For this exam, you may use your textbook, your notes and assignments, your brain, the Math/CS computers, and divine inspiration, if available. If you get stuck, talk to Jeff Ondich, but please don't talk to anyone else about the exam.

Explain and justify your answers. Submit your answers on paper.

1. (10 points) The following questions concern IEEE 754 32-bit floating point numbers. When I refer to a "representable number," I mean a number whose exact value is one of the values that has a 32-bit IEEE 754 representation.
   • What is the smallest positive representable number? (Call this number A.)
   • What is the second smallest positive representable number? (Call this number B.)
   • Suppose you represent the number x = (A+B)/2 by A. What is the relative error of x as represented by A? (Relative error of an approximation = (actual # - approximation) / (actual #).)
   • What is the largest positive representable number? (Call this number C.)
   • What is the second largest positive representable number? (Call this number D.)
   • Suppose you represent the number y = (C+D)/2 by C. What is the relative error of x as represented by C?
   • What is the smallest positive integer S that is not representable? Are there any representable integers larger than S?
   
   
   
2. (15 points) Suppose we have a display that consists of four line segments forming a square. The display has four input lines, one for each side of the square. When any of the input lines is asserted, the corresponding side of the square lights up.

   Your goal for this problem is to design a clock-driven circuit that will create a simple animation on the four-line display. The "frames" of the animation will be:

   
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   and
   
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   That is, there's a little right-angle shape that goes around in a circle.

   Your job is to design a clock-driven circuit that has one input (other than the clock), which we will call A, and four outputs, T, R, L, and B (top, right, left, and bottom). When A=0, the little right angle should go around in a circle clockwise as the clock ticks. When A=1, the shape should go around counter-clockwise.

   You may find it helpful to design a state diagram and use logical formulas to express T, R, L, and B in terms of state numbers. If so, show me your diagram and formulas.
   
   

3. (2 points) Tell me a joke, if you wish. Otherwise, go ahead and collect your 2 points humorlessly.
   
   
4. (12 points) Do Problem 5.11 on page 428 of Patterson and Hennessy.
   
   
5. (12 points) Carry lookahead versus ripple carry.
   • Suppose each gate (AND, OR, XOR, NOT, NAND, and NOR) has a 1 ns propagation delay. What is the propagation delay of a 32-bit ripple carry adder?
   • How many gates does a 32-bit ripple carry adder require?
   • Read about carry lookahead addition starting on page 241 of Patterson and Hennessy.
   • Still assuming 1 ns per gate, what is the propagation delay of a 32-bit "infinite hardware" adder as described on page 241 and 242?
   • How many gates are in a 32-bit infinite hardware adder?
   • Finaly, what are the propagation delay and gate count for a 32-bit adder using the "one level of abstraction" approach of pages 242 and 243?