William Stallings Computer Organization and Architecture 6 th Edition

1. William Stallings Computer Organization and Architecture6th Edition Chapter 9 Computer Arithmetic

2. Arithmetic & Logic Unit • Performs arithmetic and logical operations on data. • Everything else in the computer is there to service this unit • Handles integers • May handle floating point (real) numbers • May be separate FPU (maths co-processor) • May be on chip separate FPU (486DX +)

3. ALU Inputs and Outputs • Data are presented to the ALU in registers, and the results of an operation are stored in registers. • The ALU may also set flags as the result of an operation. • The control unit provides signals that control the operation of the ALU and the movement of the data into and out of the ALU.

4. Integer Representation • Only have 0 & 1 to represent everything • Positive numbers stored in binary • e.g. 41=00101001 • No minus sign • No period • Negative number representation • Sign-Magnitude • Two’s complement

5. Sign-Magnitude • Left most bit is sign bit • 0 means positive • 1 means negative • +18 = 00010010 • -18 = 10010010 • Problems • Need to consider both sign and magnitude in arithmetic • Two representations of zero (+0 and -0)

6. Two’s Complement • Two’s complement • Take the Boolean complement of each bit of the integer. i.e., set each 1 to 0 and each 0 to 1. • Treating the result as an unsigned binary integer, add 1. • Example: • 3 = 00000011 • Boolean complement gives 11111100 • Add 1 to LSB 11111101 • +3 = 00000011 • +2 = 00000010 • +1 = 00000001 • +0 = 00000000 • -1 = 11111111 • -2 = 11111110 • -3 = 11111101

7. Benefits • One representation of zero • Arithmetic works easily (see later) • Negating is fairly easy

8. Geometric Depiction of Twos Complement Integers 若加上k，則往順時針方向移動k個位置 若減去k，則往逆時針方向移動k個位置

9. Negation Special Case 1 • 0 = 00000000 • Bitwise not 11111111 • Add 1 to LSB +1 • Result 1 00000000 • Overflow is ignored, so: • - 0 = 0  • Overflow • When the result is larger than can be held in the word size being used.

10. Negation Special Case 2 • -128 = -10000000 • bitwise not 01111111 • Add 1 to LSB +1 • Result 10000000 • So: • -(-128) = -128 X

11. Range of Numbers • 8 bit 2s complement • +127 = 01111111 = 27 -1 • -128 = 10000000 = -27 • 16 bit 2s complement • +32767 = 011111111 11111111 = 215 - 1 • -32768 = 100000000 00000000 = -215 • N bits 2s complement • +(2n-1-1) • -(2n-1)

12. Conversion Between Lengths • Positive number pack with leading zeros • +18 = 00010010 • +18 = 00000000 00010010 • Negative numbers pack with leading ones • -18 = 10010010 • -18 = 11111111 10010010 • i.e. pack with MSB (sign bit) • Sign extension

13. Addition and Subtraction • Normal binary addition • Monitor sign bit for overflow • Take twos compliment of subtrahend and add to minuend • i.e. a - b = a + (-b) • So we only need addition and complement circuits

14. Addition of Numbers in Twos Complement Representation

15. Subtraction of Numbers in Twos Complement Representation (M-S)

16. Subtraction of Numbers in Twos Complement Representation (M-S)

17. Overflow • Overflow • The result of addition is larger than can be held in the word size. • No overflow when adding a positive and a negative number • No overflow when signs are the same for subtraction • Overflow occurs when the value affects the sign: • overflow when adding two positives yields a negative • or, adding two negatives gives a positive • or, subtract a negative from a positive and get a negative • or, subtract a positive from a negative and get a positive • Effects of Overflow • An exception (interrupt) occurs • Control jumps to predefined address for exception

18. Hardware for Addition and Subtraction

19. Multiplication • Complex • Work out partial product for each digit • Take care with place value (column) • Add partial products

20. 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

21. Unsigned Binary Multiplication

22. Execution of Example

23. Flowchart for Unsigned Binary Multiplication

24. Multiplying Negative Numbers • Solution 1 • Convert to positive if required • Multiply as above • If signs were different, negate answer • Solution 2 • Booth’s algorithm

25. Comparison of Multiplication of Unsigned and Twos Complement Integer 部分積以2n bit表示 可解決被乘數為負值的問題，但若乘數為負值時，仍無解

26. Booth’s Algorithm

27. Example of Booth’s Algorithm

28. Examples Using Booth’s Algorithm

29. Examples Using Booth’s Algorithm (cont.)

30. Division • More complex than multiplication • Negative numbers are really bad! • Based on long division

31. Division of Unsigned Binary Integers Quotient 00001101 Divisor 1011 10010011 Dividend 1011 001110 Partial Remainders 1011 001111 1011 Remainder 100

32. Flowchart for Unsigned Binary Division

33. Real Numbers • Numbers with fractions • Could be done in pure binary • 1001.1010 = 23 + 20 +2-1 + 2-3 =9.625 • Where is the binary point? • Fixed-point notation • Very limited • Very large numbers cannot be represented, nor can very small fractions. • Floating-point notation • Scientific notation • Sign: plus or minus • Significand S • Exponent E • Base B

34. Floating Point • +/- .significand x 2exponent • Point is actually fixed between sign bit and body of mantissa • There is one bit to the left of the radix point. • Exponent value is biased representation • A fixed value is subtracted from the field to get the true exponent value • The bias equals (2k-1-1) Biased Exponent Significand or Mantissa Sign bit

35. Floating Point Examples

36. Signs for Floating Point • Exponent is in excess or biased notation • e.g. Excess (bias) 127 means • 8 bit exponent field • Pure value range 0-255 • Subtract 127 to get correct value • Range -127 to +128 • Significand also called the mantissa

37. Normalization • FP numbers are usually normalized • i.e. exponent is adjusted so that leading bit (MSB) of mantissa is 1 • Since it is always 1 there is no need to store it • e.g. 3.123 x 103 • The normalized nonzero number is one in the form • See Fig. 9.18b • The sign is stored in the first bit of the word • The first bit of the true significand is always 1 and need not be stored in the significand field. • The value 127 is added to the true exponent to be stored in the exponent field. • The base is 2.

38. FP Ranges • For a 32 bit number • Using 2s complement integer representation • Using 8 bit exponent, 23 bit significand • Negative number –(2-2-23)*2128~-2-127 • Positive number: 2-127~(2-2-23)*2128 • Five regions on the number line are not included in these ranges: • Negative overflow • Negative underflow • Zero • Positive underflow • Positive overflow • Accuracy • The effect of changing lsb of mantissa • 23 bit mantissa 2-23  1.2 x 10-7 • About 6 decimal places

39. Expressible Numbers

40. IEEE 754 • Standard for floating point storage • 32 bit single precision • 64 bit double precision • 8 and 11 bit exponent respectively • 23 and 52 bit significand respectively • The exponent is biased, • the rang of exponents is -126~+127，-1022~+1023 • exponent 與significand均為零時， 由 sign決定為 positive or negative zero. • Exponent均為1，significand為零時,由 sign決定為正負無窮大

41. IEEE 754 Formats

42. IEEE 754 format parameter

43. Interpretation of IEEE 754 FP numbers

44. Example 1 • (a) Convert the decimal number -1.5 to a 32-bit floating-point number with IEEE 754 format.(b) Convert a 32-bits IEEE 754 format floating-point number 43A0C000 to the decimal number. • Solution: • (a) 1.5=1.12=1.1*20S=1, C=E+127=127=01111111,F=10000000000000000000000(b)43A0C000=0100 0011 1010 0000 1100 0000 0000 00002S=0, C=10000111=135=E+127, =>E=8,F= 010000011000000000001.01000001102*28=101000001.12= 321.5

45. Example 2 • Convert the decimal number -1.5 to a 32-bit floating-point number with IBM S/390 format. • Solution: • Base of 16, 7 bit exponent, 24 bit significand • -1.5=-1.8(16)=-0.18*161 • S=1, C=E+64=1+64=65=100 0001F=0001 1000 0000 0000 0000 0000 • The IBM S/390 format is1 100 0001 0001 1000 0000 0000 0000 0000

46. Floating-point Arithmetic • A FP operation may produce one of these conditions: • Exponent overflow • Exponent underflow • Significand underflow • Significand overfolw

47. FP numbers and arithmetic operations

48. FP Arithmetic +/- • Check for zeros • Align significands (adjusting exponents) • Add or subtract significands • Normalize result

49. FP Addition & Subtraction Flowchart

50. FP Arithmetic x/ • Check for zero • Add/subtract exponents • Multiply/divide significands (watch sign) • Normalize • Round • All intermediate results should be in double length storage