Free web hosting music - 210 ARITHMETIC 4.2.1 The use of floating binary
210 ARITHMETIC 4.2.1 The use of floating binary arithmetic was seriously considered in 1944-1946 by researchers at the Moore School in their plans for the first electronic digital computers, but it turned out to be much harder to implement floating point circuitry with tubes than with relays. The group realized that scaling was a problem in programming; but at the time it was only a very small part of a total programming job, and it seemed to be worth the time and trouble it took, since it tended to keep a programmer aware of the numerical accuracy he was getting. Furthermore, they argued that floating point representation would take up valuable memory space, since the exponents must be stored, and that it would be difficult to adapt floating point arithmetic to multiple-precision calculations. [See von Neumann s Collected Works 5 (New York: Macmillan, 1963), 43, 73-74.1 At this time, of course, they were designing the first stored- program computer and the second electronic computer, and their choice had to be either fixed point or floating point arithmetic, not both. They anticipated the coding of floating binary routines, and in fact shift left and shift right instructions were put into their machine primarily to make such routines more efficient. The first machine to have both kinds of arithmetic in its hardware was apparently a computer developed at General Electric Company [see Proc. 2nd Symp. Large-Scale Digital Calculating Machinery (Cambridge: Harvard University Press, 1951), 65-691. Floating point subroutines and interpretive systems for early machines were coded by D. J. Wheeler and others, and the first publication of such routines was in The Preparation of Programs for an Electronic Digital Computer by Wilkes, Wheeler, and Gill (Reading, Mass.: Addison-Wesley, 1951), subroutines Al-All, pp. 35-37, 105-117. It is interesting to note that floating decimal subroutines are described here, although a binary computer was being used; in other words, the numbers were represented as lO f, not aef, and therefore the scaling operations required multiplication or division by 10. On this particular machine such decimal scaling was about as easy as shifting, and the decimal approach greatly simplified input/output conversions. Most published references to the details of floating point arithmetic routines are scattered in technical memorandums distributed by various computer man- ufacturers, but there have been occasional appearances of these routines in the open literature. Besides the reference above, the following are of historical interest: R. H. Stark and D. B. MacMillan, Math. Comp. 5 (1951), 86-92, where a plugboard-wired program is described; D. McCracken, Digital Computer Pro- gramming (New York: Wiley, 1957) 121-131; J. W. Carr III, CACM 2,5 (May 1959), 10-15; W. G. Wadey, JACM 7 (1960), 129-139; D. E. Knuth, JACM 8 (1961), 119-128; 0. Kesner, CACM 5 (1962) 269-271; F. P. Brooks and K. E. Iverson, Automatic Data Processing (New York: Wiley, 1963), 184-199. For a discussion of floating point arithmetic from a computer designer s standpoint, see Floating point operation by S. G. Campbell, in Planning a computer System, ed. by W. Buchholz (New York: McGraw-Hill, 1962), 92-121. A set of algorithms by J. Coonen, W. M. Kahan, and H. S. Stone, submitted to the IEEE Micro- processor Floating-Point Standards Committee during 1978-1980, represented