Floating-point arithmetic is governed by the IEEE 754 standard, which defines a mathematical representation for real numbers composed of three distinct fields: a sign bit, a biased exponent representing magnitude range via powers of two, and a normalized significand (or mantissa) constrained to an interval $[1.0, <2.0)$ through implicit leading bits. This theoretical framework addresses the domain limitations of fixed-point integer arithmetic by enabling the representation of extremely large or small values with fractional precision, while formally defining conditions for overflow (exceeding representable magnitude), underflow (non-zero subnormal values approaching zero), and rounding errors due to finite bit widths. The mechanism relies on normalized representation rules where operations such as addition require exponent alignment before significand arithmetic, followed by hardware-based normalization and specific rounding modes like "round to nearest even" to minimize systematic bias in error accumulation across computational domains like scientific computing and digital signal processing.
Floating-point arithmetic is governed by the IEEE 754 standard, which defines a mathematical representation for real numbers composed of three distinct fields: a sign bit, a biased exponent represent…