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Chapter 8 Computer Arithmetic

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Presentation on theme: "Chapter 8 Computer Arithmetic"— Presentation transcript:

1 Chapter 8 Computer Arithmetic
2/23/2019 Chapter 8 Computer Arithmetic 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

2 Arithmetic & Logic Unit
2/23/2019 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 +) 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

3 Created by Vivi Sahfitri
2/23/2019 ALU Inputs and Outputs 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

4 Integer Representation
2/23/2019 Integer Representation Only have 0 & 1 to represent everything Positive numbers stored in binary e.g. 41= No minus sign No period Sign-Magnitude Two’s compliment 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

5 Created by Vivi Sahfitri
Sign-Magnitude 2/23/2019 Left most bit is sign bit 0 means positive 1 means negative +18 = -18 = Problems Need to consider both sign and magnitude in arithmetic Two representations of zero (+0 and -0) 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

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2/23/2019 Two’s Compliment +3 = +2 = +1 = +0 = -1 = -2 = -3 = 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

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2/23/2019 Benefits One representation of zero Arithmetic works easily (see later) Negating is fairly easy 3 = Boolean complement gives Add 1 to LSB 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

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2/23/2019 Negation Special Case 1 0 = Bitwise not Add 1 to LSB Result Overflow is ignored, so: - 0 = 0  2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

9 Created by Vivi Sahfitri
2/23/2019 Negation Special Case 2 -128 = bitwise not Add 1 to LSB Result So: -(-128) = X Monitor MSB (sign bit) It should change during negation 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

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2/23/2019 Range of Numbers 8 bit 2s compliment +127 = = 27 -1 -128 = = -27 16 bit 2s compliment = = = = -215 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

11 Conversion Between Lengths
2/23/2019 Conversion Between Lengths Positive number pack with leading zeros +18 = +18 = Negative numbers pack with leading ones -18 = -18 = i.e. pack with MSB (sign bit) 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

12 Addition and Subtraction
2/23/2019 Addition and Subtraction Normal binary addition Monitor sign bit for overflow Take twos compliment of substahend and add to minuend i.e. a - b = a + (-b) So we only need addition and complement circuits 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

13 Hardware for Addition and Subtraction
2/23/2019 Hardware for Addition and Subtraction 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

14 Created by Vivi Sahfitri
2/23/2019 Multiplication Complex Work out partial product for each digit Take care with place value (column) Add partial products 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

15 Multiplication Example
2/23/2019 Multiplication Example Multiplicand (11 dec) x Multiplier (13 dec) Partial products Note: if multiplier bit is 1 copy multiplicand (place value) otherwise zero Product (143 dec) Note: need double length result 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

16 Unsigned Binary Multiplication
2/23/2019 Unsigned Binary Multiplication 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

17 Created by Vivi Sahfitri
2/23/2019 Execution of Example 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

18 Flowchart for Unsigned Binary Multiplication
2/23/2019 Flowchart for Unsigned Binary Multiplication 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

19 Multiplying Negative Numbers
2/23/2019 Multiplying Negative Numbers This does not work! Solution 1 Convert to positive if required Multiply as above If signs were different, negate answer Solution 2 Booth’s algorithm 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

20 Created by Vivi Sahfitri
2/23/2019 Booth’s Algorithm 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

21 Example of Booth’s Algorithm
2/23/2019 Example of Booth’s Algorithm 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

22 Created by Vivi Sahfitri
2/23/2019 Division More complex than multiplication Negative numbers are really bad! Based on long division 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

23 Division of Unsigned Binary Integers
2/23/2019 Division of Unsigned Binary Integers Quotient Divisor 1011 Dividend 1011 001110 Partial Remainders 1011 001111 1011 Remainder 100 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

24 Flowchart for Unsigned Binary Division
2/23/2019 Flowchart for Unsigned Binary Division 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

25 Created by Vivi Sahfitri
2/23/2019 Real Numbers Numbers with fractions Could be done in pure binary = =9.625 Where is the binary point? Fixed? Very limited Moving? How do you show where it is? 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

26 Created by Vivi Sahfitri
2/23/2019 Floating Point +/- .significand x 2exponent Misnomer Point is actually fixed between sign bit and body of mantissa Exponent indicates place value (point position) 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

27 Floating Point Examples
2/23/2019 Floating Point Examples 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

28 Signs for Floating Point
2/23/2019 Signs for Floating Point Mantissa is stored in 2s compliment Exponent 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 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

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2/23/2019 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 x 103) 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

30 Created by Vivi Sahfitri
2/23/2019 FP Ranges For a 32 bit number 8 bit exponent +/  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 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

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2/23/2019 Expressible Numbers 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

32 Density of Floating Point Numbers
2/23/2019 Density of Floating Point Numbers 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

33 Created by Vivi Sahfitri
2/23/2019 FP Arithmetic +/- Check for zeros Align significands (adjusting exponents) Add or subtract significands Normalize result 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

34 FP Addition & Subtraction Flowchart
2/23/2019 FP Addition & Subtraction Flowchart 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

35 Created by Vivi Sahfitri
2/23/2019 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 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

36 Floating Point Multiplication
2/23/2019 Floating Point Multiplication 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

37 Floating Point Division
2/23/2019 Floating Point Division 2/23/2019 Created by Vivi Sahfitri Created by Vivi Sahfitri

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