattern 7FF0000000000000
• Negative infinity is represented by the bit pattern FFF0000000000000
• A signalling NaN is represented by any bit pattern
  between 7FF0000000000001 and 7FF7FFFFFFFFFFFF or
  between FFF0000000000001 and FFF7FFFFFFFFFFFF
• A quiet NaN is represented by any bit pattern
  between 7FF8000000000000 and 7FFFFFFFFFFFFFFF or
  between FFF8000000000000 and FFFFFFFFFFFFFFFF

last updated: 27-Nov-03 Ian Harries <ih@doc.ic.ac.uk>