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Topic: Precision (arithmetic)

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	Arbitrary-precision arithmetic - Wikipedia, the free encyclopedia
		On a computer, arbitrary-precision arithmetic, also called bignum arithmetic, is a technique that allows computer programs to perform calculations on integers and rational numbers with an arbitrary number of digits of precision, limited only by the available memory of the host system.
		Perhaps the earliest widespread implementation of arbitrary precision arithmetic was in Maclisp.
		Arbitrary precision arithmetic is also used to compute fundamental mathematical constants such as pi to millions or more digits and to analyze their properties.
	en.wikipedia.org /wiki/Arbitrary-precision_arithmetic (575 words)

	Precision (arithmetic) - Wikipedia, the free encyclopedia
		The precision of a value describes the number of digits that are used to express that value.
		For example, in floating-point arithmetic, a result is rounded to a given or fixed precision, which is the length of the resulting significand.
		If insufficient precision is available then the number is rounded in some manner to fit the available precision.
	en.wikipedia.org /wiki/Precision_(arithmetic) (229 words)

	Encyclopedia :: encyclopedia : Fundamental theorem of arithmetic (Site not responding. Last check: 2007-10-13)
		In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers.
		The fundamental theorem ensures that additive and multiplicative arithmetic functions are completely determined by their values on the powers of prime numbers.
		Another proof of the uniqueness of the prime factorization of a given integer uses infinite descent: Assume that a certain integer can be written as (at least) two different products of prime numbers, then there must exist a smallest integer s with such a property.
	www.hallencyclopedia.com /Fundamental_theorem_of_arithmetic (915 words)

	DC, An Arbitrary Precision Calculator
		The precision of the result is determined only by the values of the arguments, and is enough to be exact.
		The default precision value is zero, which means that all arithmetic except for addition and subtraction produces integer results.
		The precision is always measured in decimal digits, regardless of the current input or output radix.
	www.cs.utah.edu /dept/old/texinfo/dc/dc.html (1178 words)

	Using the Symbolic Math Toolbox (Symbolic Math Toolbox)
		The floating-point operations used by numeric arithmetic are the fastest of the three, and require the least computer memory, but the results are not exact.
		The symbolic operations used by rational arithmetic are potentially the most expensive of the three, in terms of both computer time and memory.
		Variable-precision arithmetic falls in between the other two in terms of both cost and accuracy.
	www.technion.ac.il /guides/matlab/toolbox/symbolic/ch210.html (266 words)

	[No title]
		For single precision input, the computer's native double precision is a way to achieve these benefits easily on all commercially significant computers, at least when only a few extra-precision operations are needed.
		The last question remaining is how much precision would be needed to perform an entire eigencalculation by conventional means, without refinement, to obtain results as good and as soon as are obtained from refinement with a little extra-precise arithmetic in the program; attempts to answer this question empirically have occasionally generated astonishment.
		One possible exception to the rule that the system may use precision at least as high as requested by the user might be T. We cannot think of a situation where the user would explicitly specify traditional evaluation, as well as differing individual input precisions, without wanting this specific evaluation scheme.
	www.netlib.org /utk/papers/xblas.txt (5094 words)

	Numerics Report
		Mathematica's bigfloat arithmetic model is a variant of interval arithmetic where a single floating point number is used to represent the error.
		The primary goal of significance arithmetic is to give a good estimate of the number of correct digits in a result, whilst being efficient and accurate most of the time.
		The arithmetic model works well provided that the length of the interval representing the error is short relative to the magnitude of the numbers represented.
	documents.wolfram.com /v4/GettingStarted/NumericsReport.html (4555 words)

	Go4Expert - Arbitrary Precision Arithmetic
		In most computer programs and computing environments, the precision of any calculation (even including addition) is limited by the word size of the computer, that is, by largest number that can be stored in one of the processor's registers.
		Arbitrary-precision arithmetic consists of a set of algorithms, functions, and data structures designed specifically to deal with numbers that can be of arbitrary size.
		This can be either in the form of a failsafe, or a configurable 'maximum precision' at which the computation will always stop when it gets to a particular very small number.
	www.go4expert.com /forums/showthread.php?t=84 (305 words)

	Project Ideas: Multi-Precision Arithmetic
		While the standard 64-bit IEEE arithmetic system is adequate for almost all scientific applications, a small but growing set of applications requires more.
		DHB has an arbitrary precision package, but it was designed for a more general floating-point arithmetic system, and is not nearly as efficient as it could be if designed specifically to take advantage of IEEE arithmetic.
		A full-blown arbitrary precision package may not be possible in the scope of a CS267 project, but some subset (say the basic arithmetic routines) would be appropriate.
	www.cs.berkeley.edu /~ejr/GSI/2000-spring/cs267/final-project-ideas/high-precision.html (543 words)

	[No title]
		Many of the functions are supplied in six different arithmetic precisions: 32 bit single (24-bit significand), 64 bit IEEE double (53-bit), 64 bit DEC (56-bit), 80 or 96 bit IEEE long double (64-bit), and extended precision formats having 144-bit and 336-bit significands.
		For DEC and IEEE arithmetic, numerical constants and approximation coefficients are supplied as integer arrays in order to eliminate conversion errors that might be introduced by the language compiler.
		Higher precision may be realized if an arithmetic unit such as the 8087 or 68881 is used in conjunction with an optimizing compiler.
	cm.bell-labs.com /netlib/cephes/cephes.doc (544 words)

	Floating Point
		Arithmetic with integers is exact, unless the answer is outside the range of integers that can be represented (overflow).
		In contrast, floating point arithmetic is not exact since some number require an infinite number of digits to be represented, e.g., the mathematical constants e and π and 1/3.
		With single precision, when N = 10,000, the sum is accurate to 5 decimal digits, when N = 1,000,000 it is accurate to only 3 decimal digits, when N = 10,000,000 it is accurate to only 2 decimal digits.
	www.cs.princeton.edu /introcs/91float (6340 words)

	Arbitrary-Precision Arithmetic
		High- but not arbitrary-precision arithmetic can be conveniently performed using the Chinese remainder theorem and modular arithmetic.
		The authors use base 10 for arithmetic and arrays of digits to represent long integers, with short integers as array indices, thus limiting computations to 32,768 digits.
		Notes: Knuth [Knu81] is the primary reference on algorithms for all basic arithmetic operations, including implementations of them in the MIX assembly language.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE144.HTM (1500 words)

	The Yacas arithmetic library
		Note that the values are compared arithmetically, their internal precision may differ, and integers may be compared to floats.
		The precision is either set automatically (to enough digits to hold the integer), or explicitly to a given number of bits.
		A "poor man's interval arithmetic" is proposed where the precision is represented by the "number of correct bits".
	homepage.mac.com /yacas/manual/essayschapter7.html (8724 words)

	[No title]
		Converting double precision (16 decimal digit precision) to quadruple precision(33 decimal digit precision) involves execution time expense considerably greater than converting from single precision (7 decimal digit precision) to double precision.
		There are at least two reasons for this: (1) Fortran Hx's generated code has little or no register optimization; that is a QP number fills two of the System 370's four floating-point registers; this necessitates many more machine instructions (stores and fetches).
		The combination of optimization loss and software simulation of QP divide may cause a program whose arithmetic is all QP to execute several times slower than an equivalent all DP program.
	ftp.cac.psu.edu /pub/ger/fortran/hdk/qpnote.for (446 words)

	Fortran Software for Multiple Precision Arithmetic
		When the sign bit for FM numbers was moved from element (2) of the array to element (-1) in version 1.2, one of the cases was not updated properly.
		When the FM number in a call to FMADDI was negative and between 1 and the base being used for the arithmetic, a wrong result could be returned.
		The precision and base for the arithmetic can be set by the user, and routines are available for floating-point arithmetic, conversion and input/output operations, trigonometric, exponential, logarithmic, and hyperbolic functions.
	myweb.lmu.edu /dmsmith/FMLIB.html (519 words)

	Introduction to MPFI
		The basic principle of interval arithmetic consists in enclosing every number by an interval containing it and being representable by machine numbers: for instance it can be stored as its lower and upper endpoints and these bounds are machine numbers, or as a centre and a radius which are machine numbers.
		The arithmetic operations are extended for interval operands in such a way that the exact result of the operation belongs to the computed interval.
		The purpose of an arbitrary precision interval arithmetic is on the one hand to get guaranteed results, thanks to interval computation, and on the other hand to obtain accurate results, thanks to multiple precision arithmetic.
	pauillac.inria.fr /cdrom/www/mpfi/eng.htm (724 words)

	[No title]
		There is a tradeoff between precision and compression performance, but nearly optimal results are obtained with a precision of six bits, and precisions of as low as three bits give reasonable results.
		A precision of at least two bits is required for correct operation.
		Its coding efficiency appears similar to that which would be obtained with this method if divisions are performed to a precision of two bits.
	www.cs.toronto.edu /~radford/ftp/lowp-ac-doc (903 words)

	Publications of the SPACES team
		More precisely, from a partition of the parameters' space, such that in any connected component of this partition the number of triple roots is constant, we need to compute one sample point by cell, in order to have a full description, in terms of cuspidality, of the different possible configurations.
		It was shown in a previous work that this ability to change posture without meeting a singularity is equivalent to the existence of a point in the workspace, such that a polynomial of degree four depending on the parameters of the manipulator and on the cartesian coordinates of the effector has a triple root.
		The key idea is to precise the relations between the successive Sylvester matrix of A and B in one hand and of A and XB on the other hand, using the notion of G-remainder we introduce.
	www-calfor.lip6.fr /~safey/Spaces/publications.html (13078 words)

	The Ultimate Computer Arithmetic Specification (Site not responding. Last check: 2007-10-13)
		Obviously, using arithmetic that is not implemented in hardware will run slower, but given all of the other overhead in modern computers, you may or may not even notice any change in speed.
		The bits in each of the two vectors are the various properties of the arithmetic format of that precision.
		Subsequent trapped arithmetic instructions would first perform a computed GOTO on the static variable to jump to the appropriate incarnation of the simulator code.
	www.smart-life.net /FP (1673 words)

	[No title]
		The precision of a calculation is defined as follows: Compute the requested operation exactly (with "infinite precision"), and truncate the result to the destination variable precision.
		The precision of X is undefined unless a default precision has already been established by a call to `mpf_set_default_prec'.
		Since changing the precision involves calls to `realloc', this routine should not be called in a tight loop.
	www.math.psu.edu /Doc/emacs/info/gmp.info-2 (2339 words)

	CARRY FACTS AND INFORMATION (Site not responding. Last check: 2007-10-13)
		In elementary arithmetic a carry is a digit that is transferred from one column of digits to another column of more significant digits during a calculation algorithm.
		When speaking of a digital circuit like an adder, the word ''carry'' is used in a similar sense.
		In most computers, the carry from the most significant bit of an arithmetic operation (or bit shifted out from a shift operation) is placed in a special ''carry bit'' which can be used as a carry-in for multiple precision arithmetic or tested and used to control execution of a computer program.
	www.factagent.com /carry (108 words)

	High-Precision Software Directory
		This package supports a flexible, arbitrarily high level of numeric precision -- the equivalent of hundreds or even thousands of decimal digits (up to approximately ten million digits if needed).
		One enters expressions in a Mathematica-style syntax, and the operations are performed using the ARPREC package, with a level of precision that can be set from 100 to 1000 decimal digit accuracy.
		This program supports all basic arithmetic operations, common transcendental and combinatorial functions, multi-pair PSLQ (one-, two- or three-level versions), high-precision quadrature, i.e.
	crd.lbl.gov /~dhbailey/mpdist/mpdist.html (939 words)

	IEEE Arithmetic
		Rounding precision; for example, if a system delivers results in double extended format, the user should be able to specify that such results are to be rounded to the precision of either the single or double format.
		A second question refers to the precision (not to be confused with the accuracy or the number of significant digits) of the numbers represented in a given format.
		The presence of subnormal numbers in the arithmetic means that untrapped underflow (which implies loss of accuracy) cannot occur on addition or subtraction.
	docs.sun.com /source/806-3568/ncg_math.html (5560 words)

	The GNU MP Bignum Library
		GMP is a free library for arbitrary precision arithmetic, operating on signed integers, rational numbers, and floating point numbers.
		There is no practical limit to the precision except the ones implied by the available memory in the machine GMP runs on.
		The speed is achieved by using fullwords as the basic arithmetic type, by using fast algorithms, with highly optimized assembly code for the most common inner loops for a lot of CPUs, and by a general emphasis on speed.
	www.swox.com /gmp (1864 words)

	Comparison of two arbitrary-precision arithmetic packages (Site not responding. Last check: 2007-10-13)
		As we discuss in more detail later, radix conversions tend to be significantly more expensive than arithmetic operations such as multiplication or division.
		In order to compare the costs of the arithmetic operations themselves, we should instead use a program that does not display the results (or at least does not display the result of each operation).
		More precisely, we wanted to implement some efficient algorithms for multiplication, division, and radix conversion, and we felt like a large-number calculator using a class that represents large integers in arbitrary radix would be a nice way to show off the implementations.
	www.cis.ksu.edu /~howell/calculator/comparison.html (3122 words)

	What Every Computer Scientist Should Know About Floating-Point Arithmetic
		Guard digits were considered sufficiently important by IBM that in 1968 it added a guard digit to the double precision format in the System/360 architecture (single precision already had a guard digit), and retrofitted all existing machines in the field.
		IEEE 754 single precision is encoded in 32 bits using 1 bit for the sign, 8 bits for the exponent, and 23 bits for the significand.
		In the case of single precision, where the exponent is stored in 8 bits, the bias is 127 (for double precision it is 1023).
	docs.sun.com /source/806-3568/ncg_goldberg.html (8200 words)

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