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A VL of 1 implies that the computation is essentially scalar, and only one complex element can fit in a vector register. An example of such a scenario is when using interleaved double-precision floating-point arithmetic on an SSE2 machine: one 128-bit XMM register is used to store two 64-bit floats that represent the real and imaginary parts of a complex number.
When $VL=1$ , the process of generating a program for a hard-coded FFT is as follows:
[link] is a function, written in C++, that performs the first task in the process.
As mentioned earlier, elaborating a topological ordering of nodes with a depth-first recursive structure is much likeactually computing an FFT with a depth-first recursive program (cf.
Listing 3 in Appendix 2 ).
[link] lists the nodes contained in the list `
ns
' after elaborating a size-8
transform by invoking
elaborate(8, 0, 0, 0)
.
CSplitRadix::elaborate(int N, int ioffset, int offset, int stride) {
if(N > 4) {
elaborate(N/2, ioffset, offset, stride+1); if(N/4 >= 4) {
elaborate(N/4, ioffset+(1<<stride), offset+(N/2), stride+2);
elaborate(N/4, ioffset-(1<<stride), offset+(3*N/4), stride+2);
}else{ CNodeLoad *n = new CNodeLoad(this, 4, ioffset, stride, 0);
ns.push_back(assign_leaf_registers(n)); }
for(int k=0;k<N/4;k++) {
CNodeBfly *n = new CNodeBfly(this, 4, k, stride); ns.push_back(assign_body_registers(n,k,N);
} }else if(N==4) {
CNodeLoad *n = new CNodeLoad(this, 4, ioffset, stride, 1); ns.push_back(assign_leaf_registers(n));
}else if(N==2) { CNodeLoad *n = new CNodeLoad(this, 2, ioffset, stride, 1);
ns.push_back(assign_leaf_registers(n)); }
}
Elaborate function for hard-coded conjugate-pair FFT
A transform is divided into sub-transforms with recursive calls at lines 4, 6 and 7, until the base cases of size 2 or size 4 are reached at the leaves of the elaboration. As well as the size-2 and size-4 base cases, which are handled at lines 20-21 and 17-18 (respectively), there is a special case where two size-2 base cases are handled in parallel at lines 9-10. This special case of handling two size-2 base cases as a larger size-4 node ensures that larger transforms are composed of nodes that are homogeneous in size – this is of little utility when emitting $VL=1$ code, but it is exploited in "Other vector lengths" where the topological ordering of nodes is vectorized. The second row of [link] is just such a special case, since two size-2 leaf nodes are being computed, and thus the size is listed as 2(x2).
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