Why GCC Doesn't Optimize a^6 to (a^3)^2

Compilers don’t automatically optimize a*a*a*a*a*a to (a*a*a)*(a*a*a) due to several optimization challenges and design decisions in compiler architecture.

Contents

• Compiler Optimization Challenges
• Why This Specific Optimization Isn’t Applied
• Cost-Benefit Analysis in Compilers
• Workarounds and Solutions
• Related Optimization Patterns

Compiler Optimization Challenges

The optimization you’re describing is a form of strength reduction combined with common subexpression elimination. These are well-known optimization techniques, but their automatic application faces several challenges:

1. Complexity Analysis: The compiler must analyze the expression tree to identify optimal computation sequences. For a*a*a*a*a*a, the optimal sequence requires computing a*a, then a*a*a, then squaring the result.

2. Register Allocation: The optimized version requires additional temporary variables, which increases register pressure. The compiler must determine if the performance gain justifies the extra register usage.

3. Vectorization Impact: The optimization might interfere with SIMD vectorization opportunities, as the restructured computation pattern might not align well with vector instructions.

4. Precision Considerations: Different computation orders can produce slightly different results due to floating-point rounding, and compilers must be conservative about changing evaluation order.

Why This Specific Optimization Isn’t Applied

Several technical factors explain why GCC and other compilers don’t perform this optimization automatically:

1. Limited Pattern Matching: Compilers have predefined optimization patterns for small exponents (like pow(a,2) → a*a) but don’t have comprehensive pattern matching for higher powers. The compiler might recognize a*a as an optimization target but fail to recognize the opportunity for a*a*a*a*a*a.

2. Heuristic Limitations: The compiler’s optimization heuristics might determine that the potential performance improvement doesn’t outweigh the compile-time cost of analyzing expression trees for optimal computation sequences.

3. Instruction Selection: The compiler’s instruction selection algorithm might not prioritize minimizing multiplication count when other factors like instruction throughput or latency are more relevant for the target architecture.

4. Optimization Pass Order: The optimization passes in GCC might be ordered in a way that this particular transformation occurs too early or too late in the compilation process, missing the opportunity.

Cost-Benefit Analysis in Compilers

Compilers perform sophisticated cost-benefit analysis before applying optimizations. For your specific case:

// Original: 5 multiplications
a*a*a*a*a*a
= ((a*a)*a)*a*a*a
= temp1 = a*a
= temp2 = temp1*a
= temp3 = temp2*a
= temp4 = temp3*a
= temp5 = temp4*a

// Optimized: 3 multiplications  
(a*a*a)*(a*a*a)
= temp1 = a*a
= temp2 = temp1*a
= result = temp2*temp2

The compiler must consider:

• Performance gain: 40% reduction in multiplication operations (5→3)
• Register usage: Increased from 1-2 temporaries to 2-3 temporaries
• Code size: Slightly increased due to additional instructions
• Numerical precision: Potentially different rounding behavior

For modern CPUs with pipelined multiplication units, the performance difference might be less dramatic than expected, leading the compiler to deem the optimization not worthwhile.

Workarounds and Solutions

Since automatic optimization isn’t reliable, you can use several approaches:

1. Manual Optimization: As you’ve discovered, rewriting a*a*a*a*a*a as (a*a*a)*(a*a*a) is the most reliable method.

2. Helper Functions: Create inline functions for common power calculations:

   c

   static inline double cube(double x) { return x*x*x; }
   // Then use cube(a) * cube(a)
   

3. Compiler-Specific Attributes: Use compiler attributes to hint about optimization opportunities:

   c

   __attribute__((always_inline, hot)) 
   static inline double pow6(double a) {
       double cube_a = a*a*a;
       return cube_a * cube_a;
   }
   

4. Pragma Directives: Some compilers support optimization pragmas:

   c

   #pragma GCC optimize ("unroll-loops")
   

5. Template Metaprogramming: In C++, you can create compile-time power calculation templates that generate optimal sequences at compile time.

Related Optimization Patterns

This optimization is part of a broader category of multiplication sequence optimization. Similar opportunities exist:

1. Exponentiation by Squaring: For arbitrary powers, compilers could use this algorithm:

   pow(a, n) = {
       if n == 0: return 1
       if n == 1: return a
       if n is even: return pow(a*a, n/2)
       if n is odd: return a * pow(a*a, (n-1)/2)
   }
   

2. Horner’s Method: For polynomial evaluation, compilers can optimize sequences like a*a*a + b*a*a + c*a + d to more efficient forms.

3. Strength Reduction: Replacing multiplication with addition in loops, though this is more commonly applied to array indexing.

The fact that compilers don’t automatically apply these optimizations highlights the challenge of balancing between comprehensive optimization and compile-time performance. As hardware continues to evolve, compilers must constantly adapt their optimization strategies to target specific processor characteristics.

Sources

1. GCC Optimization Options
2. Compiler Optimization Techniques - Strength Reduction
3. Common Subexpression Elimination in Compilers
4. Floating-Point Optimization Considerations
5. Intel Compiler Optimization Guide

Conclusion

Compilers don’t optimize a*a*a*a*a*a to (a*a*a)*(a*a*a) primarily due to heuristic limitations, cost-benefit analysis, and the complexity of comprehensive expression tree optimization. While the performance improvement is significant (40% reduction in multiplications), compilers must balance this against increased register usage, potential numerical precision differences, and compile-time costs.

For scientific computing applications where performance is critical, manual optimization remains the most reliable approach. Creating helper functions or using template metaprogramming can help maintain code readability while ensuring optimal performance. As compiler technology evolves, we may see more sophisticated automatic optimization for multiplication sequences, but for now, explicit optimization is necessary for the best results.