CO212 Lecture 10:Arithmetic & Logical Unit

Shobhanjana Kalita,Shobhanjana Kalita,

Dept. of CSE, Tezpur University

Slides courtesy: Computer Architecture and Organization, 9th Ed, W. Stallings

Integer Representation For purposes of computer storage and processing, however,

we do not have the benefit of special symbols for the minus sign and radix point.

Only binary digits (0 and 1) may be used to represent numbers.

If we are limited to nonnegative integers If we are limited to nonnegative integers

8-bit word can represent the numbers from 0 to 255

If we are to include negative numbers as well, there are different conventions such as

Sign-Magnitude representation

1’s complement representation

2’s complement representation

Integer Representation Sign and Magnitude

The simplest form of representation employs the MSB as a sign bit

If the sign bit is 0, the number is positive; if the sign bit is 1, thenumber is negative.number is negative.

In an n-bit word, the rightmost (n – 1) bits hold the magnitude of the integer.

Example:

Disadvantages: Addition and subtraction become complex

Two representations of 0

Integer Representation 1’s complement

1’s complement representation also uses the MSB as a sign bit –making it easy to test whether an integer is positive or negative

Negative of a number is obtained by inverting all of the bits of the positive binary valuethe positive binary value

Range of values: –(2^(n – 1) – 1) to +(2^(n – 1) – 1)

Example: +18 = 00010010

– 18 = 11101101

Addition and subtraction is simple addition of two binary numbers

Disadvantage: Two representation for zero: 00000000 is +0 and 11111111 is –0

Integer Representation 2’s complement

2’s complement representation also uses the MSB as a sign bit –making it easy to test whether an integer is positive or negative

Negative of a number is obtained by inverting all of the bits of the positive binary value and adding a 1 to the resultthe positive binary value and adding a 1 to the result

Range of values: –(2^(n – 1)) to +(2^(n – 1) – 1)

Example: +18 = 00010010

– 18 = 11101110

Addition and subtraction is simple addition of two binary numbers

One representation for zero: 00000000 is 0

Integer Representation Most computer integer representations of computer uses 2’s

complement

Other signed integer representations may be used -

Biased representation: add a bias value to the number and represent using the corresponding binary equivalentrepresent using the corresponding binary equivalent

Fixed Point Representation

Aforementioned representation are called fixed point representations because the radix point is assumed to have fixed position – right of the rightmost digit (LSB)

Integer Arithmetic–2’s complement Negation

Take the Boolean complement of each bit of the integer, add 1 to the result

Negative of negative is positive

Negative of -128 ??

Addition and Subtraction

Addition proceeds as simple addition of two unsigned integers

In some instances, if there is a carry bit beyond the end of the word it is ignored.

If the result is larger than can be held in the word size

Integer Arithmetic–2’s complement

If the result is larger than can be held in the word sizebeing used – it is called overflow. When overflow occurs, the ALU must signal so that the result is not used for processing OVERFLOW RULE: If two numbers are added, and they are both

positive or both negative, then overflow occurs if and only if the result has the opposite sign.

Subtraction is achieved using addition. SUBTRACTION RULE: To subtract one number (subtrahend) from

another (minuend), take the twos complement (negation) of the subtrahend and add it to the minuend.

Multiplication

Multiplication of positive numbers is straightforward 4-bit multiplied to 4-bit results in 8-bits

For negative numbers, simple multiplication of 2’s complement value will not give the correct result (Eg. -5 * -3 = -113)

Integer Arithmetic–2’s complement

value will not give the correct result (Eg. -5 * -3 = -113) Filling up the left most values in the partial product with binary 1s.

Division

Division of unsigned binary integers is straightforward

Integer Arithmetic–2’s complement

2’s complement division for negative numbers can be done by converting the operands into unsigned values perform division and then account for the sign on the resultant quotient D = Q * V + R

sign(R) = sign(D)

sign(Q) = sign(D) * sign(V)

Floating Point Representation By assuming a fixed binary or radix point, numbers with a

fractional component may be represented - but limited in the range of numbers

Solution: Use scientific notation (floating point)

976,000,000,000,000 = 9.76 * 10^14, and 976,000,000,000,000 = 9.76 * 10^14, and

0.0000000000000976 = 9.76 * 10^(-14)

Any number can be represented in the form

Sign: + or –; Significand S; Exponent E

The base B is implicit and is the same for all numbers

Floating Point Representation Sign

0 for positive; 1 for negative numbers

Exponent

Exponent is stored using a biased representation

A fixed value, called the bias, is subtracted from the field to get A fixed value, called the bias, is subtracted from the field to get the true exponent value

Typically, bias is (2^(k – 1) – 1), where k is the number of bits in the binary exponent

For an 8-bit field the range of unsigned numbers is 0 to 255

Bias = 2^7 – 1 = 127,

actual exponent values are in the range -127 to +128

Advantage of biased representation: Comparison is easier

Significand

A floating point number can have many representations

To simplify operations need a standard representation –normalization

For a base 2 representation a normal number is one in which

Floating Point Representation

For a base 2 representation a normal number is one in which the MSB is of the significand is 1 i.e.

In such normal numbers – MSB is implicit, not stored – 23bit field is used to store 24 bit significand

The representation as presented will not accommodate a value of 0 – include a special bit-pattern of 0

Floating Point Representation

Overflow – when an arithmetic operation results in an absolute value greater than can be expressed with an exponent of 128

Underflow – when the fractional magnitude is too small – less serious problem because result can generally be approximated by 0

Trade offs

Numbers represented in floating-point notation are notspaced evenly along the number line, unlike fixed-point numbers

Range determines maximum and minimum number of values

Floating Point Representation

Range determines maximum and minimum number of values that can be represented

Precision determines the minimum value fractional value that can be represented accurately

Trade-off between range and precision:

more number of bits in significand => more precision

more number of bits in exponent => greater range

Floating Point – IEEE standard IEEE standard 754 for floating point numbers was developed

to facilitate portability of programs from one processor to another

3 basic binary formats defined with lengths of 32, 64, and 128 bits128 bits

Exponents of length 8, 11, 15 respectively

The standard defines extended precision formats, whichextend a supported basic format by providing additional bits in the exponent (extended range) and in the significand(extended precision)

Defines only standards; implementation dependent

Floating Point – IEEE standard

Floating Point Arithmetic For addition and subtraction – both operands must have the

same exponent value

This may require shifting the radix point on one of the operands to achieve alignment

Multiplication and division are more straightforward. Multiplication and division are more straightforward.

Floating Point Arithmetic A floating-point operation may produce one of these

conditions:

Exponent overflow: A positive exponent exceeds the maximum possible exponent value; may be designated as +∞or –∞.

∞or –∞.

Exponent underflow: A negative exponent is less than the minimum possible exponent value; it may be reported as 0.

Significand underflow: While aligning significands, digits may flow off the right end of the significand; requires rounding

Significand overflow: Addition of two significands of the same sign may result in a carry out of the most significant bit; requires realignment

In floating-point arithmetic, addition and subtraction are more complex than multiplication and division

need for alignment.

There are four basic phases of the algorithm for addition and subtraction

Floating Point Arithmetic

subtraction

1. Check for zeros

2. Align the significands

3. Add or subtract the significands

4. Normalize the result.

Floating Point Guard Bits

For a floating-point operation, the exponent and significand of each operand are loaded into ALU registers

In case of significand, the length of the register is greater than the length of the significand plus an implied bitthe length of the significand plus an implied bit

The register contains additional bits, called guard bits, which are used to pad out the right end of the significand with 0s

Important: to maintain precision

Example: x = 1.00...00 * 2^1 and y = 1.11...11 * 2^0

Rounding

The result of any operation on the significands is generally stored in a longer register

When the result is put back into the floating-point format, the extra bits must be eliminated in such a way as to produce a

Floating Point

extra bits must be eliminated in such a way as to produce a result that is close to the exact result – rounding.

Round to nearest: The result is rounded to the nearest representable number

Round toward +∞ : The result is rounded up toward plus infinity.

Round toward –∞: The result is rounded down toward negative infinity

Round toward 0: The result is rounded toward zero.

Special cases: Denormalized numbers

When exponent is all 0’s, it is considered a special case and the floating point number thus represented is assumed to be a denormalized number – i.e. the implicit integral part of the significand is assumed to be 0 (not 1)

Floating Point

the significand is assumed to be 0 (not 1)

To represent 0: Exponent all 0’s, significand all 0’s

To signal error conditions: overflow, underflow, division by 0

NaN (not a number)

Exponent of all 0’s and non=zero significand