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Arithmetic & Logic Unit. Arithmetic & Logic Unit. Arithmetic & logic unit hardware implementation Booth’s Algorithm Restoring division & non restoring division algorithm IEEE floating point number representation & operations. Arithmetic & Logic Unit. Does the calculations - PowerPoint PPT Presentation
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Arithmetic & Logic Arithmetic & Logic UnitUnit
Arithmetic & Logic UnitArithmetic & Logic Unit
Arithmetic & logic unit hardware Arithmetic & logic unit hardware implementationimplementation
Booth’s AlgorithmBooth’s AlgorithmRestoring division & non Restoring division & non
restoring division algorithmrestoring division algorithm IEEE floating point number IEEE floating point number
representation & operationsrepresentation & operations
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 +)
ALU Inputs and Outputs
Integer Representation
• Only have 0 & 1 to represent everything• Positive numbers stored in binary
—e.g. 41=00101001
• No minus sign• No period• Sign-Magnitude• Two’s compliment
Sign-Magnitude Representation
• 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)
Two’s Compliment Representation
• Two's compliment representation uses the MSB as a sign bit.
• Range: -2n-1 through 2n-1 -1
• Twos compliment A= -2n-1an-1 + Σ 2iai
• In case of positive integers, an-1 =0.
Two’s Compliment
• +3 = 00000011• +2 = 00000010• +1 = 00000001• +0 = 00000000• -1 = 11111111• -2 = 11111110• -3 = 11111101
Benefits
• One representation of zero• Arithmetic works easily (see later)• Negating is fairly easy
—3 = 00000011—Boolean complement gives 11111100—Add 1 to LSB 11111101
Geometric Depiction of Twos Complement Integers
Negation Special Case 1
• 0 = 00000000• Bitwise not 11111111• Add 1 to LSB +1• Result 1 00000000• Overflow is ignored, so:• - 0 = 0
Negation Special Case 2
• -128 = 10000000• bitwise not 01111111• Add 1 to LSB +1• Result 10000000• So:• Monitor MSB (sign bit)• It should change during negation
Range of Numbers
• 8 bit 2s compliment—+127 = 01111111 = 27 -1— -128 = 10000000 = -27
• 16 bit 2s compliment—+32767 = 011111111 11111111 = 215 - 1— -32768 = 100000000 00000000 = -215
Conversion Between Lengths
• Positive number pack with leading zeros• +18 = 00010010• +18 = 00000000 00010010• Negative numbers pack with leading ones• -18 = 10010010• -18 = 11111111 10010010• i.e. pack with MSB (sign bit)
Integer Arithmetic
Addition and Subtraction
• Normal binary addition• Monitor sign bit for overflow
• Take twos compliment of subtrahend and add to minuend—i.e. a - b = a + (-b)
• So we only need addition and complement circuits
Hardware for Addition and Subtraction
Multiplication
• Complex• Work out partial product for each digit• Take care with place value (column)• Add partial products
Multiplication Example
• 1011 Multiplicand (11 dec)• x 1101 Multiplier (13 dec)• 1011 Partial products• 0000 Note: if multiplier bit is 1 copy
• 1011 multiplicand (place value)• 1011 otherwise zero• 10001111 Product (143 dec)• Note: need double length result
Unsigned Binary Multiplication
Execution of Example(11*13) (product in A,Q)
Flowchart for Unsigned Binary Multiplication
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
Booth’s Algorithm
Example of Booth’s Algorithm (7*3)
Division
• More complex than multiplication• Negative numbers are really bad!• Based on long division
001111
Division of Unsigned Binary Integers
1011
00001101
100100111011001110
1011
1011100
Quotient
Dividend
Remainder
PartialRemainders
Divisor
Flowchart for Unsigned Binary Division
Floating-Point Representation
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
• Moving?—How do you show where it is?
Floating Point
• +/- Significand x 2Exponent
• Point is actually fixed between sign bit and body of mantissa
• Exponent indicates place value (point position)• Known as ‘biased representation’.• A fixed value called ‘bias’, is subtracted from the
field to get true exponent value.
Floating Point Examples
Signs for Floating Point
• Mantissa/Significand 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
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• ( Scientific notation where numbers are
normalized to give a single digit before the decimal point e.g. 3.123 x 103)
IEEE Standard For Binary Floating-point
Representation
IEEE 754
• Standard for floating point storage• 32 and 64 bit double format,• 8 and 11 bit exponent respectively• Extended formats (both mantissa and
exponent) for intermediate results
IEEE 754 Formats
Special classes of numbers
• For single format 1 through 254
double format 1 through 2046
normalized nonzero floating-point numbers are
represented.
Exponent is biased.
For single format : -126 through +127
double format: -1022 through +1023.
A normalized number requires a 1-bit to the left of the binary point, this bit is implied giving 24-bit or 53-bit significand/fraction.
Exponent values
Special classes of numbers
• Exponent: All zero
Fraction : All zero
• Exponent: All one
Fraction : All zero
Represents +ve or –ve zero depending on sign bit
Represents +ve or –ve infinity depending on sign bit
Special classes of numbers
• Exponent: All zero
Fraction : Nonzero
• Exponent: All one
Fraction : Nonzero
Represents denormalized number depending on sign bit
Represents NaN, Not a Number & is used to signal various exception conditions.
Floating-Point Arithmetic
Floating-point numbers & Arithmetic Operations
Floating-Point numbers
Arithmetic Operations
X= Xs * BXE
Y= Ys * BYE
X+Y= (Xs * BXE-YE + Ys)*BYE
X-Y= (Xs * BXE-YE - Ys)*BYE
X*Y= (Xs *Ys)* BXE+YE
X/Y= (Xs/Ys) * BXE-YE
XE<=YE
X=0.3*102=30 Y=0.2*103=200
X+Y=(0.3*102-3 +0.2) *103 =230X-Y=(0.3*102-3 -0.2) *103 =-170X*Y=(0.3*0.2) *102+3 =6000X/Y=(0.3/0.2) *102-3 =0.15
Conditions produced in floating-point operations
• Exponent overflow : +ve/-ve infinity• Exponent underflow: zero• Significand underflow: In aligning
significands, digits may flow off the right end of the significand. Rounding is required.
• Significand overflow: In addition of significands, carry out of MSB, realignment is required.
FP Arithmetic +/-
• Check for zeros• Align significands (adjusting exponents)• Add or subtract significands• Normalize result
Algorithm for addition & subtraction of FP Numbers
• Step 1: Zero Check Addition & Subtraction are same except
for sign change. If subtraction, change the sign of
subtrahend. If either operand is 0, the other is
reported as result.
Algorithm for addition & subtraction of FP Numbers
• Step 2: Significant Alignment Manipulate the numbers so that two
exponents are equal.Eg. 123 * 100 + 456 * 10-2
=123 * 100 + 4.56 * 100
= 123 + 4.56
= 127.56
Algorithm for addition & subtraction of FP Numbers
• Step 3: AdditionAdd two significant; result may be zero If significand overflow by 1 digit,
significand of result is shifted right & exponent is incremented.
If exponent overflow occurs as a result, this would be reported & operation is halted.
Algorithm for addition & subtraction of FP Numbers
• Step 3: NormalizationNormalize the result.Shift significant digits left till MSB is
nonzero. Decrement exponent, may cause underflow.
Round off the result & report.
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
A Multiplication Algorithm
• Step 1: Zero check If either operand is 0, the other is reported
as result.Add exponent. If exponent are stored in
bias format, sum would have double bias, so subtract bias value from sum.
If exponent overflow or underflow , report & end algorithm
A Multiplication Algorithm
• Step 2: MultiplyMultiply significands taking into account
their signs.
• Step 3: NormalizeNormalize & round the result
Floating Point Multiplication
A Division Algorithm
• Step 1: Zero check If divisor is 0, an error report is issued or
result is set to infinity.A dividend of 0 results in 0.A divisor exponent is subtracted from
dividend exponent. This removes bias which must be added back in.
Check for exponent overflow or underflow.
• Step 2: DivideDivide significands & normalize & round
result.
Floating Point Division
QuestionnaireQuestionnaire
• Explain restoring division method. Hence perform -7 ÷ -3.
• Write algorithm for floating point addition.
• Explain with an example an algorithm for 2’s compliment multiplication.
• Explain Booth Algorithm for 2’s compliment multiplication using flow-chart.
• Explain Booth Algorithm for 2’s compliment multiplication using flow-chart. Hence multiply 7 * -3 using Booth algorithm.
• Explain floating point representation IEEE standard 754.
• Explain the principle of Booth Algorithm, when it is less efficient.
• A 8-bit register holds number in 2’s compliment form with leftmost bit as sign bit.– i) What is the largest positive number can be
stored, express answer in binary & decimal format.
– ii) What is the largest negative number can be stored, express answer in binary & decimal mformat.