Web host 4 life - 4.2.1 SINGLE-PRECISION CALCULATIONS 209 Similarly, suppose MIX had
4.2.1 SINGLE-PRECISION CALCULATIONS 209 Similarly, suppose MIX had a FADD operation but not FIX. If we wanted to round a number u from floating point form to the nearest fixed point integer, and if we knew that the number was nonnegative and would fit in at most three bytes, we could write FADD FUDGE where location FUDGE contains the constant + Q+4 1 0 0 0 ; the result in rA would be + Q+4 I iz$j-. (13) I 1 D. History and bibliography. The origins of floating point notation can be traced back to Babylonian mathematicians (1800 B.C. or earlier), who made extensive use of radix-60 floating point arithmetic but did not have a notation for the exponents. The appropriate exponent was always somehow understood by the man doing the calculations. At least one case has been found in which the wrong answer was given because addition was performed with improper alignment of the operands, but such examples are very rare; see 0. Neugebauer, The Exact Sciences in Antiquity (Princeton, N. J.: Princeton University Press, 1952), 26-27. Another early contribution to floating point notation is due to the Greek mathematician Apollonius (3rd century B.C.), who apparently was the first to explain how to simplify multiplication by collecting powers of 10 separately from their coefficients, at least in simple cases. [For a discussion of Apollonius s method, see Pappus, Mathematical Collections (4th century A.D.).] After the Babylonian civilization died out, the first significant uses of floating point notation for products and quotients did not emerge until much later, about the time logarithms were invented (1600) and shortly afterwards when Oughtred invented the slide rule (1630). The modern notation z~ for exponents was being introduced at about the same time; separate symbols for x squared, x cubed, etc., had been in use before this. Floating point arithmetic was incorporated into the design of some of the ear- liest computers. It was independently proposed by Leonardo Torres y Quevedo in Madrid, 1914; by Konrad Zuse in Berlin, 1936; and by George Stibitz in New Jersey, 1939. Zuse s machines used a floating binary representation that he called semi-logarithmic notation ; he also incorporated conventions for dealing with special quantities like 00 and undefined. The first American computers to operate with floating point arithmetic hardware were the Bell Laboratories Model V and the Harvard Mark II, both of which were relay calculators designed in 1944. [See B. Randell, The Origins of Digital Computers (Berlin: Springer, 1973), 100, 155, 163-164, 259-260; Proc. Symp. Large-Scale Digital Calculating Machinery (Harvard, 194i ), 41-68, 69-79; Datamation 13 (April 1967), 35-44 (May 196i ), 45-49; Zeit. fiir angew. Math. und Physik 1 (1950), 345-346.1