FastPow

This package provides a macro @fastpow that can speed up the computation of integer powers in any Julia expression by transforming them into optimal sequences of multiplications, with a slight sacrifice in accuracy compared to Julia's built-in x^n function. It also optimizes powers of the form 1^p, (-1)^p, 2^p, and 10^p.

In particular, it uses optimal addition-chain exponentiation for (literal) integer powers up to 255, and for larger powers uses repeated squaring to first reduce the power to ≤ 255 and then use addition chains.

For example, @fastpow z^25 requires 6 multiplications, and for z = 0.73 it gives the correct answer to a relative error of ≈ 1.877e-15 (about 8 ulps), vs. the default z^25 which gives the correct answer to a relative error of ≈ 6.03e-16 (about 3 ulps) but is about 10× slower.

Note that you can apply the @fastpow macro to a whole block of Julia code at once. For example,

@fastpow function foo(x,y)
    z = sin(x)^3 + sqrt(y)
    return z^7 - 4x^5.3 + 3y^12
end

applies the @fastpow transformation to every literal integer exponent (^3, ^7, and ^12) in the function foo.

An alternative to @fastpow is to use Julia's built-in @fastmath macro, which enables various LLVM optimizations including, in some cases, faster integer powers using repeated multiplication. The advantages of @fastpow are that it guarantees optimal addition-chain exponentiation and that it works for exponentiating any Julia type (e.g. complex numbers, matrices, …), whereas LLVM will only optimize a small set of hardware numeric types.