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Arbitraryprecision arithmetic
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In computer science, arbitraryprecision arithmetic, also called bignum arithmetic, multiple precision arithmetic, or sometimes infiniteprecision 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 fixedprecision arithmetic found in most arithmetic logic unit (ALU) hardware, which typically offers between 8 and 64 bits of precision.
Several modern programming languages have builtin support for bignums, and others have libraries available for arbitraryprecision integer and floatingpoint math. Rather than store values as a fixed number of binary bits related to the size of the processor register, these implementations typically use variablelength 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.
Contents
Applications
A common application is publickey cryptography, whose algorithms commonly employ arithmetic with integers having hundreds of digits.^{[1]}^{[2]} Another is in situations where artificial limits and overflows would be inappropriate. It is also useful for checking the results of fixedprecision 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^{[3]} 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.
Arbitraryprecision arithmetic can also be used to avoid overflow, which is an inherent limitation of fixedprecision arithmetic. Similar to a 5digit odometer's display which changes from 99999 to 00000, a fixedprecision 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 16bit 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 arbitraryprecision 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, arbitraryprecision 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 arbitraryprecision 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.
Implementation issues
Arbitraryprecision 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 floatingpoint 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 fixedpoint format, or in a floatingpoint 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 arbitraryprecision 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(N^{2}) 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 realworld 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 x86assembly, see: External links.
Preset precision
In some languages such as REXX the precision of all calculations must be set before doing a calculation. Other languages, such as Python and Ruby extend the precision automatically to prevent overflow.
Example
The calculation of factorials can easily produce very large numbers. This is not a problem for their usage in many formulae (such as Taylor series) because they appear along with other terms, so that—given careful attention to the order of evaluation—intermediate calculation values are not troublesome. If approximate values of factorial numbers are desired, Stirling's approximation gives good results using floatingpoint arithmetic. The largest representable value for a fixedsize integer variable may be exceeded even for relatively small arguments as shown in the table below. Even floatingpoint numbers are soon outranged, so it may help to recast the calculations in terms of the logarithm of the number.
But if exact values for large factorials are desired, then special software is required, as in the pseudocode that follows, which implements the classic algorithm to calculate 1, 1×2, 1×2×3, 1×2×3×4, etc. the successive factorial numbers.
Constant Limit = 1000; %Sufficient digits. Constant Base = 10; %The base of the simulated arithmetic. Constant FactorialLimit = 365; %Target number to solve, 365! Array digit[1:Limit] of integer; %The big number. Integer carry,d; %Assistants during multiplication. Integer last,i; %Indices to the big number's digits. Array text[1:Limit] of character;%Scratchpad for the output. Constant tdigit[0:9] of character = ["0","1","2","3","4","5","6","7","8","9"]; BEGIN digit:=0; %Clear the whole array. digit[1]:=1; %The big number starts with 1, last:=1; %Its highestorder digit is number 1. for n:=1 to FactorialLimit do %Step through producing 1!, 2!, 3!, 4!, etc. carry:=0; %Start a multiply by n. for i:=1 to last do %Step along every digit. d:=digit[i]*n + carry; %The classic multiply. digit[i]:=d mod Base; %The loworder digit of the result. carry:=d div Base; %The carry to the next digit. next i; while carry > 0 %Store the carry in the big number. if last >= Limit then croak("Overflow!"); %If possible! last:=last + 1; %One more digit. digit[last]:=carry mod Base; %Placed. carry:=carry div Base; %The carry reduced. Wend %With n > Base, maybe > 1 digit extra. text:=" "; %Now prepare the output. for i:=1 to last do %Translate from binary to text. text[Limit  i + 1]:=tdigit[digit[i]]; %Reversing the order. next i; %Arabic numerals put the low order last. Print text," = ",n,"!"; %Print the result! next n; %On to the next factorial up. END;
With the example in view, a number of details can be discussed. The most important is the choice of the representation of the big number. In this case, only integer values are required for digits, so an array of fixedwidth integers is adequate. It is convenient to have successive elements of the array represent higher powers of the base.
The second most important decision is in the choice of the base of arithmetic, here ten. There are many considerations. The scratchpad variable d must be able to hold the result of a singledigit multiply plus the carry from the prior digit's multiply. In base ten, a sixteenbit integer is certainly adequate as it allows up to 32767. However, this example cheats, in that the value of n is not itself limited to a single digit. This has the consequence that the method will fail for n > 3200 or so. In a more general implementation, n would also use a multidigit representation. A second consequence of the shortcut is that after the multidigit multiply has been completed, the last value of carry may need to be carried into multiple higherorder digits, not just one.
There is also the issue of printing the result in base ten, for human consideration. Because the base is already ten, the result could be shown simply by printing the successive digits of array digit, but they would appear with the highestorder digit last (so that 123 would appear as "321"). The whole array could be printed in reverse order, but that would present the number with leading zeroes ("00000...000123") which may not be appreciated, so we decided to build the representation in a spacepadded text variable and then print that. The first few results (with spacing every fifth digit and annotation added here) are:
Factorial numbers  Reach of computer integers  

1 =  1!  
2 =  2!  
6 =  3!  
24 =  4!  
120 =  5!  8bit  255 
720 =  6!  
5040 =  7!  
40320 =  8!  16bit  65535 
3 62880 =  9!  
36 28800 =  10!  
399 16800 =  11!  
4790 01600 =  12!  32bit  42949 67295 
62270 20800 =  13!  
8 71782 91200 =  14!  
130 76743 68000 =  15!  
2092 27898 88000 =  16!  
35568 74280 96000 =  17!  
6 40237 37057 28000 =  18!  
121 64510 04088 32000 =  19!  
2432 90200 81766 40000 =  20!  64bit  18446 74407 37095 51615 
51090 94217 17094 40000 =  21!  
11 24000 72777 76076 80000 =  22!  
258 52016 73888 49766 40000 =  23!  
6204 48401 73323 94393 60000 =  24!  
1 55112 10043 33098 59840 00000 =  25!  
40 32914 61126 60563 55840 00000 =  26!  
1088 88694 50418 35216 07680 00000 =  27!  
30488 83446 11713 86050 15040 00000 =  28!  
8 84176 19937 39701 95454 36160 00000 =  29!  
265 25285 98121 91058 63630 84800 00000 =  30!  
8222 83865 41779 22817 72556 28800 00000 =  31!  
2 63130 83693 36935 30167 21801 21600 00000 =  32!  
86 83317 61881 18864 95518 19440 12800 00000 =  33!  
2952 32799 03960 41408 47618 60964 35200 00000 =  34!  128bit  3402 82366 92093 84634 63374 60743 17682 11455 
1 03331 47966 38614 49296 66651 33752 32000 00000 =  35! 
We could try to use the available arithmetic of the computer more efficiently. A simple escalation would be to use base 100 (with corresponding changes to the translation process for output), or, with sufficiently wide computer variables (such as 32bit integers) we could use larger bases, such as 10,000. Working in a powerof2 base closer to the computer's builtin integer operations offers advantages, although conversion to a decimal base for output becomes more difficult. On typical modern computers, additions and multiplications take constant time independent of the values of the operands (so long as the operands fit in single machine words), so there are large gains in packing as much of a bignumber as possible into each element of the digit array. The computer may also offer facilities for splitting a product into a digit and carry without requiring the two operations of mod and div as in the example, and nearly all arithmetic units provide a carry flag which can be exploited in multipleprecision addition and subtraction. This sort of detail is the grist of machinecode programmers, and a suitable assemblylanguage bignumber routine can run much faster than the result of the compilation of a highlevel language, which does not provide access to such facilities.
For a singledigit multiply the working variables must be able to hold the value (base1)² + carry, where the maximum value of the carry is (base1). Similarly, the variables used to index the digit array are themselves limited in width. A simple way to extend the indices would be to deal with the bignumber's digits in blocks of some convenient size so that the addressing would be via (block i, digit j) where i and j would be small integers, or, one could escalate to employing bignumber techniques for the indexing variables. Ultimately, machine storage capacity and execution time impose limits on the problem size.
History
IBM's first business computer, the IBM 702 (a vacuum tube machine) of the mid1950s, implemented integer arithmetic entirely in hardware on digit strings of any length from one to 511 digits. The earliest widespread software implementation of arbitrary precision arithmetic was probably that in Maclisp. Later, around 1980, the operating systems VAX/VMS and VM/CMS offered bignum facilities as a collection of string functions in the one case and in the languages EXEC 2 and REXX in the other.
An early widespread implementation was available via the IBM 1620 of 1959–1970. The 1620 was a decimaldigit machine which used discrete transistors, yet it had hardware (that used lookup tables) to perform integer arithmetic on digit strings of a length that could be from two to whatever memory was available. For floatingpoint arithmetic, the mantissa was restricted to a hundred digits or fewer, and the exponent was restricted to two digits only. The largest memory supplied offered sixty thousand digits, however Fortran compilers for the 1620 settled on fixed sizes such as ten, though it could be specified on a control card if the default was not satisfactory.
Software libraries
Arbitraryprecision arithmetic in most computer software is implemented by calling an external library that provides data types and subroutines to store numbers with the requested precision and to perform computations.
Different libraries have different ways of representing arbitraryprecision numbers, some libraries work only with integer numbers, others store floating point numbers in a variety of bases (decimal or binary powers). Rather than representing a number as single value, some store numbers as a numerator/denominator pair (Rationals) and some can fully represent computable numbers, though only up to some storage limit. Fundamentally, Turing machines cannot represent all real numbers, as the cardinality of <math>\mathbb{R}</math> exceeds the cardinality of <math>\mathbb{Z}</math>.
See also
References
 ^ Researchers: 307digit key crack endangers 1024bit RSA
 ^ http://www.rsa.com/rsalabs/node.asp?id=2218 recommends important RSA keys be 2048 bits (roughly 600 digits).
 ^ R. K. Pathria (1962). "A Statistical Study of the Randomness Among the First 10,000 Digits of Pi". Mathematics of Computation 16 (78): 188–197. doi:10.1090/s00255718196201444437. Retrieved 20140110. 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 twentyeight times in one block of a thousand digits.
 Knuth, Donald (2008). Seminumerical Algorithms. The Art of Computer Programming 2 (3rd ed.). AddisonWesley. ISBN 0201896842{{inconsistent citations}}, Section 4.3.1: The Classical Algorithms
External links
 Chapter 9.3 of The Art of Assembly by Randall Hyde discusses multiprecision arithmetic, with examples in x86assembly.
 Rosetta Code task Arbitraryprecision integers Case studies in the style in which over 47 programming languages compute the value of 5**4**3**2 using arbitrary precision arithmetic.
