University of California at Berkeley

College of Engineering

Department of Electrical Engineering and Computer Science

EECS 61C, Spring 2005

Lab 6: Floating Point and Integer Representation

# Purpose

To examine how computers store integer and floating point values.

Floating Point in P&H

### Description

Recall that the single precision floating point number is stored as:

```SEEE EEEE EIII IIII IIII IIII IIII IIII```

where:

S is the sign bit, 0 for positive, 1 for negative
E is the exponent, bias 127
I is the significant, with an implicit 1

For example, the floating point representation of 1.0 would be 0x3F800000. Verify to yourself that this is correct.

Exercise 1: Integer numbers

Find the shortest sequence of MIPS instructions to determine if there is a carry out from the addition of two registers, say \$t3 and \$t4. Place a 0 or 1 in register \$t2 if the carry out is 0 or 1, respectively. (This can be done in just two instructions). Verify that your code works for the following values:

 Operand Operand Carry out? `0x7fffffff` `0x80000000` no `0xffffffff` `0` no `0xffffffff` `1` yes

Exercise 2: Floating point numbers

Find a positive floating point value x, for which x+1.0=x. Verify your result in a MIPS assembly language program, and determine the stored exponent and fraction for your x value (either on the computer or on paper).

Note: The provided MIPS program
p2.s (in ~cs61c/labs/lab06) will allow you to experiment with adding floating point values. It leaves the output in \$f12 and also \$s0, so you can examine the hex representation of the floating point value by printing out \$s0.

Exercise 3: Floating point numbers revisited

Next, find the smallest positive floating point value x for which x+1.0=x. Again, determine the stored exponent and fraction for x.

Exercise 4: Floating point associativity

Finally, using what you have learned from the last two parts, determine a set of positive floating point numbers such that adding these numbers in a different order can yield a different value. You can do this using only three numbers. (Hint: Experiment with adding up different amounts of the x value you determined in part 3, and the value 1.0).

This shows that for three floating point numbers `a`, `b`, and `c`, `a+b+c` does not necessarily equal `c+b+a`.

If time permits, you should write a program to add these three values in different orders. It should be a straightforward modification of the program from part 2-3.