Computer Arithmetic

Sep 10, 2009 SYSC 2001 - F09. SYSC2001-Ch9.ppt 1

See Stallings Chapter 9

Computer Arithmetic


Arithmetic & Logic Unit (ALU)

ALU does the calculations “Everything else in the computer is there to service this

unit” (!) Handles integers May handle floating point (real) numbers

• there may be a separate floating point unit (FPU) (“math co-processor”)

• or an on-chip separate FPU (486DX +)• on Pentium (earlier slides) multiple ALUs and FPUs


ALU Inputs and Outputs

requested arithmetic operation

operands (in) operands (out)

Status e.g overflow?


Integer Representation Positive numbers stored in binary

• e.g. 41=00101001 Only have 0 & 1 to represent everything

• No minus sign!• No period!• Exponent?

Two approaches to integers• Sign-Magnitude

• Twos complement

unsigned integers?

“counting numbers”?

Is zero positive?

(NO!)


Sign-Magnitude Approach

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)

• Leftmost bit is most significant bit (msb)• Rightmost bit is least significant bit (lsb)


Two’s Complement

+3 = 00000011 +2 = 00000010 +1 = 00000001 +0 = 00000000 00000000 -1 = 11111111 11111111 -2 = 11111110 -3 = 11111101

subtract 1

subtract 1 ???? add 1


Aside re Binary ArithmeticYou should already know that 02+12 = 12

012+ 012 = 102 (or 12+12 = 02 carry 1)

112 + 12 = 002 carry 1

12 – 12 = 02 (no borrow), 02 – 12 = 12 borrow 1

00..002 – 12 = 11..112 borrow 1

000112 = 0•24 + 0•23 + 0•22 + 1•21 + 1•20 = bi 2i

i=0

K-1

K bit binary number, e.g. K = 5

b2


Two’s Complement Benefits

One representation of zero Arithmetic works easily (see later) Negating is fairly easy:

• 310 = 000000112

• bitwise complement gives 11111100

• Add 1 11111101 = –310

one’s complementtwo’s complement


Two’s Complement Benefits

What about negating a negative number ?

• – 310 = 111111012

• bitwise complement gives 00000010

• Add 1 00000011 = 310

GOOD! – ( – 3 ) = 3


Two’s Complement Integers

In n bits, can store integers from

–2n-1 to + 2n-1 – 1

neg pos


Range of Numbers

8 bit 2s complement• +127 = 01111111 = 27 -1• -128 = 10000000 = -27

16 bit 2s complement• +32767 = 01111111 11111111 = 215 - 1• -32768 = 10000000 00000000 = -215

N bit 2s complement• 011111111..11111111 = 2N-1 - 1• 100000000..00000000 = -2N-1

Largest positiveSmallest (?) negative


Negation Special Case 1

0 = 00000000

Bitwise not 11111111

Add 1 +1

Result 1 00000000

if this bit is ignored:

– 0 = 0 OK

hmmm . . .


Carry vs. Overflow So . . . what about the ignored bit on the last slide? CARRY: is an artifact of performing addition

• always happens: for binary values, carry = 0 or 1• exists independently of computers!

OVERFLOW: an exception condition that results from using computers to perform operations• caused by restricting values to fixed number of bits• occurs when result exceeds range for the number of bits• important limitation of using computers!

e.g. Y2K bug!


Overflow is Implementation Dependent!

recall example: negation of 0 binary addition is performed – regardless of interpretation

11111111

+1

1 00000000

carryfixed number

of bits torepresent

values

for this additionthe answer is:

0 with carry = 1


Unsigned Interpretation Overflow

consider values as unsigned integers

11111111

+1

1 00000000 but . . . 255 + 1 = 256 ?? answer = 0 ??? OVERFLOW occurred! WHY? cannot represent 256 as an 8-bit unsigned integer!

25510


Signed Interpretation No Overflow

consider values as signed integers

11111111

+1

1 00000000 –1 + 1 = 0 answer is correct! OVERFLOW did not occur (even though carry =1!) WHY? can represent 0 as an 8-bit signed integer!

– 110


Negation Special Case 2

– 128 = 10000000

bitwise not 01111111

Add 1 to lsb +1

Result 10000000 Whoops!

– (– 128) = – 128 Need to monitor msb (sign bit) It should change during negation?

Problem! OVERFLOW!

what about negating zero?


Conversion Between Lengths, e.g. 8 16

Positive number: add leading zeros• +18 = 00010010• +18 = 00000000 00010010

Negative numbers: add leading ones• -18 = 11101110• -18 = 11111111 11101110

i.e. pack with msb (sign bit) called “sign extension”


Addition and Subtraction

a – b = ? Normal binary addition Monitor sign bit of result for overflow

Take twos complement of b and add to a• i.e. a – b = a + (– b)

So we only need addition and 2’s complement circuits


Hardware for Addition and Subtraction

A – B = ?2’s

CF

Adders

Fall 2008 SYSC 5704: Elements of Computer Systems

21

Murdocca, Figure 3-2Ripple-Carry Adder

Murdocca, Figure 3-6Addition/Subtraction Unit


Multiplication

Complicated

1. Work out partial product for each digit

2. Take care with place value (column)

3. Add partial products

Aside: Multiply by 2? (shift?)


Multiplication Example

1011 Multiplicand (1110)

x 1101 Multiplier (1310)

1011 Partial products

0000 Note: if multiplier bit is 1

1011 copy multiplicand (place value)

1011 otherwise zero

10001111 Product (14310)

Note: need double length result – could easily overflow single word


Unsigned Binary Multiplication


Execution of 4-Bit ExampleM x Q = A,Q

1 add

0 no add

1 add

1 add


Flowchart for Unsigned Binary Multiplication

number of bits


27

SystolicArray Multiplier

Murdocca,Figure 3-23


Multiplying Negative Numbers

Multiplying negative numbers by this method does not work! Solution 1

• Convert to positive if required• Multiply as above• If signs were different, negate answer

Solution 2• Booth’s algorithm – not in scope of course? (Page 322)


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Computer Architecture: Arithmetic

Booth’s Algorithm : Increase speed of a multiplication when there are consecutive ones or zeros in the multiplier.

• Based on shifting of multiplier from right-to-left1. If at boundary of a string of 0’s, add

multiplicand to product (in proper column)2. If at boundary of a string of 1’s, subtract

multiplicand to product (in proper column)3. If within string, just shift.

Booth’s Algorithm

Example of Booth’s Algorithm


Division

More complex than multiplication Negative numbers are really bad! Based on long division

Aside: Divide by 2? (shift?)


001111

Division of Unsigned Binary Integers

101100001101100100111011001110

1011

1011100

Quotient

Dividend

Remainder

PartialRemainders

Divisor


Flowchart for Unsigned Binary Division

differences with multiply?

shift left

subtract?

no “C”

add

Quotient

Remainder

Divisor

Subtract

Shift left

/ Dividend


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• Division is somewhat more complex than multiplication; based on the same general principles


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?

• Very limited representation ability Moving?

• How do you show where it is?

fixed point

floating point


Normalization

Floating Point (FP) numbers are usually normalized• i.e. exponent is adjusted so that leading bit (msb) of

significand is always 1 Since msb is always 1 there is no need to store it

Recall scientific notation where numbers are normalized to give a single digit before the decimal point

e.g. 3.123 x 103


Floating Point (Scientific Notation)

+/ – 1.significand x 2exponent

Floating point misnomer: Point is actually fixed after first digit of the significand: 1.xxxxxx • xxxxxx is stored signficand

Exponent indicates place value (point position)

Sign

bit

(Biased)exponent

(stored) significand


Floating Point Examples

Typo here!What's wrong?


Signs for Floating Point

Significand: explicit sign bit (msb) • rest of bits are magnitude (sort of)

Exponent is in excess or biased notation• 8 bit exponent field• binary value range 0-255• Excess (bias) 127 means

– Subtract 127 to get correct value– Range -127 to +128

for k bit exponent:bias = 2k–1 –1


FP Ranges

Range: The range of numbers that can be represented• For a 32 bit number

• 8 bit exponent • +/- 2127 +/- 1.5 x 1038

Accuracy: The effect of changing lsb of significand• 23 bit significand 2-23 1.2 x 10-7

• About 6 decimal places


Expressible Numbers

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IEEE 754

Standard for floating point storage 32 and 64 bit standards 8 and 11 bit exponent respectively Extended formats (both significand and exponent) for

intermediate results special cases: (table 9.4, page 318)

• 0 (how to normalize?)• denormalized (for arithmetic)• (how to quantify?)• Not a number (NaN exception)

How?use exponent = all 0’s zero, denormalized use exponent = all 1’sinfinity, NAN reduces range! 2 –126 to 2127


IEEE 754 Formats


FP Arithmetic +/-

Check for zeros Align significands (adjusting exponents)

denormalize! Add or subtract significands Normalize result


FP Addition & Subtraction Flowchart


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


Floating Point Multiplication

1.xxxxx x 2bias+a

1.yyyyy x 2bias+bx


Floating Point Division

Computer Arithmetic

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