Assignment 10, due Apr 6

On every assignment, write your name legibly as it appears on your University ID card! Homework is due on paper at the start of class on the day indicated (usually Friday). Exceptions will be made only by advance arrangement (excepting "acts of God"). Late work must be turned in to the TA's mailbox (ask the CS receptionist in 14 MLH for help). Never push homework under someone's door!

1. Background: Here is a floating-point representation that might have been of some small use on a 16-bit minicomputer. It is designed to explicitly avoid advanced concepts common in floating point representations today:

   |_ _ _ _|_ _ _ _|_ _ _ _|_ _ _ _|
   |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
   |s|   exp   |      mantissa     |
   

   • s -- the sign bit; 1 = negative, 0 = positive.
   • exp -- the exponent; biased so 00000 = -16 and 11111 = 15.
   • mantissa -- a binary fraction, so that 10 0000 0000 = 0.5

     If the exponent is greater than 00000, the mantissa should always be normalized in the range from 0.5 to just less than 1.0.

   Each part is worth 0.2 points.

   a) What is the approximate decimal equivalent of FFFF₁₆ in this number system?

   b) What is the exact decimal equivalent of 1234₁₆ in this number system?

   c) What is the normalized binary representation of of 1 in this number system?

   d) What is the normalized binary representation of of 10₁₀ in this number system?

   e) What is the normalized approximate binary representation of of 0.1₁₀ in this number system?

   Background: Here is a floating-point representation that might have been of some small use on a 16-bit minicomputer. It is designed to make use of all of the advanced concepts typical of modern floating-point numbers

   |_ _ _ _|_ _ _ _|_ _ _ _|_ _ _ _|
   |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
   |s|   exp   |      mantissa     |
   

   • s -- the sign bit; 1 = negative, 0 = positive.

   • exp -- the exponent; biased so 00001 = -15 and 11110 = 14.

     The special exponent value 00000 also represents -15, but indicates that the mantissa is not normalized.

     The special exponent value 11111 means not a number or infinity.

   • mantissa -- a binary fraction.

     For normalized numbers, there is a hidden bit just to the left of the point, so the mantissa 00 0000 0000 = 1.0 and 10 0000 0000 = 1.5 (the hidden bit is always one as a consequence of normalization).

     For non-normalized numbers, the hidden bit is zero, so 00 0000 0000 = 0.0 and 10 0000 0000 = 0.5.

   Each part is worth 0.2 points.

   a) What is the binary equivalent of 1.0₁₀ in this number system.

   b) What is the binary representation of the largest non-infinite positive number in this system?

   c) What is the approximate decimal equivalent of your answer to part a? (That was a typo, this should have asked about part b!)

   d) What is the binary representation of the smallest non-zero positive number in this system?

   e) What is the approximate decimal equivalent of your answer to part c? (That was a typo, this should have asked about part d!)

2. A Problem: Write a SMAL Hawk subroutine that takes, as an argument, a floating point number in the least significant 16 bits of R3 encoded in the format defined for problem 1. The subroutine should convert this number to the smallest integer greater than or equal to that number and return the result in R3 when it is done. Write textbook-quality code. Insufficient or excessive comments will be penalized. (1 point)