What specific IEEE standard defines how floating-point numbers are stored and operated upon in most modern computers?

The specific IEEE standard that defines how floating-point numbers are stored and operated upon in most modern computers is IEEE 754. This standard, officially known as 'IEEE Standard for Floating-Point Arithmetic', provides a uniform way to represent and compute with floating-point numbers, which are approximations of real numbers, ensuring consistent behavior across different computer architectures. It specifies several formats for representing these numbers, the most common being single-precision (32-bit) and double-precision (64-bit). In these formats, a floating-point number is composed of three primary parts: a sign bit, which indicates whether the number is positive or negative; an exponent, which determines the magnitude or scale of the number (like where the decimal point is located); and a significand (also called the mantissa), which represents the precision or the significant digits of the number. Beyond these numerical representations, IEEE 754 also defines specific bit patterns for special values, including positive and negative infinity (representing results that exceed the maximum representable value, such as division by zero), and Not a Number (NaN), which is used for undefined or unrepresentable results (like 0/0 or the square root of a negative number). The standard also addresses denormalized (or subnormal) numbers, which are very small numbers close to zero that allow for a gradual underflow, preventing immediate flush-to-zero. Crucially, IEEE 754 precisely defines the behavior of arithmetic operations such as addition, subtraction, multiplication, division, and square root, along with rules for numerical comparisons and various rounding modes. These rounding modes specify how inexact results are adjusted to fit within the available precision, ensuring that the same calculation yields bit-for-bit identical results on any system adhering to the standard, which is vital for the portability and reliability of numerical software.