Computer Organization and Design MIPS Edition, Fifth Edition: The Hardware/Software Interface (The Morgan Kaufmann Series in Computer Architecture and Design)
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Chapter B, Problem 27E

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Computer Organization and Design MIPS Edition, Fifth Edition: The Hardware/Software Interface (The Morgan Kaufmann Series in Computer Architecture and Design), Chapter B, Problem 27E

  • The above diagram describes about the 64 bit carry look ahead adder that includes 64 numbers of 1-bit full adders to add the 64 bits.
  • It includes carry generator, carry propagator and sum.
    • Carry generator denoted by G is used for generating the output query without depending on input query.
    • Carry propagator denoted by P is used for propagating the next state.
    • The sum of each adder is denoted by S.

From the above diagram, the carry propagator (P) and carry generator (G) are calculated as follows

Pi=AiBi

Gi=AiBi

Then, the sum of the each adder can be calculated as follows

Sum(Si)=PiCi

Thus, the required equation is Ci+1=Gi+(PiCi) (1)

Substitute, “0” for “I” in the equation (1)

The first equation can be calculated as follows

C1=G0+(P0C0)

The first equation with the new notation is c4=G3,0+(P3,0c0).

Thus, the first equation with the new notation for 64 bit is c16=G15,0+(P15,0c0).

The second equation can be calculated as follows

C2=G1+(P1C1) (2)

Substitute, “G0+(P0C0)” for “C1” in the equation (2)

C2=G1+P1G0+P1P0C0

The second equation with new notation is c8=G7,4+(P7,4G3,0)+(P7,4G3,0)+(P7,4P3,0c0).

Thus, the second equation with new notation for 64 bit is c32=G31,16+(P31,16G15,0)+(P31,16P15,0c0).

The third equation can be calculated as follows

C3=G2+(P2C2) (3)

Substitute, “G2+(P2C2)” for “C2” in the equation (3)

C3=G2P2G1+P2P1G0+P2P1P0C0

The third equation with new notation is c12=G11,8+(P11,8G7,4)+(P11,8P7,4G3,0)+(P11,8P7,4P3,0c0).

Thus, the third equation with new notation for 64 bit is c48=G47,32+(P47,32G31,16)+(P47,32P31,16G15,0)+(P47,32P31,16P15,0c0).

The fourth equation can be calculated as follows

C4=G3+(P3C3) (4)

Substitute, “G3+(P3C3)” for “C3” in the equation (4)

C4=G3P3G2P3P2G1+P3P2P1G0+P3P2P1P0C0

The fourth equation with new notation is c16=G15,12+(P15,12G11,8)+(P15,12P11,8G7,4)+(P15,12P11,8P7,4G3,0)+(P15,12P11,8P7,4P3,0c0)

Thus, the fourth equation with new notation for 64 bit is c64=G63,48+(P63,48G47,32)+(P63,48P47,32G31,16)+(P63,48P47,32P31,16G15,0)+(P63,48P47,32P31,16P15,0c0)

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