Questions for Self-Testing

1. How is multiplication performed in a positional number system?

2. What are the peculiarities of fast multiplication algorithm realization?

3. What does the multiplication speed depend on?

4. What is the idea of the Booth algorithm?

5. Why, with appearing systems-on-chip, has the return to old methods of multiplication operation performing without using the method of increasing been become?

2.4. Integer Division

We discussed positive-number multiplication by relating the way in which the multiplication operation can be done on paper to the way a logic circuit can do it. It is instructive to follow the same strategy here. We will only discuss positive-number division in detail. Some general comments on the signed-operand case will be made later.

F ig. 2.21 shows an example of decimal division along with the binary-coded division of the same values. Consider the decimal version. The 2 in the quotient is determined by the following reasoning. We first "try" 13 divided into 2 and it "doesn't go". Next we "try" 13 divided into 27. We go through the trial exercise of multiplying 13 by 2 to get 26, and knowing that 27-26 = 1 is less than 13, we finally enter 2 as the quotient and perform the required subtraction. The next digit of the dividend, 4, is "brought down" and we finish by deciding that 13 goes into 14 once, with the remainder being determined as 1. A similar discussion can be given for the binary case, with the simplification that the only possibilities for the quotient bits are 0 and 1.

T he important point to conclude from the above division "algorithm" is that the process of trial determination of the quotient digits is more difficult to automate in a logic circuit than the selection and addition of summands in the multiplication case. The simplest circuit that implements binary division must methodically position the divisor with respect to the dividend and perform a subtraction. If the remainder is zero or positive, a quotient bit of 1 is determined, the remainder is extended by another bit of the dividend, the divisor is repositioned, and another subtraction is performed. On the other hand, if the remainder is negative, a quotient bit of 0 is determined, the dividend is restored by adding back the divisor, and the divisor is repositioned for another subtraction. Fig. 2.22 shows a logic circuit arrangement that implements the above restoring-division technique. Note its similarity to the structure for multiplication that was shown in Fig. 2.8. An n-bit positive divisor is loaded into register M and an n-bit positive dividend is loaded into register Q at the start of the operation. Register A is set to 0. After the division is complete, the n-bit quotient will be in register Q with the remainder in A.

The required subtractions are facilitated by using 2's-complement arithmetic. The extra bit position at the left end of both A and M is for the sign bit for the subtractions. The following algorithm performs the division:

S1: DO n times

Shift A and Q left one binary position.

Subtract M from A, placing the answer back in A.

If the sign of A is 1, set q₀ to 0 and add M back to A (restore A); otherwise, set q₀ to 1.

Fig. 2.23 shows a 4-bit example as it would be processed by the circuit in Fig. 2.22. It is possible to improve on this algorithm by avoiding the need for restoring A after an unsuccessful subtraction. (We define a subtraction to be unsuccessful if the result is negative.) Consider the sequence of operations that takes place after the subtraction operation in the above algorithm. If A is positive, we shift it left and subtract M; that is, we perform 2A - M. If A is negative, we restore it by performing A + M, and then we shift it left and subtract M. This is equivalent to performing 2A + M. The q₀ bit is appropriately set to 0 or 1 after the correct operation has been performed. We can summarize this into the following nonrestoring-division algorithm:

S1: DO n times

If the sign of A is 0, shift A and Q left one binary position and subtract M from A; otherwise, shift A and Q left and add M to A.

If the sign of A is 0, set q₀ to 1; otherwise, set q₀ to 0.

S2: If the sign of A is 1, add M to A.

The S2 step is needed to leave the proper positive remainder in A at the end of n cycles. It is clear that the logic circuitry of Fig. 2.22 can also be used to perform this algorithm as well as the original one. Note that the restore operations are not used and exactly one add or subtract operation is performed per cycle. Fig. 2.24 shows the division example of Fig. 2.23 is executed by the nonrestoring-division algorithm.

There are no simple algorithms for performing signed division that are comparable to the multiplication situation. In division, preprocessing of the operands and/or postprocessing of the results are usually required. These extra operations depend on the sign of the operands in conjunction with the desired signs of the results, and complementation of one or more numbers is usually involved. It is useful to note that we can always transform the operands to positive values, use one of the algorithms discussed above, and transform the results to the correct signed values as necessary.