1- Percentage of Vectorization
a-
Let f=fraction speedup , s=speedup on that fraction
New_time=(1-f)*old_time+(f/s)*old_time
Speedup = old_time / new_time
Amdahl’s Law ;
Speedup = 1/[(1-f)+(f/s)]
Label the Y axis “Net speedup” and label the X axis “Percent Vectorization” ;
for s=40 (times faster) ;
X : the percentage vectorization, f = 0 to 1 as following
When we calculate Net Speedup with Amdahl’s Law , we get following results ;
X : Percent Vectorization
0
0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90
1,00
Y : Net Speedup
1
1,11 1,24 1,41 1,64 1,95 2,41 3,15 4,55 8,16 40,00
Vector Mode Speedup Graph
45
Y : Net Speedup;
40,00
40
Net Speedup
35
30
25
20
15
10
5
Percent Vectorization
0
0
b-
0,10
0,20
0,30
0,40
0,50
Speedup = 2 and s =40 , f = ?
Speedup = 1/[(1-f)+(f/s)]  2= 1/[(1-f)+(f/40)] ,
we can calculate f value from this equation ,
percent vectorization , f= %51
0,60
0,70
0,80
0,90
1,00
c-
Since the maximum speedup attainable would be 40, one-half of this value would be
Speedup =20 and s =40 , f= ?
Speedup = 1/[(1-f)+(f/s)]  20= 1/[(1-f)+(f/40)] ,
we can also calculate f value from this equation ,
percent vectorization , f= %97
The percentage vectorization needed to gain one-half of the maximum speedup
attainable is %97.
d- Using the hardware doubling the following equation can be generated:
f=%55 and s =80 , Speedup = ?
Speedup = 1/[(1-f)+(f/s)]  = 1/[(1-0,55)+(0,55/80)] ,
Speedup = 2,189
To find what percentage of vectorization would need to be achieved to match the
hardware doubling ; Speedup =2,189 and s =40 , f= ?
Speedup = 1/[(1-f)+(f/s)]  2,189= 1/[(1-f)+(f/40)] ,
we can also calculate f value from this equation ,
percent vectorization , f= %55,72
2- Iron Law
Assume code size is 100. CPI can be calculated as
Assume cycle time is 1 for initial case. Then performance will be
If ALU instructions are removed from branch instructions, then 20% of ALU instructions will be
removed. We assumed code size is 100 for simplicity. Then 20 ALU should be removed. New code size is
reduced to 80 with 30 ALU, 30 Memory and 20 branch instructions. New CPI can be calculated as
Then performance can be calculated as
Even though ALU instructions are removed, performance of the new design is worse.