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Kumar Sambhav Pandey, Hitesh Shrimali, Dinesh Kumar B School of Computing and Electrical Engineering, Indian Institute of Technology, Mandi Neeraj Goel Department of Computer Science & Engineering, Indian Institute of Technology, Ropar

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Kumar Sambhav Pandey, Hitesh Shrimali, Dinesh Kumar B

School of Computing and Electrical Engineering, Indian Institute of Technology, Mandi

Neeraj Goel

Department of Computer Science & Engineering, Indian Institute of Technology, Ropar Friday, June 28, 2019

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In the given instance unless the carry out from the least significant bit adder is produced, more significant bits cannot be computed (Carry Ripple Adders). Can we compute all the sum bits in parrallel

?

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Define two signals, carry generate (𝑕𝑗) and carry propagate (π‘žπ‘—), which being local to each bit position can all be computed in parallel:

𝑕𝑗 = 𝑏𝑗 β‹… 𝑐𝑗 π‘žπ‘— = 𝑏𝑗 + 𝑐𝑗

and one ancillary signal carry transmit (𝑒𝑗):

𝑒𝑗 = 𝑏𝑗⨁𝑐𝑗

With these signals, it is trivial to compute Carry at bit position i+1 (𝐷𝑗+1) in terms

  • f carry at bit at position i (𝐷𝑗) as:

𝐷𝑗+1 = 𝑕𝑗 + π‘žπ‘— β‹… 𝐷𝑗

and the sum at bit position i as:

𝑑𝑗 = 𝑒𝑗⨁𝐷𝑗

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The first and the last stages are purely local in nature as they operate on signals

  • nly at their respective bit positions. Hence all of the bits can be operated upon
  • concurrently. However, there is a data dependence between 𝐷𝑗+1 and 𝐷𝑗.

Define a special binary prefix operator (∘) on pairs of operands as:

𝑕𝑗 π‘žπ‘— ∘ π‘•π‘˜ π‘žπ‘˜ = 𝑕𝑗 + π‘žπ‘— β‹… π‘•π‘˜ π‘žπ‘— β‹… π‘žπ‘˜

Thus,

𝐷𝑗+1 π‘žπ‘— = π‘•π‘˜ π‘žπ‘— ∘ 𝐷𝑗 1

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Obviously, therefore, Input carry at any bit position as a function of π·π‘—π‘œ can thus be trivially computed using a sequence of the prefix operations (∘)s introduced before as:

𝐷𝑗+1 π‘žπ‘— β‹… π‘žπ‘—βˆ’1 β‹…β‹…β‹… π‘ž0 = 𝑕𝑗 π‘žπ‘— ∘ π‘•π‘—βˆ’1 π‘žπ‘—βˆ’1 β‹…β‹…β‹… 𝑕0 π‘ž0 ∘ π·π‘—π‘œ 1

Define group generate signal π‘•π‘—β‹…β‹…β‹…π‘˜ and a group propagate signal π‘žπ‘—β‹…β‹…β‹…π‘˜ as:

π‘•π‘—β‹…β‹…β‹…π‘˜ π‘žπ‘—β‹…β‹…β‹…π‘˜ = 𝑕𝑗 π‘žπ‘— ∘ π‘•π‘—βˆ’1 π‘žπ‘—βˆ’1 β‹…β‹…β‹… π‘•π‘˜+1 π‘žπ‘˜+1 ∘ π‘•π‘˜ π‘žπ‘˜

Thus,

𝐷𝑗+1 π‘žπ‘—β‹…β‹…β‹…0 = 𝑕𝑗⋅⋅⋅0 π‘žπ‘—β‹…β‹…β‹…0 ∘ π·π‘—π‘œ 1

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Two Important Properties:

  • Associativity:

𝑕𝑗 π‘žπ‘— ∘ π‘•π‘˜ π‘žπ‘˜ ∘ 𝑕𝑙 π‘žπ‘™ = 𝑕𝑗 π‘žπ‘— ∘ π‘•π‘˜ π‘žπ‘˜ ∘ 𝑕𝑙 π‘žπ‘™

  • Idempotency:

π‘•β„Žβ‹…β‹…β‹…π‘— π‘žβ„Žβ‹…β‹…β‹…π‘— ∘ π‘•π‘˜β‹…β‹…β‹…π‘™ π‘žπ‘˜β‹…β‹…β‹…π‘™ = π‘•β„Žβ‹…β‹…β‹…π‘™ π‘žβ„Žβ‹…β‹…β‹…π‘™

provided, β„Ž > 𝑗, 𝑗 ≀ π‘˜ + 1 and π‘˜ > 𝑙

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βˆ€π‘— β‰₯ 0 𝑕𝑗, π‘žπ‘—and 𝑒𝑗are all computed in parallel.

[1] A. Weinberger and J. Smith, β€œA logic for high-speed addition,” Nat. Bur. Stand. Circ., vol. 591, pp. 3–12, 1958.

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Following are all computed in parallel:

𝑕1β‹―0 π‘ž1β‹―0 = 𝑕1 π‘ž1 Β° 𝑕0 π‘ž0 𝑕2β‹―1 π‘ž2β‹―1 = 𝑕2 π‘ž2 Β° 𝑕1 π‘ž1 𝑕3β‹―2 π‘ž3β‹―2 = 𝑕3 π‘ž3 Β° 𝑕2 π‘ž2

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Following are all computed in parallel:

𝑕2β‹―0 π‘ž2β‹―0 = 𝑕2β‹―1 π‘ž2β‹―1 Β° 𝑕0 π‘ž0 𝑕3β‹―0 π‘ž3β‹―0 = 𝑕3β‹―2 π‘ž3β‹―2 Β° 𝑕1β‹―0 π‘ž1β‹―0

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Following are all computed in parallel:

𝐷1 π‘ž0 = 𝑕0 π‘ž0 Β° π·π‘—π‘œ 1 𝐷2 π‘ž1β‹―0 = 𝑕1β‹―0 π‘ž1β‹―0 Β° π·π‘—π‘œ 1 𝐷3 π‘ž2β‹―0 = 𝑕2β‹―0 π‘ž2β‹―0 Β° π·π‘—π‘œ 1 𝐷4 π‘ž3β‹―0 = 𝑕3β‹―0 π‘ž3β‹―0 Β° π·π‘—π‘œ 1

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βˆ€π‘— β‰₯ 0 𝑇𝑗 are all computed in

parallel.

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A pseudo carry 𝐼𝑗 = 𝐷𝑗 + π·π‘—βˆ’1 is propagated in lieu of the conventional carry 𝐷𝑗. Conventional carries can be extracted from them by 𝐷𝑗 = π‘žπ‘—βˆ’1 βˆ™ 𝐼𝑗 as proved below:

π‘žπ‘—βˆ’1 βˆ™ 𝐼𝑗 = π‘žπ‘—βˆ’1 βˆ™ 𝐷𝑗 + π·π‘—βˆ’1 = π‘žπ‘—βˆ’1 βˆ™ π‘•π‘—βˆ’1 + π‘žπ‘—βˆ’1 βˆ™ π·π‘—βˆ’1 + π·π‘—βˆ’1 = π‘žπ‘—βˆ’1 βˆ™ π‘•π‘—βˆ’1 + π·π‘—βˆ’1 = π‘žπ‘—βˆ’1 βˆ™ π‘•π‘—βˆ’1 + π‘žπ‘—βˆ’1 βˆ™ π·π‘—βˆ’1 = π‘•π‘—βˆ’1 + π‘žπ‘—βˆ’1 βˆ™ π·π‘—βˆ’1 = 𝐷𝑗

[2] H. Ling, β€œHigh-speed binary adder,” IBM Journal of Research and Development, vol. 25, no. 3, pp. 156–166, March 1981.

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Define two more local signals β„Žπ‘— and π‘Ÿπ‘— as:

β„Žπ‘— = 𝑕𝑗 + π‘•π‘—βˆ’1

and

π‘Ÿπ‘— = π‘žπ‘— βˆ™ π‘žπ‘—βˆ’1

Starting with the definition of 𝐼𝑗+1and the fact that 𝑕𝑗 βˆ™ π‘žπ‘— = 𝑕𝑗:

𝐼𝑗+1 = 𝐷𝑗+1 + 𝐷𝑗 = 𝑕𝑗 + π‘žπ‘— βˆ™ 𝐷𝑗 + 𝐷𝑗 = 𝑕𝑗 + 𝐷𝑗 = 𝑕𝑗 + π‘•π‘—βˆ’1 + π‘žπ‘—βˆ’1 βˆ™ π‘•π‘—βˆ’2 + π‘žπ‘—βˆ’2 βˆ™ π·π‘—βˆ’2 = 𝑕𝑗 + π‘•π‘—βˆ’1 + π‘žπ‘—βˆ’1 βˆ™ π‘žπ‘—βˆ’2 π‘•π‘—βˆ’2 + π·π‘—βˆ’2 = β„Žπ‘— + πΌπ‘—βˆ’1

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In terms of the binary prefix operator defined before:

𝐼𝑗+1 π‘Ÿπ‘—βˆ’1 = β„Žπ‘— π‘Ÿπ‘—βˆ’1 ∘ πΌπ‘—βˆ’1 1

Ling carry (pseudo carry) at any bit position as a function of πΌπ‘—π‘œ = π·π‘—π‘œcan thus be trivially computed by a sequence of prefix operations as:

𝐼𝑗+1 π‘Ÿπ‘—βˆ’1 β‹… π‘Ÿπ‘—βˆ’3 β‹…β‹…β‹… π‘Ÿ0 = β„Žπ‘— π‘Ÿπ‘—βˆ’1 ∘ β„Žπ‘—βˆ’2 π‘Ÿπ‘—βˆ’3 β‹…β‹…β‹… πΌπ‘—π‘œ 1

It is noted that ,

𝐼4 = β„Ž3 + π‘Ÿ2 βˆ™ β„Ž1 + π‘Ÿ2 βˆ™ π‘Ÿ0 βˆ™ πΌπ‘—π‘œ = 𝑕3 + 𝑕2 + π‘ž2 βˆ™ 𝑕1 + π‘ž2 βˆ™ π‘ž1 βˆ™ 𝑕0 + π‘ž2 βˆ™ π‘ž1 βˆ™ π‘ž0 βˆ™ π·π‘—π‘œ

which is logically much more simpler than the corresponding expression for the conventional carry 𝐷4:

𝐷4 = 𝑕3 + π‘ž3 βˆ™ 𝑕2 + π‘ž3 βˆ™ π‘ž2 βˆ™ 𝑕1 + π‘ž3 βˆ™ π‘ž2 βˆ™ π‘ž1 βˆ™ 𝑕0 + π‘ž3 βˆ™ π‘ž2 βˆ™ π‘ž1 βˆ™ π‘ž0 βˆ™ π·π‘—π‘œ

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Following is an expansion of Ling recurrence relation for Ling pseudo carries for a radix-4 adder:

𝐼4 π‘Ÿ2 βˆ™ π‘Ÿ0 = β„Ž3 π‘Ÿ2 Β° β„Ž1 π‘Ÿ0 Β° π·π‘—π‘œ 1 𝐼3 π‘Ÿ1 = β„Ž2 π‘Ÿ1 Β° β„Ž0 1 𝐼2 π‘Ÿ0 = β„Ž1 π‘Ÿ0 Β° π·π‘—π‘œ 1 𝐼1 1 = β„Ž0 1 𝐼0 1 = π·π‘—π‘œ 1

Dimitrakopoulos and Nikolos observed that in the above expansion the even and

  • dd subscripted pseudo Ling carries are independent of each other.
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[3] G. Dimitrakopoulos and D. Nikolos, β€œHigh-speed parallel prefix VLSI Ling adders,” IEEE Transactions on Computers, vol. 54,

  • no. 2, pp. 225–231, Feb 2005.
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Jackson and Talwar generalized the Ling factorization to further speed up multi- bit addition and proved that the relations for carries can be factorized even beyond what was established by Ling. For example, the expression for the conventional carry (𝐷4 ) can be further factorized as given below:

𝐷4 = 𝑕3 + π‘ž3 βˆ™ 𝑕2 + π‘ž3 βˆ™ π‘ž2 βˆ™ 𝑕1 + π‘ž3 βˆ™ π‘ž2 βˆ™ π‘ž1 βˆ™ 𝑕0 + π‘ž3 βˆ™ π‘ž2 βˆ™ π‘ž1 βˆ™ π‘ž0 βˆ™ π·π‘—π‘œ = π‘ž3 βˆ™ 𝑕3 + 𝑕2 + π‘ž2 βˆ™ 𝑕1 + π‘ž2 βˆ™ π‘ž1 βˆ™ 𝑕0 + π‘ž2 βˆ™ π‘ž1 βˆ™ π‘ž0 βˆ™ π·π‘—π‘œ = 𝑕3 + π‘ž3 βˆ™ π‘ž2 βˆ™ 𝑕3 + 𝑕2 + 𝑕1 + π‘ž1 βˆ™ 𝑕0 + π‘ž1 βˆ™ π‘ž0 βˆ™ π·π‘—π‘œ = 𝑕3 + π‘ž3 βˆ™ 𝑕2 + π‘ž3 βˆ™ π‘ž2 βˆ™ π‘ž1 βˆ™ 𝑕3 + 𝑕2 + 𝑕1 + 𝑕0 + π‘ž0 βˆ™ π·π‘—π‘œ

[4] R. Jackson and S. Talwar, β€œHigh speed binary addition,” in Conference Record of the Thirty-Eighth Asilomar Conference on Signals, Systems and Computers, 2004., vol. 2, Nov 2004, pp. 1350–1353 Vol.2.

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The 2nd equality in the previous slide is the Ling factorization and the quantity in the second factor is actually the Ling pseudo group carry (𝐼4). The 3rd and the 4th equalities are the higher order factorizations. The quantities in the second factors in these equalities may be called the Jackson-Talwar pseudo carries of order 3 and 4 respectively. In order to understand the theory define 2 more signals (𝑐𝑗) and (𝑒𝑗) as:

𝑐𝑗 = 𝑕𝑗 + π‘•π‘—βˆ’1 + π‘•π‘—βˆ’2 + π‘•π‘—βˆ’3 𝑒𝑗 = 𝑕𝑗 + π‘žπ‘— βˆ™ π‘•π‘—βˆ’1 + π‘žπ‘— βˆ™ π‘žπ‘—βˆ’1 βˆ™ π‘žπ‘—βˆ’2

Define 4th order Jackson-Talwar group carry (𝐾𝑗) as:

𝐾𝑗 = 𝐷𝑗 + π·π‘—βˆ’1 + π·π‘—βˆ’2 + π·π‘—βˆ’3

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Starting with this definition for (𝐾𝑗+1), we can define it in terms of (πΎπ‘—βˆ’3) as:

𝐾𝑗+1 = 𝐷𝑗+1 + 𝐷𝑗 + π·π‘—βˆ’1 + π·π‘—βˆ’2 = 𝑕𝑗 + π‘žπ‘— βˆ™ 𝐷𝑗 + 𝐷𝑗 + π·π‘—βˆ’1 + π·π‘—βˆ’2 = 𝑕𝑗 + 𝐷𝑗 +π·π‘—βˆ’1 + π·π‘—βˆ’2 = 𝑕𝑗 + π‘•π‘—βˆ’1 + π‘žπ‘—βˆ’1βˆ™ π·π‘—βˆ’1 + π·π‘—βˆ’1 + π·π‘—βˆ’2 = 𝑕𝑗 + π‘•π‘—βˆ’1 + π·π‘—βˆ’1 + π·π‘—βˆ’2 = 𝑕𝑗 + π‘•π‘—βˆ’1 + π‘•π‘—βˆ’2 + π‘žπ‘—βˆ’2 βˆ™ π·π‘—βˆ’2 + π·π‘—βˆ’2 = 𝑕𝑗 + π‘•π‘—βˆ’1 + π‘•π‘—βˆ’2 + π·π‘—βˆ’2 = 𝑕𝑗 + π‘•π‘—βˆ’1 + π‘•π‘—βˆ’2 + π‘•π‘—βˆ’3 + π‘žπ‘—βˆ’3 βˆ™ π·π‘—βˆ’3 = 𝑐𝑗 + π‘’π‘—βˆ’3 βˆ™ πΎπ‘—βˆ’3

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Thus the above equation and the definition of (π‘’π‘—βˆ’3) can be collected together using the binary prefix operator (Β°) as:

𝐾𝑗+1 π‘’π‘—βˆ’3 = 𝑐𝑗 π‘’π‘—βˆ’3 Β° πΎπ‘—βˆ’3 1

Strikingly similar!

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β–ͺ

The discussions in the above subsections are based on the assumption that the generate signals (𝑕𝑗) and the propagate signals (π‘žπ‘—) are computed for each bit.

β–ͺ

There is no reason why group generate signals (π‘•π‘—β‹―π‘˜) and group propagate signals (π‘žπ‘—β‹―π‘˜) can not be computed for groups of adjacent bits in place of individual bits.

β–ͺ

These signals defined over such groups can then be reduced in similar treelike structures as discussed earlier.

β–ͺ

Such parallel prefix adders are known as higher valency adders.

β–ͺ

However, a parallel prefix adder with valency 2 is not the same as a Ling adder

  • r that with valency 4 is not the same as a Jackson-Talwar adder which are

different and architecturally more efficient.

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  • In case of Ling adders, the generate signals (𝑕𝑗) are combined as conjunctions

and the propagate signals (π‘žπ‘—) are combined as disjunctions respectively. These combinations are, however, limited to only 2 adjacent signals.

  • Arguably Jackson-Talwar adders are motivated by the fact that more than 2

adjacent generate signals can be combined as conjunctions (Reduced Generate) and corresponding propagate signals (Hyper Propagate) were calculated in such a way that the overall addition of multibit integers remains correct.

  • In case more than 2 propagate signals are combined as disjunctions and the

corresponding generate signals are calculated in similar way to preserve the multi-bit addition semantics, one can create a new family of adders which when looked from the Dimitrakopoulos-Nikolos perspective, can be decoupled in more than 2 subtrees and are consequently faster and more efficient.

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A pseudo carry 𝐿𝑗 = 𝐼𝑗 + π‘Ÿπ‘—βˆ’2 βˆ™ πΌπ‘—βˆ’2 = 𝐷𝑗 + π·π‘—βˆ’1 + π‘žπ‘—βˆ’2 βˆ™ π‘žπ‘—βˆ’3 βˆ™ π·π‘—βˆ’2 + π·π‘—βˆ’3 is propagated in lieu of the conventional carry 𝐷𝑗 or pseudo Ling carry 𝐼𝑗. Conventional carries can be extracted from them by 𝐷𝑗 = π‘žπ‘—βˆ’1 βˆ™ 𝐿𝑗 as proved below:

π‘žπ‘—βˆ’1 βˆ™ 𝐿𝑗 = π‘žπ‘—βˆ’1 βˆ™ π‘•π‘—βˆ’1 + π‘žπ‘—βˆ’1 βˆ™ π‘•π‘—βˆ’2 + π‘žπ‘—βˆ’1 βˆ™ π‘Ÿπ‘—βˆ’2 βˆ™ π‘•π‘—βˆ’3 + π‘žπ‘—βˆ’1 βˆ™ π‘Ÿπ‘—βˆ’2 βˆ™ π‘•π‘—βˆ’4 + π‘žπ‘—βˆ’1 βˆ™ π‘Ÿπ‘—βˆ’2 βˆ™ π‘žπ‘—βˆ’4 βˆ™ π·π‘—βˆ’4 = π‘•π‘—βˆ’1 + π‘žπ‘—βˆ’1 βˆ™ π‘•π‘—βˆ’2 + π‘žπ‘—βˆ’1 βˆ™ π‘žπ‘—βˆ’2 βˆ™ π‘•π‘—βˆ’3 + π‘žπ‘—βˆ’1 βˆ™ π‘žπ‘—βˆ’2 βˆ™ π‘žπ‘—βˆ’3 βˆ™ π‘•π‘—βˆ’4 + π‘žπ‘—βˆ’1 βˆ™ π‘žπ‘—βˆ’2 βˆ™ π‘žπ‘—βˆ’3 βˆ™ π‘žπ‘—βˆ’4 βˆ™ π·π‘—βˆ’4 = 𝐷𝑗

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Define two more local signals (𝑙𝑗) and (𝑠𝑗) as:

𝑙𝑗 = 𝑕𝑗 + π‘•π‘—βˆ’1 + π‘Ÿπ‘—βˆ’2 βˆ™ π‘•π‘—βˆ’2 + π‘•π‘—βˆ’3 = β„Žπ‘— + π‘Ÿπ‘—βˆ’1 βˆ™ β„Žπ‘—βˆ’2

and

𝑠𝑗 = π‘žπ‘— βˆ™ π‘žπ‘—βˆ’1 βˆ™ π‘žπ‘—βˆ’2 βˆ™ π‘žπ‘—βˆ’3 = π‘Ÿπ‘— βˆ™ π‘Ÿπ‘—βˆ’2

Starting with the definition of (𝐿𝑗+1)we can define it in terms of (πΏπ‘—βˆ’3) as:

𝐿𝑗+1 = 𝑕𝑗 + π‘•π‘—βˆ’1 + π‘Ÿπ‘—βˆ’1 βˆ™ π‘•π‘—βˆ’2 + π‘Ÿπ‘—βˆ’1 βˆ™ π‘•π‘—βˆ’3 + π‘Ÿπ‘—βˆ’1 βˆ™ π‘žπ‘—βˆ’3 βˆ™ π·π‘—βˆ’3 = 𝑙𝑗 + π‘Ÿπ‘—βˆ’1 βˆ™ π‘žπ‘—βˆ’3 βˆ™ π·π‘—βˆ’3 = 𝑙𝑗 + π‘Ÿπ‘—βˆ’1 βˆ™ π‘žπ‘—βˆ’3 βˆ™ π‘žπ‘—βˆ’4 βˆ™ πΏπ‘—βˆ’3 = 𝑙𝑗 + π‘ π‘—βˆ’1 βˆ™ πΏπ‘—βˆ’3

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Thus the above equation and the definition of (π‘ π‘—βˆ’1) can be collected together using the binary prefix operator (Β°) as:

𝐿𝑗+1 𝑠

π‘—βˆ’1

= 𝑙𝑗 𝑠

π‘—βˆ’1 Β° πΏπ‘—βˆ’3

1

Strikingly similar!

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Recurrence Relations Number of Bits Logic Levels in Critical Path Number of Gates Weinberger- Smith 8 11 107 16 13 259 32 15 611 64 17 1,411 128 19 3,203

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Recurrence Relations Number of Bits Logic Levels in Critical Path Number of Gates Ling 8 10 118 16 12 274 32 14 646 64 16 1,478 128 18 3,334

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Recurrence Relations Number of Bits Logic Levels in Critical Path Number of Gates Jackson- Talwar 8 9 212 16 11 460 32 13 1,004 64 15 2,188 128 17 5,260

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Recurrence Relations Number of Bits Logic Levels in Critical Path Number of Gates Our Proposal 8 9 164 16 11 364 32 13 812 64 15 1,804 128 17 3,980

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  • The number of logic levels in the critical path for all the adders based on Ling

recurrence are 1 less than the values for the corresponding adders based on Weinberger-Smith recurrence. These levels in case of adders based on Jackson- Talwar recurrence as well as those based on our proposed novel recurrence are still lower by 1, as expected.

  • The total gate count for all the adders are increasing from Weinberger-Smith

adder to Ling adder to Jackson-Talwar adder.

  • The comparison between Jackson-Talwar adder and our proposed adder is

particularly interesting. Though the speeds achieved by both the adders is the same, yet the total gate count in case of our proposed adder is much lower as compared to the former.

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May I answer any of your questions? Thanks for your Attention!