Average number of bits processed in a cycle = (4+4+4+2+4+2+3+3+4+3+3+2+4+2+4+4)/16 = 3.25

1. -3.25
2. 1.25 x 2^(-128)
3. Negative Infinity

The bias for the 1-8-23 IEEE format is 127.

  3EE00000h = 0 01111101 11000000000000000000000 = 1.11 * 2^-2
  3D800000h = 0 01111011 00000000000000000000000 = 1.0 * 2^-4

Adjusting exponents, we get

        1.11 * 2^(-2)
        0.01 * 2^(-2)


Since the exponents are now the same, we can add the two
fractions. The result has the same exponent as the operands. The
result is: 10.00 * 2^(-2). When normalized to IEEE format, this
becomes 1.00 * 2^(-1),

IEEE format: 0 01111110 00000000000000000000000 which in hex is
3F000000h.

Model 0.001: 1-7-8
Smallest positive normalized number is 0 0000001 00000000 = 1.0 * 2^(1-63) = 2^-62
Largest postive normalized number is 0 1111110 11111111 = 1.11111111 * 2^(126-63) = 1.11111111 * 2^63

This model has (8+1) binary bits of precision. Since 1024 (2^10) is
approximately 1000 (10^3); 10 binary bits of precision correspond to
about 3 decimal digits of precision. Hence, this model has 9 * (3/10)
= 2.7 decimal digits of precision

Model 0.002: 1-5-10
Smallest positive normalized number is 0 00001 0000000000 = 1.0 * 2^(1-15) = 2^-14
Largest postive normalized number is 0 11110 1111111111 = 1.1111111111 * 2^(30-15) = 1.1111111111 * 2^15

This model has (10+1) binary bits of precision. Hence, this model has
11 * (3/10) = 3.3 decimal digits of precision

The model chosen would depend on the application - if a higher range
is desired, then model 0.001 is better; if precision is more
important, then model 0.002 will be preferred.

1. 5 bits
2. 1

3.  
         48       0 110 10000
         19.5     0 101 00111
         5/16     0 000 01010
         0        0 000 00000
        -1        1 001 00000
      -infinity   1 111 00000
         

a) Single processor system - using Horner's rule

(((((ax + b)x + c)x + d)x + e)x + f)x + g

Number of operations = 12 (6 multiplies & 6 adds)

Number of time-steps = 12

b) Using 4 processors: time steps = 5, operations = 15.

One way of doing this is:

An intuitive proof for why it can't be done in less than 5 time steps:

Lets assume it takes x time steps. In the xth time step, only one operation can be performed (either an add or a multiply) to get the final result. In the (x-1)th time step, at the most two operations can be performed, one each to get each of the operands for the operation in step x. In the (x-2)nd time step & other below, possibly 4 units of work can be found. If x = 4, then theres no way to compute all the operands such that only 3 operations are left to be performed after 2 time steps.

c)  Speedup = T₁/ T_p = 12/5 = 2.4

Reasonable choice depends on whether lower cost, latency, or contention is desired.

For n=64, Using k=64 yields the lowest latency, and k=2 or k=4 yields the lowest cost. If we would like low latency we would choose k to be 64. If we would like lowest cost, we would choose k to be 2 or 4. If we would like to minimize contention, we would choose k to be 64.

Another approach could be to minimize the cost-latency product, in which case we would choose k=8 for n=64.