multipleprecision

Multiple-precision arithmetic, also called arbitrary-precision arithmetic, is the area of numerical computation that handles numbers with precision beyond the native word size of a computer. It allows exact integers of unbounded size and floating-point numbers with as much precision as needed, at the cost of greater memory use and slower computations.

Implementations typically represent numbers as arrays of limbs in a fixed base (for example 2^32 or 2^64). A sign bit accompanies the integer part, and floating-point numbers store an exponent and a precision along with the significand. All arithmetic is performed on these limbs, with carry handling, normalization, and sometimes interval or error tracking.

Algorithms for basic operations include naive methods for small operands and advanced techniques for large ones. Multiplication may use Karatsuba, Toom-Cook, or FFT-based Schönhage-Strassen. Division, modular reduction, and square roots employ specialized algorithms. Floating-point arbitrary precision requires careful rounding or directed rounding modes to guarantee results within a chosen error bound.

Common software tools include GMP for general integer and rational arithmetic and MPFR for correctly rounded floating-point arithmetic. Many programming languages provide built-in or external multi-precision support (for example BigInteger or BigDecimal in Java, and Python’s long integers).

Applications span computational number theory, cryptography, exact scientific computation, and symbolic mathematics, where fixed precision cannot guarantee correctness or reproducibility. The field emphasizes accuracy guarantees, performance trade-offs, and scalable implementations on modern hardware.