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CBNST PPT, Floating point arithmetic,Normalization

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Representation of floating point arithmetic, Operation, Normalization, PIt Falls of floating point arithmetic

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Floating point arithmetic Representation of floating point arithmetic Operations Normalization Pit-Falls of floating point arithmetic
There are two types of arithmetic operations :- Integer Arithmetic Real floating point arithmetic Integer Arithmetic :- deals with integer operands. i.e. - num without fractional parts Real Arithmetic :- use number with fractional parts as operands and is use Real no. = mantissa * 10^exponent
COMPUTER REPRESENTATION OF FLOATING POINT NUMBERS In the CPU, a 32-bit floating point number is represented using IEEE standard format as follows: S | EXPONENT | MANTISSA where S is one bit, the EXPONENT is 8 bits, and the MANTISSA is 23 bits.
The mantissa represents the leading significant bits in the number. It is made less than 1 and greater than or equal to 0.1 The exponent is used to adjust the position of the binary point (as opposed to a "decimal" point) In power of 10 and multiplies with mantissa
Normalization :- Mantissa and Exponent have their own independent signs While storing no. of the leading digit in the mantissa, mantissa is always made non- zero by appropriately shifting it and adjusting the value of the exponent The shifting of mantissa to the left till its most significant digit is non- zero is called normalization.
Arithmetic operations with normalized floating point numbers Addition Subtraction Multiplication Division
Addition :- Consider a 4-digit decimal example 9.999 101 + 1.610 101 1. Align decimal points Shift number with smaller exponent 9.999 101 + 0.016 101 2. Add significant 9.999 101 + 0.016 101 = 10.015 101 3. Normalize result & check for over/underflow 4. Round and renormalize if necessary
Subtraction :- Consider a 4-digit decimal example 9.999 101 - 1.610 101 1. Align decimal points Shift number with smaller exponent 9.999 101 - 0.016 101 2. Subtract significant 9.999 101 - 0.016 101 = 9.983 101 3. Normalize result & check for over/underflow 4. Round and renormalize if necessary
Multiplication :- Consider a 4-digit decimal example 1.110 1010 9.200 105 1. Add exponents New exponent = 10 + 5 = 5 2. Multiply significant 1.110 9.200 = 10.212 10.212 105 3. Normalize result & check for over/underflow 1.0212 106 4. Round and renormalize if necessary 1.021 106 5. Determine sign of result from signs of operands +1.021 106
Division :- Consider a 4-digit decimal example 1.110 1010 / 9.200 105 1. Subtract exponents New exponent = 10 (5) = 15 2. Divide significant 1.110 / 9.200 = 0.12065 0.12065 1015 3. Normalize result & check for over/underflow 1.2065 1014 4. Round and renormalize if necessary 5. Determine sign of result from signs of operands
CBNST PPT, Floating point arithmetic,Normalization

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CBNST PPT, Floating point arithmetic,Normalization

  • 1. Floating point arithmetic Representation of floating point arithmetic Operations Normalization Pit-Falls of floating point arithmetic
  • 2. There are two types of arithmetic operations :- Integer Arithmetic Real floating point arithmetic Integer Arithmetic :- deals with integer operands. i.e. - num without fractional parts Real Arithmetic :- use number with fractional parts as operands and is use Real no. = mantissa * 10^exponent
  • 3. COMPUTER REPRESENTATION OF FLOATING POINT NUMBERS In the CPU, a 32-bit floating point number is represented using IEEE standard format as follows: S | EXPONENT | MANTISSA where S is one bit, the EXPONENT is 8 bits, and the MANTISSA is 23 bits.
  • 4. The mantissa represents the leading significant bits in the number. It is made less than 1 and greater than or equal to 0.1 The exponent is used to adjust the position of the binary point (as opposed to a "decimal" point) In power of 10 and multiplies with mantissa
  • 5. Normalization :- Mantissa and Exponent have their own independent signs While storing no. of the leading digit in the mantissa, mantissa is always made non- zero by appropriately shifting it and adjusting the value of the exponent The shifting of mantissa to the left till its most significant digit is non- zero is called normalization.
  • 6. Arithmetic operations with normalized floating point numbers Addition Subtraction Multiplication Division
  • 7. Addition :- Consider a 4-digit decimal example 9.999 101 + 1.610 101 1. Align decimal points Shift number with smaller exponent 9.999 101 + 0.016 101 2. Add significant 9.999 101 + 0.016 101 = 10.015 101 3. Normalize result & check for over/underflow 4. Round and renormalize if necessary
  • 8. Subtraction :- Consider a 4-digit decimal example 9.999 101 - 1.610 101 1. Align decimal points Shift number with smaller exponent 9.999 101 - 0.016 101 2. Subtract significant 9.999 101 - 0.016 101 = 9.983 101 3. Normalize result & check for over/underflow 4. Round and renormalize if necessary
  • 9. Multiplication :- Consider a 4-digit decimal example 1.110 1010 9.200 105 1. Add exponents New exponent = 10 + 5 = 5 2. Multiply significant 1.110 9.200 = 10.212 10.212 105 3. Normalize result & check for over/underflow 1.0212 106 4. Round and renormalize if necessary 1.021 106 5. Determine sign of result from signs of operands +1.021 106
  • 10. Division :- Consider a 4-digit decimal example 1.110 1010 / 9.200 105 1. Subtract exponents New exponent = 10 (5) = 15 2. Divide significant 1.110 / 9.200 = 0.12065 0.12065 1015 3. Normalize result & check for over/underflow 1.2065 1014 4. Round and renormalize if necessary 5. Determine sign of result from signs of operands