What Are Subnormal Floating Point Numbers and Why Are They Important?

What Are Subnormal Floating Point Numbers?

When discussing floating point numbers, we often encounter values such as zero, infinity, NaN (Not a Number), normal numbers, and subnormal numbers. Subnormal numbers are a special category within floating point numbers.

IEEE 754 Basics

IEEE 754 is the standard for floating point numbers used in most computer systems. It specifies the organization of these numbers as follows:

• 1 bit: sign (0 for positive, 1 for negative)
• 8 bits: exponent (range from 0 to 255)
• 23 bits: fraction (mantissa)

The Leading Bit Convention

IEEE 754 employs the leading bit convention, which assumes that all numbers, except 0.0, have a leading 1 in binary representation. By default, the 1 is not explicitly written or stored, saving one precision bit.

Exception for Zero and Subnormal Numbers

However, there are exceptions to this rule:

• When the exponent is 0, and the fraction is 0, the number represents either positive or negative 0.0, maintaining that 0.0 is still a subnormal number.
• If the exponent is 0 and the fraction is non-zero, the number is a subnormal number.

Defining Subnormal Numbers

Based on these exceptions, subnormal numbers are defined as follows:

• Exponent: 0 (except for 0.0)
• Leading Bit: 0 (instead of the usual 1)

Significance of Subnormal Numbers

Subnormal numbers enable the representation of numbers that are very close to zero while using the same number of bits as normal numbers. They serve as fillers in the range where normal numbers would overflow.

Main Features of Subnormal Numbers

• They denormalize the leading bit convention by setting it to 0.
• They extend the exponent range, allowing representation of even smaller numbers.
• They double the length of the exponent 0 range, allowing more points to be represented accurately.
• They trade off precision for range, allowing for greater representation accuracy at the expense of reduced precision.

The above is the detailed content of What Are Subnormal Floating Point Numbers and Why Are They Important?. For more information, please follow other related articles on the PHP Chinese website!