Consider these two functions for computing the 4-th power:
power4(x) = x^4
power31(x) = x * x^3
The second is much speedier than the first because (looking at code_llvm or code_native) the first calls a general helper function and the second just does two multiplies. Indeed, the second is intelligently compiled and essentially does this:
y = x*x
return y*y
Running Julia with -O3 optimization doesn’t help.
It appears that fixed exponents from -1 to 3 are handled well. Consider this:
julia> code_llvm(x -> x^3 * x^3 * x^3, (Int,))
; @ REPL[9]:1 within `#15'
define i64 @"julia_#15_176"(i64 signext %0) {
top:
; ┌ @ operators.jl:560 within `*' @ int.jl:88
%1 = mul i64 %0, %0
%2 = mul i64 %1, %1
%3 = mul i64 %2, %0
%4 = mul i64 %3, %2
; └
ret i64 %4
}
That’s terrific: Computing x^9 is reduced to four multiplies.
It may be worthwhile for exponentiation by small, fixed exponents (larger than 3) be compiled directly into multiplications (including for other types such as matrices and polynomials).