Handling Floating Point Operands

1
Handling Floating Point Operands

• Type of Processing
• Addition .
• Subtraction.
• Multiplication.
• Division.
• Associated Operations ( always required )
• i) Compare Biased Exponents.
• ii) Bias / Un bias Exponents.
• iii) Adjust Significand / Mantissa by modifying
  Biased Exponent portion.
• iv) Normalize a De Normalized Number extension
  of iii) .

2
Handling Floating Point Operands-2

• The Operands In Base r
• A ? As n digits ? r AX(AeBa)
• AX Biased Exponent (m digits), Ae Actual
  Exponent , Ba Bias
  
• B ? Bs k digits ? n ? r BX(BeBa)
• BXBiased Exponent (L digits ? m )Be
  Actual Exponent , Ba Bias
• The Basic Circuit
• Signed Magnitude Arithmetic Circuit with
  separate ADDER , SUBTRACTOR , COMPARATOR
  SHIFTER section.

3
The Addition Subtraction - 1

• Step 1. Check for either A or B ?? OR NaN .
• Step 2. If any Operand happens to be invalid then
  Exit
• with proper message else go to Step
  3.
• Step 3. Check for either A or B 0 , if it is
  then
• I) Generate Result Z Other
  Operand A / B.
• ii) EXIT
• Step 4. If the Operation is to perform A MINUS B
  then change
• the sign of B.
• Step 5. Compare the signs of the two Operands A
  B . If they happen to be the same then activate
  ADDER circuit else activate SUBTRACTOR section.

4
The Addition Subtraction - 2

• Step 6. Compare the two exponents AX BX .
• Step 7. If the two exponents are equal i.e. AX
  BX go to Step 9.
• Step 8. AX ? BX . Adjust Exponents
• Step 8A. Increment smaller
  Exponent.
• Step 8B. Shift its Significant
  Right (may get De Normalized)
• Step 8C. If Significant 0 then
  Result Other Operand
• EXIT ELSE go to
  Step 6 .
• Step 9. Exponents are Equal , ADD/ SUBTRACT
  Signed Significands
• Step 10. If result Significand 0 treat as ZERO
  RESULT EXIT.

5
The Addition Subtraction - 3

• Step 11. If the result significant produces Carry
  OR Borrow then
• Step 11a. Shift Result Significant
  Right.
• Step 11b. Increment Result
  Exponent.
• Step 11c. If Exponent Overflows
  Report EXIT
• Step 12. If the Result is De Normalized then
• Step 12a. Shift Significant Left.
• Step 12 b. Decrement Exponent.
• Step 12 c. If Exponent Underflows ,
  Report EXIT.
• Step 13. Round Off Result (if needed).
• EXIT

6
Addition / Subtraction Example

• A - 0.23967 e 95 , B 0.4543102 e 90
  Decimal
• To Compute C A MINUS B
• Step 1 , 2 3. Passed.
• Step 4. Change sign of B i.e. B 9 0.4543102 e
  90
• A
  9 0. 23967 e 95
• Step 5. Activate ADDER Section ( Signs are Equal
  ).
• Step 6. Exponents AX 95 gt BX 90 adjust
  Exponents.
• Step 8. Step 8A. Increment Exponent BX ? 90 ..
  95
• Step 8B. Right shifted B
  significant 90.000004543
• Step 9. ADD Significant C 9 0.239674543 e 95
  Normalized.

7
Multiplication Algorithm

• To compute Z A X B in Base r
• Step 1A . Check for either A or B ?? OR NaN .
  In such case Report Outside Range Exit.
• Step 1B. If A OR B 0 then Result Z 0 EXIT.
  
• Step 2. Check Note the signs of the two
  Significands Asign Bsign.
• Step 3a. Add the two Exponents Ax Bx to
  produce the result Exponent Zx. Ignore any
  overflow now.
• Step 3b. Subtract Bias from Cx. This this will
  remove the effect of the Bias which has been
  added twice in the Result Exponent Zx.
• Step 4. If Exponent Underflow / Overflow still
  persists in Zx, Report EXIT.
• Step 5. Multiply Significands Zs As ? Bs using
  Shift ADD Meta Step.
• Step 6. Normalize Result Significand Zs. Report
  Exponent Underflow/Overflow.
• Step 7. Adjust sign of the result Zsign Round
  off Result (if necessary)..

8
Division Algorithm

• To compute Q A / B in Base r
• Step 1A. Check for either A or B ?? OR NaN .
  In such case Report Outside Range EXIT.
• Step 1B. If A 0 then Result Q 0 EXIT.
• Step 1C. If B 0 then Report Divisor too small
  EXIT.
• Step 2. Check Note the signs of the two
  Significands Asign Bsign.
• Step 3. Subtract the two Exponents Ax Bx this
  will cause Bias removed from the Result Exponent
  Qx.
• Step 4. Restore Bias in result exponent by adding
  Bias to Zx.
• Step 5. If Exponent Underflow / Overflow occurs
  in Zx, Report EXIT.
• Step 6. Divide Significands Qs As / Bs using
  Shift SUBTRACT Meta Step to generate the
  Quotient .
• Step 7. Set up Quotient Sign Qsign .
• Step 8. Normalize Quotient Significand Qs.
  Report Exponent Underflow/Overflow.
• Step 9. Round off Result (if necessary)..