Please see question below Assume you use the following 3 digit decimal arithmeti

Question

Please see question below

Assume you use the following 3 digit decimal arithmetic plusminus X.YZ x 10E where X is any decimal digit 1,... ,9 (0 is admitted only for true zero and unnormalized numbers) Y.Z are any decimal digits 0,..., 9, E is any small decimal integer digit -4,..., 0,..., +4. Assume that we use the "round-to-even" rule, that overflows become inf 's and that underflows become zeroes. The "round-to-even" rule says that any result that is not already representable should be rounded to the nearest representable number, ignoring exponent limits. If the resulting number has an exponent beyond the exponent limits, then the answer is one of the exceptional values. If the result is exactly half way between two representable numbers, it should be rounded to the representable number that ends in an even digit. Answer the following questions (a ) Compute 2.04 x 10+0 + 4.50 x 10-22 (b ) Compute 9.08 x 10+4 + 9.19 x 10+3 (c ) Compute 1.04 x 10-3 - 9.98 x 10-4 (d ) What is the unit roundoff mach for this system, defined as the smallest representable floating point such that fl(l + ) > 1. (e ) Find the positive real number R. such that any real number x strictly larger than R would round to a nonzero floating number in this system, while any positive x strictly less than R would round to zero (underflow). (f ) Give an example of a positive floating point number of the form a = 1.000 x 10k such that the floating product fl(a * a) > 0, but fl(a * a * a) = 0. (g ) How many distinct positive numbers can be represented in this system?

Explanation / Answer

a)2.04*10^0+4.50*10^(-2)=2.04*10^0+.045*10^0

=2.085*10^0=2.08*10^2(after rounding)

b)9.08 *10^4+9.19*10^3=9.08*10^4+.919*10^4=9.999*10^4=1.00*10^5

c)1.04*10^(-3)-9.98*10^(-4)=1.04*10^(-3)-.998*10^(-3)=.042*10^(-3)=.04*10^(-3)

d).06*10^(-1)

e).051*10^(-5)

f)(0.06*10^(-5))^.5=7.14*10^-4

g)10*10*10*9-8 as when 0.00 is base all the power give same numbers.

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