Unnormalized Floating Point Arithmetic

Ashenhurst, Robert L.; Metropolis, N.

doi:10.1145/320986.320996

Cited by 49 publications

(12 citation statements)

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“…In some sense, our internal representation is similar to the one proposed in the late 1950s [1]. However, our approach is very different as computers only manipulate unique machine representations.…”

Section: Floating Point Representationmentioning

confidence: 97%

Representable correcting terms for possibly underflowing floating point operations

Boldo

Daumas

16th IEEE Symposium on Computer Arithmetic, 2003. Proceedings.

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Studying floating point arithmetic, authors have shown that the implemented operations (addition, subtraction, multiplication, division and square root) can compute a result and an exact correcting term using the same format as the inputs. Following a path initiated in 1965, many authors supposed that neither underflow nor overflow occurred in the process. Overflow is not critical as this kind of exception creates persisting non numeric quantities. Underflow may be fatal to the process as it returns wrong numeric values with little warning. Our new conditions guarantee that the correcting term is exact when the result is a number. We have validated our proofs against Coq automatic proof checker. Our development has raised many questions, some of them were expected while other ones were surprising.

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Section: Floating Point Representationmentioning

confidence: 97%

Representable correcting terms for possibly underflowing floating point operations

Boldo

Daumas

16th IEEE Symposium on Computer Arithmetic, 2003. Proceedings.

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“…For a # 0, this means that f, is normalized, so that ,} =< ]f~ I < 1, and for a = 0, one should have f~ = 0 and eo at its minimum representable value. Now, if x is added to or subtracted from a using "significant digit" arithmetic rules such as are described in [1], the adjustment of the result (e2, f2) is determined by the rule e2 = Max (e~, o).…”

Section: Simple Functionsmentioning

confidence: 99%

“…Iff~ is again taken to be normalized, so tha, t m~ = O, the product has rn2 = m~, and this adjustment is given by the formulas (see [1] for the explicit formulation from which this is derived). Hence, using the earlier conventions for ~2, 62 2xt'~f,~ --2x(f~ --6~).G = -.Ol and so…”

Section: Simple Functionsmentioning

confidence: 99%

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Function Evaluation in Unnormalized Arithmetic

Ashenhurst

1964

J. ACM

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The evaluation of a function of one argument is a standard computationaI task. When an unnormMized number representation is used, it is appropriate that function evaluation be subject to certain "adjustrcmnt" criteria, defined independently of the computing method. In this paper some such criteria are developed, and their application described.In particular, consideration is given to questions of the extent to which general principles apply (i.e. for large classes of functions of a single argument) and the manner in which particular properties of the functions involved (e.g. the functions evaluated by standard library subroutines) may be invoked. The analysis, although framed in terms of unnormalized arithmetic, gives insight into the general nature of significance propagation through function evaluation, and provides a means for analyzing calculations carried out in the usual normalized arithmetic, particularly in the case where an "index of signifi. canoe" is employed.

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“…The question of radix conversion of variable-precision binary numbers arises naturally in the context of unnormalized number representation [1], but may be of interest in other situations where it is desired to have number representations carry a reflection of significance. The present paper discusses a method for binary-decimal conversion of unnormalized numbers; this method differs in certain respects from one previously developed, and described elsewhere [2], for use with the maniac III computer.…”

mentioning

confidence: 99%

Radix conversion in an unnormalized arithmetic system

Metropolis¹,

Ashenhurst²

1965

Math. Comp.

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Introduction. The question of radix conversion of variable-precision binary numbers arises naturally in the context of unnormalized number representation [1], but may be of interest in other situations where it is desired to have number representations carry a reflection of significance. The present paper discusses a method for binary-decimal conversion of unnormalized numbers; this method differs in certain respects from one previously developed, and described elsewhere [2], for use with the maniac III computer. The question of decimal-binary conversion, taking into account explicit "uncertainty" in the decimal representation, is also investigated from the significance viewpoint.Binary-Decimal Conversion. First, consider the task of converting a number in unnormalized floating point binary form to a decimal in such a way that a discrepancy in the lowest-order binary digit corresponds to a discrepancy in the lowestorder decimal digit in the result (the standard maniac III routine for doing this has a "guard bit" provision for specifying a position other than the lowest-order one, but this feature can be incorporated by a preliminary transformation and so is neglected in the development here). The question of whether the decimal exponent of the result is represented explicitly or by the insertion of a decimal point character is irrelevant to the present discussion ; it will be assumed that the desired output is a string of decimal characters, representing an integer, and a second integer specifying an associated power of 10.Since 2~l = 10~ , equivalent precision in binary and decimal is given by numbers of digits in roughly the ratio 10 to 3; one could, of course, simply keep a count of decimal characters generated in a standard conversion procedure and stop at some approximately appropriate point. It seems not unreasonable, however, to ask for a conversion procedure which affords the user a more precise statement of the relation between the binary form and the decimal result. The observation that the exact conversion of integers gives also a true estimate of precision (i.e., the 10-for-3 criterion is naturally achieved), suggests that binary-decimal conversion can be accomplished by first transforming the floating point number to an integer expressed with the same significance, which differs from the original number only by a power of 10, and then taking the converted representation of this integer as the desired

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Unnormalized Floating Point Arithmetic

Cited by 49 publications

References 1 publication

Representable correcting terms for possibly underflowing floating point operations

Representable correcting terms for possibly underflowing floating point operations

Function Evaluation in Unnormalized Arithmetic

Radix conversion in an unnormalized arithmetic system

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