Floating-point arithmetic and stable formulas

In a radix-based format, a finite normal number stores a sign, a limited significand, and an exponent. Near larger values, the absolute gap between representable numbers is larger. Under round-to-nearest, a basic operation is often modeled as its exact real result multiplied by one plus a small relative perturbation. Unit roundoff bounds that perturbation for normal results under the usual assumptions. It is an analysis tool, not an assertion that errors are random or independent.

Overflow, underflow, subnormal results, division by zero, and nonfinite inputs need separate handling. Relative-error bounds can fail near zero even though an absolute-error bound still makes sense. In common binary64 round-to-nearest arithmetic, unit roundoff is 2⁻⁵³, while the gap from one to the next larger representable number is 2⁻⁵². These quantities are often both informally called epsilon, so state the convention. Decimal display length does not reveal how many trustworthy digits survived the calculation.

Subtracting nearby rounded quantities can leave an answer far smaller than either operand. The subtraction may itself be correctly rounded, even exact, while the earlier operand errors now dominate the answer. This distinction matters: cancellation is not proof that the final machine instruction malfunctioned. It is a warning that the chosen route to the answer carried uncertainties on a scale larger than the desired difference. Sometimes that sensitivity is inherent in measured inputs; sometimes algebra can avoid it.

Consider the difference between the square root of one plus a small positive x and one. Multiplying numerator and denominator by the conjugate gives an equal expression with no subtraction of nearby unit-sized numbers. The small quantity x is retained explicitly. This reformulation addresses algorithmic cancellation for the specified function; it does not improve genuinely uncertain input data. Related standard library functions such as log1p and expm1 exist because evaluating an elementary function first and subtracting afterward can discard useful information near zero.

Floating-point addition is not associative. Adding many small terms to a large running total may lose contributions that would survive if grouped first. Pairwise summation reduces the depth over which rounding errors accumulate, while compensated methods track information lost during additions. Neither approach promises exact results for arbitrary ranges and cancellation patterns. A sum with a tiny true value compared with the sum of absolute terms is itself relatively sensitive, so algorithm choice and conditioning must be assessed together.

Scaling can prevent overflow in intermediate expressions whose final answer is representable. For example, robust norm routines avoid blindly squaring every large component before summing. Reproducibility across parallel execution orders also requires a deliberate reduction strategy, since changing grouping can change rounded results. Validate against an analytically tractable case, a higher-precision reference, or a backward-error estimate. A result that is unchanged after printing fewer decimals has not passed an accuracy test; the comparison must target the underlying quantity and its allowed error.