W. S. Brown 1981. "A simple but realistic model of floating-point computation." ACM Transactions on Mathematical Software 7 :4 (December 1981). pp. 445--480.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... than to prevent the exception from occurring in the first place [Demmel and Li 1994; Hull et al. 1994] Conversely, when exception handling is not available, it is sometimes necessary to artfully evade exceptions, resulting in programs that exhibit no exceptional behavior but waste time doing so [Brown 1981; Parlett 1979] In recent years, processor manufacturers have become increasingly suspicious that arithmetic exception handling is an unneeded nicety, of little value to their customers. Without a doubt, the main concerns of heavy users of computer arithmetic are accuracy and speed. If a poll ....

....(e.g. 0 #) Thus when substitution is misapplied, # has a greater chance than# of either visibly propagating through the calculation or causing an exception trap. The arithmetic of # is discussed in greater depth in Section 4. Note that there is no strict guarantee that this will happen [Brown 1981; Lynch and Swartzlander 1991] Simply, # is more likely to be noticed than# . The IEEE Standard requires that # be substituted on overflow, but it mitigates the trouble this substitution may cause by raising an overflow exception flag that can be tested by the program. If substitution of ....

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Brown, W. S. 1981. A simple but realistic model of floating-point computation. ACM Trans. Math. Softw. 7, 4 (Dec.), 445--480.

....into double precision representation, the intersection with the battery of hexadecimal double precision vectors from UCBTEST is rather large. While formal verification methods have also been applied to floating point systems, we shall not discuss such methods here but refer, among others, to [Brown 1981], Barrett 1989] and [Popova 1994] 2.2 Main features of the new precision independent tool As one can see from the above overview, the available test tools are rather complementary in nature and each implement a different approach to testing: explicit testing of specific floating point ....

Brown, W. 1981. A simple but realistic model of floating-point computation. ACM Transactions on Mathematical Software 7, 445--480.

....Wilkinson shows that it is often simpler to view the inexact result of floating point operations as the exact result of perturbed arguments. Several researchers have also proposed models of floating point operations and used them to justify claims about sequences of floating point operations. [6, 8, 14, 17] These efforts work toward putting floatingpoint arithmetic on more solid ground. Even so, the complexity of proofs using these systems, and the apparent gap between the floating point axioms and realistic numerical programs has discouraged application of work in this area. Much of the effort ....

W. S. Brown. "A Simple but Realistic Model of Floating-Point Computation". ACM Transactions on Mathematical Software 7, 4 (December 1981).

....extensions of Fortran 77, would require every declaration to be changed if a change of precision were needed. There are many inquiry and manipulation intrinsic functions that return information on the representation or manipulate parts of a data value. They are described in terms of the model of Brown (1981) for the representation and behaviour of numbers on a processor. The model set for real X is determined by the parameters b (base) n (number of digits) e (maximum exponent) and e max min (minimum exponent) which must be chosen by the implementor to best fit the machine. The set consists of the ....

Brown, W.S. (1981). A simple but realistic model of floating-point computation. ACM Trans. Math. Softw., 7, 445-480.

....were resolved by using mathematical notation for LIA 1, and providing this rationale in English to explain the notation. c fl ISO IEC ISO IEC 10967 1:1994(E) There are various notations for giving a formal definition of arithmetic. In [36] a formal definition is given in terms of the Brown model [22]. Since the current proposal differs from the Brown model, the definition in [36] is not appropriate for LIA 1. The production of a formal definition using VDM [27] would nevertheless be useful. A.4.1 Symbols LIA 1 uses the conventional notation for sets and operations on sets. The set Z denotes ....

....(giving a few axioms which essentially every floating point system already satisfies) ISO IEC 10967 1:1994(E) c fl ISO IEC IEEE 754 [1] takes the highly prescriptive approach, allowing relatively little latitude for variation. It even stipulates much of the representation. The Brown model [22] comes close to the other extreme, even permitting non deterministic behavior. There are (at least) five interesting points on the range from a strong specification to a very weak one. These are a) Specify the set of representable values exactly; define the operations exactly; but leave the ....

W S Brown, A Simple but Realistic Model of Floating-Point Computation, ACM Transactions on Mathematical Software, Vol. 7, 1981, pp.445-480.

....operators. Function names can be overloaded in Ada. Each derived floating point type does not need its own set of library functions; the Ada type system can determine the appropriate library function to call for a derived floating point type. Ada floating point numbers follow the Brown model [9] of floating point numbers. The Brown model distinguishes between model numbers, which adhere to Brown s axioms, and machine numbers, which may not. Essentially, in order to represent a variety of floating point arithmetics with different computational peculiarities, the model numbers have reduced ....

W. S. Brown, "A Simple but Realistic Model of Floating-Point Computation," ACM Transactions on Mathematical Software, vol. 7, no. 4, pp. 445-480, 1981.

....of the strict mode at little or no cost to the user has no real incentive to offer the user another choice, and the two modes might be identical. 2. 2 Accuracy Requirements for Predefined Floating Point Operations Ada 83 had a model of floating point arithmetic based on the Brown model [1], which served as the basis for the accuracy requirements for the predefined floating point arithmetic operations. Experience has shown that only certain aspects of the model have been used to any great extent, other than by implementors; that the model fostered confusion as often as insight; and ....

W. S. Brown. A Simple but Realistic Model of Floating-Point Computation. TOMS 7(4):445--480, December 1981.

....Arithmetic and Attributes The strict mode accuracy requirements for predefined floating point arithmetic operations are based on the same kind of model that was used in Ada 83, but with several changes. The Ada 83 model of floating point arithmetic was a two level adaptation of the Brown Model [Brown 81] and defined both model numbers and safe numbers. The Ada 95 model is closer to a one level, classical Brown Model that defines only model numbers, although it innovates slightly in the treatment of the overflow threshold. The existence of both model numbers and safe numbers in Ada 83 caused ....

....may end up being ignored by the vendor. Such matters are best left to the judgment of the marketplace and not dictated by the language. The particular minimum range required in Ada 83 (as a function of precision) is furthermore about twice that deemed minimally necessary for numeric applications [Brown 81] Among implementations of Ada 83, the only predefined types whose characteristics are affected by the relaxation of the 4B Rule are DEC VAX D format and IBM Extended Precision, both of which have a narrow exponent range in relation to their precision. In the case of VAX D format, even though the ....

W. S. Brown. "A simple but realistic model of floating-point computation". Transactions on Mathematical Software 7(4): 445-480, December, 1981.

....the smallest nonzero number of the appropriate sign when a result would otherwise underflow. More importantly, the goal of an axiomatic approach is the unification of hypotheses, and Dekker s distinction between different families of properties contributes little to that goal. The Brown model [11], subsequently expressed as a formal specification of floating point arithmetic by Wichmann [63] also specifically includes a definition of float27 ing point arithmetic. Brown defines model numbers to be floating point numbers within a given exponent range, although he allows an arithmetic to ....

Brown, W. S., A Simple but Realistic Model of Floating-Point Computation, ACM Trans. Math. Soft. 7 (1981), 445--480.

....to some readers. These problems were resolved by using mathematical notation for the LIA 1, and providing this rationale in English to explain the notation. There are various notations for giving a formal definition of arithmetic. In [40] a formal definition is given in terms of the Brown model [24]. Since the current proposal differs from the Brown model, the definition in [40] is not appropriate for the LIA 1. The production of a formal definition using VDM [30] would nevertheless be useful. c fl ISO IEC ISO IEC DIS 10967 1:1993 A.4.1 Symbols The LIA 1 uses the conventional notation for ....

....every last detail) to loosely descriptive (giving a few axioms which essentially every floating point system already satisfies) IEEE 754 [1] takes the highly prescriptive approach, allowing relatively little latitude for variation. It even stipulates much of the representation. The Brown model [24] comes close to the other extreme, even permitting non deterministic behavior. There are (at least) five interesting points on the range from a strong specification to a very weak one. These are a) Specify the set of representable values exactly; define the operations exactly; but leave the ....

W S Brown, A Simple but Realistic Model of Floating-Point Computation, ACM Transactions on Mathematical Software, Vol. 7, 1981, pp.445-480

....machine. Thus, 3.1) is not suitable to define correct FP arithmetic. Clearly, at least some FP results are going to have to be exactly right, like 1 1 = 2, but which ones A model of the dynamic FP behavior of a machine, which takes this and many other things into account, is given in [13]. That model will be used to define correct FP arithmetic. In the interest of completeness and conciseness, an outline of the axioms and results of that paper is presented below. First, we need to define some terms. The numbers defined by (2.1) are called m mooddeell n nuummbbeerrss, and the ....

....Any operation that conforms only to Axiom 1a or 2a will be called w weeaakkllyy ssuuppppoorrtteedd. The parameters must be chosen so that the basic operations satisfy Axioms 1 and 2, and so that division at least satisfies the weaker Axiom 2a. 6 Axioms We now present the axioms of [13] that define correct arithmetic. Let x x and y y be l bounded machine numbers. We first present the strong versions of the Axioms. Axiom 1 (For , and possibly . Let be a strongly supported binary operator. Then ff ll ( x x y y ) x x y y ) provided ....

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W. S. Brown, "A Simple but Realistic Model of Floating-Point Computation", Bell Laboratories Computing Science Technical Report 83, November, 1980.

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W. S. Brown 1981. "A simple but realistic model of floating-point computation." ACM Transactions on Mathematical Software 7 :4 (December 1981). pp. 445--480.

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W. S. Brown, A simple but realistic model of floating-point computation,ACM Trans. Math. Software, vol. 7, pp. 445--480, 1981.

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W. S. Brown, "A Simple but Realistic Model of Floating-Point Computation," ACM Transactions on Mathematical Software, vol. 7, no. 4, pp. 445480, Dec. 1981.

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W.S. Brown. "A Simple But Realistic Model of Floating-Point Computation, " Computer Science Technical Report no. 83, May 1980.

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Brown, W. S.: A Simple But Realistic Model of Floating-Point Computation. ACM Trans. Math. Software 7, 4, pp. 445-480, 1981.

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Brown, W. S. 1981. A Simple but Realistic Model of Floating-Point Computation, ACM Trans. on Math. Software 7(4), pp. 445-480.

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