Arbitrary-precision arithmetic

In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are potentially limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most arithmetic logic unit (ALU) hardware, which typically offers between 8 and 64 bits of precision.

Several modern programming languages have built-in support for bignums, and others have libraries available for arbitrary-precision integer and floating-point math. Rather than storing values as a fixed number of bits related to the size of the processor register, these implementations typically use variable-length arrays of digits.

Arbitrary precision is used in applications where the speed of arithmetic is not a limiting factor, or where precise results with very large numbers are required. It should not be confused with the symbolic computation provided by many computer algebra systems, which represent numbers by expressions such as π·sin(2), and can thus represent any computable number with infinite precision.

A common application is public-key cryptography, whose algorithms commonly employ arithmetic with integers having hundreds of digits. Another is in situations where artificial limits and overflows would be inappropriate. It is also useful for checking the results of fixed-precision calculations, and for determining optimal or near-optimal values for coefficients needed in formulae, for example the ${\textstyle {\sqrt {\frac {1}{3}}}}$ that appears in Gaussian integration.

Arbitrary precision arithmetic is also used to compute fundamental mathematical constants such as π to millions or more digits and to analyze the properties of the digit strings or more generally to investigate the precise behaviour of functions such as the Riemann zeta function where certain questions are difficult to explore via analytical methods. Another example is in rendering fractal images with an extremely high magnification, such as those found in the Mandelbrot set.

Arbitrary-precision arithmetic can also be used to avoid overflow, which is an inherent limitation of fixed-precision arithmetic. Similar to an automobile's odometer display which may change from 99999 to 00000, a fixed-precision integer may exhibit wraparound if numbers grow too large to represent at the fixed level of precision. Some processors can instead deal with overflow by saturation, which means that if a result would be unrepresentable, it is replaced with the nearest representable value. (With 16-bit unsigned saturation, adding any positive amount to 65535 would yield 65535.) Some processors can generate an exception if an arithmetic result exceeds the available precision. Where necessary, the exception can be caught and recovered from—for instance, the operation could be restarted in software using arbitrary-precision arithmetic.

In many cases, the task or the programmer can guarantee that the integer values in a specific application will not grow large enough to cause an overflow. Such guarantees may be based on pragmatic limits: a school attendance program may have a task limit of 4,000 students. A programmer may design the computation so that intermediate results stay within specified precision boundaries.

Some programming languages such as Lisp, Python, Perl, Haskell, Ruby and Raku use, or have an option to use, arbitrary-precision numbers for all integer arithmetic. This enables integers to grow to any size limited only by the available memory of the system. Although this reduces performance, it eliminates the concern of incorrect results (or exceptions) due to simple overflow. It also makes it possible to almost guarantee that arithmetic results will be the same on all machines, regardless of any particular machine's word size. The exclusive use of arbitrary-precision numbers in a programming language also simplifies the language, because a number is a number and there is no need for multiple types to represent different levels of precision.