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5/5 - (1 vote)

To get full credit, you must write down sufficient intermediate steps, only giving the final answer earns you no credit!
Please make sure that your handwriting is recognizable, otherwise you only get partial credit for the recognizable part.

Let x = β^e, e ∈ Z, L < e < U be a normalized FPN in F and x_L,x_R∈ F the two normalized FPNs adjacent to x such that x_L< x < x_R. Prove x_R− x = β(x − x_L).

By reusing your result of II, find out the two nor-malized FPNs adjacent to x = 3/5 under the IEEE 754 single-precision protocol. What is fl(x) and the relative roundoff error?
If the IEEE 754 single-precision protocol did notround off numbers to the nearest, but simply dropped excess bits, what would the unit roundoff be?
How many bits of precision are lost in the subtraction

1 − cosx when?

Suggest at least two ways to compute 1 − cosx to avoid catastrophic cancellation caused by subtraction.

The above eight questions weigh 3, 4, 7, 6, 3, 3, 4 points, respectively, totaling 30 points.

1 C++ programming

(10 points) By programming in C++, print values of the functions in (1) at 101 equally spaced points covering the interval [0.99,1.01]. Calculate each function in a straightforward way without rearranging or factoring. Note that the three functions are theoretically the same, but the computed values might be very different. Plot these functions near 1.0 using a magnified scale for the function values to see the variations involved. Discuss what you see. Which one is the most accurate? Why?
(10 points) Consider a normalized FPN system F with the characterization β = 2,p = 3,L = −1,U = +1. Answer the following by programming in C++
- compute UFL(F) and OFL(F) and output them as decimal numbers;
- enumerate all numbers in F and verify the corollary on the cardinality of F in the summary handout;
- plot F on the real axis; • enumerate all the subnormal numbers of F;
- plot the extended F on the real axis.

Thus the total point of this homework is 50