suppose we wish to write a procedure that computes the inner product of two vectors. An abstract version of the function has a CPE of 54 for both integer and floating point data.

void inner4(vec_ptr u, vec_ptr v, data t *dest)
{
int i;
int length = vec_length(u);
data_t *udata = get_vec_start(u);
data_t *vdata = get_vec_start(v);
data_t sum = (data_t) 0;

for (i = 0; i
sum = sum + udata[i] * vdata[i];
}

*dest = sum;
}

Our measurements show that this fucntion requires 3.11 cycles per iteration for integer data. The assembly code for the inner loop is as follows:



.L24:
movl (%esi,%edx,4),%eax Get udate[i]
imull (%ebx,%edx,4),%eax Multiply by vdata[i]
addl %eax,%ecx Add to sum
incl %edx i++
cmpl %edi,%edx Compare i:length
jl .L24 If


Assume that integer multiplcation is performed by the general integer functional unit and that this unit is pipelined. This means that one cycle after a multiplication has started, a new integer operation (multiplication or otherwise) can begin. Assume also that the Integer/Branch function unit can perform simple integer operations.

A) show the translation of these lines of assembly code into a sequence of operations. The movl instruction translates into a single load operation. Register %eax gets updated twice in the loop. Label the different versions %eax.1a and %eax.1b.

B) Explain how the function can go faster than the number of cycles required for integer muiltiplication.

C) Explain what factor limits the performance of this code to at best a CPE of 2.5.

D) For floating-point data, we get a CPE of 3.5. Without needing to examine the assembly code, describe a factor that will limit the performance to at best 3 cycles per iteration.

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Write a version of the inner product procedure described in the previous problem that uses four-way loop unrolling.
Our measurement for this procedure gives a CPE of 2.20 for integer data and 3.50 for floating point.

A) explain why any version of any inner product procedure cannoy achieve a CPE greater than 2.
B) Explain why the performance for floating point did not improve with loop unrolling.


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Write a version of the inner product procedure described in the first problem that uses four-way loop unrolling and two-way parrallelism.
Our measurements for this procedure give a CPE of 2.25 for floating-point data. Describe two factors that limit the performance to a CPE of at best 2.0