Uncertaintyaware Circuit Optimization

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Uncertainty-aware Circuit Optimization

• Xiaoliang Bai, Chandu Visweswariah,
• Philip N. Strenski, David J. Hathaway
• University of California, San Diego
• IBM T. J. Watson Research Center
• IBM Microelectronics

2
Outline

• Why optimization is good
• Why optimization is bad
• Uncertainty-aware tuning
• Experimental results
• Alternative formulations
• Conclusions

3
Why optimization is good

• Background EinsTuner automatic transistor
  resizing tool
• based on static timing all paths implicitly
  considered
• uses nonlinear optimization optimality
  guaranteed
• uses fast simulation and sensitivity analysis

• Benefits
• better circuits (delay, area, power, noise,
  combinations thereof)
• enhanced designer productivity

4
Optimizing circuit delay
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What tuning does the wall
signals
slack
6
What tuning does the wall
signals
untuned
slack
20 ps
-10 ps
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What tuning does the wall
signals
untuned
slack
20 ps
-10 ps
8
What tuning does the wall
signals
The expected value of the worst slack is
adversely impacted by the height of the wall
untuned
slack
20 ps
-10 ps
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Why optimization is bad

• Disadvantages of the wall
• extreme sensitivity to uncertainties
• manufacturing
• environmental
• inaccuracy in modeling and tools
• further optimization is very difficult
• manual
• automatic
• high delay-testing cost

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Key idea

• The wall is an artifact of mathematical
  optimization the optimizer will increase the
  height of the wall even to save 0.001 ps
• Define separation of a path as the difference
  between the slack of the path and the slack of
  the most critical path
• We need to give the optimizer incentives to
  increase separation
• To accomplish this, we add penalty terms to the
  objective function to avoid the wall
• The penalty must be in relation to the most
  critical path, which changes dynamically

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Uncertainty-aware tuning
z
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In mathematical terms

• Nominal tuning
• Uncertainty-aware tuning
• k is a weight factor the penalty function forces
  separation

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Observation

• Optimizers often convert inequalities to
  equalities by introducing a non-negative
  constraint slack variable
• (z-ATi) is the value of the constraint slack
  introduced by the nonlinear optimizer for the
  inequality z ? ATi
• We express the penalty in terms of this
  constraint slack

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Choice of penalty function
Penalty Box
z
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Choice of parameters
Penalty Box
z
To guarantee the downward pressure on z
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Uncertainty-aware tuning histogram
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Estimated manufacturing result
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Area tradeoff minimization modes

• Area minimization mode
• Tradeoff mode

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Pruning

• EinsTuner prunes the timing graph
• reduces problem size
• reduces degeneracy and redundancy
• reduces CPU time
• But a primary output arrival time may be pruned
• In this case, we add a penalty term corresponding
  to every sub-path that ends at a primary output
• Again, we re-use the constraint slack variable

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Experimental results
Similar slacks, better manufacturability
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Nominal tuning
z
ATi
Arrival time
Delay
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Nominal tuning with constraint slacks
Constraint Slacks
Critical path
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Uncertainty-aware tuning
Separation
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Pruned
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Internal separation AT/RAT separation
Arrival Time
Required Arrival Time
Delays
Constraint Slacks
Same delay
Separation
Critical Path (shown for AT constraints only)
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Conclusions

• Optimizer gratuitously creates a wall of
  equally critical paths
• Uncertainty-aware tuning reduces the height of
  the wall
• The resulting design is less susceptible to
  manufacturing and other sources of variation
• Easier downstream restructuring, delay testing
• Side benefit optimizer is more efficient by
  eliminating degeneracy