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An SMT Based Method for Optimizing Arithmetic Computations in Embedded Software Code. Presented by: Kuldeep S. Meel Adapted from slides by Hassan Eldib and Chao Wang (Virginia Tech). A Robotic Dream. Having a tool that automatically synthesizes the optimum version of a software program.

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an smt based method for optimizing arithmetic computations in embedded software code

An SMT Based Method for Optimizing Arithmetic Computations in Embedded Software Code

Presented by: Kuldeep S. Meel

Adapted from slides by Hassan Eldib and Chao Wang (Virginia Tech)

a robotic dream
A Robotic Dream
  • Having a tool that automatically synthesizes the optimum version of a software program.
some bugs matter more than others
Some bugs matter more than others

https://www.youtube.com/watch?v=AjUpe7Oh1j0#t=75

some even more
Some even more ……

Cost: US $350 million

the age of sci fi cs
The Age of Sci-Fi CS
  • Verification
    • Verify if there is overflow/underflow bug in the code
  • Synthesis
    • Given a reference implementation, synthesize an implementation that does not have any of these bugs
objective
Objective
  • Synthesizing an optimal version of the C code with fixed-point linear arithmetic computation for embedded devices.
    • Minimizing the bit-width.
    • Maximizing the dynamic

range.

motivating example
Motivating Example
  • Compute average of Aand B on a microcontroller with signed 8-bit fixed-point
  • Given: A, B ∈ [-20, 80].
    • (A +B)/2 may have overflow errors
    • A/2 + B/2 may have truncation errors
    • A + (B-A)/2 has neither overflow nor truncation errors .
bit width versus range
Bit-width versus Range
  • Larger range requires a larger bit-width.
  • Decreasing the bit-width, will reduce the range.
fixed point representation
Fixed-point Representation

Representations for 8-bit fixed-point numbers

Range ∝ Bit-width

Resolution ∝ Bit-width

  • Range: -128 ↔ 127
  • Resolution = 1
  • Range : -16 ↔ 15.875
  • Resolution = 1/8
problem statement
Problem Statement

Program:

Optimized program:

Range & resolution of the input variables:

A -1000 3000

res. 1/4

B -1000 3000

res. 1/4

problem statement1
Problem Statement
  • Given
    • The C code with fixed-point linear arithmetic computation
      • Fixed-point type: (s,N,m)
    • The range and resolution of all input variables
  • Synthesize the optimized C code with
    • Reduced bit-width with same input range, or
    • Larger input range with the same bit-width
restrictions
Restrictions
  • Bounded loops (via unrolling)
  • Same bit-width for all the variables and constants
    • Can have different bit representations
step 1 finding a candidate program
Step 1: Finding a Candidate Program
  • Create the most general ASTthat can represent any arithmetic equation, with reduced bit-width.
  • Use SMT solver to find a solution such that
    • For some test inputs (samples),
    • output of the AST is the same as the desired computation
smt based solution
SMT-based Solution
  • SMT encoding for the general equation AST structure
    • Each Op node can any operation from *, +, -, >> or <<.
    • Each L node can be an input variable or a constant value.
  • SMT Solver finds a solution by equating the AST output to that of the desired program

Fig. General Equation AST.

smt encoding
SMT Encoding
  • : Desired input program to be optimized.
  • : General AST with reduced bit-width.
  • : Same input values.
  • Same output value.
  • : Test cases (inputs).
  • : Blocked solutions.
step 2 verifying the solution
Step 2: Verifying the Solution
  • Is the program good for all possible inputs?
    • Yes, we found an optimized program
    • No, block this (bad) solution, and try again
smt encoding1
SMT Encoding
  • : Desired input program to be optimized.
  • : Found candidate solution.
  • : Same input values.
  • : Different output value.
  • : Ranges of the input variables.
  • : Resolution of the input variables.
scalability problem
Scalability Problem
  • Advantage of the SMT-based approach
    • Find optimal solution within an AST depth bound
  • Disadvantage
    • Cannot scale up to larger programs
      • Sketch tool by Solar-Lezama& Bodik (5 nodes)
      • Our own tool based on YICES (9 nodes)
incremental optimization
Incremental Optimization
  • Combine static analysis and SMT-based inductive synthesis.
  • Apply SMT solver only to small code regions
    • Identify an instruction that causes overflow/underflow.
    • Extract a small code region for optimization.
    • Compute redundant LSBs (allowable truncation error).
    • Optimize the code region.
    • Iterate until no more further optimization is possible.
region for optimization
Region for Optimization
  • BFS order (Tree has return as root)
    • Parent node doesn’t have overflow or underflow
    • Parent node restores the value created in the overflowing node back to the normal range in orig.
    • Larger extracted region allows for more opportunities
    • AST with at most 5 levels, 63 AST nodes
region for optimization1
Region for optimization

Detecting Overflow Errors

  • The addition of a and b may overflow

The parent nodes

Some sibling nodes

Some child nodes

truncation error margins
Truncation Error margins

Computing Redundant LSBs

  • The redundant LSBs of a are computed as 4 bits
  • The redundant LSBs of b are computed as 3 bits.
truncation error margins1
Truncation Error margins
  • step(x*y) = step(x) + step(y)
  • step(x+y) = min(step(x),step(y))
  • step(x-y) = min(step(x),step(y))
  • step(x << c) = step(x) + c
  • step(x >> c) = max(step(x)-c,0)
extracting code region
Extracting Code Region
  • Extract the code surrounding the overflow operation.
  • The new code requires a smaller bit-width.
inductive generation of new regions
Inductive Generation of New Regions
  • : Extracted region of bit width (bw1).
  • : Desired solution with bit wi.
  • : Same input values.
  • : Same output value.
  • : random tests.
  • : Blocked programs
verification
Verification
  • : Desired input program to be optimized.
  • : Found candidate solution.
  • : Same input values.
  • : Different output value.
  • : Ranges of the input variables.
  • : Resolution of the input variables.
implementation
Implementation
  • Clang/LLVM + Yices SMT solver
  • Bit-vector arithmetic theory
  • Evaluated on a set of public benchmarks for embedded control and DSP applications
benchmarks embedded control software
Benchmarks(embedded control software)

All benchmark examples are public-domain examples

experiment increase in range
Experiment (increase in range)
  • Average increase in range is 307%

(602%, 194%, 5%, 40%, 32%, 1515%, 0% , 103%)

experiment decrease in bit width
Experiment (decrease in bit-width)
  • Required bit-width: 32-bit 16-bit

64-bit 32-bit

experiment scaling error
Experiment (scaling error)

Original program New program

If we reduce microcontroller’s bit-width, how much error will be introduced?

related work sketch
Related Work: Sketch
  • CEGIS based
  • Synthesis is over the whole program
  • Solver not capable of efficiently handling linear fixed-point arithmetic computations (?)
related work gulwani et al pldi 2011
Related Work: Gulwani et al. (PLDI 2011)
  • User specified logical specification (e.g., unoptimized program)
  • Uses CEGIS to find optimized version
  • Slow (This technique performs worse than enumeration/stochastic techniques)
related work rupak et al ices 12
Related Work: Rupak et al. (ICES 12)
  • Optimization of bit-vector representations
  • Does not change structure of the program
  • Uses Mixed Integer Linear Programming (MILP) Solver
conclusions
Conclusions
  • We presented a new SMT-based method for optimizing fixed-point linear arithmetic computations in embedded software code
    • Effective in reducing the required bit-width
    • Scalable for practice use
  • Future work
    • Other aspects of the performance optimization, such as execution time, power consumption, etc.
more on related w ork
More on Related Work
  • Solar-Lezamaet al. Programming by sketching for bit-streaming programs, ACM SIGPLAN’05.
    • General program synthesis. Does not scale beyond 3-4 LoC for our application.
  • Gulwaniet al. Synthesis of loop-free programs, ACM SIGPLAN’11.
    • Synthesizing bit-vector programs. Largest synthesized program has 16 LoC, taking >45mins. Do not have incremental optimization.
  • Jha. Towards automated system synthesis using sciduction, Ph.D. dissertation, UC Berkeley, 2011.
    • Computing the minimal required bit-width for fixed-point representation. Do not change the code structure.
  • Rupaket al. Synthesis of minimal-error control software, EMSOFT’12.
    • Synthesizing fixed-point computation from floating-point computation. Again, only compute minimal required bit-widths, without changing code structure.
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