# Multi-output pointwise fusion: evidence and limits ## Technical summary The compiler change admits same-domain, independent multi-output graphs to the existing fused element grid. It removes the serial program-per-worker boundary without kernel-name/shape rules, new DSL primitives, a score refit, or parameter search. This is an execution/resource realization extension; it does not improve the CPU or reduction emitters by itself. On September 6, 2026, Apple M1 Max/macOS 26.6.2, six frozen fresh-JIT rounds validate 192 complete value/derivative pairs (384 planes). All 48 new/old pairs improve in batched GPU and E2E time. The eight median new/Torch GPU ratios are 0.309–0.952; small GELU loses one GPU batch round and ragged GELU does not beat Torch's single-call E2E latency. This is fused Tile versus **preallocated eager Torch**, not compiled Torch, direct MPS or a full training step. See the [reader-facing route report](../../../../../docs/source/performance/tile/results.md#multi-output-pointwise-fusion-removes-a-mapping-boundary). ## Measurement definitions and population - Graphs: sigmoid and tanh-approximate GELU, each returning value and derivative in separate native parameters. Both are named `benchmark_activation_pair` in the capture API; the compiler uses IR structure, not this label. - Dimensions: 1×127, 37×1537, 1024×4096 and 4096×4096 for both graphs. The existing element block is fixed at 1×256; both plans request automatic execution and the same preserved shared-SSA policy. No tuning occurs. - Native: FP32 input/output, full FP64 formula check with `atol=2e-6, rtol=2e-5`. Value and derivative arrays are concatenated after timing. Torch: 2.14.0, output preallocation on both sides. Sigmoid uses three out calls; GELU uses forward and backward out calls, with a preallocated unit gradient. Input/compilation/allocation/upload/download are outside timing. - Timing: nine samples per phase, 30 ms calibrated batch window, 100 ms warmup. Mapper order, native/Torch order and starting case are rotated across six rounds. Every invocation group is recaptured/JIT-compiled. - Reported GPU comparison: no-counter completed command-buffer intervals, including GPU work/gaps, with a separate repetition denominator. This is not an isolated instruction/kernel timestamp. Compute-pass instrumentation is retained separately and never substituted into this comparison. - E2E throughput is synchronized host batch time per invocation; single-call E2E and GPU latency are separate phases. Do not subtract phase medians to invent CPU dispatch cost. Desktop activity is not controlled. - FP64 inputs are deterministic non-dyadic values over approximately [-1,1]. Python oracle tests additionally compare analytic derivatives against independent autograd at saturation and near zero. The native benchmark does not establish every input distribution, dtype or device. ## Evidence map and independent validation The [pre-change protocol](protocol.md) predates the compiler edit. [Old automatic](old-auto/results.json) and [old reference](old-reference/results.json) pilot sources match on the four small cases. A frozen copy of the old executable and Tile/bridge libraries also reproduces the four large reference sources in [large controls](control-large-old-old/results.json). Thus the current disabled-fusion control reproduces the old automatic Metal code on all eight cases, not merely an identical option name. Preliminary pilots do not have a full before/after loaded-library receipt; their times are not final rankings. [Current automatic](new-auto/results.json) and [current reference](new-reference/results.json) freeze the schedules used by the [six-round replay](replay/results.json). The accompanying [four-metric report](replay/results.md) includes paired medians, full ranges, and single-call results. All 23 recorded compiler/runtime/helper/executable artifacts are unchanged. Twelve distinct generated Metal sources are saved; each case/variant retains one source across all six JITs. Aligned large shapes differ in launch size, not shader source. [audit.py](audit.py) independently recomputes p50s from raw samples, verifies phase denominators, exact coverage/order, both-output counts, tolerances, source hashes and realized scalar/grid plans. Its [receipt](audit.json) also records eight adversarial checks: missing, duplicate, failed, partial output, changed artifact, wrong denominator, missing source and unbalanced framework order. Invalid variants must be rejected. Output arrays themselves are transient after complete runtime validation; the saved receipt is not a second independent rerun of those original numerical arrays. ## Scope controls and failure investigations [controls.py](controls.py) executes 32 additional native/Torch cases under frozen old and current bridge libraries, with 27 unchanged artifacts: - Add and single-output GELU(A+B), plus Softmax/RMSNorm/LayerNorm, at 37×1537 and 1024×4096: all ten old/new Metal sources are byte-identical. - CPU value/derivative graphs at the two small shapes: all eight native/Torch comparisons validate. Raw LLVM hashes differ because TVM writes object addresses into TBAA metadata labels, not because the CPU emitter changed. - Four large old-automatic sources match the current disabled-fusion control. The raw [controls receipt](controls.json) retains its failed byte-identity check for CPU. [audit_controls.py](audit_controls.py) permits only bijective renaming of those specific complete TBAA object-label definitions, preserving alias equivalence classes, every instruction and all other metadata. Its [separate receipt](controls-audit.json) verifies four normalized CPU pairs and fourteen raw Metal pairs, with mutations of instructions/alias classes rejected. These short pilots support source/correctness control, not new CPU or reduction speed claims. The [full-build correctness receipt](correctness/receipt.json) covers the execution, matrix, basic PoC, neural PoC and algorithm PoC suites on both CPU and Metal. All ten pass with unchanged test/compiler libraries. Coverage includes three interleaved outputs, nonzero minima, ragged and negative input origins, conditional and permuted output maps, explicit-scope/noalias controls, and RAW/WAR/WAW/different-domain rejection. Existing losses, attention, convolution/filter and sorting/Top-K PoCs are numerical controls only. The [initial source-assertion failure](constant-source-assertion.log) is preserved. A test required exactly one textual `exp` even when all inputs were zero. TIRx substituted constant `exp(0)` expressions for the Metal compiler to fold; input storage was entirely absent. The first hypothesis mistakenly attributed the failure to a missing `exp`. Printing the exact source showed repeated constant expressions instead. The final test checks input removal for that case and exactly one nonconstant `exp` for the other two shapes; all numerical checks are unchanged. 99 Python tests pass, including complete second-output rejection and FP64 analytic derivatives checked against autograd. clangd/clang-tidy report no errors in the three changed C++ translation units; style/performance hints remain, including by-value DSL signature parameters. ## Interpretation and next questions Legality generalizes across the tested expression graphs and beyond two outputs. Profitability is established only for this finite Metal cohort: constant/shared-producer graphs, more output/register pressure, irregular layouts and different devices need their own checks. Every output still requires one injective compact store and no global reads/escapes of written buffers. Unknown cases retain their original mapping. Neither `parallel` independence nor shape equality is used to erase inter-domain dependencies. The most useful next step is an IR-level fusion/partition and live-state candidate model, then reuse of those facts in CPU vector and reduction realizations. A future solver should compare emit-able distributions with traffic/reuse/spill costs on held-out graphs; it should not learn a table of the eight measurements here. Direct XIR/SIMD GEMM, large-MPS parity, native MPP epilogues and performance of attention/CNN/sort remain separate open work. Validation assessment: **share with caveats**, not general-performance completion. No MPS capture or new hardware-counter attribution is claimed. ## Reproduce and report QA Complete the selected CMake build and run `check.py --tag NEW_TAG` before native timing. `repeat.py` accepts `new-reference/results.json` and `new-auto/results.json`, `--operations sigmoid_pair,gelu_pair`, `--rounds 6 --samples 9 --sample-ms 30 --warmup-ms 100 --capture-sources`, the prebuilt Metal timing helper and five `--compiler-artifact` paths for TVM compiler/runtime/runtime_metal/runtime_extra/ffi. Its saved metadata and per-row commands are the exact argument/source/loader specification. Run `audit.py --check-local-artifacts` on the original machine, or omit that flag for a source-and-receipt-only audit on another machine. The reader-facing surface is the user's existing Sphinx documentation, not a second report app. The technical-report roles are integrated into the route section: answer, definitions, exact comparison, limitations and next work. Detailed methods/failures remain here. The visualization contract is an eight-row exact-lookup table: neutral colors, µs units, paired-ratio denominators and near-table caveats. A chart spanning the 2–2045 µs range would obscure the small-case lookup; no extra chart is added. Final rendered Sphinx QA checks the new table, lowering map and navigation at desktop and narrow widths. The fresh build renders 48 HTML pages; 3,665 local targets and 199 compatibility anchors pass. It has exactly ten existing missing-Doxygen- XML warnings in `api_reference.rst`, no new Tile warnings. Doxygen is not available locally, so the warnings-as-errors build exits 1: full API docs are **not** certified. No warnings are suppressed to present an all-green build.