Abstract:
A binary adder of the carry multiplex signal selection type wherein multiple levels of multiplexing between parallel carry paths is used to achieve improved adder performance as measured by adder fabrication area requirements and other performance criteria. The resulting adder employs a plurality of different adder stages of successively increasing complexity and achieves performance time that can be characterized as being of the order of Log 2  (n), wherein n represents bit count, and as requiring a gate count that is of the order of n. Both internal arrangement of the adder stages and interconnection arrangements therefor are disclosed.

Description:
RIGHTS OF THE GOVERNMENT 
     The invention described herein may be manufactured and used by or for the Government of the United States for all governmental purposes without the payment of any royalty. 
    
    
     CROSS REFERENCE TO RELATED APPLICATION 
     This application is somewhat related to a co-pending patent application &#34;Carry Multiplexed Adder&#34;, Ser. No. 07/649,781, which is in the name of the present inventor, Michael W. Scriber, and other coinventors, and was filed of even data herewith, and is hereby incorporated by reference herein. 
     BACKGROUND OF THE INVENTION 
     This invention relates to the field of binary adders of the signal selection type. 
     Numerous adder architectures are used today. Three of the most popular of these architectures are the ripple adder, the carry look-ahead adder, and the carry select adder. One of the latest adder architectures is the carry multiplexed adder, which is described in the above referred-to copending patent application. 
     The ripple adder is the simplest adder architecture. It uses full adder cells with the carry out of one cell connected to the carry in of the next cell. It requires very few gates, but it is extremely slow, being of the order of n relationship [O(n)] between propagation delay time and number of adder cells. 
     The carry look-ahead adder uses combinational logic to determine propagate terms and generate terms that are used to produce the carry and sum output. The carry look-ahead adder processes data very quickly and is of time O(log 2  (n)) with respect to n (the number of bits processed), but its gate growth is very high, O(n 3 ). Carry look-ahead adders are therefore not useful for large bit applications. 
     The carry select adder uses parallel carry chains to produce two possible summation outputs. The correct summation is selected by the carry input to the carry select stage. The carry select adder requires about twice as many gates as the ripple adder, however, it processes numbers on the order of O(n 1/2 ) time, which is faster than the ripple adder but not as fast as the carry look-head adder. 
     The carry multiplexed adder calculates four possible carry outputs with a minimal set of hardware components. It then multiplexes the carries using the carry results from previous substages and stages to determine the correct carry output. Then the carry output is used to calculate the summation output. The carry multiplexed adder requires slightly fewer gates than the carry select adder and processes n bits in order O(n 1/3 ) time, but it is still slower than the carry look-ahead adder. The Carry Multiplexed Adder is moreover described in the above referred-to and incorporated by reference patent application of Scriber et al, herein referred to as the Scriber et al application or more simply as Scriber et al. 
     The high order carry multiplexed adder (HOCMA) is an extension of the carry multiplexed adder. This extension provides an adder form capable of processing addend and augend inputs of larger bit sizes while also retaining the advantages of small delay time and small circuit area requirement. 
     It is therefore an object of the invention to provide a desirable binary adder architecture for adders of larger bit capacity. 
     It is another object of the invention to provide a large bit adder architecture that is capable of desirable propagation delay time performance. 
     It is another object of the invention to provide a large bit adder architecture capable of fabrication within desirable circuit area requirements. 
     It is another object of the invention to provide an adder capable of adding two binary numbers in the time equivalent to that of a full carry look-ahead adder. 
     It is another object of the invention to provide a large bit adder which is characterized by a desirable delay time and circuit area performance product. 
     Additional objects and features of the invention will be understood from the following description and claims and the accompanying drawings. 
     These and other objects of the invention are achieved by high order carry multiplexed adder apparatus which includes a plurality of lower ordered adder stages each comprised of plural bit slice cells having addend, augend, and carry input signal ports and sum and carry output signals ports, the lower ordered adder stage input-ports being connected with the lowest ordered and respective successive consecutive increasing ordered bits of the addend and augend signals, the lower ordered adder stages also including second and third level substage interconnection of the bit slice cells wherein presumed zero and presumed one related carry signals are interconnected between predetermined of the bit slice cells therein resident, and a higher order adder stage connected to the next successive consecutive increasing order bits of the addend and augend signals following the lower ordered adder stages, the higher order adder stage including a group of substages having a plurality of interconnected bit slice cells wherein at least one of the interconnected cells is a fourth level cell inclusive of a next addend and augend signal connected sum signal circuit, exclusive OR, and exclusive NOR signal generating circuits and also inclusive of presumed zero and presumed one determined carry signal generating circuits and a three-leveled tree of multiplexer circuit means for selecting the correct carry signal output thereof. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 shows a level four adder cell according to the invention including cell synthesis produced redundant circuitry. 
     FIG. 2 shows a reduced level four adder cell. 
     FIG. 3 shows a further reduced or optimized level four adder cell. 
     FIG. 4 shows an optimized level five adder cell. 
     FIG. 5 shows an optimized level six adder cell. 
     FIG. 6 shows the first five stages of a sixty-four bit adder using the cells of FIGS. 3 and 4. 
     FIG. 7 which includes the views of FIG. 7A and FIG. 7B shows a sixth stage for the same sixty-four bit adder. 
     FIG. 8 shows the relationship between adder processing time and number of adder bits for the present and two other types of adders. 
     FIG. 9 shows the FIG. 8 relationship for smaller increments of delay time. 
     FIG. 10 compares the number of logic gates required for adders of up to 128 bits, using three different architectures. 
     FIG. 11 compares the number of gates per bits processed for three adder architectures. 
     FIG. 12 compares the product of logic gate count and total delay time for three different adder types. 
     FIG. 13 shows the FIG. 12 relationship for smaller values of gate count and delay time product. 
    
    
     DETAILED DESCRIPTION 
     FIG. 1 in the drawings show how two of the level three adder bit slice cells described in the above incorporated by reference Scriber et al application may be joined in a somewhat crude or first blush combination to achieve a new level four adder cell. 
     As implied by this commencement of describing the present high order carry multiplexed adder (HOCMA), this adder uses the same XOR/XNOR units, carry units, sum units, and multiplexers as the Scriber et al carry multiplexed adder. These structures are therefore not described herein. The present HOCMA moreover may be achieved in two steps wherein the first step involves the creation of HOCMA bit-slice cells and the second step involves the accomplishment of a desirable interconnect arrangement for the cells--to create an optimum structure that processes the most number of bits within a given time. 
     The process of achieving HOCMA cells in addition may be considered to somewhat relate to the approach used in expanding the Scriber et al carry multiplexed adder cells from level two to level three capability. The FIG. 1 level four adder cell herein therefore replicates two level three adder cells within the same bit slice. The resulting structure of FIG. 1, however contains redundant XOR/XNOR units, carry units, and sum units which can be eliminated. A reduced or simplified level four bit slice cell arrangement is therefore shown in FIG. 2. 
     The FIG. 2 cell in actuality looks just like a level three bit slice cell, except there are four additional carry signal multiplexers or muxes or switches in commutating circuits included. The added muxes receive inputs from the two carry units and are controlled by the Coupt0 and Coutp1 signals from the level three substage carry units. The remaining three muxes are controlled by the same respective signals as in a level three cell. A careful examination of the four added muxes shows that two of them are a redundant and can be eliminated with the result shown in FIG. 3. The final level four structure is identical to a level three cell with the addition of two muxes that are controlled by the Coutp0 and Coutp1 of the level three substage. 
     Reduction of the four added muxes down to only two is a key as to how higher level carry multiplexed cells can be used to make an efficient adder. If this reduction could not be made, it would mean that a level 5 cell would require eight more muxes than a level 4 cell for a total of 15 muxes. Level 6 cells would then require a total of 31 muxes. The resulting large number of muxes would prohibit the use of higher level cells because the cells would be large and inefficient. 
     The process of creating HOCMA cells can be repeated to produce a level 5 cell as shown in FIG. 4. Each successive higher level cell is the same as the previous or lower level cell with the addition of two muxes that receive inputs from the two carry units and are controlled by the Coupt0 and the Coutp1 of the previous level substage. FIG. 5 shows a level 6 cell structure according to this relationship. 
     The previous figures have shown circuitry for the first bit of a substage. Successive bits within a substage use the same structure as in the figures, except the carry-in for the carry units comes from the output of the carry unit of the previous bit slice, instead of being the assumed &#34;0&#34; and &#34;1&#34; values shown in FIGS. 2-5. The mux control signals extend through the substage and follow through to the higher level (sub)stages which may be identified successively as, i.e., subsubstages, subsubsubstages, etc. 
     A benefit from employing higher level order cells and substages in an adder includes the ability to process more bits in a given amount of time. This is accomplished in the present higher order carry multiplex adder by way of a significantly increased degree of parallel processing in which carry signal computations are accomplished in parallel in an early part of the addition process and prior to concern with sum computations. The present invention particularly maximizes the degree of possible parallel processing in the adder. 
     The number of gate delays present in a bit slice cell circuit is easy to calculate. The XOR/XNOR unit, for example, produces its two output signals in parallel, uses two muxes, and takes one gate delay of time. Since all of the bits of an addend or augend number arrive at the adder at the same time, all of the XOR/XNOR units within an entire adder will process in parallel and produce the XOR and XNOR signals after one gate delay. The carry unit in each bit slice cell circuit is actually another mux circuit and also uses one gate delay from the time that it receives its XOR, XNOR, and carry in signals. The carry selection multiplexers in each cell also take one gate delay time interval from the time they receive their input signals. 
     For a level 3 cell therefore, the XOR and XNOR signals are produced after one gate delay, the Coutp0 and Coutp1 signals are produced after two gate delays, the Cout0 and Cout1 signals are produced after three gate delays, and the final Cout signal is produced after a total of four gate delays. Therefore, level 3 cells can be used whenever the gate delay for the stage is at least four--since four gate delays are required to produce the carry out signal in a level 3 cell. This means that the previous stage will generate a Cout mux control signal in four gate delays. Likewise, a level 4 cell can be used whenever the gate delay for the stage is at least five because level 4 cells have one more layer of muxes than level 3 cells. 
     With the creation of efficient HOCMA cells, it also becomes desirable to achieve the best way to interconnect such cells. The herein-disclosed HOCMA cells can be interconnected in any number of ways as long as the needed mux control signals for a cell have been timely generated by lower level cells in previous bit slices of the adder. The following illustration demonstrates the bit capacity growth for a fixed delay time and also the employed interconnections associated with the higher level structured adders. In each case of this illustration the final carry out is present after eight delays. 
     For the case of using up to level 1 cells, the adder and its time determined performance can be represented as: 
     
         1111111=7 bits                                             (1) 
    
     where a 1 represents a level 1 adder cell. The 1&#39;s in this &#34;equation 1&#34; adder are placed next to each other to show that the Cout signal from a carry unit goes directly to the Cin port of the next successive carry unit. Here therefore, for the price of eight delay units, an adder of seven bits capabity can be achieved. 
     For the case of using up to the level 2 cells that are disclosed in the incorporated by reference Scriber et al patent: 
     
         222222|22222|2222|222|22|2.vertline.1=22 bits                                            (2) 
    
     where a 2 represents a level 2 cell and the symbol &#34;|× represents a stage boundary. At the stage boundary, the carry out (Cout) from the stage is connected to the mux control (Mux) for the next successive stage. The Mux signal is used by all of the cells within the stage. The first two stages here alternately can be combined into one stage with two level 1 cells and still produce Cout at the appropriate time. 
     
         222222|22222|2222|222|22|11=22 bits                                                      (3) 
    
     In either example of this case, therefore, the use of both level 1 and level 2 cells enables a 22-bit adder capacity for the price of eight delay units. 
     For the case of using up to the also Scriber et al disclosed level 3 cells: 
     
         33333,3333,333,33,3,2|3333,333,33,3,2|333,33,3,2.vertline.33,3,2|3,2|11=42 bits                   (4) 
    
     where a 3 represents a level 3 cell and the symbol &#34;,&#34; represents a substage boundary. At the substage boundary, Coutp0 and Coutp1 from a level 2 substage are connected to Mux0 and Mux1 for the next successive level 3 substage. For a substage boundary between level 3 substages than Cout0 and Cout1 are connected to Mux0 and Mux1 of the next successive substage. The Mux0 and Mux1 signals are used by all of the cells within the substage. The first two substages of each stage can be combined into one substage with two level 2 cells and still produce Coutpu0 and Coutp1 at the appropriate time. This combining step will be true for each new level and will be assumed from here on in the present illustration. 
     
         33333,3333,333,33,22|3333,333,33,22|333,33,22|33,22|22|11=42 bits                        (5) 
    
     For the case of using up to the herein-disclosed level 4 cells: 
     
         4444:444:44:33,444:44:33,44:33,33,22|444:44:33,44:33,33;22.vertline.44:33,33,22|33,22|22|11=57 bits (6) 
    
     where a 4 represents a level 4 cell and the new symbol &#34;:&#34; represents a subsubstage boundary (underlining here is used only to call attention to the introduced symbology scheme, the previously used term &#34;substage&#34; is, however, herein considered generic to the terms substage, subsubstage, subsubsubstage, etc.--including in the appended claims). At the subsubstage boundary, Coutp0 and Coutp1 from a level 3 subsubstage are connected to Mux00 and Mux11 for the next successive level 4 subsubstage. For a subsubstage boundary between level 4 subsubstages, Cout00 and Cout11 are connected to Mux00 and Mux11 of the next successive subsubstage. The Mux00 and Mux11 signals are used by all of the cells within the subsubstage. The availability of level 4 cells therefore increases the adder bit capacity from 42 to 57 bits while yet remaining within the 8 units of delay. 
     For the case of using up to level 5 cells: 
     
         555;55;44;55;44:44:33,55;44:44:33,44:33,33;22|55;44:44:33,44:33,33,22|44:33,33,22|33,22|22|11=63 bits (7) 
    
     where a 5 represents a level 5 cell and the new symbol &#34;;&#34; represents a subsubsubstage boundary. At the subsubsubstage boundary, Coutp0 and Coutp1 from a level 4 subsubsubstage are connected to Mux000 and Mux111 for the next successive level 5 subsubsubstage. For a subsubsubstage boundary between level 5 subsubsubstages then Cout000 and Cout111 are connected to Mux000 and Mux111 of the next successive subsubsubstage. The Mux000 and Mux111 signals are used by all of the cells within the subsubsubstage. 
     For the case of using up to level 6 cells: 
     
         66!55;55;44:55;44:44:33,55;44:44:33,44:33,33,22|55;44:44:33,44:33,33,22|44:33,33,22|33,22|22|11=64 bits (8) 
    
     where a 6 represents a level 6 cell and the new symbol &#34;!&#34; represents a subsubsubsubstage boundary. At the subsubsubsubstage boundary, Coutp0 and Coutp1 from a level 5 subsubsubsubstage are connected to Mux0000 and Mux1111 for the next successive level 6 subsubsubsubstage. For a subsubsubsubstage boundary between level 6 subsubsubsubstages then Cout0000 and Cout1111 are connected to Mux0000 and Mux1111 of the next successive subsubsubsubstage. The Mux0000 and Mux1111 signals are used by all of the cells within the subsubsubsubstage. 
     A diagram of an eight delay unit adder in accordance with the invention and the case of the present 64-bit illustration is shown in FIGS. 6 and 7. The first five stages of the adder are shown in FIG. 6; the sixth stage is shown in FIG. 7. This illustration has six stages and therefore can only employ up to level 6 cells. This six stage HOCMA can process 64 bits. From the illustration it can be seen that the FIG. 6 five stages can process 32 bits, while four stages can process 16 bits, etc. When extended, this bit capacity in fact means that a HOCMA which uses the highest level cells possible for a stage can process n bit in log 2  (n) stages. Since each such stage adds one mux of delay, n bits can therefore be processes in O(log 2  (n)) time. 
     With respect to the underlying timing considerations in the equation (8) adder, it is interesting to note that all of the exclusive or/nor units of the adder are settled after the first gate delay. Additionally, half of the carry units, the carry units in the first bit slice cell of each (sub)stage, are settled after the second gate delay. The remaining second bit slice carry units in each (sub)stage are settled after three gate delays. The remainder of the adder processing time is spent in making carry decisions and sum computation (after the final carry decision in each cell). Notably, the present high order carry multiplex adder devotes a majority of its processing time to parallel decision making in the carry signal selection circuitry rather than awaiting carry propagation within a substage, as occurs in the adder of the Scriber et al patent document. The higher order cells and their efficient interconnection in the present invention in essence enable the productive use of time devoted to awaiting carry settle out in the Scriber et al adder substages. 
     If a 128-bit adder according to the invention is needed, then a seven stage HOCMA could be created by using the six stage adder above with a 64-bit seventh stage. The seventh stage is built by using the structures of the six stage adder. The following five-step sequence will illustrate: 
     Step 1: Take the last stage. 
     
         66!55;55;44;55;44:44:33,55;44:44:33,44:33,33,22|  (9) 
    
     Step 2: Add the last substage (66!55;55;44:55;44:44:33,) at the high order bit end. 
     
         66!55;55;44:55;44:44:33, 66!55;55;44:55;44:44:33,55;44:44:33,44:33,33,22| (10) 
    
     Step 3: Add the last subsubstage (66!55;55;44:) also at the high order bit end. 
     
         66!55;55;44:66!55;55;44:55;44:44:33, 66!55;55;44:55;44:44:33,55;44:44:33,44:33,33,22| (11) 
    
     Step 4: Add the last subsubsubstage (66!55;) also at the higher order bit end. 
     
         66!55;66!55;55;44:66!55;55;44:55;44:44:33, 66!55;55;44:55;44:44:33,55;44:44:33,44:33,33,22| (12) 
    
     Step 5: Add a new subsubsubsubstage consisting of two subsubsubsubsubstages. The first subsubsubsubsubstage is made from two level 6 cells directly connected. The second subsubsubsubsubstage is made from two levels 7 cells directly connected. The new subsubsubsubstage has the appearance of 77#66!. Where a 7 represents a level 7 cell and &#34;#&#34; represents a subsubsubsubsubstage boundary. 
     The final 64-bit stage has the arrangement of: 
     
         77#66!66!55;66!55;55;44:66!55;55;44:55;44:44:33, 66!55;55;44:55;44:44:33,55;44:44:33,44:33,33,22| (13) 
    
     At the subsubsubsubsubstage boundary, Coutp0 and Coutp1 from a level 6 subsubsubsubsubstage are connected to Mux00000 and Mux11111 for the next successive level 7 subsubsubsubsubstage. If it is a subsubsubsubsubstage boundary between level 7 subsubsubsubsubstages when Cout00000 and Cout11111 are connected to Mux00000 and Mux11111 of the next successive subsubsubsubsubstage. The Mux00000 and Mux11111 signals are used by all of the cells within the subsubsubsubsubstage. 
     A 256-bit, eight-stage HOCMA can be created by making an eighth stage from seventh stage structures in the same way as the seven stage HOCMA was made. The process can be repeated to create HOCMAs with as many stages as desired. Each stage will process all of its bits in just one more gate delay than that required by the previous stage. The above five-step procedure in fact illustrates how an adder in accordance with the invention can be assembled. 
     In the above &#34;equations&#34;, it may be noted that the length of the indicated second stage is changed in comparison with the lengths recited in the Scriber et al application and in Scriber&#39;s thesis publication which is identified and incorporated by reference therein (and also incorporated by reference in the present application). This second stage length different results from the later realization that for optimum adder performance, successive third level substages further increase in length by one bit, however, this increase is not possible in the first third level substage because of the extra multiplexer delay of a third level bit slice cell in comparison with a second level bit slice cell. This realization also contributes to the improved performance of the present invention over that disclosed in the Scriber et al application--which is entirely functional and a useful embodiment, but not appreciated to be slightly less than optimum because of this consideration. 
     Propagation delays determine the pattern of interconnections between bits slice cells and in fat, the assignment of (sub)stage and stage boundaries in the FIG. 6 and 7 drawings. FIG. 7 includes the FIG. 7A and FIG. 7B portions which may be joined together for viewing. The rules for accomplishing the interconnection of bit slice cells within any (sub)stage or within any stage of the present adder may be expressed in mathematical symbols as follows. 
     There are four possible cases for cell interconnections in the present adder. In every case, Cout from the preceeding cell connects to Cin of the next cell. 
     In the first of the four cases, an i th  level cell is connected to an i th  level cell within a (sub)stage, where i is a whole number greater than 0 and (sub)stage means stage, substage, subsubstage, subsubsubstage, etc. For this case, Coutp0 connects to Cinp0 and Coutp1 connects to Cinp1. All of the Mux Ctr1 signals connect from one cell to the next cell. None of the other Cout(0) and Cout(1) signals connect to the next cell. 
     The second case is an i th  level cell connected to an i th  level cell where the two cells are in different (sub)stages. This is a (sub) i-2  stage boundary, where the notation (sub) 0  stage=stage, (sub) 1  stage=substage, (sub) 2  =stage=subsubstage, etc. For this case, Cout(0) i-2  connects to Mux(0) i-2  and Cout(1) i-2  connects to Mux(1) i-2 . All other Mux Ctr1 signals connect across to the next cell. Cin0 and Cin1 are connected to logic 0 and logic 1 respectively. Mux(0) i-2 , Mux(1) i-2 , Coutp0, Coutp1, and all other Cout(0) and Cout(1) signals terminate and do not connect across to the next cell. 
     The third case is an i th  level cell connected to an (i+1) th  level cell. Adjacent cells cannot exceed a level difference of one in ascending bit progression in such a connection. The connection of this case occurs at a (sub) i-1  stage boundary. For this case, Coutp0 connects to Mux(0) i-1 , Coutp1 connects to Mux(1) i-1 , and all Mux Ctr1 signals from the i th  cell connect to the remaining Mux Ctr1 signals of the (i+1) th  cell. Cin0 and Cin1 of the (i+1) th  cell are connected to logic 0 and logic 1 respectively. None of the Cout(0) and Cout(1) signals are connected across the boundary. 
     The fourth case is an i th  level cell connected to a k th  level cell, where k is less than i. This is a (sub) k-2  stage boundary situation. For this case, Cout(0) k-2  connects to Mux(0) k-2  and Cout(1) k-2  connects to Mux(1) k-2 ). Mux(0) j  connects to Mux(0) j  and Mux(1) j  connects to Mux(1) j  for 0≦j≦k-3. Cin0 and Cin1 of the k th  cell are connected to logic 0 and logic 1 respectively. Coutp0, Coutp1, and all other Mux(0), Mux(1), Cout(0), and Cout(1) signals from the i th  cell do not connect across the boundary. 
     In the illustrations, diagrams, and examples that have been given so far, the first stage consists of two ripple adder level 1 cells. Level 1 cells have been used to help show the pattern that exits between stages, however they are not required. The first stage can, in fact, make use of any adder architecture with any number of bits so long as the carry out signal therefrom is generated in the equivalent of three mux gate delays. A carry look-ahead adder could therefore be used for the first stage and would make the HOCMA even more efficient because a carry look-ahead adder can more than two bits in the time of three gate delays, however, all of the higher stages of such an adder must use level 2 and higher cells. 
     These analyses and illustrations have concentrated on the carry propagation time and have quoted the time delay for the final carry out. After Cout has been determined, the sum unit of the present adder cells, however, also uses the Cout signals to calculate the sum result. The sum unit takes an additional gate delay. This extra gate delay can, in fact, also be used to accommodate the delay of an additional stage containing a single level 1 adder cell because a level 1 cell requires one gate delay from the time that the Cin signal is available. Therefore, such a 65-bit adder can produce a sum output in nine gate delays; such an adder would appear as: 
     
         1|66!55;55;44:55;44:44:33,55;44:44:33,44:33,33,22|55;44:44:33,44:33,33,22|44:33,33,22|33,22|22|11(14) 
    
     The nine gate delays come from the three delays in the first stage plus one delay for each of the next five stages plus one delay for the combination of sum units and the last stage. The 1-bit level 1 last stage is not required, but can be significant value because it is an extra bit that requires no additional time and requires only a few gates to implement. 
     By way of further explanation of the time relationships in a carry multiplexed adder, each exclusive or/exclusive nor circuit can be described as two parallel multiplexers and therefore take one unit of propagation time to settle. The carry unit, the sum unit and each multiplexer layer within the bit slice also take one multiplexer gate delay to settle. Since addend and augend bits all arrive simultaneously at t=0 every exclusive or/exclusive nor unit in the adder settles after one gate delay. After this first delay, the carry unit outputs Coutp0 and Coutp1 for the first bit slice of each (sub)stage are settled--because the fixed input thereto are unchanging and the circuits await only the exclusive or/nor inputs before commencing to settle. Each successive carry unit within the (sub)stage settles after one additional gate delay per carry unit as the carry propagates through the (sub)stage. At the stage level, the first stage, once its carry output is settled, provides the mux control for all of the bits of the following stage. 
     Similarly, after the carry from a (sub)stage is settled it is used to determine the carry output of the following (sub)stage. Each successive (sub)stage requires one additional gate delay because of the multiplexing of the carry from the carry unit therein. Significantly, since each (sub)stage in the adder requires an additional gate delay, time is provided in which that (sub)stage can process an extra bit of addend and augend data. This is an underlying concept of the adder. 
     Similarly, each stage of the adder uses the carry out from the previous stage to determine its carry out. Since the carry out determination takes one additional gate delay, time is provided in which the stage can process an additional (sub)stage of addend and augend data. This addition of an entire (sub)stage of bit slice cells is in fact, a major advantage of the carry multiplex adder in comparison with the carry select adder, since the carry select adder could add only one extra bit for an equivalent gate delay. 
     In the present high level carry multiplex adder (HOCMA), moreover, this addition of a substage is even more productive of extra bit capacity because the adder substage can be composed of subsub stages that in turn may be composed of subsubsubstages and so on. This expansion is in fact only limited by the total time required for a cell to fully settle through all of its multiplex layers--each layer of multiplexing requires one time delay. 
     For example, if we have available a five-unit time, it is not possible to employ a level five substage because the carry out signal is not determined until after six units of gate delay time in a level five substage. The amount of time required for a cell&#39;s carry output to settle is the sum of the times of one unit each for the exclusive or unit and the carry unit plus one unit of time for each layer of multiplexer. 
     A significant advantage of the present invention high order carry multiplexed adder resides in its efficient performance. FIG. 8 shows the time (in gate delays) required to process n bits with three different types of adders, including a HOCMA. The FIG. 8 graph actually extends out to an adder of 128 bits. Because of the HOCMA&#39;s log 2  (n) performance, however, the achievable performance of this adder is even better for larger bit sizes not shown in FIG. 8--i.e., as the number of bits increases. FIG. 9 is a scaled version of FIG. 8 that gives a better distinction between the HOCMA and the carry select adder, particularly for smaller values of total delay time. 
     Usually increased speed results in a significant increase in the amount of required hardware and the physical size of this hardware. FIG. 10, however, compares the number of gates required to create an adder on n bits where n has values up to 128, and shows that the HOCMA is comparable to a carry select adder. The gate growth for the HOCMA is of order O(n). FIG. 11 compares the number of gates per bits processed; it shows that the HOCMA uses fewer gates per number of bits than a carry select adder when the adder is of less than about 80 bits capacity, for example. 
     A good metric used to compare adder architectures is to multiply the processing time by the number of gates required in the adder. The adder with the lowest gate time metric is, of course, the most efficient. FIG. 12 compares the gate time metric for HOCMA and two other adders. FIG. 13 is an ordinate scaled version of FIG. 12 to illustrate the differences between the HOCMA and the carry select adder. FIG. 13 shows the HOCMA is a significant improvement over other adder architectures. 
     Alternative Arrangements 
     The above illustrations have shown how to make adders whose bit lengths are powers of 2 plus one. If a HOCMA is needed whose length is not a power of 2 plus one then the number of stages required is log 2  (n-1) rounded up to the next whole number. The final full stage of such an adder may not require the use of the highest level cells in its fabrication. For example, if a 64-bit adder is required then a seven stage HOCMA can be used with the last full stage using only up to level 5 cells: 
     
         1|555;55;44:55;44:44:33,55;44:44:33,44:33,33,22|55;44:44:33,44:33,33,22 44:33,33,22|33,22|22|11=64 bits (15) 
    
     A comparison of this adder with the adders of equations (8) and (13) further illustrates this point. The equation (15) adder also employs the one extra level one cell concept introduced in equation (14) above. 
     If a 39-bit adder is required then a seven stage HOCMA can be used with the last full stage consisting of six level 2 cells as follows: 
     
         1|222222|55;44:44:33,44:33,33,22|44:33,33,22.vertline.33,22|22|11                        (16) 
    
     If a 49-bit adder is needed then the adder could appears as: 
     
         1|33333,3333,333,33,22|55;44:44:33,44:33,33,22|44:33,33,22|33,22|22|11          (17) 
    
     By this arrangement of using lower level cells in the last full stage of an adder, the total gate count of a HOCMA can be decreased, since the required total number of multiplexer circuits is reduced. FIGS. 10-13 have not, however, taken into account this reduced gate count from using lower level cells in the final full stage, nor do they include the use of a one bit level 1 last stage, as shown in the equation (14)-(17) adders. The figures do, however, show the worst case performance for bit counts that are not powers of 2. 
     HOCMA cells can also be used in several different configurations within an adder stage, with the different configurations producing variations in performance results. Shown herein are what is believed to be the most efficient configuration, but not the only configuration. 
     The purpose of this invention is therefore to produce an efficient adder architecture that can add two numbers together quickly without requiring an overabundance of hardware. The high order carry multiplexed adder is very efficient, since it can process n bits in the order of O(log 2  (n) time, yet it requires in the order of O(n) gates. 
     While the apparatus and method herein described constitute a preferred embodiment of the invention, it is to be understood that the invention is not limited to this precise form of apparatus or method and that changes may be made therein without departing from the scope of the invention which is defined in the appended claims. ##SPC1##