[Solved] CIS450 Computer Organization and Architecture Homework #1

(25 points) Problems:

(2.43/2.61) Using only bit-level and logic operations, write C expressions that yield 1 for the described condition, and 0 otherwise. Assume that x is of type int. You may not use any equality (==) or inequality (!=) tests.

1. Any bit of x equals 1. _______________________________________________

1. Any bit of x equals 0. _______________________________________________
2. Any bit in the least significant byte of x equals 1. _________________________
3. Any bit in the least significant byte of x equals 0. _________________________

Hint: One solution to part (a.) is: !!x (and, yes, you can use this as your solution :-).

2. (2.81) Write C expressions to generate the bit patterns that follow, where a^k represents k repetitions of symbol a. Assume a w-bit data type. Your code may contain references to j and k, representing the values of j and k, but not a parameter representing w.

1. 1w-k 0k
2. 0w-k-j 1k 0 j

For example, if we are dealing with chars, an 8-bit data type, and k = 3, then for a. we want an expression that results in 11111000.

3. (2.50/2.76/2.77) Suppose we are given the task of generating code to multiply integer variable x by various different constant factors K. To be efficient, we want to use only the operations +, -, and <<. For the following values of K, write C expressions to perform the multiplication using at most three operations per expression.

1. K = 31: ________________________________________________________
2. K = -7: ________________________________________________________
3. K = 80: ________________________________________________________
4. K = 144: ________________________________________________________

Hint: One solution to part (a.) is: (x<<5) x.

4. (2.55/2.82/2.83) Consider the numbers having a binary representation consisting of an infinite string of the form 0.yyyyyy, where y is a k-bit sequence. For example, the binary representation of 1/3 is 0.0101010101 (that is, y = 01, and k = 2), while the representation of 1/5 is 0.001100110011 (that is, y = 0011, and k = 4).
   1. Give a formula in terms of y and k for the value represented by the infinite string. Hint: Consider the effect of shifting the binary point k positions to the right, and do part b. first

1. What is the numeric value of the string for the following values of y? Note that the value of k is implied; e.g., for case i., k = 3, etc.

1. 110 ____________________________________________________ ii. 01001 ____________________________________________________ iii. 010111 ____________________________________________________

5. Consider a 16-bit twos complement representation for signed integers. Fill in the empty boxes in the following table. Spaces in the binary representation are just added to enhance readability.

Show work. Hint: 2¹⁵ = 32768, 2¹⁴ = 16384, .. , 2⁵ = 32, 2⁴ = 16, .. , 2¹ = 2, 2⁰ = 1