# Arithmetic Logic Unit

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1 Arithmetic Logic Unit A.R. Hurson Department of Computer Science Missouri University of Science & Technology A.R. Hurson 1

2 Arithmetic Logic Unit It is a functional bo designed to perform "basic" arithmetic, logic, and shift operations on the data. Implementation of the basic operations such as logic, program control, and data transfer operations are easier than arithmetic and I/O operations. Therefore, in this section we concentrate on arithmetic operations. A.R. Hurson 2

3 Arithmetic Logic Unit For a simple machine, the ALU should at least be able to perform operations such as: add increment subtract decrement A simple ALU is basically an adder and some control circuits augmented by special circuits to carry out the logic and shift operations. A.R. Hurson 3

4 Arithmetic Logic Unit An ALU can be of three types: Serial Parallel Functional (Modular) A.R. Hurson 4

5 Arithmetic Logic Unit Similar to the definition of serial and parallel adders, one can define serial and parallel ALUs. In a serial ALU during each clock pulse one bit of operand(s) participates in the operation. In a parallel ALU operation on all the bits of operand(s) is initiated simultaneously. In a simple word a parallel ALU can be looked at as a cascade of identical units forming a one dimensional array of cells. A.R. Hurson 5

6 Arithmetic Logic Unit Parallel ALU ALU operation is determined by the control signals. In a very simple form, the bit-pattern of the control signals is determined by the operation code. In a parallel ALU, one needs to determine the design of a unit and then replicate it. Control Signals B n A n B 2 A 2 B 1 A 1 C n+1 C n C 3 C 2 C 1 F n F 2 F 1 A.R. Hurson 6

7 Arithmetic Logic Unit Parallel ALU (Eample) S 2, S 1, S, and M are the control signals. A i, B i, and C i are the operand bits and carry-in, respectively. F i and C i+1 are the result bit and carry-out, respectively. A.R. Hurson 7

8 A i X i S 2 C i+1 B i S Y F Arithmetic 1 Logic i Unit Parallel ALU i (Eample) S C i Z i M A.R. Hurson 8

9 Arithmetic Logic Unit Parallel ALU (Eample) The function of each stage can be defined as: F i = X i Y i Z i C i+1 = X i Y i +(X i Y i )Z i = X i Y i +X i Z i +Y i Z i By appropriate setting of the control signals one can initiate a variety of the operations. A.R. Hurson 9

10 Arithmetic Logic Unit Parallel ALU (Eample) M = S 2 =1 S 1 =1 F S =1 M=1 S 2 =1 S 1 =1 S =1 C 1 = F A - 1 A.R. Hurson 1

11 Arithmetic Logic Unit As discussed before a parallel ALU offers a higher speed relative to a serial ALU. How can one improve the performance (speed) of ALU further? In other words; is it possible to build (design) an ALU faster than a parallel ALU? A.R. Hurson 11

12 Arithmetic Logic Unit Functional (modular) ALU In this model, ALU is a collection of independent units each tailored (specialized) for a specific operation. As a result, eecution of independent operations can be overlapped. This approach allows an additional degree of concurrency relative to a parallel ALU, since it allows several operations to be performed on data simultaneously. However, this speed improvement comes at the epense of etra overhead which is needed to detect data independent operations. A.R. Hurson 12

13 Arithmetic Logic Unit Functional (modular) ALU Adder Adder 1 2 Subtractor Multiplier A.R. Hurson 13

14 Arithmetic Logic Unit Question Is it possible to improve the performance of an ALU further? Naturally, we can improve performance (physical speed) by taking advantage of the advances in technology. How can we improve the logical speed of the ALU further? In a functional ALU, is it possible to devise algorithms which allow one to improve the performance of the basic operations? If this is a valid direction, then the question of "How to design a fast ALU?" will change to "How to design a fast adder, a fast multiplier,...?" A.R. Hurson 14

15 Arithmetic Logic Unit Arithmetic Algorithm The goal is to develop a set of algorithms which allow the eecution of basic arithmetic operations on data values presented in different formats and notations (i.e., fied point, floating point, signed numbers). In general fied point operations are easier and faster than floating point operations. Signed numbers could be in signed magnitude or signed complement format. A.R. Hurson 15

16 Arithmetic Logic Unit Comparison of 2-Unsigned Numbers (Direct Comparison) Two binary bits (a, b) are equal iff: Consequently two registers of n-bits are equal iff: = n-1 n-2... = 1 where i = A i B i i n-1 In other words ( a b ) ( a ) = 1 = b n 1 A = B = i i= = 1 A.R. Hurson 16

17 Arithmetic Logic Unit Comparison of 2-Unsigned Numbers (Direct Comparison) Similarly A >B A n-1 B n-1 + n-1 A n-2 B n n-1 n A B = 1 A <B A n-1 B n-1 + n-1 A n-2 B n n-1 n A B = 1 A.R. Hurson 17

18 Arithmetic Logic Unit Comparison of 2-Unsigned Numbers (Direct Comparison) Eample: A= B= = = A B A.R. Hurson 18

19 Arithmetic Logic Unit Comparison of 2-Unsigned Numbers (Direct Comparison) = = A B A B A B A B A B A B A B A.R. Hurson = = A B A > B = = A B A B A B A B A B A B A B A < B

20 Arithmetic Logic Unit Comparison of 2-Unsigned Numbers (Direct Subtraction) Assume E is the borrow-out of a parallel subtractor then: if A B then E = if A < B then E =1 if A = B then E = and result all s In other words, to check the relative magnitude of two unsigned numbers say, A and B. Perform A-B and then check the borrow-out. The relative magnitude of A and B is determined according to the above relationships. A.R. Hurson 2

21 Arithmetic Logic Unit Comparison of 2-Unsigned Numbers (2 s Complement Addition) Assume E is the carry-out of a parallel adder then: if A < B then E = if A B then E = 1 if A = B then E = 1 and result all s A.R. Hurson 21

22 Arithmetic Logic Unit Comparison of 2-Unsigned Numbers (2 s Complement Addition) Proof: ~ A B = A + B = A + n 2 B = 2 if A B A - B 2 n + A - B 2 n E = 1 if A < B A B < 2 n + A - B < 2 n E = if A = B A - B = 2 n + A - B = 2 n E = 1 and result all s To determine the relative magnitude of two unsigned numbers, perform A + B ~ and then check the carry-out. A.R. Hurson 22 n + A B

23 Arithmetic Logic Unit Comparison of 2-Unsigned Numbers (1 s Complement Addition) Assume E is the carry-out of a parallel adder then: if A B then E = if A = B then E= and result all 1s if A > B then E =1 To determine the relative magnitude of two unsigned numbers, perform A + B and then check the carry-out. A.R. Hurson 23

24 Arithmetic Logic Unit Comparison of 2-Unsigned Numbers (Direct Subtraction) Eample Borrow-out A - B E = 1 A < B A.R. Hurson 24

25 Arithmetic Logic Unit Comparison of 2-Unsigned Numbers (2 s Complement Addition) Eample Carry-out A B E = A < B A.R. Hurson 25

26 Arithmetic Logic Unit Comparison of 2-Unsigned Numbers (1 s Complement Addition) Eample Carry-out A B E = A < B.. A.R. Hurson 26

27 Questions Prove the aforementioned formulas for the 1 s complement addition operation. How can one determine the relative magnitude of signed numbers? A.R. Hurson 27

28 Arithmetic Logic Unit Addition Algorithm (2 s Complement Numbers) Add numbers including the sign bits. If there is a carry-out of the last digit, then disregard it. A.R. Hurson 28

29 Arithmetic Logic Unit Addition Algorithm (1 s Complement Numbers) Add numbers including the sign bits. If there is a carry-out then increment the result by 1 (wrap around the carry). A.R. Hurson 29

30 Arithmetic Logic Unit Addition Algorithm (Signed magnitude numbers) Addition (Subtraction): When the signs of numbers (say A and B) are identical (different) add the two magnitudes. Sign of the result is the same as A. When the signs of A and B are different (identical), compare the magnitudes and subtract the smaller magnitude from the larger one. Sign of the result is the same as the sign of the larger magnitude. If result zero then set the sign of the result to. A.R. Hurson 3

31 Arithmetic Logic Unit Addition Algorithm (Signed magnitude numbers) How to reduce hardware requirements of the proposed algorithm? Subtraction can be converted into addition. Magnitudes can be compared using an adder. A.R. Hurson 31

32 Arithmetic Logic Unit Addition Algorithm (Signed magnitude numbers) B s B Register End Carry Complementor Parallel Adder cc A s E A Register cc = if (addition and A s = B s ) or (subtraction and A s B s ) cc = 1 if (addition and A s B s ) or (subtraction and A s = B s ) A.R. Hurson 32

33 Arithmetic Logic Unit Addition Algorithm (Signed magnitude numbers) Eample A (+29) +B (+14) Same signs and addition E Add the magnitudes E = A = Final result (+43). A.R. Hurson 33

34 Arithmetic Logic Unit Addition Algorithm (Signed magnitude numbers) Eample A (-14) +B (-7) E Same signs and addition Add the magnitudes. 111 E = A = 1111 Final result (-21). A.R. Hurson 34

35 Arithmetic Logic Unit Addition Algorithm (Signed magnitude numbers) Eample A (-14) -B (-7) E Same signs and subtract Add 2 s complete of B to A E = 1, A>B A = 1111 Final result (-7). A.R. Hurson 35

36 Arithmetic Logic Unit Addition Algorithm (Signed magnitude numbers) Eample A (+14) -B (+2) Same signs and subtract E Add 2 s complete of B to A. 111 E =, A<B, take 2 s complement of result A = 111 Final result (-6). A.R. Hurson 36

37 Arithmetic Logic Unit Addition Algorithm (Signed magnitude numbers) Eample A (-14) -B (-14) E Same signs and subtract Add 2 s complete of B to A. 1 E = 1, and result zero A = Final result (+). A.R. Hurson 37

38 Arithmetic Logic Unit Addition Algorithm (Signed magnitude numbers) Eample A +B E Different signs and addition Take 2 s complement of B and add E = 1 and result A > B A = 111 Final result. A.R. Hurson 38

39 Arithmetic Logic Unit Addition Algorithm (Signed magnitude numbers) Eample A +B Different signs and addition Take 2 s complement of B and add. E 11 1 E = 1 and result A > B A = 1 1 Fin al r esu lt. A.R. Hurson 39

40 Arithmetic Logic Unit Addition Algorithm (Signed magnitude numbers) Eample A - B Same signs and subtraction. E Take 2 s complement of B and add. E = B > A A = 1 1 Final result, take 2 s complement of result. A.R. Hurson 4

41 Questions What is an overflow? How can we detect overflow for: Signed magnitude addition? 1 s complement addition? 2 s complement addition? A.R. Hurson 41

42 Arithmetic Logic Unit Addition Overflow Overflow occurs when two numbers of n digits each are added and the sum occupies n+1 digits. An overflow cannot occur after an addition if one number is positive and the other is negative. In general, overflow can only occur when adding numbers of the same sign or subtracting numbers of different signs. A.R. Hurson 42

43 Arithmetic Logic Unit Addition Overflow (Signed magnitude numbers) Detection of overflow in signed magnitude addition is simple. Carry-out of the magnitude bits represents an overflow: B s B Overflow Bit AVF Complementor CC Parallel-Adder A s E A If (cc = and E = 1) then AVF 1 otherwise AVF A.R. Hurson 43

44 Arithmetic Logic Unit Addition Overflow (2 s Complement Numbers) Overflow is occurred when adding numbers of the same sign results in a number of a different sign. C n B s B FA Parallel-Adder C n-1 A s Overflow occurs if C n C n-1 = 1 A.R. Hurson 44 A

45 Arithmetic Logic Unit Multiplication Algorithm Multiplication can be performed as a sequence of additions. B * Q add B, Q times Two general approaches have been adapted for multiplication. Software approach Hardware approach A.R. Hurson 45

46 Arithmetic Logic Unit Multiplication Algorithm Software approach AC ; For i = 1 to Q do; AC (AC) + (B); End; Time compleity of this algorithm is (m) where m is the magnitude of Q. A.R. Hurson 46

47 Arithmetic Logic Unit Multiplication Algorithm Software approach Time compleity of the algorithm can be improved if the number of iterations is based on the multiplier length: AC ; For i = 1 to n Do; if Q 1 = 1 then C out AC (AC) + (B); AC n...ac 2 AC 1 Q n...q 2 Q 1 C out AC n...ac 2 AC 1 Q n...q 2 ; End; A.R. Hurson 47

48 Arithmetic Logic Unit Multiplication Algorithm Hardware Approach The simplest hardware approach is based on the previous algorithm - e.g., add and shift. In each iteration the least-significant bit of multiplier (Q) is checked; If one, then B is added to the accumulator and the contents of accumulator and Q is shifted right one position. If zero, just shift accumulator and Q to the right. A.R. Hurson 48

49 Arithmetic Logic Unit Multiplication Algorithm (Signed magnitude numbers) Hardware Configuration A double register (AC and Q) holds the final result. Sign of the result is the EX-OR of the signs of multiplier and multiplicand. B s B Parallel Adder AC s Q s AC Q C out Least Significant Bit A.R. Hurson 49

50 Arithmetic Logic Unit Multiplication Algorithm (Signed magnitude numbers) Eample E B s B AC Q s Q 1 Q1=1 then EAC (AC)+(B) 1) ) Shift E(AC)(Q) to the right Q1=1 then EAC (AC)+(B) Shift E(AC)(Q) to the right Q1= shift E(AC)(Q) to the right 3) 4) Q1= shift E(AC)(Q) to the right Q1=1 then EAC (AC)+(B) 5) Shift E(AC)(Q) to the right A.R. Hurson 5 AC s Q s

51 Arithmetic Logic Unit Multiplication Algorithm (2 s Complement numbers) Pre-Post Complementation Convert negative numbers to positive numbers. Apply "add and shift" algorithm. Convert the result - if necessary. A.R. Hurson 51

52 Arithmetic Logic Unit Multiplication Algorithm (2 s Complement numbers) Etension of "add and shift" Convert the multiplier to a positive number - i.e., if multiplier is negative then multiply the multiplier and the multiplicand by -1. Apply a modified "add and shift" algorithm in which addition and shift operations are done for 2 s complement numbers. A.R. Hurson 52

53 Arithmetic Logic Unit Multiplication Algorithm (2 s Complement numbers) Booth's Algorithm Instead of checking one bit of the multiplier at a time, check two bits at a time and take proper actions according to the following table: no action, shift right. 1 add multiplicand, shift right. 1 subtract multiplicand, shift right. 11 no action, shift right. A.R. Hurson 53

54 Arithmetic Logic Unit Multiplication Algorithm (2 s Complement numbers) Booth's Algorithm It is suitable for 2 s complement numbers. On average, Booth's Algorithm is faster than "add and shift". Always, etend the multiplier by a zero on the right. A.R. Hurson 54

55 Arithmetic Logic Unit Multiplication Algorithm (2 s Complement numbers) Booth's Algorithm (Eample) Etension st 2 bits are 1 subtract (add 2 s complement) A.R. Hurson 55 B * Q Shift right 1 st 2 bits are 11 shift right 1 st 2 bits are 11 shift right 1 st 2 bits are 1 add Shift right 1 st 2 bits are shift right 1 st 2 bits are shift right 1 st 2 bits are shift right

56 Questions Why is Booth's algorithm faster than "add and shift" algorithm (in general)? Why does Booth's Algorithm work? A.R. Hurson 56

57 Arithmetic Logic Unit Division Algorithm Similar to multiplication, one can develop a software routine which performs division as a sequence of subtractions. Naturally, such an algorithm is very inefficient and slow. A.R. Hurson 57

58 Arithmetic Logic Unit Division Algorithm Similar to the pencil and paper routine, one can develop an algorithm which performs division as a sequence of Compare, Shift, and Subtract operations. One should note that division in a binary system is much simpler than the division in decimal system, since the quotient digits are either or 1. A.R. Hurson 58

59 Arithmetic Logic Unit Division Algorithm To minimize the hardware requirements for division, we should remember that: Comparison can be performed via arithmetic operation(s). Subtraction can be performed via complement-addition. In other words; division requires almost the same hardware modules as multiplication does. A.R. Hurson 59

60 Arithmetic Logic Unit Division Algorithm (Signed magnitude numbers) Division can be carried out as a sequence of n (n is the length of divisor) iterations. Dividend is a double register. One bit of the quotient is generated in each iteration. At the end of the operation, the quotient is in the 1 st half part of the double register (low-order part), and remainder is in the 2 nd half part. Sign of the quotient is the XOR of the signs of dividend and divisor. Sign of the remainder is the same as the sign of the dividend. A.R. Hurson 6

61 Arithmetic Logic Unit Division Algorithm (Divide Overflow) In the division algorithm for signed magnitude numbers, if the 1 st half part of dividend is larger than or equal to the divisor, then the quotient overflows to the remainder part. This phenomenon is called divide overflow. As a result, a wrong answer will be generated. To avoid divide overflow and also due to the fact that division is a time consuming operation, the divide overflow condition is checked at the beginning of the operation. In case of divide overflow, proper actions will be taken (in the form of say an interrupt). A.R. Hurson 61

62 Arithmetic Logic Unit Methods of Division There are several different algorithms for division: Restoring Method Non-Restoring Method Direct Comparison A.R. Hurson 62

63 Arithmetic Logic Unit Methods of Division Restoring Method: In the restoring method, the contents of the partial remainder is restored whenever it is detected that the divisor is larger than the partial remainder. A.R. Hurson 63

64 Arithmetic Logic Unit Methods of Division Non-Restoring Method: In non-restoring method, if it is detected that the divisor is larger than the partial remainder, its contents are not restored. However, in the net iteration, instead of subtracting the divisor from the partial remainder, it will be added to the partial remainder. A.R. Hurson 64

65 Arithmetic Logic Unit Methods of Division Direct Comparison: divisor is compared against partial remainder. If it is smaller than or equal to partial remainder, it will be subtracted and quotient digit is set to 1. Otherwise, just the quotient digit is set to zero. A.R. Hurson 65

66 Arithmetic Logic Unit Methods of Division Naturally, restoring and non-restoring techniques are equivalent. Consider the consecutive sequence of iterations for both restoring and non-restoring techniques (net slide). As can be seen, both schemes generate the same value as the partial remainder. However, the non-restoring technique should be faster than the restoring method since it requires fewer number of operations. A.R. Hurson 66

67 Arithmetic Logic Unit Methods of Division Restoring Non-Restoring Subtract, Restore, and Shift in the first iteration Subtract in the second Iteration (2 ( (A-B)+B) -B) =2A-B Addition in the second Iteration 2 (A-B) +B =2A-B Subtraction and shift in the first iteration A.R. Hurson 67

68 Arithmetic Logic Unit Division Algorithm (Eample) Restoring Method E Etended A Q B Sign Bit Shift left EAQ. Subtract B. E=1 Q 1. Restore A. Shift left EAQ. Subtract B. E=1 Q 1. Restore A. Shift left EAQ. Subtract B. E= Q 1 1. Shift left EAQ. Subtract B. E= Q A.R. Hurson R Q Q n

69 Arithmetic Logic Unit Division Algorithm (Eample) Non-restoring Method E A Q B R Shift left EAQ. Subtract B. E=1 Q 1. Shift left EAQ. Add B. E=1 Q 1. Shift left EAQ. Add B. E= Q 1 1. Shift left EAQ. Subtract B. E= Q 1 1. A.R. Hurson 69 Q

70 Arithmetic Logic Unit Floating Point Numbers A number N in base r can be represented in many ways. For eample, decimal number can be represented as:.2945 * * 1-1 A scientific or floating point format of a number is a unique representation in which a number is represented as the multiple of 2 parts: a power of the base (called eponent)* a fraction (called mantissa) For eample, = * 1 +2 A.R. Hurson 7

71 Arithmetic Logic Unit Floating Point Numbers Such a representation in the computer requires two registers, one to represent the eponent and one to represent the mantissa. Note that each part is a signed number which could be represented in signed magnitude or signed complement format. A floating point number is normalized if the most significant digit of the mantissa is non-zero. In most systems the eponent is represented in ecess base. This means positive eponent. Therefore, a fied value (e.g., biased value) will be added to the eponent. For eample, IBM uses ecess 64 for the power, and DEC1 uses ecess 128 to represent the power. A.R. Hurson 71

72 Arithmetic Logic Unit Floating Point Numbers Convert (-25.75) 1 to a floating point number for IBM: (-25.75) 1 = -(111.11) 2 = -( * 2 5 ) 2 = ( * 2 8 ) 2 = * 16 2 A.R. Hurson 72

73 Arithmetic Logic Unit Floating Point Addition and Subtraction Check for zeroes. Align the mantissas (smaller eponent should be equated to the larger one). Add or subtract the mantissas. Normalize the result. If ep 1 ep 2 then If ep 1 < ep 2 then ( mantissa ) ( mantissa ) ( ep ep ) r 2 fl1 fl2 = ± 1 ± 2 * * b fl ( mantissa ) ( ep ep ) 1 ep A.R. Hurson 73 r ( mantissa ) r 2 r 1 = 1 * ± b 2 ep 1 ± fl2 * b b 1 2

74 Arithmetic Logic Unit Floating Point Multiplication Check for zeros. Add the eponents. Adjust the eponent of the result. Multiply the mantissas. Normalize the result. fl * fl 2 = ( mantissa 1 ) * (mantissa) 2 * r b (ep 1 + ep 2 ) 1 A.R. Hurson 74

75 Arithmetic Logic Unit Floating Point Division Check for zeroes. Align the dividend (for divide overflow condition). Subtract the eponents. Adjust the eponent of the result. Divide the mantissas. fl 1 / fl 2 = ( mantissa 1 ) / (mantissa) 2 * r b (ep 1 - ep 2 ) A.R. Hurson 75

76 Arithmetic Logic Unit Singed magnitude number: A method to represent signed numbers. 2 s complement number: A method to represent signed numbers. 1 s complement number: A method to represent signed numbers. Addition overflow: A case where the addition result is too big to fit in the destination register. A.R. Hurson 76

77 Arithmetic Logic Unit Divide overflow: A case when the quotient result is too big to fit in the quotient register. Add-and-shift algorithm: A method to perform multiplication. Booth's algorithm: A method to perform multiplication for 2 s complement numbers. Restoring technique: A method to perform division. Non-restoring technique: A faster method than restoring technique to perform division. A.R. Hurson 77

78 Arithmetic Logic Unit Biased value: A fied and positive value added to the eponent of a floating point number. The goal is to deal with positive (unsigned) eponents. Normalized number: A floating point number is normalized if the first digit in the mantissa is a non zero digit. Aligning the mantissa: A process by which the mantissa is shifted to the right by the number of differences between two eponents. Mantissa: The fraction portion of a floating point number. A.R. Hurson 78

79 Questions: Prove that two unsigned numbers can be compared by means of 1s complement addition. How do we detect overflow for signed magnitude addition? How do we detect overflow for 2s complement addition (note I am looking for another technique than the one discussed in this unit)? Prove that the add-and-shift algorithm, as implemented in this unit, does not generate an overflow. Justify the following statement. On average, Booth s algorithm is faster than add-and-shift algorithm. A.R. Hurson 79

80 Questions: On average, Booth s algorithm is faster than add-and-shift algorithm, this means that in some cases Booth s algorithm would be slower than add-and-shift algorithm. Can you give an eample that supports this? Add-and-shift checks one bit of multiplier during each iteration, Booth s algorithm checks two bits of multiplier during each iteration and as we discussed Booth s algorithm is faster that addand-shift. Can you come up with an algorithm that multilies two numbers faster than Booth s algorithm? A.R. Hurson 8

81 Questions: In case of a floating point addition or subtraction operation, why should we always align the mantissa of the smaller number? In the case of multiplication of two floating point numbers, after adding the eponents, why should we subtract one ecess base from the result? In the case of division of two floating point numbers, after subtracting the eponents, why should we add one ecess base to the result? A.R. Hurson 81

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