Coonen, Jerome T.: Contributions to a Proposed Standard for Binary FloatingPoint Arithmetic. PhD dissertation, Univ. of California, Berkeley, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... arithmetic: reliability [Rum83, Ham90, Rat90] experience [RRZN84, Gue87] error analysis [Wil63, Kah65, Kah72, Ste74, Kah78, Knu81, Kah83, Kah83a] IEEE standards [IEEE85, IEEE87] older versions of these standards [ACM79, Cod81, IEEE81] discussion of these standards [Coo80, Kah81, Coo84, Cod84, Cod88, Lyn92, Pra93] implementation: 8087 coprocessor [Pal80, Rei81] language independent standards [ISO93, ISO93a, ISO93b] Ada model arithmetic [Wal82, Wal83] software and hardware implementation [Hwa79, Spa76, Spa81, ARITH, Gol90, Gol91, Swa91, Her94c, Sch95, Sch95a] conversion of ....

Coonen, Jerome T.: Contributions to a Proposed Standard for Binary FloatingPoint Arithmetic. PhD dissertation, Univ. of California, Berkeley, 1984.

....is the floating point number (1 2 u 1 2 u ) 2 L2 1 The results of the conversion are the respective denormal numbers (2 1 2 u 1 ) 2 L2 and 2 1 2 L2 depending on the rounding mode. Note that this conversion should raise the inexact (x) and underflow exceptions. Following [Coonen 1984], we use one of three characters to denote the underflow exception in a test vector (u, v and w) corresponding to the three di#erent definitions of underflow permitted by the IEEE standard. For a detailed description of these underflow detection mechanisms, we refer the reader to [Cuyt et al. ....

....decimal# binary conversion logically di#ers from the encodings described till now. We will discuss this and the corresponding test vectors extensively in Section 6. Under revision 5 2. 2 The complete test set For conversions no test sets are available in [Hough et al. 1988] while the vectors in [Coonen 1984] refer specifically only to the hardware single, double and quadruple formats (except for the rounding of floating point numbers to integral values) Hence, for the conversion test set, the first job was to investigate which of these test vectors could be generalized for arbitrary floating point ....

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Coonen, J. 1984. Contributions to a Proposed Standard for Binary Floating-Point Arithmetic. Ph. D. thesis, University of California at Berkeley, USA.

....IEEE floating point arithmetic and gives an overview of the main features of our testing tool. In section 3 the concept of test vector and the precision independent syntax to encode floating point operands, is introduced. This syntax is a streamlined and extended version of the syntax developed by [Coonen 1984]. In section 4 through section 8 we discuss, for each 4 B. Verdonk, A. Cuyt and D. Verschaeren of the six basic operations, which aspects of the operation are tested by the new test vectors. Section 9 describes the functionality of the driver program. In section 10, we discuss the results of ....

....or boundary in the values of the operands or some part of them. Under revision: A precision and range independent tool for testing floating point arithmetic I 5 Last but certainly not least, a test suite which has been used by major manufacturers of IEEE hardware was developed by J.T. Coonen [Coonen 1984]. This test tool consists of a large database of test vectors together with a driver program. An essential feature of this tool is that the vectors are designed to be as format independent as possible. To run the tests, the driver program of [Coonen 1984] decodes the test vectors, given in a ....

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Coonen, J. 1984. Contributions to a Proposed Standard for Binary Floating-Point Arithmetic. Ph. D. thesis, University of California at Berkeley, USA.

.... Thus for P 13, the use of the single extended format enables 9 digit decimal numbers to be converted to the closest binary number (i.e. exactly rounded) If P 13, then single extended is not enough for the above algorithm to always compute the exactly rounded binary equivalent, but Coonen [1984] shows that it is enough to guarantee that the conversion of binary to decimal and back will recover the original binary number. 172 Numerical Computation Guide E If double precision is supported, then the algorithm above would be run in double precision rather than single extended, but to ....

....to be exactly rounded except conversion between decimal and binary. The reason is that efficient algorithms for exactly rounding all the operations are known, except conversion. For conversion, the best known efficient algorithms produce results that are slightly worse than exactly rounded ones [Coonen 1984]. The IEEE standard does not require transcendental functions to be exactly rounded because of the table maker s dilemma. To illustrate, suppose you are making a table of the exponential function to 4 places. Then exp(1.626) 5.0835. Should this be rounded to 5.083 or 5.084 If exp(1.626) is ....

Coonen, Jerome 1984. Contributions to a Proposed Standard for Binary FloatingPoint Arithmetic, PhD Thesis, Univ. of California, Berkeley.

....and by INTEL for their Pentium processors. At the end we also mention a multiprecision C class library developed by one of the authors, which supports a U implementation. The consistency of each implementation was tested using a large set of test vectors which is a generalization of Coonen s [3] and Hough s [7] sets of test vectors and which will be publicly available soon [4] For each of the precisions (single, double, 64 bit extended on INTEL and 113 bit quadruple on SUN) our test set for multiplication contained 1152 cases of U underflow, an extra 176 cases of V:U underflow and an ....

J.T. Coonen. Contributions to a proposed standard for binary floating-point arithmetic. University of California, Berkeley, 1984.

....IEEE floating point arithmetic and gives an overview of the main features of our testing tool. In section 3 the concept of test vector and the precision independent syntax to encode floating point operands, is introduced. This syntax is a streamlined and extended version of the syntax developed by [Coonen 1984]. In section 4 till section 8 we discuss, for each of the six basic operations, which aspects of the operation are tested by the test vectors. This discussion is restricted to the newly added test vectors. Section 9 describes the functionality of the driver program. In section 10, we discuss the ....

....which is almost correct, errors are more likely to occur as edge effects, at or near some discontinuity or boundary in the values of the operands or some part of them. Last but certainly not least, a test suite which has been used by major manufacturers of IEEE hardware was developed by J.T. Coonen [Coonen 1984]. This test tool consists of a large database of test vectors together with a driver program. An essential feature of this tool is that the vectors were designed to be as format independent as possible, so that the same cases would apply to all IEEE formats. The vectors are very useful for testing ....

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Coonen, J. T. 1984. Contributions to a Proposed Standard for Binary Floating-Point Arithmetic. Ph. D. thesis, University of California at Berkeley (UCB), Berkeley, California, USA.

..... Thus for P 13, the use of the single extended format enables 9 digit decimal numbers to be converted to the closest binary number (i.e. exactly r ounded) If P 13, then single extended is not enough for the above algorithm to always compute the exactly r ounded binary equivalent, but Coonen [1984] shows that it is enough to guarantee that the conversion of binary to decimal and back will r ecover the original binary number. 176 Numerical Computation Guide E If double pr ecision is supported, then the algorithm above would be r un in double precision rather than single extended, but to ....

....exactly rounded except conversion between decimal and binary . The r eason is that efficient algorithms for exactly r ounding all the operations ar e known, except conversion. For conversion, the best known ef ficient algorithms pr oduce results that ar e slightly worse than exactly r ounded ones [Coonen 1984]. The IEEE standard does not r equire transcendental functions to be exactly rounded because of the table maker s dilemma. T o illustrate, suppose you ar e making a table of the exponential function to 4 places. Then exp(1.626) 5.0835. Should this be r ounded to 5.083 or 5.084 If exp(1.626) is ....

Coonen, Jerome 1984.Contributions to a Proposed Standard for Binary FloatingPoint Arithmetic , PhD Thesis, Univ . of California, Berkeley.

....requiring sophisticated numerical analysis [57] see section 6. 10.1 for an example) The double extended format also allows better algorithms to be used for certain problems, such as more accurate iterative refinement techniques for systems of linear equations [62] and binary to decimal conversion [21]. 6.1.2. Specification FloatingPointType: one of float double indigenous FloatTypeSuffix: one of f F d D n N Figure 2 Modifications to Java s grammar to support Borneo s indigenous type. Borneo adds indigenous as a primitive floating point type and indigenous as a keyword. The size ....

Jerome Coonen, Contributions to a Proposed Standard for Binary Floating-Point Arithmetic, Ph.D. Thesis, University of California, Berkeley 1984.

....(fixed point notation) In Section 3 we explain several ways to speed up such calculations. Section 4 describes some computational experience, and 5 offers concluding remarks. The rest of this section introduces some notation and assumptions. For other work on conversions, see Coonen s thesis [4] and the works cited in [3, 4, 12] Since signs are easy to treat, we restrict the discussion to conversion of nonnegative numbers. We assume that nonnegative normalized internal floating point numbers have the form (1) b = i = 0 S p 1 b i b e i , where the floating point arithmetic ....

....In Section 3 we explain several ways to speed up such calculations. Section 4 describes some computational experience, and 5 offers concluding remarks. The rest of this section introduces some notation and assumptions. For other work on conversions, see Coonen s thesis [4] and the works cited in [3, 4, 12]. Since signs are easy to treat, we restrict the discussion to conversion of nonnegative numbers. We assume that nonnegative normalized internal floating point numbers have the form (1) b = i = 0 S p 1 b i b e i , where the floating point arithmetic base b and the number p of base b ....

J. T. Coonen, "Contributions to a Proposed Standard for Binary Floating-Point Arithmetic," Ph.D. Dissertation (1984), Univ. of California, Berkeley.

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