Arbitrary-precision Arithmetic - Implementation Issues

Implementation Issues

Arbitrary-precision arithmetic is considerably slower than arithmetic using numbers that fit entirely within processor registers, since the latter are usually implemented in hardware arithmetic whereas the former must be implemented in software. Even if the computer lacks hardware for certain operations (such as integer division, or all floating-point operations) and software is provided instead, it will use number sizes closely related to the available hardware registers: one or two words only and definitely not N words. There are exceptions, as certain variable word length machines of the 1950s and 1960s, notably the IBM 1620, IBM 1401 and the Honeywell Liberator series, could manipulate numbers bound only by available storage, with an extra bit that delimited the value.

Numbers can be stored in a fixed-point format, or in a floating-point format as a significand multiplied by an arbitrary exponent. However, since division almost immediately introduces infinitely repeating sequences of digits (such as 4/7 in decimal, or 1/10 in binary), should this possibility arise then either the representation would be truncated at some satisfactory size or else rational numbers would be used: a large integer for the numerator and for the denominator. But even with the greatest common divisor divided out, arithmetic with rational numbers can become unwieldy very swiftly: 1/99 - 1/100 = 1/9900, and if 1/101 is then added the result is 10001/999900.

Bounding the size of arbitrary-precision numbers is not only the total storage available, but the variables used by the software to index the digit strings. These are typically themselves limited in size.

Numerous algorithms have been developed to efficiently perform arithmetic operations on numbers stored with arbitrary precision. In particular, supposing that N digits are employed, algorithms have been designed to minimize the asymptotic complexity for large N.

The simplest algorithms are for addition and subtraction, where one simply adds or subtracts the digits in sequence, carrying as necessary, which yields an O(N) algorithm (see big O notation).

Comparison is also very simple. Compare the high order digits (or machine words) until a difference is found. Comparing the rest of the digits/words is not necessary. The worst case is O(N), but usually it will go much faster.

For multiplication, the most straightforward algorithms used for multiplying numbers by hand (as taught in primary school) require O(N2) operations, but multiplication algorithms that achieve O(N log(N) log(log(N))) complexity have been devised, such as the Schönhage–Strassen algorithm, based on fast Fourier transforms, and there are also algorithms with slightly worse complexity but with sometimes superior real-world performance for smaller N. The Karatsuba multiplication is such an algorithm.

For division, see: Division algorithm.

For a list of algorithms along with complexity estimates, see: Computational complexity of mathematical operations

For examples in x86-assembly, see: External links.