Arithmetic Unit Addition Subtraction Multiplication and Division
- 1. Multiplication and Division with Signed-Magnitude Data Module II
- 2. Eight Conditions for Signed- Magnitude Addition/Subtraction Operation ADD Magnit udes SUBTRACT Magnitudes A > B A < B A = B (+A) + (+B) + (A + B) (+A) + (-B) + (A – B ) - (B – A ) + (A – B ) (-A) + (+B) - (A – B ) + (B – A ) + (A – B ) (-A) + (-B) - ( A + B) (+A) - (+B) + (A – B ) - (B – A ) + (A – B ) (+A) - (-B) + (A + B) (-A) - (+B) - ( A + B) (-A) - (-B) - (A – B ) + (B – A ) + (A – B )
- 3. Examples Example of adding two magnitudes when the result is the sign of both operands: +3 0 011 + +2 0 010 +5 0 101 Example of adding two magnitudes when the result is the sign of the larger magnitude: -3 1 011 + +2 0 010 -( +3 011 - +2) 010
- 4. Flowchart of Addition and Subtraction with Signed-Magnitude Data
- 5. Addition and Subtraction with Signed- Magnitude Data Hardware Design A register AVF E Bs As B register Complementer Parallel Adder S Load Sum M Mode Control Input Carry Output Carry
- 6. Summary of Addition and Subtraction with Signed-Magnitude Data The signs use an exclusive OR gate where if the output is 0, then the signs are the same. Hence, add the magnitudes of the same signed numbers. If the sum is an overflow, then a carry is stored in E where E = 1 and transferred to the flip-flop AVF, add- overflow. Otherwise, the signs are opposite and subtraction is initiated and stored in A. No overflow can occur with subtraction so the AVF is cleared. If E = 1, then A > B. However, if A = 0, then A = B and the sign is made positive. If E = 0, then A < B and sign for A is
- 7. BCD Adder Is a circuit that adds two BCD digits in parallel and produces the sum in BCD. Correction logic
- 8. Derivation of BCD Adder
- 9. Block Diagram of BCD Adder
- 10. Correction logic formula C=1 add 0110 to the binary sum
- 11. Operation Decimal digits are added with input carry with binary adder. If output carry is zero, nothing is to be added to the binary sum. When output carry=1, 0110 is added to the binary sum through the next adder. the next output carry can be ignored..
- 12. Multiplication • A complex operation compared with addition and subtraction Many algorithms are used, esp. for large numbers Simple algorithm is the same long multiplication taught in grade school — Compute partial product for each digit — Add partial products
- 13. Multiplication Example 1011 Multiplicand (11 dec) x 1101 Multiplier (13 dec) 1011 Partial products 0000 Note: if multiplier bit is 1 copy 1011 multiplicand (place value) 1011 otherwise zero 10001111 Product (143 dec) Note: need double length result
- 15. looking at successive bits of the multiplier, least significant bit first. If the multiplier bit is a 1, the multiplicand is copied down; otherwise, zeros are copied down. The numbers copied down in successive lines are shifted one position to the left from the previous number. Add these numbers.
- 16. Array Multiplier Positive numbers can be implemented in a combinational two-dimensional array Main component of each cell is a full adder FA
- 19. Array multiplier is well known due to its regular structure. Multiplier circuit is based on add and shift algorithm. Each partial product is generated by the multiplication of the multiplicand with one multiplier bit. The partial product are shifted according to their bit orders and then added. The addition can be performed with normal carry propagate adder. N-1 adders are required where N is the multiplier length
- 22. Flowchart
- 24. Booths Multiplication Algorithms 1. The multiplicand is subtracted from the partial product upon encountering the first least significant 1 in a string of 1's in the multiplier. 2. The multiplicand is added to the partial product upon encountering the first 0 (provided that there was a previous 1) in a string of O's in the multiplier. 3. The partial product does not change when the multiplier bit is identical to the previous multiplier bit.
- 26. Flowchart
- 29. Hardware Implementation for Signed-Magnitude Data for restoring divioin Instead of shifting the partial product to the right,the divisor and partial product are shifted to the right. For subtraction, perform 2’s complemented addition.
- 31. Algorithms steps Do the following n times Shift A and Q left one binary position Subtract M from A, and place the answer back in A.(2’s complement addition of divisor) If E=1 A>=B set Qn=1. Quotient bit 1 is added in Qn bit of partial remainder and that is shifted to the left.
- 32. If E=0,A<B so the quotient in Qn remains a 0. The value of B is then added to the partial remainder and is shifted to the left. Finally, , the quotient is in Q and final remainder in A.
- 34. Divide Overflow Critical when implemented in hardware The length of the registers is finite and will not hold a number exceeds its length. When the dividend is twice as long as the divisor, overflow occurs: If the high-order half bits of the dividend constitute a number greater than or equal to the divisor.
- 35. Division by zero must be avoided. detected by DVF(divide overflow flipflop).
- 36. Handling Divide Overflow Duty of the programmer The occurrence of a divide overflow stopped the computer (divide stop). Not recommended(time consuming) Provide an interrupt request when DVF is set. Suspend the current program and branch to a service routine to take the corrective actions.
- 37. Corrective measure: Remove the program and type an error message stating the reason. The best way to avoid divide overflow is use floating point data.
- 38. Flowchart