finiteprecision

Finite precision refers to the limitation that digital computers use to represent numbers and perform arithmetic, using a fixed number of bits. Because most numbers cannot be represented exactly, computations incur rounding errors. This contrasts with infinite-precision arithmetic, where numbers can be represented and manipulated exactly.

The two main models are fixed-point and floating-point. Fixed-point uses a fixed scaling factor; floating-point represents numbers as a significand times a base raised to an exponent, enabling a wide dynamic range. The most widespread standard is IEEE 754, with formats such as 32-bit (single) and 64-bit (double). Key concepts include the machine epsilon, which is the maximum relative error due to rounding, and ulp, the distance to the next representable value. Rounding modes determine how results are mapped to representable values.

Arithmetic operations produce results that are rounded back to finite precision, and the errors can accumulate. Issues include cancellation when subtracting nearly equal numbers, and loss of significance. Numerical stability concerns how algorithms amplify input errors; backward error and condition number describe sensitivity of the problem to perturbations.

Finite precision imposes limits such as overflow, underflow, and denormal numbers; some values are not representable and must be approximated. Denormals allow gradual underflow but can be slow on some hardware. Designers seek algorithms that are numerically stable and use techniques like compensated summation or reformulation to reduce error.

In practice, finite precision shapes software design, testing, and verification. For high-precision needs, arbitrary-precision libraries or mixed-precision strategies are used. Understanding finite precision is essential in fields like numerical linear algebra, simulations, and signal processing.