Multiprecision

Multiprecision refers to arithmetic performed with numbers whose precision is not fixed by the hardware word size but is instead limited only by available memory. It enables exact integers of arbitrary length and floating-point numbers with user-specified precision. This contrasts with fixed-precision types (such as 32- or 64-bit integers or floats), where overflow or rounding errors are common when dealing with large or highly precise quantities.

Representations include big integers, rationals, and floating-point numbers with controlled precision. They are typically stored as arrays of digits or limbs in a base such as 2^32 or 10^9, with a sign. Operations cover addition, subtraction, multiplication, division, and gcd, as well as exact arithmetic and, for floating-point, rounding to a specified precision. Some systems offer interval or ball arithmetic to bound errors.

Prominent libraries and implementations include the GNU Multiple Precision Arithmetic Library (GMP) for integers and rationals, the MPFR library for correctly rounded floating-point numbers, and MPIR. Languages often provide native or binding support (Python's int, Java BigInteger; C++ Boost.Multiprecision). Large-number algorithms include schoolbook, Karatsuba, Toom-Cook, and Schönhage–Strassen multiplication; modular arithmetic often employs Montgomery reduction for efficiency.

Applications span cryptography (RSA, ECC), computational number theory, computer algebra systems, numerical analysis with guaranteed precision, and financial calculations that require exactness or controlled error bounds.

Limitations include higher memory use and slower performance compared with fixed-size arithmetic, as well as the complexity of maintaining correct rounding and error estimates. Ongoing work explores faster algorithms, better representations, and hardware acceleration.