# Part 5c: Addition - UMass Amherst

UNIVERSITY OF MASSACHUSETTS Dept. of Electrical & Computer Engineering Digital Computer Arithmetic CE666/Koren Part.5c.1 ECE 666 Part 5c Fast Addition - III Israel Koren Spring 2008 Copyright 2008 Koren Hybrid Adders Combination of two or more addition methods Common approach: one method for carry -

another for sum Two hybrid adders combining variation of a carryselect for sum and modified Manchester for carry Both divide operands into equal groups - 8 bits each First - uses carry-select for sum for each group of 8 bits separately Second - uses a variant of conditional-sum Group carry-in signal that selects one out of two sets of sum bits not generated in ripple-carry Instead, carries into 8-bit groups generated by a carry-look-ahead tree 64-bit adder - carries are c8,c16,c24,c32,c40,c48,c56 CE666/Koren Part.5c.2 Copyright 2008 Koren Blocking Factor in Carry Tree

Structure of carry-look-ahead tree for generating carries similar to those seen before Differences - variations in blocking factor at each level and exact implementation of fundamental carry operator Restricting to a fixed blocking factor - natural choices include 2, 4 or 8 2 - largest number of levels in tree, vs. 8 - complex modules for fundamental carry operator with high delay Factor of 4 - a reasonable compromise A Manchester carry propagate/generate module (MCC) with a blocking factor of 4 CE666/Koren Part.5c.3 Copyright 2008 Koren

64-bit Hybrid Adder CE666/Koren Part.5c.4 Copyright 2008 Koren CE666/Koren Part.5c.5 Manchester Carry Module Copyright 2008 Koren MCC - General Case MCC accepts 4 pairs of inputs: (Pi1:i0,Gi1:i0),(Pj1:j0,Gj1:j0),(Pk1:k0,Gk1:k0),(Pl1:l0,Gl1:l0) where i1 i0, j1 j0, k1 k0, l1 l0 Produces 3 pairs of outputs:

(Pj1:i0,Gj1:i0),(Pk1:i0,Gk1:i0),(Pl1:i0,Gl1:i0) where i1 j0-1, j1 k0-1, k1 l0-1 Allows overlap among input subgroups CE666/Koren Part.5c.6 Copyright 2008 Koren Carry Tree First level - 14 MCCs calculating (P3:0,G3:0),(P7:4,G7:4),,(P55:52,G55:52) only outputs P3:0 and G3:0 are utilized Second level: each MCC generates 2 pairs (P3:0, G3:0),(P1:0, G1:0) Providing (P7:0,G7:0),(P15:0,G15:0), (P23:16,G23:16),(P31:16,G31:16), (P39:32,G39:32),(P47:32,G47:32),

(P55:48,G55:48) Generates c8 & c16 - G7:0 & G15:0 c0 is incorporated into (P3:0, G3:0) CE666/Koren Part.5c.7 Copyright 2008 Koren Third level - Two MCCs Sufficient One for dashed box generating c24, c32 and c40 Second MCC for 2 remaining outputs with inputs 55:48, 47:32, 31:16 and 15:0 generating c48 and c56 MCC in dashed box must implement 2 dotted lines from 23:16 - required for generating

23:0 Above implementation of adder not unique does not necessarily minimize overall execution time Alternate implementations: variable size of carry-select groups and of MCCs at different levels of tree CE666/Koren Part.5c.8 Copyright 2008 Koren A Schematic Diagram of a 32-bit Hybrid Adder CE666/Koren Part.5c.9 Copyright 2008 Koren Grouping of Bits in a 64-bit

Adder 64 bits divided into two sets of 32 bits, each set further divided into 4 groups of 8 bits For every group of 8 bits - 2 sets of conditional sum outputs generated separately Two most significant groups combined into group of size 16 Further combined with next group of 8 to form group of 24 bits and so on principle of conditional-sum addition However, the way input carries for basic 8-bit groups are generated is different MCC generates Pm, Gm and Km and cout ,cout 0 1

for assumed incoming carries of 0 and 1 Conditional carry-out signals control multiplexers CE666/Koren Part.5c.10 Copyright 2008 Koren Dual and Regular Multiplexer Two sets of dual multiplexers (of size 8 and 16) Single regular multiplexer of size 24 CE666/Koren Part.5c.11 Copyright 2008 Koren High-Order Half of 64-bit Adder Similar structure but incoming carry c32

calculated by separate carry-look-ahead circuit Inputs are conditional carry-out signals generated by 4 MCCs Allows operation of high-order half to overlap operation of low-order half Summary: combines variants of 3 different techniques for fast addition: Manchester carry generation, carry-select, conditional-sum Other designs of hybrid adders exist - e.g., groups with unequal number of bits Optimality of hybrid adders depends on technology and delay parameters CE666/Koren Part.5c.12 Copyright 2008 Koren Carry-Save Adders (CSAs) 3 or more operands added simultaneously (e.g., in

multiplication) using 2-operand adders Time-consuming carry-propagation must be repeated several times: k operands - k-1 propagations Techniques for lowering this penalty exist - most commonly used - carry-save addition Carry propagates only in last step - other steps generate partial sum and sequence of carries Basic CSA accepts 3 n-bit operands; generates 2 n-bit results: n-bit partial sum, n-bit carry Second CSA accepts the 2 sequences and another input operand, generates new partial sum and carry CSA reduces number of operands to be added from 3 to 2 without carry propagation CE666/Koren Part.5c.13 Copyright 2008 Koren Implementing Carry Save

Adders Simplest implementation - full adder (FA) with 3 inputs x,y,z x+y+z=2c+s (s,c - sum and carry outputs) Outputs - weighted binary representation of number of 1's in inputs FA called a (3,2) counter n-bit CSA: n (3,2) in parallel links CE666/Koren Part.5c.14 counters with no carry Copyright 2008 Koren

Carry-Save Adder for four 4-bit Operands Upper 2 levels - 4-bit CSAs 3rd level - 4-bit carry-propagating adder (CPA) Ripple-carry adder - can be replaced by a carry- look-ahead adder or any other fast CPA Partial sum bits and carry bits interconnected to guarantee that only bits having same weight are added by any (3,2) counter CE666/Koren Part.5c.15 Copyright 2008 Koren Adding k Operands (k-2) CSAs + one CPA If CSAs arranged in

cascade k operands - time to add is (k-2)TCSA + TCPA TCPA ; TCSA - operation time of CPA ; CSA G ; FA delay of a single gate ; full adder TCSA = FA 2 G Sum of k operands of size n bits each can be as large asn k(2 -1) Final addition result may reach a length of n+log 2 k bits CE666/Koren Part.5c.16 Copyright 2008 Koren Six-operand Wallace Tree

Better organization for CSAs - faster operation time CE666/Koren Part.5c.17 Copyright 2008 Koren Number of Levels in Wallace Tree Number of operands reduced by a factor of 2/3 at each level - (l - number of levels) Consequently, l = Only an estimate of l - number of operands at

each level must be an integer Ni - number of operands at level i Ni+1 - at most 3/2 Ni ( x - largest integer smaller than or equal to x ) Bottom level (0) has 2 - maximum at level 1 is 3 - maximum at level 2 is 9/2 =4 Resulting sequence: 2,3,4,6,9,13,19,28, For 5 operands - still 3 levels CE666/Koren Part.5c.18 Copyright 2008 Koren Number of Levels in a CSA Tree for k operands Example: k=12 - 5 levels - delay of 5TCSA instead of 10TCSA in a linear cascade of 10 CSAs CE666/Koren Part.5c.19

Copyright 2008 Koren Most Economical Implementation (Fewer CSAs) Achieved when number of operands is element of 3,4,6,9,13,19,28, If given number of operands, k, not in sequence - use only enough CSAs to reduce k to closest (smaller than k) element Example: k=27, use 8 CSAs (24 inputs) rather than 9, in top level - number of operands in next level is 82+3=19 Remaining part of tree will follow the series CE666/Koren Part.5c.20 Copyright 2008 Koren

(7,3) and Other Counters (7,3) counter: 3 outputs - represent number of 1's in 7 inputs Another example: (15,4) counter In general: (k,m) counter - k and m satisfy m 2 -1 k or m log 2 (k+1) (7,3) counter using (3,2) counters: Requires 4 (3,2)s in 3 levels - no speed-up CE666/Koren Part.5c.21 Copyright 2008 Koren

(7,3) Counters (7,3) can be implemented as a multilevel circuit - may have smaller delay Number of interconnections affects silicon area - (7,3) preferrable to (3,2) (7,3) has 10 connections and removes 4 bits (3,2) has 5 connections and removes only 1 bit Another implementation of (7,3) - ROM of 7 size 2 x 3= 128 x 3 bits Access time of ROM unlikely to be small enough Speed-up may be achieved for ROM implementation of (k,m) counter with higher values of k CE666/Koren Part.5c.22

Copyright 2008 Koren Avoiding Second Level of Counters Several (7,3) counters (in parallel) are used to add 7 operands - 3 results obtained Second level of (3,2) counters needed to reduce the 3 to 2 results (sum and carry) added by a CPA Similarly - when (15,4) or more complex counters are used - more than two results generated In some cases - additional level of counters can be combined with first level - more convenient implementation When combining a (7,3) counter with a (3,2) counter - combined counter called a (7;2) compressor

CE666/Koren Part.5c.23 Copyright 2008 Koren (k;m) Compressor Variant of a counter with k primary inputs, i all of weight 2 , and m primary outputs of i+m-1 i i+1 weights 2 ,2 ,...,2 Compressor has several incoming carries ofi weight 2 from previous compressors, i+1 and several outgoing carries of weights 2 and up

i+1 Trivial example of a (6;2) compressor: All outgoing carries have weight 2 Number of outgoing carries = number of incoming carries = k-3 (in general) CE666/Koren Part.5c.24 Copyright 2008 Koren Implementation of a (7;2) Compressor Bottom right (3,2) - additional (3,2), while remaining four ordinary (7,3)

counter 7 primary inputs of weight i 2 and 2 carry inputs from columns i-1 and i-2 i+1 i 2 primary outputs, S2 and S2 , and 2 outgoing carries i+2 , to columns i+1 and i+2 i+1 , C2 C2 Input carries do not participate in generation of output carries - avoids slow carry-propagation Not a (9,4) counter - 2 outputs with same weight

Above implementation does not offer any speedup Multilevel implementation may yield smaller delay as long as outgoing carries remain independent of incoming carries CE666/Koren Part.5c.25 Copyright 2008 Koren Multiple-column counters Generalized parallel counter: add l input columns and produce m-bit output - (kl-1,kl2,...,k0,m) i ki - number of input bits in i-th column with weight 2 (k,m) counter - a special case Number of outputs m must satisfy If all l columns have same height k l

(km0=k1= ... =k l-1=k) 2 - 1 k(2 - 1) CE666/Koren Part.5c.26 Copyright 2008 Koren Example - (5,5,4) Counter k=5,l =2,m=4 2m -1=k(2l -1) - all 16 combinations of output bits are useful (5,5,4) counters can be used to reduce 5 operands (of any length) to 2 results that can then be added with one CPA Length of operands determines number of

(5,5,4) counters in parallel Reasonable implementation - using ROMs For (5,5,4) - 25+5x4 (=1024x4) ROM CE666/Koren Part.5c.27 Copyright 2008 Koren Number of Results of General Counters String of (k,k,,k,m) counters may generate more than 2 intermediate results requiring additional reduction before CPA Number of intermediate results: A set of (k,k,,k,m) counters, with l columns each, produces m-bit outputs at intervals of l bits Any column has at most m/l output bits k operands can be reduced to s= m/l

operands If s=2 - a single CPA can generate final sum Otherwise, reduction from s to 2 needed CE666/Koren Part.5c.28 Copyright 2008 Koren Example Number of bits per column in a 2-column counter (k,k,m) is increased beyond 5 m 5 and s= m/2 > 2 For k=7, 2m -1 7 x 3 = 21 m=5 (7,7,5) counters generate s=3 operands - another set of (3,2) counters is needed to reduce number of operands to 2 CE666/Koren Part.5c.29

Copyright 2008 Koren Reducing Hardware Complexity of CSA Tree Design a smaller carry-save tree - use it iteratively n operands divided into n/j groups operands - design a tree for operands and a CPA j+2 of j Feedback paths - must complete first pass through CSA tree before second set of j operands is applied Execution slowed down - pipelining not possible CE666/Koren Part.5c.30

Copyright 2008 Koren Pipelining of Arithmetic Operations Pipelining - well known technique for accelerating execution of successive identical operations Circuit partitioned into several subcircuits that can operate independently on consecutive sets of operands Executions of several successive operations overlap - results produced at higher rate Algorithm divided into several steps - a suitable circuit designed for each step Pipeline stages operate independently on different sets of operands Storage elements - latches - added between adjacent stages - when a stage works on one set of operands, preceding stage can work on next

set of operands CE666/Koren Part.5c.31 Copyright 2008 Koren Pipelining - Example Addition of 2 operands X,Y performed in 3 steps Latches between stages 1 and 2 store intermediate results of step 1 Used by stage 2 to execute step 2 of algorithm Stage 1 starts executing step 1 on next set of operands X,Y CE666/Koren Part.5c.32 Copyright 2008 Koren

Pipelining Timing Diagram 4 successive additions with operands X1 & Y1, X2 & Y2, X3 & Y3, X4 & Y4 producing results Z1, Z2, Z3, Z4 CE666/Koren Part.5c.33 Copyright 2008 Koren Pipeline Rate i - execution time of stage i l - time needed to store new data into latch Delays of different stages not identical - faster stages wait for slowest before switching to next

task - time interval between two successive results being produced by pipeline: k - number of stages - pipeline period ; 1/ - pipeline rate or bandwidth Clock period After latency of 3, new results produced at rate 1/ CE666/Koren Part.5c.34 Copyright 2008 Koren Design Decisions Partitioning of given algorithm into steps to be executed by separate stages Steps should have similar execution times - pipeline

rate determined by slowest step Number of steps As this number increases, pipeline period decreases, but number of latches (implementation cost) and latency go up Latency - time elapsed until first result produced Especially important when only a single pass through pipeline required Tradeoff between latency and implementation cost on one hand and pipeline rate on the other hand Extra delay due to latches, l , can be lowered by using special circuits like Earl latch CE666/Koren Part.5c.35

Copyright 2008 Koren Pipelining of Two-Operand Adders Two-operand adders - usually not pipelined Pipelining justified with many successive additions Conditional-sum adder - easily pipelined log2n stages corresponding to log2n steps execution of up to log2n additions can be overlapped Required number of latches may be excessive Combining several steps to one stage reduces latches' overhead and latency Carry-look-ahead adder cannot be pipelined - some carry signals must propagate backward Different designs can be pipelined - final carries and carry-propagate signals (implemented as Pi=xiyi) used to calculate sum bits - no need for

feedback connections CE666/Koren Part.5c.36 Copyright 2008 Koren Pipelining in Multiple-Operand Adders Pipelining more beneficial in multiple- operand adders like carry-save adders Modifying implementation of CSA trees to form a pipeline is straightforward - requires only addition of latches Can be added at each level of tree if maximum bandwidth is desired Or - two (or more) levels of tree can be combined to form a single stage, reducing overall number of latches and pipeline latency

CE666/Koren Part.5c.37 Copyright 2008 Koren Partial Tree Reduced hardware complexity of CSA tree - partial tree Two feedback connections prevent pipelining Modification - intermediate results of CSA tree connected to bottom level of tree Smaller tree with j inputs, 2 separate CSAs, and set of latches at the bottom CSAs and latches form a pipeline stage Top CSA tree for j operands can be

pipelined too - overall time reduced CE666/Koren Part.5c.38 a Copyright 2008 Koren

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