28.3 Special Floating-Point Values

IEEE floating point provides for special values that are not ordinary numbers.

infinities

+Infinity and -Infinity are two different infinite values, one positive and one negative. These result from operations such as 1 / 0, Infinity + Infinity, Infinity * Infinity, and Infinity + finite, and also from a result that is finite, but larger than the most positive possible value or smaller than the most negative possible value.

See Handling Infinity, for more about working with infinities.

NaNs (not a number)

There are two special values, called Not-a-Number (NaN): a quiet NaN (QNaN), and a signaling NaN (SNaN).

A QNaN is produced by operations for which the value is undefined in real arithmetic, such as 0 / 0, sqrt (-1), Infinity - Infinity, and any basic operation in which an operand is a QNaN.

The signaling NaN is intended for initializing otherwise-unassigned storage, and the goal is that unlike a QNaN, an SNaN does cause an interrupt that can be caught by a software handler, diagnosed, and reported. In practice, little use has been made of signaling NaNs, because the most common CPUs in desktop and portable computers fail to implement the full IEEE 754 Standard, and supply only one kind of NaN, the quiet one. Also, programming-language standards have taken decades to catch up to the IEEE 754 standard, and implementations of those language standards make an additional delay before programmers become willing to use these features.

To enable support for signaling NaNs, use the GCC command-line option -fsignaling-nans, but this is an experimental feature and may not work as expected in every situation.

A NaN has a sign bit, but its value means nothing.

See Handling NaN, for more about working with NaNs.

subnormal numbers

It can happen that a computed floating-point value is too small to represent, such as when two tiny numbers are multiplied. The result is then said to underflow. The traditional behavior before the IEEE 754 Standard was to use zero as the result, and possibly to report the underflow in some sort of program output.

The IEEE 754 Standard is vague about whether rounding happens before detection of floating underflow and overflow, or after, and CPU designers may choose either.

However, the Standard does something unusual compared to earlier designs, and that is that when the result is smaller than the smallest normalized representable value (i.e., one in which the leading significand bit is 1), the normalization requirement is relaxed, leading zero bits are permitted, and precision is gradually lost until there are no more bits in the significand. That phenomenon is called gradual underflow, and it serves important numerical purposes, although it does reduce the precision of the final result. Some floating-point designs allow you to choose at compile time, or even at run time, whether underflows are gradual, or are flushed abruptly to zero. Numbers that have entered the region of gradual underflow are called subnormal.

You can use the library functions fesetround and fegetround to set and get the rounding mode. Rounding modes are defined (if supported by the platform) in fenv.h as: FE_UPWARD to round toward positive infinity; FE_DOWNWARD to round toward negative infinity; FE_TOWARDZERO to round toward zero; and FE_TONEAREST to round to the nearest representable value, the default mode. It is best to use FE_TONEAREST except when there is a special need for some other mode.