ECEG-3202: Computer Architecture and Organization, Dept of ECE, AAU 1 Floating-Point Arithmetic...

ECEG-3202: Computer Architecture and Organization, Dept of ECE, AAU

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Floating-Point Arithmetic Operations


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Floating-Point Hardware

• Older computer systems included hardware support for integer arithmetic operations only.– For floating-point operations, they relied on

software libraries for simulating the floating-point operations using the hardware’s integer units.

• Including floating-point hardware is quite expensive. However, it is much more efficient than the software method.

• This is why almost all modern computers have hardware that is specifically designed for performing floating-point arithmetic operations.


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Range of Representation

• Floating-point representation increases the range of values that can be stored in a given register.

• Consider a 48-bit register:– The range of signed integers that can be stored is:

-247 247–1• 247 because one bit must be reserved for the sign.


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Range of Representation

• Assuming 36 bits for a fractional mantissa and 12 bits for the exponent, calculate the range of floating-point numbers that can be stored:– One bit for the mantissa sign 35 bits for the

mantissa.– Fractional mantissa 1 – 2-35

– One bit for the exponent sign 11 bits for the exponent.

– Integer exponent 22047

±(1 – 2-35) x 22047


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Alignment

• Consider the following operation:

123 X 100 + 456 X 10-2

– How can we add these two numbers?

• Before floating-point numbers can be added or subtracted, their exponents must be made equal. This process is called alignment.


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Aligning the Mantissas

• To align floating-point numbers:– Determine which has the smaller exponent.– Shift the mantissa of the number with the smaller

exponent (the smaller number) to the right and increment its exponent by 1.

– Repeat until the two exponents become equal.

• Shifting the smaller number to the right will discard lesser significant bits.– If the shifting results in a 0 mantissa, stop.


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Biased Exponent

• A “biased” exponent is an exponent to which a value has been added to make it always positive.– This makes it easier to compare exponents and

determine which is larger.• Very useful during alignment.

• Using 8 bits for the exponent we can represent values from –128 up to 127.– If we add 128 to these values, we would end up

with exponents in the range of 0 to 255.– Deciding that 129 is larger than 126 is much

easier than deciding whether 1 is larger than –2.


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Addition and Subtraction

• There are 4 main steps to adding or subtracting floating-point numbers:– Check for zeros.– Align the mantissas.– Add or subtract the mantissas.

• Since the exponents are now equal, the mantissas can be added/subtracted following the regular rules of addition/subtraction.

– Normalize the result.


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Examples

• Represent 0.5 and 0.25 using the following floating point representation:6 bit normalized signed 2’s complement fraction

mantissa, including the sign.• 1 bit for the sign, 5 bits for the mantissa.

3 bit biased exponent.• Range of representation is -4 to 3.

Add 4 to bias it.

• Follow the steps of floating point addition to find the sum of the two numbers.


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Examples (Contd.)0.5 0.1 X 20

• 0 10000 100 (Already normalized).

0.25 0.01 X 20 • 0 01000 100 • 0 10000 011 (After normalization).

0 10000 100

0 10000 011+


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Examples (Contd.)1. Check for Zero.

– None of the numbers is 0.

2. Align Mantissas

3. Add

4. Normalize– Already normalized.

0 10000 1000 10000 011+

0 10000 1000 01000 100+

0 100000 01000+0 11000 0 11000 100

The result is


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Examples (Contd.)

• Add 0.5 to –0.25

1. Check for zeros.• None.

2. Align Mantissas.• Already Aligned.

3. Add Mantissas.

4. Normalize the result.

0.5 0.1 X 20 0 10000 100

-0.25 -0.01 X 20 - 0 01000 100 1 11000 100

0 100001 11000+

1 0 01000

0 10000 011

0 01000 100Preliminary

Result

Final Result


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Examples (Contd.)

• Add -0.5 to 0.25


2. Align Mantissas.• 0 10000 011

3. Add Mantissas.

4. Normalize the result.• Already Normalized.

-0.5 -0.1 X 20 - 0 10000 100

0.25 0.01 X 20 0 01000 100

1 10000 100

1 100000 01000+1 11000 1 11000 100

Preliminary

Result

0 10000 011

0 01000 100


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Overflow in Floating-Point Operations

• What about overflow?– If adding the mantissas produces an overflow, we

shift the result’s mantissa to the right 1 position and increment the exponent by 1.

– If incrementing the exponent produces and exponent overflow, the operation is terminated and an overflow error is reported.


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Examples (Contd.)

• Add 0.5 to 0.5



3. Add Mantissas.

4. Normalize the result.• Already Normalized.

0.5 0.1 X 20 0 10000 100

0.5 0.1 X 20 0 10000 100

0 100000 10000+1 00000 0 10000 101

Overflow

Correction


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Examples (Contd.)

• Add 4 to 4



3. Add Mantissas.

• Attempting to correct the overflow will cause the exponent to exceed its limit.– Cannot correct this overflow.

4 0.1 X 23 0 10000 111

4 0.1 X 23 0 10000 111

0 100000 10000+1 00000 Overflow


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Multiplication of Floating-Point Numbers

• Multiplication of floating-point numbers is a four step process:– Check for zeros.– Add the exponents.– Multiply the mantissas.– Normalize the product.


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Division of Floating-Point Numbers

• Division is done by:– Check for zeros.– Align the dividend.– Subtract the exponents.– Divide the mantissas.

• All of the operations on floating-point numbers have been simplified so that the same hardware can be used for floating-point numbers and integers.

ECEG-3202: Computer Architecture and Organization, Dept of ECE, AAU 1 Floating-Point Arithmetic...

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