Number types

Float32 and Float64, real or complex, take the fast path: SIMD codelets, threads, plans with cached kernels. Other floating-point types take a generic path. It is slower, single-threaded and has no SIMD, but it computes in the precision of the type.

using FFTL, LinearAlgebra

x = big.(randn(ComplexF64, 24))
X = fft(x)
eltype(X)

Complex{BigFloat}

maximum(abs, ifft(X) - x) < big"1e-60"

true

The generic path covers the complex transforms, the real transforms (rfft, irfft, brfft) and the real-to-real kinds, for any length and any number of dimensions. Tests check the round trip for BigFloat to a few units in the last place of the type.

input	computed as
Float32, Float64, ComplexF32, ComplexF64	fast path, the output has the same precision
BigFloat, Double64 (DoubleFloats.jl), Float128 (Quadmath.jl) and other AbstractFloats	generic path, the output has the same precision
Float16	converted to Float64
integers, Rationals	converted to Float64 (or ComplexF64)

A Vector{Int} is therefore transformed as ComplexF64. To compute it in BigFloat, convert the input first: fft(big.(v)).