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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 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 store values as a fixed number of binary 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. ==Applications== A common application is public-key cryptography, whose algorithms commonly employ arithmetic with integers having hundreds of digits.〔(Researchers: 307-digit key crack endangers 1024-bit RSA )〕〔http://www.rsa.com/rsalabs/node.asp?id=2218 recommends important RSA keys be 2048 bits (roughly 600 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 the optimum value for coefficients needed in formulae, for example the √⅓ 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〔 A quote example from this article: "Such an extreme pattern is dangerous even if diluted by one of its neighbouring blocks"; this was the occurrence of the sequence 77 twenty-eight times in one block of a thousand digits.〕 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 a 5-digit odometer's display which changes 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 programmer can guarantee that the integer values in a specific application will not grow large enough to cause an overflow. However, as time passes and conditions change, the bounds of the guarantee can be exceeded. For example, implementations of the binary search method that employ the form (L + R)/2 may function incorrectly when the ''sum'' of L and R exceeds the machine word size, although the individual variables themselves remain valid. Some programming languages such as Lisp, Python, Perl, Haskell and Ruby use, or have an option to use, arbitrary-precision numbers for ''all'' integer arithmetic. Although this reduces performance, it eliminates the possibility of incorrect results (or exceptions) due to simple overflow. It also makes it possible to 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Arbitrary-precision arithmetic」の詳細全文を読む スポンサード リンク
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