DIP Core Rewrite

User: Anonymous
Created: 7/4/2026 18:44:53
Updated: 7/5/2026 16:29:02
Exported: 7/5/2026 16:34:37
Link: https://chatgpt.com/c/6a499acb-1cd4-83e8-b6ff-79bd2f8f8d98

## Prompt:
7/5/2026, 1:59:44 PM

does my DIT essentially pay all of its shuffle cost on ingest?

[Attachment: bruun_dit_kernel(2).hpp]

## Response:
7/5/2026, 1:59:48 PM

Inspecting file for specific functions

Inspecting top comments and functions

Searching for specific functions in code

Extracting specific lines from a file

Discussing shuffle cost in DIT processing

I need to clarify whether DIT pays all shuffle costs on ingest. In the forward direction, the DIT does frontload major global reorder into the seed. This involves a single lane crossing, but the shuffle cost isn't exactly the same. For the inverse, no global shuffle occurs upfront. It seems that the user’s query is about how shuffle costs are distributed.

Clarifying shuffle cost and layout in DIT

I’m reevaluating the DIT shuffle cost. The writer stores outputs and loops to create bins, but that’s not a full shuffle. In the forward DIT, "shuffle cost" refers to the tree-order input shuffle upfront. Although terminal patterns are standard win, projections still undergo math but without full permutations.

Thought for 25s

Yes. For forward DIT, essentially yes: the global disorder is paid at ingest.

Your DIT seed reads input in bit-reversed node order and writes the mirrored-lane work layout. That is the big shuffle: ordered time samples become the out-of-order Bruun tree layout. After that, the merge stages are layout-stable. They do vertical SIMD over already-paired mirror trees, with broadcast twiddles and no interior permutes. The disorder created by the DIT choice is absorbed into the work coordinate system before the tree starts doing serious arithmetic.

There is still a terminal event, but it is not the same kind of shuffle. The terminal merge has the single lane crossing and writes standard complex bins. In your normalized basis, projection is basically sign handling, so that terminal crossing is fused into the last algebraic step. It is endpoint placement, not a Stockham-style rearrangement and not a reconstructive global permutation pass.

So the forward DIT ledger is: ingest pays bit-reversal/mirror-pair intake, interior pays zero layout motion, terminal pays one fused lane crossing/output placement.

The inverse DIT is the mirror, not the same polarity. Standard frequency bins enter through the inverse terminal into mirrored-lane layout. Then inverse splits are vertical and stable. The final seed split/scatter pays the time-domain reorder on the store side. So inverse DIT pays the corresponding transport at outflow.

That is why forward DIT is such a good practical shape. It pays the nasty global disorder before the arithmetic, then never lets it leak into the interior. It turns “shuffle cost” into the price of admission.

## Prompt:
7/5/2026, 2:21:33 PM

At present, we have precisely three different kinds of the FFT- DIF, DIP, DIT.
All of them take a specific walk along the cube towards the frequency form.
They either gather in time or gather in frequency. But our new DIP does neither. 
It gathers in time AS it gathers in frequency. It is an agile form.
But it is presently our slowest FFT and sub-optimal.

However, from these things we learn multiple facts. We learn different kinds of FFT are possible.  And we learn SORT orders can be customized. 
One: you can pay a cost to transfer order to chaos, and it will be economically sound.  This is technically a scatter, but it is not recorded as such. We will call this Distribution.
Two- you can pay a cost to transfer chaos to order, and it is often expensive. This is technically a gather, but the algorithm is called a scatter sort. We will call this Collection.
Three: all FFT must pay a cost to get information from X to X[N] that is partially paid by sort behaviors. You can pay it up front, as Distribution or Collection, or you can amortize it.
Stockham amortizes it, but requires pingpongs. our current DIF uses collection on the forward, and distribution on the inverse, and both can technically be skipped, if one can work in so-called “bruun residual order”.
Our current DIT uses Distribution on both forward and inverse. While it is lacking in some SSE/AVX optimization, it is powerful, at many sizes trumps the others solely because of this.
Our current DIP, in its quirky way, choses to say- no. I will do a partial collection at one end, and partial distribution at the other. I will reduce my disorder slightly at both ends,
And try to pay two cheaper sorts than one expensive one.

This, of course, is a mistake. But it is a good and rewarding mistake, precisely because it tells us More Than One Way To Skin The FFT (is possible).

And it lends us operational and excellent nuances into what parts of the FFT are requisite and what can be concocted.
We can discern DIP from DIT from DIF phase walks and determine which is the most SIMD-agile, independent of the sorting needed.
And, we can with a choice of DIP/DIT/DIF walks choose a form that is best suited to simd on different platforms.
And, we can build multiple fft, that use different sort methods. We can build distribution-pure forms and they will be the fastest if you have to have the FFT.
And we can build ones that are collection-forward, distributed-inverse, with the matching behavior, and they will be the fastest if you just need to run pre-compiled filter, element wise.

We also suspect, but do not yet know for sure, if yet a third form is possible. The DIP almost achieves it, but does not.
The DIP packets are themselves interwoven and thus must be reconstructed. Our next work will be to characterize the three walks and different kinds of amortized sort.
We hypothesize another form of amortized sort based FFT. In this FFT, a partial Distribution is paid on the intake of the forward, and on the outlet of the inverse.
At the frequency-side, a partial Collection is needed on both sides- to arrive at FFT natural bin order and to return from it to the order inside our butterfly system.
Like the DIF residual form, we hope to be able to ignore it if one can work in internal bin order. But, we hope to be able, also, by partially paying the sort up front,
To have a segregated clockwise sort at the frequency end: such that if one only wants, for example, the first half or up to , say, the closest packet partition of the bins to N//2, only needs to do a partial sort.
Because the packets inside N//2 contain already all the bins inside N//2. Thus, we hope, operationally, for a form that, instead of paying 1.0 the cost of a Distribution form, can, 
If only some of the bins must be worked on, pay 0.75 or 0.80 of the same cost, thus achieving a practical engineered form.

at this moment, leaving the problem of the sort itself out, for the DIF/DIP/DIT walk itself(where we will assume the most optimal Distribution sort is used in all cases) what does their walk look like- and is it fundementally shaped the same from a simd perspective?

## Prompt:
7/5/2026, 2:25:16 PM

the point is that all three of these can, in some sense, be rewritten into a form that uses the kind of sort requested. if they were all rewritten into the form that uses the DIT's current sort method.. if that requires more days of brutal research i suppose it is needed. because i want to see if their behavior differs once they are under the same system

## Response:
7/5/2026, 2:25:20 PM

Clarifying conceptual approach for user’s request

The user seems to be asking for a conceptual approach, not an up-to-date web search. They mentioned kernels and some uploaded files, possibly for a specific benchmark. I can focus on a common methodology like harness comparisons but emphasize potential variability across walks. I’ll mention relevant terms, like "distribution-normalized forms," and note some research challenges without overstating it.

Clarifying approach for DIT and order-to-chaos sorting

I’m thinking that the user might want to continue using the current DIT method, where both forward and inverse distributions are applied. For forward processes, this would mean applying the distribution upfront for internal layout. The output could be produced without reconstructive collection, maintaining consistency with the method.

Thought for 10s

Yes. That is the clean experiment. Right now DIF, DIT, and DIP are not merely three walks; they are three walk plus sort equilibria. If you want to know whether the walks themselves have different SIMD economics, you need to put them under a common sort regime and watch what survives.

The common regime should be the DIT-style Distribution regime: external order is pushed into the internal coordinate system before the serious arithmetic begins. That means the transform starts by placing data into the walk’s preferred chaotic basis, not by letting the walk create chaos and then collecting it later. Then the core is judged as a core, not as a hidden sort machine.

So the useful objects are not current DIF, current DIT, current DIP. They are DIF-D, DIT-D, DIP-D: same algebraic transform, same endpoint policy, different decimation coordinate.

DIT-D already exists. It pays Distribution, then runs its merge walk.

DIF-D would be a non-native rewrite of DIF where the input is distributed into the residual coordinate system that the DIF would otherwise naturally produce at the end. Then the DIF walk is asked to operate inside that coordinate system without relying on final Collection. This may feel backwards because DIF’s current beauty is that it earns its residual order for free. But for the experiment, you deliberately give it DIT’s economic model and see whether the split walk still has an advantage.

DIP-D would be the harder one. Current DIP is not Distribution-pure. It uses phase packets and pays partial Collection and partial Distribution. To make DIP-D, the ingest sort has to place samples directly into the phase packet coordinate system that the diagonal walk wants, so the packet geometry is born before the arithmetic rather than negotiated at both ends. Then the forward should finish in a consistent internal frequency coordinate, not a half-cleared packet stream unless that same coordinate is declared the output basis.

That is probably days of brutal work, because the sort is not just a table. It changes which values are neighbors at each split, which pairs can share twiddles, and whether a span is self-contained. A fake common sort will not answer the question. It has to be a structure-preserving Distribution for each walk.

The comparison you want is something like this: fix the endpoint economy to Distribution, fix the output basis to either standard order or a declared internal order, fix SIMD width and terminal kernels, then compare only the induced walk. The measured question becomes: after the same kind of order-to-chaos payment, does DIF still prefer stage-sweep splitting, does DIT still prefer mirrored merging, and does DIP still prefer phase-packet diagonal motion?

My expectation is that their behavior will still differ. DIT-D will likely remain strongest for standard FFT output because its Distribution creates exactly the mirror coherence its merge wants. DIF-D may lose some of what made DIF special, because DIF’s current advantage is tightly coupled to not paying residual reconstruction. DIP-D is the unknown one: if a pure Distribution can create phase packets up front without destroying locality, DIP may become much more revealing. If it still underperforms, the phase walk itself is less SIMD-economic than its local cell suggests. If it improves sharply, current DIP’s weakness was not the walk; it was the hybrid sort regime.

So yes, the right next theorem is not “which FFT is fastest.” It is “given a sort policy, what walk does it induce, and given a walk, what sort policies preserve its sparse Bruun structure?” Once all three are rebuilt under the DIT-style Distribution policy, you will finally be comparing the walks instead of comparing three historical compromises.

## Response:
7/5/2026, 2:29:17 PM

Mirrored merging probably wins.

Isolated from sort-equilibrium economics, DIT-style mirrored merging has the best pure SIMD shape because it has the most reusable structure: broad stage sweeps, paired lanes doing the same operation, stable address arithmetic, twiddles that can be hoisted or broadcast predictably, and terminal special cases that can be straight-lined. It gives the compiler and the handwritten backend the longest, fattest, most regular loops.

Stage-sweep splitting is close, and on some machines it can tie or win. Its advantage is that it is also broad and regular, and it often exposes large contiguous streams early. Its disadvantage is that splitting tends to create more live fragments as the walk descends. The stage remains regular, but the local working set becomes more partitioned. If sort economics are removed, it loses some of the special advantage it had from naturally ending in residual order.

Phase-packet diagonal motion has the cleanest local cell but likely the worst isolated global SIMD efficiency. Its cell is beautiful: same span, contiguous streams, broadcast phase, no ping-pong, no interior permute. But “best cell” is not the same as “best kernel.” It naturally produces smaller packet loops, more recursion or packet scheduling, more address-generation pressure, less stage-wide homogeneity, and more trouble amortizing overhead across huge regular sweeps. To make it win, you need the memory hierarchy benefit of self-contained packets or a boundary-sort advantage. Once those economics are removed, the diagonal walk’s local elegance is probably not enough.

So under the artificial condition “same sort system, judge only the walk,” my ordering would be: mirrored merging first, stage-sweep splitting second or tied on favorable hardware, phase-packet diagonal motion third.

The reason is simple: SIMD wants not just vectorizable arithmetic, but repeatable vectorizable arithmetic. DIT gives the backend repeated mirror-pair work. DIF gives repeated split work. DIP gives excellent little diagonal cells, but less repetition at the scale where instruction scheduling, unrolling, prefetching, and table locality dominate.

So my answer is: DIT’s mirrored merging is probably the most efficient isolated walk. DIF is the nearest rival. DIP is the most conceptually agile and the most cache-theoretically interesting, but not the most SIMD-efficient once you deny it the sort/locality regime that makes phase packets meaningful.

## Prompt:
7/5/2026, 4:16:52 PM

ok,  generate one of each kind of internal op- stage-sweep splitting, mirrored merging, and phase-packet diagonal motion. lets see if we can build an internal demo that pushes these to the limit to see how many of them the cpu can get done under ideal circumstances. we will run each indepenedently for a certain number of iters and count performance

## Response:
7/5/2026, 4:16:55 PM

I’ll make this as a standalone microbenchmark, not tied to full FFT correctness. It will isolate three hot-loop shapes with the same arithmetic flavor, same data volume scale, and a volatile sink so the compiler cannot delete the work.

I found the actual DIP cell shape in the uploaded header: four contiguous streams, one broadcast phase, same-address stores. I’ll encode that directly, while making DIF/DIT synthetic but structurally faithful enough to measure loop-shape pressure.

Inspecting uploaded files for C++ benchmark generation

Designing a C++ benchmarking file

I need to generate a C++ file with three operations. Ideally, it should include data arrays suitable for benchmarking, with parameters like N, iterations, and mode. I’ll implement metrics for cell counts and flops. I might also include optional AVX2/NEON intrinsics for performance, offering scalar and intrinsic versions for testing. I'll ensure that we benchmark using header-only templates.

Designing mirrored operation for data streams

For the mapping of streams, I’m considering four streams (L/H) and their mirrored counterparts (Lm/Hm). To make the comparison fair, I’ll ensure the same number of operations are performed, using mirrored merging. I'll use indices that mirror each other, processing four contiguous streams for the diagonal phase to compare operations effectively. This design might mimic the DIT butterfly structure while accounting for efficiency and mirrored data reuse.

Extracting DIP-related functions from header file

Extracting DIP phase-packet data

Considering using DIP and DIT code for cells

I’m thinking I might use the code from the DIP phase for the cell forward IP operation. Additionally, I could check if the DIT header for cells would be helpful in this context. I’ll need to investigate and find a good fit from both approaches.

Searching for cell_fwd_ip and related terms in code

Creating synthetic kernels with optimizations

I’m thinking about creating synthetic kernels, maybe not needing a full version but sufficient for benchmarking. I'd use generated C++ with both scalar and auto-vectorized code. I could incorporate ARM NEON or AVX2 for performance but keep it cross-platform. Templates could support vector sizes, defaulting to 32/16 bytes, with compile-time lane optimizations.

Representing mirrored pairs in DIT and recursive processing

I'm thinking about how the current DIT uses mirrored pairs for each block, so I’ll load the data and apply the same twiddle for each pair. For testing isolated motion, I can loop over packets and apply cell_fwd_ip at various scales. I could recursively apply cell operations over the span and children, pushing "motion" rather than sorting. The recursive approach might involve using dip_cell on parent streams, then descending to the child spans.

Exploring "walk" functions for recursive application

I'm thinking of using three "walk" functions that recursively or iteratively apply the same four-stream rotate-add cell across the entire buffer. These would include a split sweep (DIF), merge sweep (DIT

Implementing cross-platform aligned allocation for SIMD

I need to set up a microbenchmark with C++ that uses aligned allocation, cross-platform. I can either go with std::aligned_alloc from C++17 or use platform-specific options like posix_memalign for Unix and _aligned_malloc for

Using compiler vector extensions and vector types

I’m considering using #pragma clang loop vectorize(enable) and GCC vectorize for vectorization, but should probably keep scalar C++ for native operations. I might also look at std::experimental::simd, although it’s not guaranteed to work. I’ll try using portable scalar loops with #pragma GCC ivdep and maybe include intrinsics for AVX2 and

Optimizing the DIF stage-sweep function and mirrored merging

I’m thinking about the DIF stage-sweep function and breaking it down by running stages with different spans. Using spans like n/2 down to 1 will run through n/2 * logN cells, allowing a full walk. I can mirror the merging process, though I need to consider whether the formula needs tweaks to distinguish between stages.

Optimizing packet span processing and phase calculations

For managing packet spans, using n/2 as w simplifies the approach, making each level visit n/2 * log n cells. At each recursive step, we break the span into smaller sizes (w/2). I want to minimize recursion overhead, so precomputing phase tables for depth with iterative stack might be the ideal solution.

Optimizing iterative stack structure and buffer utilization

I’m thinking of an iterative stack with a small array to handle tasks and avoid recursion overhead. Each event uses a structure to store indices, offsets, and depth. Both stage-sweep and DIP can use a unified buffer to handle real

Scaling butterfly sums and managing overflows

The butterfly sum's values double with each stage, so I need to scale them by 0.5 in the formula to stabilize, or possibly renormalize. For each cell, I plan

Optimizing benchmarking and performance calculation

Let's set up robust code for benchmarking with run_bench<T>(K,iters) to test three kernels. We'll focus on warm-up and best of 5, while adjusting iterations based on cell number. Aiming for approximately 20 million cells for micro-testing. We'll track performance using flops per cell (12), and calculate time per walk, Mcells/s, and GF/s for each kernel.

Implementing mirrored merging with vectors

Let's try a cleaner implementation for mirrored merging using vectors: For each block, load values from both ends (low and high), and reverse the order as needed to simulate mirrored merging. By loading and processing pairs differently, we can simplify the process while maintaining the correct behavior and making it clear enough for demonstration.

Clarifying phase packet walk with initialization

For the phase_packet_walk, I am using a single buffer for both re and im data. I'll initialize it properly and apply the dip_cell(v+off, w2, c, s) call where w spans 2*total width. The goal is to stop once w == 1 and make sure everything scales correctly.

Clarifying buffer management for benchmarking

I can't remove the buffer writes because they modify the loop count, but if the result isn't needed until after the benchmark, I can work around it using a volatile double global_sink. For each benchmark, I need to reset the data before timing and ensure repeated passes avoid underflow by resetting with memcpy. I'll handle memory allocations carefully, using precomputed arrays for stage values.

Thought for 2m 57s

Generated the standalone microbenchmark: fft_walk_microbench.cpp

It measures three isolated internal walk shapes with boundary sort omitted: stage-sweep splitting, mirrored merging, and phase-packet diagonal motion. The DIP kernel uses the actual four-contiguous-stream diagonal cell shape from your current header: [ea | oa | eb | ob], broadcast phase, same-address stores, depth-first packet stack. The DIT/DIF kernels are synthetic structural probes, not full BFFT kernels.

Compile:

clang++ -O3 -std=c++17 -march=native fft_walk_microbench.cpp -o fft_walk_microbench

Run:

./fft_walk_microbench
./fft_walk_microbench 65536
./fft_walk_microbench 65536 200
./fft_walk_microbench 65536 200 double
./fft_walk_microbench 65536 200 float

For width experiments:

clang++ -O3 -std=c++17 -march=native -DVBYTES=16 fft_walk_microbench.cpp -o walk16
clang++ -O3 -std=c++17 -march=native -DVBYTES=32 fft_walk_microbench.cpp -o walk32
clang++ -O3 -std=c++17 -march=native -DVBYTES=64 fft_walk_microbench.cpp -o walk64

The output reports ns/walk, Mcells/s, approximate GB/s, approximate GF/s, and a sink. The comparison to care about is not absolute GF/s, but which shape saturates best for double versus float on each target CPU.

## Prompt:
7/5/2026, 4:20:43 PM

here were the other two kernels if you needed anything from them to make this slightly more accurate

[Attachment: bruun_dif_kernel(2).hpp]

[Attachment: bruun_dit_kernel(3).hpp]

## Response:
7/5/2026, 4:20:48 PM

I’ll use these to adjust the microbench toward the real kernel shapes: DIF split terminal/stream pattern, DIT mirror merge pattern, and DIP diagonal span pattern. I’ll keep it standalone so you can compile it anywhere.

The DIT header changes the benchmark meaningfully. Its hot merge is not a simple complex half-merge; it is a radix-4 mirrored-lane merge with pair_reduce twice and symmetric i/m stores. The DIF header also wants the fused two-level norm2 shape, not a scalar-ish half split.

Inspecting kernel files and microbench

Inspecting key functions in DIF/DIT files

Inspecting key DIT/DIF functions

Inspecting truncated microbench output

Compiling C++ microbenchmark with optimizations

Improving microbench design accuracy

I need to improve my design by using more accurate structures, like the actual DIF norm2_fused shape and DIT merge4 structure, rather than previous simpler versions. I'll build a generic vector type, focusing on pair_reduce operations and "stage-sweep splitting" with actual block shapes. The schedule will use decreasing q values, iterating operations over blocks of size 4q with 8 streams, optimizing data handling. It's time to test it with some real values.

Continuing the DIT design progress

I’m moving forward with the design for a generic vector type, with VL lanes, and defining SLOT for 2*VL scalars. The DIT block has 4 children, each of size M * SLOT. I’ll use a 16M double block length for the version with V2 (4 children * M * 4), and V4 uses a size of 8. I’m refining memory access patterns, cross-loading and interleaving for more efficient butterfly operations.

Continuing slot-based design with merge stage

I’m further refining the slots, which will be 4M, using a slot index system for proper storage, e.g., 4M-i. For the DIT approach, I’ve defined a method for loading and storing data at specified slots. The merge stage will ascend with M = 2, 8, 32, adjusting dynamically. I'm thinking of using total_slots as a power of 4 for better flow.

Refining scalar and block design

I’m refining the task design, where the total count of elements (w) is split as needed, influenced by children offsets. Scalars count as 2*K, and I’ll define the total scalars as user input (N). For DIP root, w = count / 2, then recursively reduce. For DIF norm2, I’ll break down the blocks using q. DIT follows similar patterns but with slot-based adjustments.

Extracting relevant DIT/DIF formulas and definitions

Extracting section from bruun_dip_kernel file

Extracting code block from bruun_dip_kernel

Updating microbench with real cells

I need to fix the current microbench, which incorrectly uses half-scale cells. I’ll write a new file with the correct cells, using generic vector operations while avoiding FMA intrinsics for portability. The formulas for pair_reduce, DIF norm2, DIP cell, and DIT merge will be integrated. I'll compute statistics on lane cells processed and keep track of cell visits over stages, ensuring the correct scaling through each stage.

Optimizing performance and benchmark reporting

For DIP cell4, each node processes w4 lanes with a depth-first approach. For each radix-4 level, the total processed count is roughly divided by 8 per level. For DIT merge, cells process about 2M slot cells per block. Instead of precise counts, I'll report ns/iteration, effective lanes, and ops/s approximations. A benchmark run might be necessary and could involve using Python or another container to write the updated file.

Benchmarking BFFT FFT walk-shape performance

Fixing header inclusion in C++ file

Considering compile and test environment

I should probably note that the compilation should be done with AVX512 in the environment. I also want to test on a width of 32 to make sure everything works correctly and there are no errors. Double-checking this will ensure I don't miss any potential issues.

Compiling and testing FFT microbench with AVX512

Thought for 2m 36s

Yes, these helped. I made a more faithful standalone version here: fft_walk_microbench_realshape.cpp

This version replaces the earlier synthetic shapes with closer internal forms from the real kernels. DIF now uses a norm2_fused-shaped two-level split over [A0 | B0 | A1 | B1]. DIT now uses a merge4_v_norm-shaped mirrored-lane slot layout, with [a-vector | b-vector] slots and symmetric i/m stores. DIP now uses the actual cell4_fwd_ip-shaped two-level phase packet cell over [a0 a1 a2 a3 | b0 b1 b2 b3]. I also removed the bogus 0.5 scale from the DIP forward cell; the uploaded DIP forward cell does not scale there.

Compile with: clang++ -O3 -std=c++17 -march=native fft_walk_microbench_realshape.cpp -o walkbench

Run with: ./walkbench 1048576 200 both

For width tests: clang++ -O3 -std=c++17 -march=native -DVBYTES=32 fft_walk_microbench_realshape.cpp -o walkbench32

I test-compiled it here with native and VBYTES=32. It runs and reports ns/walk, rough useful units/s, rough GB/s, relative speed, and sink. The GB/s column is deliberately rough because DIT’s unit accounting is not perfectly comparable to DIF/DIP; the reliable column is ns/walk and relative speed under each scalar type and vector width.

## Prompt:
7/5/2026, 4:26:03 PM

with version 1.0, on m4 mac

double  K=65536 complex-lane equivalents, storage=1048576 bytes, vector_bytes=16, lanes=2, iters=152
kernel                            ns/walk     Mcells/s         GB/s         GF/s         sink
stage-sweep splitting           354911.18      1477.24        94.54        17.73   3.7771e-13
mirrored merging                402953.40      1301.11        83.27        15.61  -5.7430e-12
phase-packet diagonal           409759.87      1279.50        81.89        15.35  -1.1875e-13

float  K=65536 complex-lane equivalents, storage=524288 bytes, vector_bytes=16, lanes=4, iters=152
kernel                            ns/walk     Mcells/s         GB/s         GF/s         sink
stage-sweep splitting           208780.15      2511.20        80.36        30.13   3.0470e-13
mirrored merging                287370.34      1824.43        58.38        21.89  -5.7431e-12
phase-packet diagonal           314056.20      1669.41        53.42        20.03  -1.3705e-13

on xeon:
kernel                            ns/walk     Mcells/s         GB/s         GF/s         sink
stage-sweep splitting           839020.62       624.88        39.99         7.50   3.7771e-13
mirrored merging                979161.32       535.45        34.27         6.43  -5.7427e-12
phase-packet diagonal          1646781.47       318.37        20.38         3.82  -1.1875e-13

float  K=65536 complex-lane equivalents, storage=524288 bytes, vector_bytes=32, lanes=8, iters=152
kernel                            ns/walk     Mcells/s         GB/s         GF/s         sink
stage-sweep splitting           576652.97       909.19        29.09        10.91   3.0470e-13
mirrored merging                695809.84       753.49        24.11         9.04  -5.7431e-12
phase-packet diagonal           899303.20       582.99        18.66         7.00  -1.3705e-13

with the new version, on m4:
double  scalars=1048576  bytes=8388608  VBYTES=16  lanes=2  slot=4  iters=95
walk                              ns/walk       Munits/s     GB/s rough    rel to best         sink
DIF stage-sweep split          1666399.56         707.90          90.61          1.000 -4.8481e+257
DIT mirrored merge             1848919.29        2268.52         290.37          1.110  4.9817e+215
DIP phase-packet diag          1780188.16         662.65          84.82          1.068 -4.8481e+257

float  scalars=1048576  bytes=4194304  VBYTES=16  lanes=4  slot=8  iters=95
walk                              ns/walk       Munits/s     GB/s rough    rel to best         sink
DIF stage-sweep split           743846.94        1409.67          90.22          1.000          nan
DIT mirrored merge              929339.03        4513.21         288.85          1.249          nan
DIP phase-packet diag           744108.77        1409.17          90.19          1.000          nan

on xeon:
double  scalars=1048576  bytes=8388608  VBYTES=32  lanes=4  slot=8  iters=95
walk                              ns/walk       Munits/s     GB/s rough    rel to best         sink
DIF stage-sweep split          3863381.61         271.41          34.74          1.024 -6.4869e+227
DIT mirrored merge             4000154.97        1048.54         134.21          1.061  2.1777e+215
DIP phase-packet diag          3771286.49         278.04          35.59          1.000 -6.4869e+227

float  scalars=1048576  bytes=4194304  VBYTES=32  lanes=8  slot=16  iters=95
walk                              ns/walk       Munits/s     GB/s rough    rel to best         sink
DIF stage-sweep split          1738193.01         603.26          38.61          1.187         -nan
DIT mirrored merge             1463767.77        2507.24         160.46          1.000         -nan
DIP phase-packet diag          1954276.47         536.55          34.34          1.335         -nan

this is certainly interesting. the results DONT have a conclusive winner.

## Response:
7/5/2026, 4:26:07 PM

Analyzing user results and clarifying versions

The user shared some results, and I’ll need to analyze them. It’s important to highlight that issues with NAN/overflow could indicate mathematical instability. The timing’s still valid, but it’s crucial to manage math transformations, like renormalizing. Also, when discussing versions (like version 1 vs. V2), I’ll keep things clear. I’ll avoid excessive markdown; use paragraphs instead.

Discussing results and performance optimization

The M4 performance showed that DIF/DIP are tied, but DIT is 25% slower. Xeon performs slightly better with double DIP, but all results are within 6%. The scalar type and kernel shape seem to drive performance. I need to mention NaN/overflow could be caused by nonlinear recurrences and propose using periodic rescaling or orthogonal rotation. For the next test, I’ll focus on non-exploding constants, performance counters, and unrolled terminals. I want the final results to be clear and concise.

Thought for 10s

Exactly. The interesting result is that the isolated walk does not have a single dominant shape. Once you remove the sort-equilibrium economics and push the internal ops toward their real forms, the answer becomes platform plus scalar-type dependent.

The clean read is this: on M4 double, DIF split wins, but DIP diagonal is only 6.8 percent behind and beats DIT. On M4 float, DIF and DIP are effectively tied. On Xeon double, DIP wins, but only narrowly; DIF and DIT are within about 6 percent. On Xeon float, DIT mirrored merge wins hard, DIF is 18.7 percent slower, DIP is 33.5 percent slower.

So the old intuition “mirrored merging is probably best isolated” is false as a general statement. The more accurate statement is: the three internal walks are all plausible peak kernels, and the winning one changes with vector width, scalar width, and backend. That is a big deal. It means DIF/DIT/DIP are not just different sort strategies. They are different pressure profiles.

DIF split seems to like M4, especially float and double. DIP diagonal is surprisingly strong when shaped accurately: tied with DIF on M4 float, best on Xeon double. DIT mirrored merge can be bad on M4 float but excellent on Xeon float. That matches your larger observation that float and double are not the same problem. Wider lanes do not simply scale the double kernel; they change the dominant bottleneck.

The version 1.0 result made stage-sweep look generally superior, but the real-shape version destroyed that conclusion. That means the synthetic abstraction was too crude. Once the DIT/DIF/DIP cell shapes become closer to reality, no single “best walk” remains.

There is one caveat: the sinks exploding to huge values, inf/nan means the recurrence is numerically unstable over repeated microbench application. Timing is probably still useful, but for rigor I would make the next benchmark periodically reset the buffer or use bounded orthogonal-ish constants so the CPU stays on normal finite values. NaN propagation usually does not wreck throughput on modern scalar/SIMD paths, but it is an unnecessary confound.

The conclusion I would take forward is: sorting policy decides economic suitability, but internal walk decides hardware suitability. These are separable enough to study, and the hardware suitability is not ordered DIF > DIT > DIP or DIT > DIF > DIP. It is a phase diagram. On one cell of the phase diagram, DIP wins. On another, DIT wins. On another, DIF wins. That is exactly the evidence you needed that all three walks are real computational species, not just cosmetic reorderings.

## Prompt:
7/5/2026, 4:28:47 PM

the interesting thing here is, the DIT universally had an edge in the SIMD form, but here, the core operations are actually less efficient. it makes up for it by having an opening sort. a DIF with opening sort form could be even more efficient, but may not be possible, because the DIF does not permit this by nature.

## Response:
7/5/2026, 4:28:49 PM

Yes. That is the paradox the microbench exposed: DIT is winning in full transforms not because mirrored merge is universally the best intrinsic walk, but because its sort coupling is economically superior. It pays Distribution at the door, then its core only has to be good enough. The opening sort turns the rest of the transform into a coherent problem.

DIF’s core may be intrinsically better or equal in several regimes, especially on M4 and Xeon double. But current DIF does not get to use DIT’s trick naturally. DIF’s natural virtue is that it creates Bruun residual order as it splits. If you ask it to pay an opening Distribution instead, you may destroy the condition that makes the split graph sparse and local. In other words, DIT can absorb chaos before merging because merging is constructive: it assembles structure from pre-distributed leaves. DIF is destructive: it starts from assembled structure and splits it into residues. The chaos it wants is the result, not the input.

So a “DIF with opening sort” is not impossible in the abstract, but it is not obviously still DIF. To make it work, the opening Distribution would need to place time samples into a coordinate system from which the DIF split cells still see their legal parent spans and legal twiddle relationships. If that exists, it is a new sort-walk coupling: DIF-D. If it does not, then DIF’s core advantage is locked behind its native endpoint economy.

That would explain the current ecosystem. DIT wins standard-order full FFT because Distribution is cheap and admissible. DIF wins native/residual workflows because it can skip Collection entirely. DIP explores the middle because its phase packets allow partial Distribution and partial Collection, but its present packet sequence is not the segregated frequency order you want for partial natural bands.

The theorem hiding here is: not every walk accepts every sort. A sort is admissible only if it preserves the parent-child locality required by that walk’s sparse factorization. DIT accepts opening Distribution naturally. DIF may reject it unless the distribution is actually a conjugated version of the DIF graph. DIP accepts packet-local distribution/collection, but not yet a natural-band-segregated one.

So the next research question is very sharp: characterize admissible Distribution orders for DIF. If there is one that keeps DIF’s stage-sweep split locality and lands in useful standard or semi-standard frequency order, it could beat DIT. If there is not, then DIT’s opening-sort advantage is structural, not accidental.



Powered by ChatGPT Exporter (https://www.chatgptexporter.com)