Chapter 14 HighRadix Dividers

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Chapter 14 High-Radix Dividers

• Basics of High-Radix Division
• Radix-2 SRT Division
• Using Carry-Save Adders
• Choosing the Quotient Digits
• Radix-4 SRT Division
• General High-Radix divider

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Basics of High-Radix Division
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Radix-4 Division in Dot Notation

• Interesting dividers have radix r 2b
• Reduces the number of cycles of by a factor of b

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Difficulty of High-Radix Division

• Guessing the correct quotient digit is more
  difficult.
• Division is naturally a sequential process
• Guess a quotient digit qk-j
• Compute term qk-j(rkd)
• Compute partial remainder
• s(j) rs(j-1) ? qk-1(rkd)

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Carry-Save Remainders

• More important for speed than high-radix.
• Lead to large performance increases by replacing
  carry-propagate adder with carry-save adder.
• Key to keeping remainder in carry-save form is
  Redundancy in the representation of q.
• Allows less precise guessing of quotient digit
  based on approximate magnitude of partial
  remainder.
• More redundancy less precision required.

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Review of Non-Restoring Division (fractional
operands)
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Using q-j in -1,0,1
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A Big Problem

• Q How can you tell if shifted partial remainder
  is in -d,d) ?
• A You have to perform trial subtractions.
• Q Can you avoid trial subtractions ?
• A Sweeny, Robertson, and Tocher-- SRT division.

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Radix-2 SRT Division

• Assume d ³ 1/2 (normalized)
• Restrict partial remainder to constant range
  -1/2, 1/2) instead of -d, d )
• May require shifting dividend initial partial
  remainder so that 1/2 ³ s (0) ³ -1/2
• Once in the proper range, subsequent partial
  remainders will stay in the range

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Radix-2 SRT Division
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Simplified Digit Selection
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Final Steps

• -1,1-quotient conversion algorithm will not
  work to convert -1, 1-quotient to
  twos-complement.
• On-the-fly algorithm by Ercedovac 1987, or
• Subtract negative digits from positive digits.
• Still requires a final correction step to make
  remainder positive.

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Using Carry-Save Adders
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Carry-Save Partial Remainders

• Two numbers sum to the actual partial remainder.
• To perform exact comparison, a full CPA would be
  required.
• Overlaps in the selection regions allow us to
  perform approximate comparisons without risk of
  choosing a wrong digit.

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Carry-Save Partial Remainders
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Tolerating Truncation Error
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Digit Selection
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Radix-2 Divider with CSA
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Select Logic

• Fast 4-bit CPA, plus decode logic, or
• 256 ? 2 Lookup table, or
• 8 input, 2 output PLA

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CLA with SRT Division?
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Choosing Quotient Digits Using a P-D Plot
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Putting Both Charts Together
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Radix-4 SRT Division

• Radix r 2 b , b gt 1
• Partial remainder kept in stored-carry form.
• Requires a redundant digit set.
• Example
• radix 4
• digit set -3, 3

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New vs. Shifted Old Partial Remainder

• Radix 4 Digit Set -3, 3

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p-d Plot for Radix-4, -3, 3 SRT Division
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Radix-4 Digit Set -2, 2

• Avoids having to compute 3d as in digit set -3,
  3.
• Fewer comparisons (fewer selection regions).
• Less redundancy means less overlap in selection
  regions.
• Partial remainder must be restricted to ensure
  convergence.

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Restricting the Range of s
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p-d Plot for Radix-4, -2, 2 SRT Division
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Observations

• Restricting digit set to -2, 2 results in less
  overlap in selection regions.
• Must examine p and d in greater detail to
  correctly choose the quotient digit.
• Staircase boundaries 4 bits of p and 4 bits of d
  are required to make the selection.

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Block Diagram
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Intels Pentium Division Bug

• Intel used the Radix-4 SRT division algorithm.
• Quotient selection was implemented as a PLA.
• The p-d plot was numerically generated.
• Script to download entries into the PLA
  inadvertently removed a few table entries from
  the table.
• When hit, these missing entries resulted in digit
  0, instead of the intended digits 2.
• These entries are consulted very rarely, and thus
  the bug was very subtle and difficult to detect.

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General High-Radix Dividers

• Radix-8 is possible.
• Minimal quotient digit set -4, 4
• Partial remainder restricted to -4d/7, 4d/7)
• Requires a 3d multiple
• Digit sets with greater redundancy (such as -7,
  7 ) lead to
• Wider overlap regions
• More comparisons but simpler digit selection
• More difficult multiples (5, 7)