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  1. CH09 Computer Arithmetic • CPU combines of ALU and Control Unit, this chapter discusses ALU • The Arithmetic and Logic Unit (ALU) • Number Systems • Integer Representation • Integer Arithmetic • Floating-Point Representation • Floating-Point Arithmetic TECH Computer Science CH08

  2. Arithmetic & Logic Unit • Does the calculations • 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

  4. Number Systems • ALU does calculations with binary numbers • Decimal number system • Uses 10 digits (0,1,2,3,4,5,6,7,8,9) • In decimal system, a number 84, e.g., means84 = (8x10) + 3 • 4728 = (4x1000)+(7x100)+(2x10)+8 • Base or radix of 10: each digit in the number is multiplied by 10 raised to a power corresponding to that digit’s position • E.g. 83 = (8x101)+ (3x100) • 4728 = (4x103)+(7x102)+(2x101)+(8x100)

  5. Decimal number system… • Fractional values, e.g. • 472.83=(4x102)+(7x101)+(2x100)+(8x10-1)+(3x10-2) • In general, for the decimal representation ofX = {… x2x1x0.x-1x-2x-3 … }X =  i xi10i

  6. Binary Number System • Uses only two digits, 0 and 1 • It is base or radix of 2 • Each digit has a value depending on its position: • 102 = (1x21)+(0x20) = 210 • 112 = (1x21)+(1x20) = 310 • 1002 = (1x22)+ (0x21)+(0x20) = 410 • 1001.1012 = (1x23)+(0x22)+ (0x21)+(1x20) +(1x2-1)+(0x2-2)+(1x2-3) = 9.62510

  7. Decimal to Binary conversion • Integer and fractional parts are handled separately, • Integer part is handled by repeating division by 2 • Factional part is handled by repeating multiplication by 2 • E.g. convert decimal 11.81 to binary • Integer part 11 • Factional part .81

  8. Decimal to Binary conversion, e.g. // • e.g. 11.81 to 1011.11001 (approx) • 11/2 = 5 remainder 1 • 5/2 = 2 remainder 1 • 2/2 = 1 remainder 0 • 1/2 = 0 remainder 1 • Binary number 1011 • .81x2 = 1.62 integral part 1 • .62x2 = 1.24 integral part 1 • .24x2 = 0.48 integral part 0 • .48x2 = 0.96 integral part 0 • .96x2 = 1.92 integral part 1 • Binary number .11001 (approximate)

  9. Hexadecimal Notation: • command ground between computer and Human • Use 16 digits, (0,1,3,…9,A,B,C,D,E,F) • 1A16 = (116 x 161)+(A16 x 16o) = (110 x 161)+(1010 x 160)=2610 • Convert group of four binary digits to/from one hexadecimal digit, • 0000=0; 0001=1; 0010=2; 0011=3; 0100=4; 0101=5; 0110=6; 0111=7; 1000=8; 1001=9; 1010=A; 1011=B; 1100=C; 1101=D; 1110=E; 1111=F; • e.g. • 1101 1110 0001. 1110 1101 = DE1.DE

  10. Integer Representation (storage) • Only have 0 & 1 to represent everything • Positive numbers stored in binary • e.g. 41=00101001 • No minus sign • No period • How to represent negative number • Sign-Magnitude • Two’s compliment

  11. 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)

  12. Two’s Compliment (representation) • +3 = 00000011 • +2 = 00000010 • +1 = 00000001 • +0 = 00000000 • -1 = 11111111 • -2 = 11111110 • -3 = 11111101

  13. Benefits • One representation of zero • Arithmetic works easily (see later) • Negating is fairly easy (2’s compliment operation) • 3 = 00000011 • Boolean complement gives 11111100 • Add 1 to LSB 11111101

  14. Geometric Depiction of Twos Complement Integers

  15. Range of Numbers • 8 bit 2s compliment • +127 = 01111111 = 27 -1 • -128 = 10000000 = -27 • 16 bit 2s compliment • +32767 = 011111111 11111111 = 215 - 1 • -32768 = 100000000 00000000 = -215

  16. 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)

  17. Integer Arithmetic: Negation • Take Boolean complement of each bit, I.e. each 1 to 0, and each 0 to 1. • Add 1 to the result • E.g. +3 = 011 • Bitwise complement = 100 • Add 1 • = 101 • = -3

  18. Negation Special Case 1 • 0 = 00000000 • Bitwise not 11111111 • Add 1 to LSB +1 • Result 1 00000000 • Overflow is ignored, so: • - 0 = 0 OK!

  19. Negation Special Case 2 • -128 = 10000000 • bitwise not 01111111 • Add 1 to LSB +1 • Result 10000000 • So: • -(-128) = -128 NO OK! • Monitor MSB (sign bit) • It should change during negation • >> There is no representation of +128 in this case. (no +2n)

  20. Addition and Subtraction • Normal binary addition • 0011 0101 1100 • +0100 +0100 +1111 • -------- ---------- ------------ • 0111 1001 = overflow 11011 • Monitor sign bit for overflow (sign bit change as adding two positive numbers or two negative numbers.) • Subtraction: Take twos compliment of subtrahend then add to minuend • i.e. a - b = a + (-b) • So we only need addition and complement circuits

  21. Hardware for Addition and Subtraction

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

  23. Multiplication Example • (unsigned numbers e.g.) • 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

  24. Unsigned Binary Multiplication

  25. Flowchart for Unsigned Binary Multiplication

  26. Execution of Example

  27. Multiplying Negative Numbers • The previous method does not work! • Solution 1 • Convert to positive if required • Multiply as above • If signs of the original two numbers were different, negate answer • Solution 2 • Booth’s algorithm

  28. Booth’s Algorithm

  29. Example of Booth’s Algorithm

  30. Division • More complex than multiplication • However, can utilize most of the same hardware. • 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 = 24 + 20 +2-1 + 2-3 =9.625 • Where is the binary point? • Fixed? • Very limited • Moving? • How do you show where it is?

  34. Floating Point • +/- .significand x 2exponent • Point is actually fixed between sign bit and body of mantissa • Exponent indicates place value (point position) 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 • The relative magnitudes (order) of the numbers do not change. • Can be treated as integers for comparison.

  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 • (c.f. Scientific notation where numbers are normalized to give a single digit before the decimal point • e.g. 3.123 x 103)

  38. FP Ranges • For a 32 bit number • 8 bit exponent • +/- 2256  1.5 x 1077 • 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 and 64 bit standards • 8 and 11 bit exponent respectively • Extended formats (both mantissa and exponent) for intermediate results • Representation: sign, exponent, faction • 0: 0, 0, 0 • -0: 1, 0, 0 • Plus infinity: 0, all 1s, 0 • Minus infinity: 1, all 1s, 0 • NaN; 0 or 1, all 1s, =! 0

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

  42. 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

  43. FloatingPointMultiplication

  44. FloatingPointDivision

  45. Exercises • Read CH 8, IEEE 754 on IEEE Web site • Email to: choi@laTech.edu • Class notes (slides) online at:www.laTech.edu/~choi