Organisasi dan Arsitektur Komputer MO-08
- 1. 1 William Stallings Computer Organization and Architecture Chapter 8 Computer Arithmetic
- 2. 2 Arithmetic & Logic Unit aDoes the calculations aEverything else in the computer is there to service this unit aHandles integers aMay handle floating point (real) numbers aMay be separate FPU (maths co-processor) aMay be on chip separate FPU (486DX +)
- 3. 3 ALU Inputs and Outputs
- 4. 4 Integer Representation aOnly have 0 & 1 to represent everything aPositive numbers stored in binary `e.g. 41=00101001 aNo minus sign aNo period aSign-Magnitude aTwo’s compliment
- 5. 5 Sign-Magnitude aLeft most bit is sign bit a0 means positive a1 means negative a+18 = 00010010 a -18 = 10010010 aProblems `Need to consider both sign and magnitude in arithmetic `Two representations of zero (+0 and -0)
- 6. 6 Two’s Compliment a+3 = 00000011 a+2 = 00000010 a+1 = 00000001 a+0 = 00000000 a -1 = 11111111 a -2 = 11111110 a -3 = 11111101
- 7. 7 Benefits aOne representation of zero aArithmetic works easily (see later) aNegating is fairly easy `3 = 00000011 `Boolean complement gives 11111100 `Add 1 to LSB 11111101
- 8. 8 Geometric Depiction of Twos Complement Integers
- 9. 9 Negation Special Case 1 a 0 = 00000000 aBitwise not 11111111 aAdd 1 to LSB +1 aResult 1 00000000 aOverflow is ignored, so: a- 0 = 0 √
- 10. 10 Negation Special Case 2 a-128 = 10000000 abitwise not 01111111 aAdd 1 to LSB +1 aResult 10000000 aSo: a-(-128) = -128 X aMonitor MSB (sign bit) aIt should change during negation
- 11. 11 Range of Numbers a8 bit 2s compliment `+127 = 01111111 = 27 -1 ` -128 = 10000000 = -27 a16 bit 2s compliment `+32767 = 011111111 11111111 = 215 - 1 ` -32768 = 100000000 00000000 = -215
- 12. 12 Conversion Between Lengths aPositive number pack with leading zeros a+18 = 00010010 a+18 = 00000000 00010010 aNegative numbers pack with leading ones a-18 = 10010010 a-18 = 11111111 10010010 ai.e. pack with MSB (sign bit)
- 13. 13 Addition and Subtraction aNormal binary addition aMonitor sign bit for overflow aTake twos compliment of substahend and add to minuend `i.e. a - b = a + (-b) aSo we only need addition and complement circuits
- 14. 14 Hardware for Addition and Subtraction
- 15. 15 Multiplication aComplex aWork out partial product for each digit aTake care with place value (column) aAdd partial products
- 16. 16 Multiplication Example a 1011 Multiplicand (11 dec) a 1101 Multiplier (13 dec) a 1011 Partial products a 0000 Note: if multiplier bit is 1 copy a 1011 multiplicand (place value) a 1011 otherwise zero a 10001111 Product (143 dec) a Note: need double length result
- 19. 19 Flowchart for Unsigned Binary Multiplication
- 20. 20 Multiplying Negative Numbers aThis does not work! aSolution 1 `Convert to positive if required `Multiply as above `If signs were different, negate answer aSolution 2 `Booth’s algorithm
- 22. 22 Example of Booth’s Algorithm
- 23. 23 Division aMore complex than multiplication aNegative numbers are really bad! aBased on long division
- 24. 24 001111 Division of Unsigned Binary Integers 1011 00001101 10010011 1011 001110 1011 1011 100 Quotient Dividend Remainder Partial Remainders Divisor
- 25. 25 start A = 0 M = Divisor Q = Dividend Count = n Division Algorithm Shift Left A,Q A = A - M A<0 ? Q0 = 0 A = A + M Q0 = 1 Count = Count - 1 Count = 0 ? Stop Y N N Y
- 26. 26 Example A Q M = 0011 0000 0111 Initial value 0000 1110 Shift 1101 Subtract 0000 1110 Restore 0001 1100 Shift 1110 Subtract 0001 1100 Restore 0011 1000 Shift 0000 Subtract 0000 1001 Set Q0 = 1 0001 0010 Shift 1110 Subtract 0001 0010 Restore
- 27. 27 Real Numbers aNumbers with fractions aCould be done in pure binary `1001.1010 = 24 + 20 +2-1 + 2-3 =9.625 aWhere is the binary point? aFixed? `Very limited aMoving? `How do you show where it is?
- 28. 28 Floating Point a+/- .significand x 2exponent aMisnomer aPoint is actually fixed between sign bit and body of mantissa aExponent indicates place value (point position) Sign bit Biased Exponent Significand or Mantissa
- 30. 30 Signs for Floating Point aMantissa is stored in 2s compliment aExponent is in excess or biased notation `e.g. Excess (bias) 128 means `8 bit exponent field `Pure value range 0-255 `Subtract 128 to get correct value `Range -128 to +127
- 31. 31 Normalization aFP numbers are usually normalized ai.e. exponent is adjusted so that leading bit (MSB) of mantissa is 1 aSince it is always 1 there is no need to store it a(c.f. Scientific notation where numbers are normalized to give a single digit before the decimal point ae.g. 3.123 x 103)
- 32. 32 FP Ranges aFor a 32 bit number `8 bit exponent `+/- 2256 ≈ 1.5 x 1077 aAccuracy `The effect of changing lsb of mantissa `23 bit mantissa 2-23 ≈ 1.2 x 10-7 `About 6 decimal places
- 34. 34 IEEE 754 aStandard for floating point storage a32 and 64 bit standards a8 and 11 bit exponent respectively aExtended formats (both mantissa and exponent) for intermediate results
- 35. 35 FP Arithmetic +/- aCheck for zeros aAlign significands (adjusting exponents) aAdd or subtract significands aNormalize result
- 36. 36 FP Arithmetic x/÷ aCheck for zero aAdd/subtract exponents aMultiply/divide significands (watch sign) aNormalize aRound aAll intermediate results should be in double length storage
- 39. 39 Required Reading aStallings Chapter 8 aIEEE 754 on IEEE Web site