let10-Floating Point Arithmetic and Polynomial Functions

Computer Arithmetic: Algorithms and Hardware Designs

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CSE 246: Computer Arithmetic  Algorithms and Hardware Design Instructor: Prof. Chung-Kuan Cheng Fall 2006 Lecture 10 Floating Point  Number Rounding, Polynomial  Expression
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CSE 246 2 Topics: Rounding F.P. Numbers Polynomial Expression
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CSE 246 3 Rounding the numbers Why we need the  Guard bit Round bit Sticky bit  
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CSE 246 4 Example 1   1.00000 2 4  -1.10000 2 -3 Normalize according to exponent    1.00000       2 4  -0.00000011 2 4   0.11111 1 1  2 4 Renormalize  1.1111 1 0 1 x2 3 Result =  3 Take 5 bits after decimal Round bit Sticky Bit
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CSE 246 5 Rounding We need only one guard bit for  normalization after addition. Assumption: Operands are normalized. Why?
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CSE 246 6 Example 2   1.00001 2 3  -1.01011 2 -1 Normalize according to exponent    1.00000       2 3  -0.000101011 2 3   0.11110 0 01 2 3 Renormalize  1.1110 0 1 01  2 2 Result =  2 Take 5 bits after decimal Round bit Bit on the boundary Non-zero => round-up
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CSE 246 7 Theory behind it g r round guard Other bits OR Sticky bit When shifting right, don’t need to remember  anything more than 3 bits below This is a necessary   and  sufficient  condition
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CSE 246 8 Polynomial Approximation of Functions
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