Consider a computer that uses 8 bits to represent floating-point numbers, 1 bit for the sign...

 

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Consider a computer that uses 8 bits to represent floating-point numbers, 1 bit for the sign s, 3 bits for the exponent c (c =e +3), and 4 bits for fractional part f. In terms of s, e, and f, the base 10 numbers are given by x = (-1)' 2 (1+ f), c is non-negative, and 0 f <1. This is the same design as the IEEE floating- point representation with a much shorter machine word length. (a) What are the smallest and largest positive (non-zero) numbers that can be represented accurately (without any chopping/rounding error) on this computer? (b) How many numbers can be accurately (without any chopping/rounding error) represented on this computer? (c) By changing c to 4 bits and f to 3 bits, what do you gain and what do you lose? Consider a computer that uses 8 bits to represent floating-point numbers, 1 bit for the sign s, 3 bits for the exponent c (c =e +3), and 4 bits for fractional part f. In terms of s, e, and f, the base 10 numbers are given by x = (-1)' 2 (1+ f), c is non-negative, and 0 f <1. This is the same design as the IEEE floating- point representation with a much shorter machine word length. (a) What are the smallest and largest positive (non-zero) numbers that can be represented accurately (without any chopping/rounding error) on this computer? (b) How many numbers can be accurately (without any chopping/rounding error) represented on this computer? (c) By changing c to 4 bits and f to 3 bits, what do you gain and what do you lose?

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