Computer Organization and Architecture

1. Computer Organization and Architecture Computer Arithmetic Chapter 9

2. Arithmetic & Logic Unit • Does the calculations • operands come from registers, result go to registers • 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. Binary Number Representation • Only have 0 & 1 to represent everything • Positive numbers stored in binary • e.g. 4110 = 001010012 • -27.437510 = -11011.01112 • Computer representation: • No minus sign • No decimal point • Sign-Magnitude • Twos complement

5. Sign-Magnitude • Fixed number of digits (8 in example below) • Leftmost (MSB) bit is the sign bit • 0 means positive • 1 means negative • Negation changes only the sign bit, the others are unchanged • +27 = 00011011 • -27 = 10011011 • Problems • Need to consider both sign and magnitude in arithmetic • Two representations of zero (+0 and -0) • Rarely used

6. Twos Complement Representation n-2 • A = -2n-1 an-1 +  2i ai i=0 • +3 = 00000011 • +2 = 00000010 • +1 = 00000001 • +0 = 00000000 (considered positive, sign bit is 0) • -1 = 11111111 • -2 = 11111110 • -3 = 11111101 • Largest positive number in n bits: 0111…1 = 2n-1-1 • Smallest negative number is: 1000…0 = - 2n-1

7. Benefits • One representation of zero • Arithmetic works easily (see later) • Negating: • 3 = 00000011 • Boolean complement gives: 11111100 • Add 1 to Boolean complement: 11111101 • Boolean complement is also known as Ones complement

8. Characteristics of Twos Complement

9. Geometric Depiction of Twos Complement Integers

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

11. Negation - Special Case 2 • -128 = 10000000 • Bitwise not 01111111 • Add 1 not +1 • Result 10000000 • So: • -(-128) = -128 - ERROR • Monitor MSB (sign bit) • It should change during negation

12. Range of Numbers • 8-bit twos complement: • +127 = 01111111 = 27 -1 • -128 = 10000000 = -27 • 16-bit twos complement: • +32767 = 01111111 11111111 = 215 - 1 • -32768 = 10000000 00000000 = -215 • In general, n-bit twos complement has range -2n-1 to 2n-1-1

13. Conversion Between Lengths • Positive numbers pack with leading zeros • +18 = 00010010 • +18 = 00000000 00010010 • Negative numbers pack with leading ones • -18 = 11101110 • -18 = 11111111 11101110 • In other words, pack with MSB (sign bit) • Proof in the book; note that the proof uses n-1  2i = 2n - 1 (prove thisby induction) i=0

14. Addition and Subtraction • Normal binary addition • Monitor sign bit for overflow • Overflow occurs if two numbers with the same sign are added and the result has the opposite sign • Overflow never occurs adding two numbers of opposite sign • Take twos compliment of subtrahend and add to minuend : • i.e. a - b = a + (-b) • Only addition and complement circuits are needed

15. Hardware for Addition and Subtraction

16. Multiplication • Complex • Work out partial product for each digit • Take care with place value (column) • Add partial products • This is known as “positional multiplication” • Works for unsigned (non-negative) numbers

17. Multiplication Example • 1011 Multiplicand (decimal 11) • x 1101 Multiplier (decimal 13) • 1011 Partial products • 0000 Note: if multiplier bit is 1 copy • 1011 multiplicand (place value) • 1011 otherwise zero • 10001111 Product (decimal 143) • Note: length of result is double of the lengths of the operands

18. Unsigned Binary Multiplication

19. Flowchart for Unsigned Binary Multiplication

20. Unsigned Binary Multiplication Example Q0

21. Multiplication Error: Negative Numbers • Positional multiplication does not work for negative numbers: • 1011 Multiplicand (decimal -5) • x 1101 Multiplier (decimal -3) • 1011 • 0000 • 1011 • 1011 • 10001111 (decimal -113)

22. Multiplying Negative Numbers • Need some solution to muliply negative numbers • Solution 1 • Convert to positive if required • Multiply as above • If signs were different, negate answer • Solution 2 • Booth’s algorithm

23. Booth’s Algorithm

24. Example of Booth’s Algorithm Q0 Q-1

25. Division of Unsigned Binary Integers • More complex than multiplication • Negative numbers are problematic • Based on long division • Example: 147:11=13, remainder is 4 • Can be extended to negative numbers Quotient 00001101 Divisor 1011 10010011 Dividend 1011 001110 Partial Remainders 1011 001111 1011 Remainder 100

26. Flowchart for Unsigned Binary Division

27. Real Numbers • Numbers with fractional parts • Possible to represent in pure binary form: • 1001.1010 = 23 + 20 +2-1 + 2-3 =9.625 • Two representations in decimal system • Fixed-point numbers • Limited to a fixed number of fractional digits • E.g. 3, as in 9.625 • Limits accuracy (range) • Floating-point numbers • Based on scientific notation • E.g. 9,625,000,000,000,000 = 9.625 x 1015 0.00000000000009625 = 9.625 x 10-14

28. Floating Point Numbers • +/- .significand x 2exponent • Base is 2 (binary) instead of 10 (decimal) • Point is actually fixed between sign bit and body of mantissa • Exponent indicates place value (point position) Biased Exponent Significand or Mantissa Sign bit

29. Signs for Floating Point Numbers • Mantissa is stored in twos complement • We want to allow negative exponents by biasing • 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

30. Normalization • FP numbers are usually normalized • Exponent is adjusted so that leading digit of mantissa is 1 • E.g. instead of 24 = 0.0110 x 26 or 24 = 110 x 22 we use 24 = 1.10 x 24 • Equivalent of scientific notation • numbers are normalized to give a single digit before the decimal point • In general: +/- 1.bbb…b x 2+/-E • Leading 1 is not stored (implicitly assumed)

31. Floating Point Examples 202 = 10100, 1272 = 01111111, 1472 = 10010011, 1072 = 01101011 0.638125 = 1/2 + 1/8 +1/128 = .10100012

32. Integer and Floating Point (FP) Ranges • 32 bit integers • Integers from –231 to 231 (232 different numbers) • 32 bit floating point real numbers • With 8 bit exponent • – 2129 to 2129, excluding the endpoints and underflows • Approximately -6.8 x 1038 to 6.8 x 1038 decimal • Precision • The effect of changing LSB of mantissa • 23 bit mantissa: 2-23  1.2 x 10-7 • About 6 decimal places – rounding occurs beyond that • More bits in exponent increase range but reduce mantissa, and thereby precision (trade-off)

33. Expressible Numbers Note: 0 is not represented directly. Special conventions are used for 0 (positive and negative), infinity etc. in real FP standards.

34. Density of Floating Point Numbers • FP numbers extend the range to – 2129 to 2129 • But still only 232 different numbers can be represented • Numbers are spread out • Density is not even • higher close to 0 • less dense farther away from 0 • same number in doubling ranges, i.e. halving density

35. IEEE 754 • Standard for floating point storage • Single format: 32 bits with 8 bit exponent • Double format: 64 bits with 11 bit exponent • Extended formats (both mantissa and exponent) for intermediate results • bigger mantissa gives more precision • bigger exponent gives extended range • lessens the chance of intermediate rounding errors, since generally overflows accumulate in a calculation, resulting in a very wrong result • Includes positive and negative 0, NaN, normalized and denormalized numbers

36. IEEE 754 Formats

37. FP Arithmetic +/- • Check for zeros (in operands) • Align significands (adjust exponents) • Operands must have identical exponents for +/- • Smaller number is adjusted for higher precision • Add or subtract significands • Check for significand and exponent overflow • Normalize result • Shift significand left until MSB is 1 (i.e. remove padding 0s) • Check for exponent underflow • Round result for reporting

38. FP Addition & Subtraction Flowchart

39. FP Arithmetic x/ • Simpler than +/- due to exponents • Check for zero • Report error or return infinity on division by 0 • Add/subtract exponents • Adding exponents doubles the bias, so bias needs to be subtracted • Report exponent over- or underflow • Multiply/divide significands (watch sign) • Normalize • Round and return result • All intermediate results should be in double length storage

40. Floating Point Multiplication

41. Floating Point Division

42. Precision and Rounding Errors • Guard bits: register is longer than the representation • Pad right end of mantissa with 0s • Guard against errors introduced by loss of digits • See book’s example (difference of close numbers) • IEEE 754 Rounding policy • Round to nearest • In case of 2 nearest, in half the cases round down, half the cases up to avoid accumulation of systematic errors • Round toward +/Round toward - • Used in interval arithmetic • Round toward 0 • Simplest (truncation) but introduces systematic errors

43. IEE754 Special Entities • Infinity • Limiting cases of real numbers • Operations extended in mathematical sense • Quiet NaN • Indicates mathematically undefined results; propagates through calculations w/o throwing an exception • Signaling NaN • Indicates mathematically undefined results and signals an exception • Denormalized Numbers • Handle exponent underflow by right shifting the decimal point and increasing the exponent