Understanding Precision in IEEE 754 Floating-Point Representation

Floats are a common format for representing real numbers in computing, and the IEEE 754 standard is the de facto standard for floating-point arithmetic. This article explores the concept of the smallest distance between two consecutive representable floating-point numbers—also known as machine epsilon, and how it varies depending on the precision of the format.

Single Precision Floating-Point Representation (32-bit)

The IEEE 754 standard defines two primary types of floating-point numbers: single precision and double precision. Single precision uses 32 bits, with a layout as follows:

1 bit for the sign 8 bits for the exponent 23 bits for the significand (mantissa)

The smallest distance between two consecutive representable floating-point numbers in single precision is determined by the machine epsilon. For single precision, this is approximately (1.19209 times 10^{-7}), which can also be expressed as (2^{-23}). This value is based on the number of bits in the significand, which is 23 bits in single precision.

Double Precision Floating-Point Representation (64-bit)

Double precision floating-point numbers, as defined by IEEE 754, use 64 bits. The layout is as follows:

1 bit for the sign 11 bits for the exponent 52 bits for the significand (mantissa)

The smallest distance between two consecutive representable floating-point numbers in double precision is approximately (2.22045 times 10^{-16}), or (2^{-52}). This is due to the 52 bits allocated for the significand in double precision.

General Formula for Calculating Machine Epsilon

The distance between two consecutive floating-point numbers can be calculated using the following general formula:

[text{Distance} text{ulp}(x) 2^{text{exponent}(x) - p - 1}]

(text{ulp}(x)) is the unit in the last place at (x) (text{exponent}(x)) is the exponent of the floating-point number (p) is the number of bits in the significand

This formula takes into account the scale of the number being represented, ensuring that the distance between consecutive numbers varies depending on the magnitude of the number.

Normalized Binary Scientific Notation and Significand

A floating-point number in IEEE 754 format is represented in normalized binary scientific notation. This notation consists of two parts: the significand (which contains the significant digits of the number) and the power of two that places the “floating” radix point. For example, the number (1.00000110001001001101111 times 2^{-10}) is a normalized binary scientific notation with a significand of (1.00000110001001001101111) and an exponent of (-10).

Unit in the Last Place (ulp) in IEEE 754 Floating-Point

The unit in the last place (ulp) is the smallest difference between two consecutive representable floating-point numbers. For a given floating-point number (x), the ulp of (x) is defined as the distance to the next floating-point number. This concept is crucial for understanding the precision limits of floating-point arithmetic.

In a single precision IEEE 754 floating point, the distance between two consecutive representable numbers is determined by the 23 bits in the significand. The distance is therefore (2^{-23}), while in double precision, the 52 bits in the significand determine the distance as (2^{-52}).

Conclusion

The precision of floating-point numbers in IEEE 754 is a fundamental concept for understanding the behavior of numerical computations. By knowing the value of machine epsilon for different floating-point formats, developers can better predict and manage the accuracy of their calculations.