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Gradual and tapered overflow and underflow: A functional differential equation and its approximation. (English) Zbl 1089.65041
Summary: A modified structure for floating-point representation, which can essentially eliminate overflow from floating-point calculations, is analyzed. The main part of the paper deals with a computational examination of a model for floating-point exponents and the probabilities of overflow and underflow. Earlier results in the absence of gradual underflow are presented for comparison but the primary focus is on the effect of gradual underflow, a version of gradual overflow, and of a proposal for an extended treatment of these exceptions which we call tapered overflow and underflow. The latter virtually eliminates the exceptions; its overflow threshold is \(10^{600000000}\).
This paper reviews several models for the distribution of exponents of floating-point numbers and their evolution in the presence of repeated multiplicative operations. A continuous model, which closely approximates the discrete case, is seen to satisfy a functional differential equation with both delay and advance terms.
MSC:
65G50 Roundoff error
68P01 General topics in the theory of data
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[1] Arnold, M.G.; Bailey, T.A.; Cowles, J.R.; Winkel, M.D., Applying features of IEEE754 to signed logarithmic arithmetic, IEEE trans. comput., 41, 1040-1050, (1992)
[2] Barlow, J.L.; Bareiss, E.H., On roundoff distribution in floating-point and logarithmic arithmetic, Computing, 34, 325-364, (1985) · Zbl 0556.65036
[3] Benford, F., The law of anomalous numbers, Proc. amer. philos. soc., 78, 551-572, (1938)
[4] Brent, R., A Fortran multiple precision arithmetic package, ACM toms, 4, 57-70, (1978)
[5] Clenshaw, C.W.; Olver, F.W.J., Level-index arithmetic operations, SIAM J. numer. anal., 24, 470-485, (1987) · Zbl 0624.65016
[6] Clenshaw, C.W.; Turner, P.R., The symmetric level-index system, IMA J. numer. anal., 8, 517-526, (1988) · Zbl 0668.68018
[7] Feldstein, A.; Goodman, R.H., Some aspects of floating-point computation, (), 169-181
[8] Feldstein, A.; Turner, P.R., Overflow, underflow and severe loss of significance in floating-point addition and subtraction, IMA J. numer. anal., 6, 241-251, (1986) · Zbl 0593.65029
[9] Feldstein, A.; Turner, P.R., Overflow and underflow in multiplication and division, Appl. numer. math., 21, 221-239, (1996) · Zbl 0860.65034
[10] Hamada, H., URR: universal representation of real numbers, New generation computing, 1, 205-209, (1983)
[11] Hamada, H., A new real number representation and its operation, (), 153-157
[12] Hull, T.E.; Cohen, M.S.; Hall, C.B., Specifications for a variable precision arithmetic coprocessor, (), 127-131
[13] IEEE, Standard for binary floating-point arithmetic, ANSI/IEEE std, vol. 754, (1985), IEEE New York
[14] Koren, I., Computer arithmetic algorithms, (1998), Brookside Court Amherst, MA
[15] Lewis, D.M., An architecture for addition and subtraction of long wordlength numbers in the logarithmic number system, IEEE trans. comput., 39, 1326-1336, (1990)
[16] Lozier, D.W., An underflow-induced graphics failure solved by SLI arithmetic, (), 10-17
[17] Lozier, D.W.; Turner, P.R., Robust parallel computation in floating-point and sli arithmetic, Computing, 48, 239-257, (1992) · Zbl 0757.68020
[18] Matsui, S.; Iri, M., An overflow/underflow-free floating-point representation of numbers, J. inform. proc., 4, 123-133, (1981)
[19] Morris, R., Tapered floating-point: A new floating-point representation, IEEE trans. comput., 20, 1578-1579, (1971) · Zbl 0226.68019
[20] Turner, P.R., The distribution of leading significant digits, IMA J. numer. anal., 2, 407-412, (1982) · Zbl 0503.65029
[21] Turner, P.R., Further revelations on l.s.d., IMA J. numer. anal., 4, 225-231, (1984) · Zbl 0564.65028
[22] Yokoo, H., Overflow/underflow-free floating-point number representations with self-delimiting variable length exponent field, (), 110-117
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