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Topic: Exact inexact arithmetic

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	Numeric Operations
		Inexact real numbers may also be represented by inexact complexnums with imaginary parts equal to zero, which cannot be used as input to the flonum-specific operators.
		Inexact complex numbers with imaginary parts equal to zero (in other words, inexact real numbers) may be represented as either inexact complexnums or flonums.
		The argument must be an exact integer (fixnum or bignum) and is treated as a two's complement integer, regardless of the underlying representation.
	www.scheme.com /csug7/numeric.html (2193 words)

	Operations on Objects (Site not responding. Last check: 2007-09-10)
		A Scheme number may also be classified as exact or inexact, depending upon the quality of operations used to derive the number and the inputs to these operations.
		Exact integer and rational arithmetic is typically supported to arbitrary precision; the size of an integer or of the denominator or numerator of a ratio is limited only by system storage constraints.
		Exact and inexact real numbers are written as exact or inexact integers or rational numbers; no provision is made in the syntax of Scheme numbers for nonrational real numbers, i.e., irrational numbers.
	www.cs.indiana.edu /l/www/chezscheme/tspl2d/objects.html (4407 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		This is generally not true of computations involving inexact quantities since approximate methods such as floating point arithmetic may be used, but implementations should make the result as close as practical to the mathematically ideal result.
		An implementation shall support exact integers throughout the range of numbers that may be used for indexes of lists and strings or that may result from computing the length of a list or string.
		If an implementation encounters an exact numerical constant that it cannot represent as an exact quantity, then it may either report a violation of an implementation restriction, or it may silently represent the constant by an inexact quantity.
	www.y12.doe.gov /sgml/wg8/DSSSL/clexpr.sgm (4752 words)

	SRFI 77: Preliminary Proposal for R6RS Arithmetic
		The alternative described in section Generic Exact Arithmetic has exactness be contagious, with the advantage of greater transparency and the validity of standard algebraic laws over the R5RS-style generic arithmetic.
		For inexact numbers, it is the programmer's responsibility to avoid using complex numbers with magnitude too large to be represented in the implementation.
		This is generally not true of computations involving inexact numbers because approximate methods such as floating point arithmetic may be used, but it is the duty of each implementation to make the result as close as practical to the mathematically ideal result.
	srfi.schemers.org /srfi-77/srfi-77.html (9619 words)

	Larceny Note #3: Generic arithmetic implementation
		The optimized cases were to be same-representation arithmetic on fixnums, flonums, and compnums (all operations that are not unary are binary; the compiler translates from e.g.
		In addition, in-line code is generated for the arithmetic ordering predicates and for some other predicates, although the exact set of optimized primitive operations has been changing over time; there is also the question of what kinds of optimizations the compiler does.
		Mixed-representation arithmetic is implemented in Scheme by first converting the operands to a common representation and then applying the original operator again.
	www.ccs.neu.edu /home/will/Larceny/notes/note3-arithmetic.html (845 words)

	Quantities and Numbers
		If the operation is unable to produce an exact result, then it may either report the violation of an implementation restriction, or it may silently coerce its result to an inexact value.
		It is desirable (but not required) for potentially inexact operations such as sqrt, when applied to exact arguments, to produce exact answers whenever possible (for example the square root of an exact 4 ought to be an exact 2).
		If min or max is used to compare quantities of mixed exactness, and the numerical value of the result cannot be represented as an inexact quantity without loss of accuracy, then the procedure may report a violation of an implementation restriction.
	www.jclark.com /dsssl/IS/dsssl49.htm (2479 words)

	Revised(5) Report on the Algorithmic Language Scheme - Semantics (Site not responding. Last check: 2007-09-10)
		Scheme's model of arithmetic is designed to remain as independent as possible of the particular ways in which numbers are represented within a computer.
		In its place is a distinction between exact arithmetic, which corresponds to the mathematical ideal, and inexact arithmetic on approximations.
		As in Common Lisp, exact arithmetic is not limited to integers.
	www.math.grin.edu /courses/Scheme/r5rs-html/r5rs_5.html (532 words)

	[No title]
		This is generally not true of computations involving inexact numbers since approximate methods such as floating point arithmetic may be used, but it is the duty of each implementation to make the result as close as practical to the mathematically ideal result.
		The prefixes are `#e' for exact, and `#i' for inexact.
		All exact integers that are small enough to be encoded as fixnums are always encoded as fixnums -- in other words, any exact integer that is not a fixnum is too big to be encoded as such.
	www.math.psu.edu /Doc/emacs/info/scheme-3 (5110 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		If these integers were derived from inexact reals (say, as the result of FLOOR) then it is appropriate for them to be inexact because the implementation cannot prove that they are the correct integer.
		Write a procedure that takes an integer x (whether exact or inexact) and returns (INEXACT->EXACT x) if x is representable as an exact integer but reports a violation of an implementation restriction is x is not representable as an exact integer.
		William Clinger PS: The mixed exactness case is similar for the arithmetic predicates such as = and <, but if every exact number is representable as an inexact number then you can use the same trick except you convert to inexact before retrying the predicate.
	martigny.ai.mit.edu /pipermail/rrrs-authors/1994-May.txt (1257 words)

	DrScheme's Numbers (Site not responding. Last check: 2007-09-10)
		Scheme, in contrast, supports both exact and inexact numbers and arithmetic.
		it also prints inexact reals with just a dot, implying that all such numbers are inexact and possibly distant from the actual result.
		For example, numbers in financial statements should always be interpreted as exact numbers; arithmetical operations on such numbers should be as precise as possible.
	www.htdp.org /2001-09-22/Book/node173.htm (368 words)

	[No title]
		A number is inexact if it was written as an inexact constant, if it was derived using inexact ingredients, or if it was derived using inexact operations.
		Exact numbers include both integers and rational numbers, there is no theoretical limit to the range of the values that may be represented (1).
		Inexact numbers are parsed from one of two representations: decimal point form, which is simply a decimal number containing a decimal point, and exponential form, which is a decimal number followed by the letter `e' and a decimal exponent multiplying the first part of the number by that power of ten.
	www.mit.edu:8001 /afs/sipb/project/sawfish/info/librep.info-1 (5244 words)

	Java theory and practice: Where's your point?
		There are some cases where arbitrary-precision decimal arithmetic is still not sufficient to maintain exact results.
		data type, which is an exact numeric type for representing fixed point decimal numbers and performs basic arithmetic operation on decimal numbers.
		It is best to reserve the use of floating point arithmetic for calculations that involve fundamentally inexact values, such as measurements.
	www-128.ibm.com /developerworks/java/library/j-jtp0114 (1845 words)

	[e-lang] Complex, rational, and other sorts of numbers in E
		This is less explicit: you get whatever definition of inexact arithmetic is used by that Scheme implementation, rather than having to choose one.
		In particular, a floating point number always corresponds to an exact rational, and operations on these rationals should be "exactly rounded" according to some well-defined rule.
		So different arithmetics can support different subsets of operations or argument/result types, and the correct feedback on type errors will be given (even if the arithmetic is selected dynamically from more than one possibility).
	www.eros-os.org /pipermail/e-lang/2004-August/009949.html (643 words)

	[No title] (Site not responding. Last check: 2007-09-10)
		In most compilers with which we are familiar, all static (compile-time) evaluation is performed exactly, because the compilers are designed to be at least somewhat rehostable and/or retargetable, and it is simply easier to use exact arithmetic than to simulate (unknown) target hardware with (unknown) host hardware.
		Furthermore, since all universal static expressions require exact evaluation, all of the machinery is in place to do the exact evaluation, and target simulation is more work.
		Here is another case: the expression: 1.0 (-5) is always equivalent to: 0.00001 in compilers that use exact arithmetic** for all static calculations, whereas for the XYZ compilers, these could produce different answers.
	archive.adaic.com /standards/95lsn/lsn-028.txt (676 words)

	[No title]
		In order to catch uses of inexact numbers where exact numbers are required, Scheme explicitly distinguishes exact from inexact numbers.
		If `min' or `max' is used to compare numbers of mixed exactness, and the numerical value of the result cannot be represented as an inexact number without loss of accuracy, then the procedure may report a violation of an implementation restriction.(2) - procedure: + Z1...
		If an inexact argument has no reasonably close exact equivalent, then a violation of an implementation restriction may be reported; in MIT Scheme this case does not occur because all inexact numbers are representable as exact numbers.
	www.mit.edu /afs/athena/software/scheme/info/scheme-3 (4919 words)

	Exact and inexact numbers (Site not responding. Last check: 2007-09-10)
		The problem is that if you give max (or any other arithmetic function) mixed exact and inexact arguments, it will return an inexact answer:
		I'll argue that scheme does a better job than most languages of shielding users from such silliness.
		Anyway, the problem is that 8 (exact) and 8.
	www.cs.rpi.edu /courses/fall00/ai/assignments/assign5/exact-inexact.html (153 words)

	Quantities and Numbers
		procedures shall return an exact integer, and it shall be an error to use anything but an exact integer as an index.
		, when applied to exact arguments, to produce exact answers whenever possible (for example the square root of an exact 4 ought to be an exact 2).
		is used to compare quantities of mixed exactness, and the numerical value of the result cannot be represented as an inexact quantity without loss of accuracy, then the procedure may report a violation of an implementation restriction.
	www.cs.berkeley.edu /~wilensky/CS294/dsssl/html/h3-29.htm (2318 words)

	John Stuart Mill
		Insofar as the premises are exact descriptions -- referring, e.g., to exactly straight lines -- they describe material attributes taken to their limits.
		The problem is that arithmetic seems to have a necessity which is at once more than verbal, as Mill correctly held, but also more than that which attaches to the inductive truths of, say, physics or botany.
		What Mill does argue about the necessity of geometry and arithmetic, and, for that matter, the basic axioms of other sciences such as physics and chemistry, is that these principles, while from the point of view of their truth are inductive generalizations, are from the point of view of the thinker matters of psychological necessity.
	plato.stanford.edu /entries/mill (17600 words)

	CGAL dD Kernel Reference Manual:
		Naively, often the exact real arithmetic is replaced by inexact floating-point arithmetic in the implementation.
		There are many approaches to this problem, one of them is to compute exactly (compute so accurate that all decisions made by the algorithm are exact) which is possible in many cases but more expensive than standard floating-point arithmetic.
		The exact computation paradigm is discussed by Yap and Dubé [YD95] and Yap [Yap97].
	www.cs.rpi.edu /courses/spring02/robotic/CGAL-2.3/doc_html/kernel_d/Chapter_introduction_d.html (486 words)

	Numbers (Site not responding. Last check: 2007-09-10)
		Since there's no great advantage in obtaining an answer quickly if it may be incorrect, we shall avoid using inexact numbers in this course except when the data for our problems are themselves products of inexact measurement (as in exercise #1).
		Since Chez Scheme uses fractional notation to print out exact numbers, but renders inexact numbers as decimals, invoking this procedure is a simple way to determine the general format in which results are printed.
		A number expressed in scientific notation is also inexact unless preceded by the exactness prefix.
	www.math.grin.edu /courses/Scheme/spring-1998/numbers.html (1070 words)

	Comp210 Lab 12: Arithmetic Imprecision
		The idea is that the answer might be inexact, but hopefully the error is small.
		While there is no bound on how big or tiny a mathematical number can be, unfortunately every computer does have a bound on how big or tiny a number it can represent internally.
		In Scheme, floating-point numbers are called "inexact" numbers because they aren't exact.
	www.owlnet.rice.edu /~comp210/98spring/Labs/lab12 (1398 words)

	SRFI 73: Exact Infinities
		The exact rational operations of addition, subtraction, multiplication, and division, plus a variety of other useful operations (comparison, reading, writing) form a useful suite of routines and is defined in Scheme spec.
		For any operation that involves both exact (infinity) and inexact numbers, it is a common arithmetic that all exact numbers will first be coerced into inexact and then the computation continues.
		Here is my implementation, which is based on a Scheme implementation that supports arbitrary-big integer arithmetic as well as exact rational number computation.
	srfi.schemers.org /srfi-73/srfi-73.html (1723 words)

	[No title]
		This note may help them distinguish between nearly exact trisection, which is easy, and exact trisection, which was long ago proved impossible using only an UNMARKED straightedge and a compass.
		Over a dozen commercially significant arithmetics boasted diverse wordsizes, precisions, rounding procedures and over/underflow behaviors, and more were in the works.
		Owners of these machines are mostly unaware that their floating-point arithmetic hardware is capable of delivering routinely better results than can be expected from the more prestigious and more expensive workstations preferred by much of the academic Computer Science community.
	www.cs.berkeley.edu /~wkahan/abstrcts.txt (5171 words)

	Arnold Toynbee -- Statement on the Armenian Genocide
		We are compelled to base our estimates on the statements of private persons, who were excluded from detailed investigation by the jealous suspicion of the Government officials and were seldom able to observe events in more than a limited section of the field.
		We must make out computations by piecing together these isolated data form private sources, and since Oriental arithmetic is notoriously inexact (and this is scarcely less true of the Nearer than of the Further East), we shall only make use of testimony from foreign witnesses of neutral nationality.
		The exact quantitative scale of the crime thus remains uncertain, but there is no uncertainty as to the responsibility for its perpetration.
	www.armenian-genocide.org /toynbee.html (6510 words)

	Scheme - Standard procedures
		Implementations of Scheme are not required to implement the whole tower of subtypes given in section Numerical types, but they must implement a coherent subset consistent with both the purposes of the implementation and the spirit of the Scheme language.
		procedures must return an exact integer, and it is an error to use anything but an exact integer as an index.
		The valid indexes of a vector are the exact non-negative integers less than the length of the vector.
	www.cs.indiana.edu /scheme-repository/R4RS/r4rs_8.html (8070 words)

	REDUCE Bibliography
		Exact solutions of the Poincaré gauge theory from its linearized field equations.
		On the dynamics of the torsion of spacetime: Exact solutions in a gauge theoretical model of gravity.
		Two-loop gluon-condensate contributions to heavy-quark current correlators: exact results and approximations.
	www.reduce-algebra.com /bibliography.htm (7590 words)

	Comp210 Lab 6: Arithmetic Imprecision
		Mathematical programs in Scheme can sometimes be unacceptably slow because they calculate exact answers.
		This really means multiplying twice, once each for the denominators and numerators, plus also simplifying the result.) The idea is that the answer might be inexact, but hopefully the error is small.
		These are all part of the IEEE floating point standard, which describes the appropriate results for lots of unusual questions.
	www.owlnet.rice.edu /~comp210/98fall/Labs/lab06 (1440 words)

	Numbers
		All multi-argument arithmetic procedures operate pairwise on arguments from right to left.
		When an arithmetic operation is applied to a combination of exact and inexact values, the exact values are first converted to inexact values.
		All numbers are either inexact or exact in both real and imaginary parts (i.e., an inexact ``real'' number is not even exactly real because its imaginary part is
	www.cs.rice.edu /CS/PLT/packages/53/doc/mzscheme/node26.htm (315 words)

	Revised(5) Scheme
		Implementations of Scheme are not required to implement the whole tower of subtypes given in section section Numerical types, but they must implement a coherent subset consistent with both the purposes of the implementation and the spirit of the Scheme language.
		The reader is referred to section section Entry format for a summary of the naming conventions used to specify restrictions on the types of arguments to numerical routines.
		If any argument is inexact, then the result will also be inexact (unless the procedure can prove that the inaccuracy is not large enough to affect the result, which is possible only in unusual implementations).
	www.mazama.net /scheme/doc/r5rs.htm (13529 words)

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