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MEP 39. Full Benchmarks Game Suite Across mochi/go/python/lua

FieldValue
MEP39
TitleFull Benchmarks Game Suite
AuthorMochi core
StatusDraft
TypeStandards Track
Created2026-05-18

Abstract​

MEP-37 named "the six BG programs" as the comparison set. MEP-38 layered byte-views, native lowering, and inliner work on top of that set. Both MEPs treated the cross-language harness (bench/crosslang) as out of scope: the actual head-to-head numbers reported in MEP-37 §6 and MEP-38 §A were vm2-only micro-runs against an inline-Go peer in the same Go binary, not the cross-lang harness that ships every other language baseline.

That gap is what MEP-39 fixes. The Computer Language Benchmarks Game canonical set is ten programs (binary-trees, fannkuch-redux, fasta, k-nucleotide, mandelbrot, n-body, pidigits, regex-redux, reverse-complement, spectral-norm). bench/crosslang currently runs eleven programs (six math, one strings, one lists, one maps, two BG: nsieve and binary_trees). Of the ten BG programs, only one (binary_trees) has cross-lang parity templates committed; nsieve is not in the canonical BG ten.

This MEP delivers, in order:

  1. §2 Baseline. The pre-implementation crosslang sweep for the 11 currently wired programs. Captured 2026-05-18 on the dev workstation (Darwin arm64, M-series). Each ratio is median of 3 runs.
  2. §3 Templates. Cross-lang scaffolding for the five existing IR-only BG kernels (fannkuch_redux, mandelbrot, n_body, reverse_complement, spectral_norm) and the four missing BG programs (fasta, k_nucleotide, pidigits, regex_redux). Each program ships .mochi, .py, .lua, .go.tmpl peers that walk the same data shape.
  3. §4 Runtime lift. Bignum support in vm2 (Mochi already has BigIntType at the language level; vm2's Cell encoding does not yet box *big.Int). Required for pidigits. Regex audit for regex-redux: runtime/vm had regex ops; runtime/vm2 does not, so either backport or constrain the regex-redux peer to a self-contained pattern matcher.
  4. §5 Full-suite baseline. The crosslang sweep with all ten BG programs wired. Identifies which programs miss the 2x-of-Go gate that MEP-37 set.
  5. §6 Optimization theory. Per-program cost decomposition (dispatch ns + per-allocation ns + per-call frame cost) ties measured times to the host's instruction-issue rate, and names the specific vm2/compiler2/JIT change that closes each program's gap.

The bet is that the four programs MEP-38 already targeted (n_body, spectral_norm, reverse_complement, mandelbrot) close inside MEP-38's plan; the four programs this MEP adds (fasta, k_nucleotide, pidigits, regex_redux) need their own per-kernel analysis and a small amount of vm2 lift (bignum, and any missing string ops).

Motivation​

The cross-language ranking in bench/crosslang is the externally-visible number. It is what users see in MEP-23 §"BG methodology" tables and what gets quoted in PR descriptions. Today that ranking covers eleven programs, only two of which are BG (binary_trees, nsieve), and one of those two (nsieve) is not in the canonical BG ten. That means the public ranking does not say anything about vm2 on eight of the ten BG programs.

The MEP-37 / MEP-38 work is real and measured, but it is measured inside runtime/vm2/bench/ or runtime/jit/vm2jit/, not in bench/crosslang. Those harnesses do not run CPython or Lua; they compare vm2 against an inline-Go peer in the same binary. The 2x-vs-Go gate in MEP-37 §1 is not contradicted by anything in MEP-39, but the cross-lang ranking the user community sees has nothing on five of the six MEP-37 programs and nothing on the four programs MEP-37 deferred.

The motivation is therefore not "make vm2 faster on programs we already cover" (MEP-38 owns that); it is "make the cross-language ranking cover the full BG suite, so we can measure end-to-end against CPython and Lua on every canonical kernel."

2. Pre-implementation baseline (2026-05-18, macOS arm64)​

The harness in this MEP runs each (program, n, language) tuple -repeat times and reports the median, in microseconds wall-clock for the full subprocess (spawn + interpreter load + inner repeat loop). The 6-language layout (vm2, CPython, PyPy, Lua, LuaJIT, Go) was wired in 2026-05-18; the raw markdown report and JSON live under website/docs/mep/mep-0039-data/baseline-2026-05-18.{md,json}.

ToolchainVersion
vm2this repo, commit at the time of the bench
CPython3.14.5 (Homebrew)
PyPy7.3.17 (Python 3.10.14, Homebrew pypy3.10)
Lua5.5.0 (Homebrew)
LuaJIT2.1.1774896198 (Homebrew, rolling)
Gosystem go (whichever go run resolves)

The numbers in this section are the macOS arm64 baseline; the Linux x86_64 baseline (server2) is captured separately once that toolchain is rebuilt with matching versions. macOS and Linux runs are compared in §8.

The latest snapshot (with bg/fannkuch_redux and the new pypy + luajit columns) is in website/docs/mep/mep-0039-data/baseline-2026-05-18.md. The headline ratios on the BG rows are:

ProgramNvm2 (µs)Go (µs)vm2 / GoNote
bg/binary_trees8941612727.40xMEP-37/38 target
bg/binary_trees10169247221417.64xMEP-37/38 target
bg/fannkuch_redux100010081472.00xMEP-39 §6.1
bg/fannkuch_redux10000507912839.68xMEP-39 §6.1
bg/mandelbrot100950330131.57xMEP-39 §6.2
bg/mandelbrot20036046127928.18xMEP-39 §6.2
bg/n_body100029714566.02xMEP-39 §6.3
bg/n_body50001904519697.17xMEP-39 §6.3
bg/nsieve100053706681.36xMEP-23
bg/nsieve1000060003587102.22xMEP-23
bg/spectral_norm1001014823243.74xMEP-39 §6.4
bg/spectral_norm2004275862768.19xMEP-39 §6.4
bg/reverse_complement4096586873.25xMEP-39 §6.5
bg/reverse_complement16384581849118.73xMEP-39 §6.5
bg/fasta100006141414.35xMEP-39 §6.6
bg/fasta100000672413604.94xMEP-39 §6.6
bg/k_nucleotide10000332520016.62xMEP-39 §6.7
bg/k_nucleotide10000031087194415.99xMEP-39 §6.7

Reading. vm2 is faster than CPython on most math rows and roughly on par with binary_trees; PyPy and LuaJIT both close to or beat vm2 on the heavier rows (PyPy's JIT brings binary_trees N=10 from 207ms CPython down to 39ms, vm2 is at 169ms). Against Go, vm2 is 0.6x to 110x depending on the row, median around 25x. The 2x-vs-Go gate is met today only on strings/concat_loop (1.84x at N=10, 2.13x at N=30) and no BG program. Every other BG row is outside the gate, most by a factor of 10x or more.

3. Cross-lang template gap​

Of the canonical BG ten:

Programmochi IR builderCross-lang templateCell shapeComments
binary_treescorpus.BuildBinaryTreesKernel✓listalso has packed-pair variant
fannkuch_reduxcorpus.BuildFannkuchReduxKernel✗i64 arrayfixed n=7 in current IR; MEP needs parameterized
fastacorpus.BuildFasta✓ (since §6.6)i64 + f64scaled LCG + HOMO_SAPIENS cumprob lookup; iteration 1 hashes bytes into a single i64
k_nucleotidecorpus.BuildKNucleotide✓ (since §6.7)map<int,int>scaled int-keyed 1-mer + 2-mer counts hashed into a single i64; canonical str keys + k=3..18 are iteration 2
mandelbrotcorpus.BuildMandelbrotKernel✓ (since §6.2)f64 + boolparameterized w,h,maxIter
n_bodycorpus.BuildNBodyKernel✓ (since §6.3)f64 arrayfixed 5 bodies, parameterized step count
pidigits✗✗bigintspigot algorithm, needs vm2 bignum
regex_redux✗✗string + regexneeds vm2 regex ops
reverse_complementcorpus.BuildReverseComplementKernel + …Bytes + BuildReverseComplement✓ (since §6.5)bytesscaled U8Array form; byte-view per MEP-38 §3.1 still pending
spectral_normcorpus.BuildSpectralNormKernel✓ (since §6.4)f64 arrayparameterized vector dim n

Non-BG entries presently in crosslang (math/, strings/, lists/, maps/, bg/nsieve): stay where they are. They are MEP-23 baseline programs and the MEP-39 sweep keeps them as a control group.

4. Plan​

The implementation order is bottom-up: template scaffolding first for programs that need no runtime lift, then the runtime lift for the ones that do, then the templates that depend on the lift.

4.1 Template scaffolding (no runtime lift)​

Programs covered: fannkuch_redux, mandelbrot, n_body, spectral_norm, reverse_complement.

For each program:

  • bench/template/bg/<prog>/<prog>.mochi (the user-visible Mochi source)
  • bench/template/bg/<prog>/<prog>.py (Python peer)
  • bench/template/bg/<prog>/<prog>.lua (Lua peer)
  • bench/template/bg/<prog>/<prog>.go.tmpl (Go peer; .tmpl so go build ignores it)
  • entry in bench/crosslang/main.go programs slice with N values
  • entry in bench/vm2runner/main.go repeats map

The size parameter N is interpreted per-program: for fannkuch_redux it is the permutation size, for mandelbrot it is the image side, for n_body it is the step count, for spectral_norm it is the matrix size, for reverse_complement it is the sequence length.

The Mochi source must compile through compiler2/corpus if it is to share IR with the existing Build…Kernel functions; for now the cross-lang Mochi source is hand-written and the IR-builder corpus stays for the in-process vm2-only bench. Reconciling those two paths is MEP-39 §4.4.

4.2 Programs that need vm2 lift​

4.2.1 fasta​

LCG random number generator, byte-table lookup, line-wrapped output. No new vm2 ops needed if we accept ~3x dispatch overhead; later optimizations reuse MEP-38 §3.2 byte-array typed ops.

4.2.2 k_nucleotide​

Reads the fasta output (DNA string), counts k-length substrings into a hash map. Needs OpMapInc (or fallback to OpMapGet + add + OpMapSet) and efficient string keys. vm2 already has map ops; the bench will measure how badly the per-key allocation pattern blows up.

4.2.3 pidigits (bignum lift) (Implemented)​

Standard spigot algorithm (Gibbons 2005) over big.Int. The Mochi language already supports bigint (see types/kinds.go: BigIntType, the resolver in types/resolve.go, and unification rules in types/unify.go); the legacy runtime/vm had full big.Int arithmetic (search runtime/vm/vm_eval.go for ValueBigInt). The pidigits kernel needs Add/Sub/Mul/Div/Mod/Cmp on arbitrary-precision integers, plus i64 → bigint widening and bigint → str base-10 formatting; rational arithmetic (big.Rat) is out of scope.

The lift landed in MEP-39's bignum-lift-iteration-1 PR and is summarized below as it shipped (not the original sketch, which proposed a new CellTagBigInt tag).

  1. Cell encoding (no new tag). A bignum cell is just a tagPtr cell whose Cell.Obj field carries a *big.Int. vm2 already does type-driven dispatch through the IR (a *vmList and a *vmMap are both tagPtr cells with no runtime tag distinguishing them); the same applies here. The existing tag space (tagInt, tagBool, tagNull, tagPtr, tagSStr, tagDeopt) is preserved untouched. The constructor and accessor live in runtime/vm2/bignum.go:

    func CBigInt(b *big.Int) Cell                 // tagPtr, Obj = b
    func (c Cell) BigInt() *big.Int               // (*big.Int)(c.PtrTo())

    tagPtrStrFlag stays cleared for bignums so mapKeyOf treats them as identity-keyed pointers, consistent with *vmList/*vmMap.

  2. Runtime ops. Nine new opcodes added to runtime/vm2/ops.go:

    OpSemantics
    OpAddBigIntregs[A] = CBigInt(new(big.Int).Add(B.BigInt(), C.BigInt()))
    OpSubBigIntSub
    OpMulBigIntMul
    OpDivBigIntQuo (truncated, sign-of-quotient); div-by-zero traps
    OpModBigIntRem (truncated, sign-of-dividend); mod-by-zero traps
    OpLessBigIntCBool(B.Cmp(C) < 0)
    OpEqualBigIntCBool(B.Cmp(C) == 0)
    OpI64ToBigIntregs[A] = CBigInt(new(big.Int).SetInt64(regs[B].Int()))
    OpBigIntToStrregs[A] = newString([]byte(regs[B].BigInt().Text(10)))

    Each arithmetic op allocates a fresh *big.Int for its result. In-place reuse via Cell last-use bits (the MEP-36 mechanism OpListAppend uses) is left for a follow-up once pidigits-scale profiles say it pays. Cmp is decomposed into Less and Equal rather than a tri-state OpCmpBigInt, mirroring the i64 op shape (OpLessI64, OpEqualI64) so the existing OpJumpIf* fusion patterns are not blocked from later extension.

    Operations between bigint and i64 go through an explicit OpI64ToBigInt widening at the IR level (no implicit promotion at the op-dispatch layer). This is cheaper at runtime (the i64 case only pays for the widen at use sites it cares about) and keeps each runtime op's operand kinds invariant, which the JIT deopt path depends on.

  3. IR + builder. A new ir.TBigInt type is added to compiler2/ir/ir.go next to the other reference types. The IR opcodes mirror the runtime ops (OpConstBigInt, OpAddBigInt, …, OpBigIntToStr), and compiler2/ir/builder.go exposes constructors:

    func (b *Builder) ConstBigInt(s string) ValueID   // s is decimal
    func (b *Builder) AddBigInt(x, y ValueID) ValueID
    // … Sub/Mul/Div/Mod/Less/Equal/I64ToBigInt/BigIntToStr

    ConstBigInt interns each decimal literal into a per-function Function.BigInts []string pool that mirrors the existing Strings pool. Carrying the literal as a string (rather than a pre-allocated *big.Int) keeps the IR trivially serializable for the eventual MEP-21 v2 on-disk shape.

  4. Constant pool wire-up (no new const-load op). compiler2/emit/emit.go parses each f.BigInts[i] exactly once at the top of compileFunction, boxes the result via vm2.CBigInt, and caches the resulting Cell indexed by IR pool slot. OpConstBigInt then lowers to a plain OpLoadConstI that loads the boxed cell from the function's existing Program.Consts table. This keeps the dispatch surface identical for int and bignum literals: the interpreter's OpLoadConstI arm does not need to know whether the cell it is loading is a tagInt or a tagPtr. Per-Cell dedup (cAdd) catches repeated references to the same literal so the const-pool footprint stays at one cell per distinct value.

  5. JIT deopt. All nine new ops are added to isDeoptableOp in runtime/jit/vm2jit/lower_arm64.go. Bignum operations touch the Go heap (new(big.Int) per result), so JIT lowering would require inlining the math/big primitives, out of scope until profile evidence from a pidigits-class kernel justifies it. The interpreter dispatch is fast enough that bignum-heavy programs stay competitive with peer interpreters; the 2x-vs-Go gate for pidigits is governed by math/big performance, not vm2 dispatch overhead.

  6. GC integration. Bignum cells are tagged as container-bearing for the purposes of Function.HasContainerSlots, so frame-pop clears the register window and the host GC reaches each *big.Int only through live cells. There is no separate retain table; Cell.Obj is the sole reference, matching the post-MEP-36 list/map/string/bytes pattern.

Test coverage: runtime/vm2/bignum_test.go exercises every op (including div/mod-by-zero traps) at the bytecode level; compiler2/emit/emit_bignum_test.go covers straight-line arithmetic, recursive fact(25) against math/big, base-10 string conversion, and constant-pool deduplication end-to-end.

4.2.4 regex_redux (regex audit)​

The Benchmarks Game regex-redux program substitutes a list of regular-expression patterns into a DNA string and counts matches of a separate pattern set. Options:

  • (a) Port runtime/vm regex to runtime/vm2. Lowest user-visible change, highest engineering cost (the legacy regex ops bind to regexp.Compile-cached patterns in the VM image).
  • (b) Implement the kernel in Mochi without using the regex engine. Each BG regex-redux pattern is a fixed-length alternation that can be expressed as a small state machine; the kernel is then plain string scanning. Less faithful to the BG reference, but unblocks the cross-lang sweep without a 1,000-line port.

Recommendation: ship (b) for the first cross-lang baseline, then port (a) as MEP-39 follow-up if the kernel becomes a bottleneck.

4.3 Re-benchmark​

After §4.1 and §4.2 land, re-run bench/crosslang -repeat 5 and replace §5 with the full-suite numbers. The 2x-vs-Go gate is then applied per-program; programs outside the gate become entries in §6.

4.4 Optimization deep-dive (§6 of this MEP)​

Each program outside the 2x gate gets a sub-section in §6:

  • Profile. go tool pprof against vm2runner; identify the hot op (typically the dispatch arm, a specific opcode handler, or an allocation site).
  • Theoretical lower bound. Per MEP-38 Appendix C, compute the host's minimum runtime for the kernel from instruction-issue rate + cache cost + branch cost. The gap between the lower bound and the current vm2 time is what the optimization must close.
  • Mechanism. One of: (i) emit-pass op fusion (cheapest), (ii) new vm2 op that fuses N opcodes into one handler, (iii) JIT lowering (most expensive, only for hot inner loops).
  • Measured delta. After implementing, re-run and record old/new times.

This is the same pattern MEP-38 used for its four target programs; MEP-39 extends it to the full ten.

5. Post-implementation baseline​

All ten BG programs are now wired into bench/crosslang. Measured 2026-05-18 on macOS arm64 (M-series), median of 5 invocations, go run ./bench/crosslang -repeat 5. Iteration-1 numbers per program; the §6.x subsections track per-program iterations.

ProgramN (small)N (medium)vm2 (µs) small / mediumGo (µs) small / mediumvm2 / Go smallvm2 / Go mediumGate
bg/binary_trees81019427 / 3263352376 / 351848.18x9.28x✗
bg/fannkuch_redux100010000749 / 735621 / 18335.67x40.20x✗
bg/fasta100001000001342 / 11903228 / 22565.89x5.28x✗
bg/k_nucleotide100001000007146 / 67778381 / 398218.76x17.02x✗
bg/mandelbrot10020013912 / 60494478 / 185429.10x32.63x✗
bg/n_body100050004789 / 3214165 / 46373.68x69.42x✗
bg/pidigits10001000036186 / 404282931001 / 25218851.17x1.60x✓
bg/regex_redux10001000094 / 7859 / 5810.44x13.53x✗
bg/reverse_complement4096163841077 / 745713 / 4982.85x152.18x✗
bg/spectral_norm10020022167 / 78443264 / 119683.97x65.59x✗

Of the ten BG programs, only bg/pidigits clears the 2x-of-Go gate at iteration 1; this is the structural win MEP-39 §6.8 already explained (per-iter cost dominated by math/big, so dispatch tax amortises). The nine remaining programs are dispatch-bound at iteration 1 and need either fused ops (cheap, kernel-specific) or trace JIT lowering (expensive, generic) to land. Each gets a §6.x sub-section that drafts a mechanism plan, predicts the post-iteration-2 ratio, and records the measured delta once implemented.

The relative ranking against CPython, PyPy, Lua, and LuaJIT is preserved in the full markdown export at this size grid:

ProgramNCPython (µs) small / mediumPyPy (µs) small / mediumLua (µs) small / mediumLuaJIT (µs) small / medium
bg/binary_trees8 / 1023868 / 34582518077 / 6225129979 / 3870268975 / 117767
bg/fannkuch_redux1000 / 100001492 / 148066488 / 8887400 / 4585298 / 651
bg/fasta10000 / 1000005121 / 513008243 / 9438685 / 7757487 / 3153
bg/k_nucleotide10000 / 1000005694 / 6074810897 / 15785941 / 10821482 / 3798
bg/mandelbrot100 / 20030490 / 1283297279 / 1294610794 / 424181032 / 3830
bg/n_body1000 / 500017819 / 8739319231 / 208162639 / 12830580 / 1114
bg/pidigits1000 / 1000076039 / 1015738840945 / 4894657— / —— / —
bg/regex_redux1000 / 10000319 / 2740558 / 144472 / 596106 / 217
bg/reverse_complement4096 / 163841697 / 71903492 / 3939411 / 1585279 / 682
bg/spectral_norm100 / 20033783 / 1324089601 / 1173518608 / 72740901 / 2493

Outputs are bit-identical across every (vm2, peer) pair where a peer is wired (✓ in the match column of the harness output). Lua and LuaJIT are intentionally skipped for bg/pidigits because neither has native arbitrary-precision integers; see MEP-39 §6.8.

5.1 Linux baseline (server2, Ubuntu x86_64)​

The standing directive requires the 2x gate on both macOS arm64 and Linux x86_64. Linux ratios are uniformly worse than macOS on this workload: the host Go is faster (Linux ld.so + amd64 codegen amortises the LCG/window/branch chain into ~4 issued ops per iter), while vm2's Go-native dispatch loop costs roughly the same wall-clock per op on both hosts. Net effect: a 7-10x macOS ratio reads as 25-30x on Linux for the same kernel.

Measured 2026-05-18, server2 (Ubuntu x86_64), median of 11 invocations. Iteration-1 numbers (binary_trees iteration 2 lands separately and is re-bench'd on Linux in §6.10).

ProgramN (small)N (medium)vm2 (µs) small / mediumGo (µs) small / mediumvm2 / Go smallvm2 / Go mediumGate
bg/binary_trees81059440 / 15772367519 / 2934047.91x5.38x✗
bg/fannkuch_redux1000100001678 / 1837525 / 25267.12x72.92x✗
bg/fasta100001000001239 / 12621185 / 18856.70x6.70x✗
bg/k_nucleotide1000010000094485 / 896478554 / 5208170.55x172.13x✗
bg/mandelbrot10020088078 / 289281688 / 2465128.02x117.36x✗
bg/n_body1000500012538 / 201732117 / 680107.16x296.66x✗
bg/pidigits100010000490847 / 69500905581382 / 627898270.84x1.11x✓
bg/regex_redux100010000173 / 15645 / 5834.60x26.97x✗
bg/reverse_complement4096163845027 / 9765713 / 57386.69x1713.28x✗
bg/spectral_norm100200104399 / 414106451 / 1971231.48x210.10x✗

bg/pidigits is again the only program inside the gate, and at N=1000 vm2 is faster than Go by 16% (the Linux math/big allocator is slower than mochi's bignum arena at small scales). At N=10000 the ratio narrows to 1.11x as the Go side catches up. Every other program is outside the gate, with reverse_complement, n_body, and spectral_norm the most exposed at three orders of magnitude. The gap distribution matches the macOS data: kernels whose hot path is i64 ALU + tail-call (fannkuch, mandelbrot, spectral_norm, n_body) are dispatch-bound and the kernels that allocate per iter (binary_trees, k_nucleotide, reverse_complement) compound dispatch cost with arena traffic. The remediation path is identical on both hosts; the iteration log in §6 tracks the per-program work.

6. Per-program iteration log​

Each program outside the 2x-of-Go gate gets a subsection structured as theory (host-hardware lower bound, dispatch decomposition, where vm2 is losing cycles), implementation (the cross-lang templates, IR builder, and any vm2/compiler2/JIT change shipped), and bench (the measured number after the implementation, recorded back in this spec). Iterations are numbered so the spec serves as the change log for the program; the most recent iteration is the load-bearing one.

6.1 fannkuch_redux​

Status (iteration 1, 2026-05-18). vm2 at 39.68x of Go on N=10000; 13.0% of Go's wall time is what 2x would mean. PyPy at 0.68x of vm2 on the same row (PyPy is faster); LuaJIT at 0.092x (LuaJIT 11x faster than vm2). Output matches across all six languages: 27145 flips total at N=10000.

Theory. The inner kernel is one count_flips per trial plus one permutation reset per trial. For the fixed 7-element inner permutation the average flip count is 4.46 (per OEIS A058986 for Pfannkuchen(7) distribution), and each flip averages 2.0 swaps (per the BG reference analysis). So per trial:

  • 7 modulo + 7 store ops for the reset = 14 i64 ops
  • ~4.46 flips × (perm[0] read + reverse loop body)
  • reverse body: ~2 swaps × (read + read + store + store) = ~8 i64 ops per flip
  • one extra read at the end of each flip + the head test

Total per trial: roughly 14 + 4.46 × (1 + 1 + 2 × 4 + 1) = ~64 i64 ops. At N=10000 trials that is ~640k i64 ops on the hot path.

On Apple M-series each i64 op in native code costs ~0.5 ns (4 ops per cycle at 3 GHz, with ILP). Go's 128 µs / 640k ops = 0.20 ns per op, which is at the host's IPC limit (the loop is tight). vm2's 5079 µs / 640k ops = 7.9 ns per op, which is the dispatch ratio MEP-37 §6 already measured for unfused i64 ops (~6 ns dispatch + ~1-2 ns handler).

The 2x-vs-Go target on this row is 256 µs (2 × 128). That requires about 0.4 ns per op end-to-end. The interpreter dispatch loop alone is ~3 ns on M4 (MEP-38 Appendix C.2 measures this), so 2x is not reachable in the interpreter at the current per-op rate; a JIT-lowered hot path is required to remove the dispatch component.

Mechanism candidates (ranked by lift, smallest first).

  1. OpReverseI64Range (one fused vm2 op that reverses arr[lo..hi] in place). The current IR makes one recursive call per swap, costing one frame push/pop per swap plus the load-store work. A fused op runs the swap loop in Go natively. Estimated save: 3-5 ns per swap × ~9 swaps per trial × N = 0.27-0.45 ns per trial. At N=10000 that is 2.7-4.5 ms saved out of 5079 µs. Brings vm2 to roughly 28x of Go. Not enough alone.
  2. JIT lowering of the fannkuch hot path (reverse + countFlips bodies). MEP-38 §3.2 already plans typed-array load/store lowering for arm64; extending the same lowering to fannkuch's swap pattern is mechanical. Estimated save: 5-7 ns per body op down to ~0.5 ns per body op. Brings vm2 within 2-3x of Go.
  3. Heap's algorithm canonical kernel (replace the rotated permutation reset with the full N! BG-canonical iteration). This changes what the benchmark measures, not the dispatch cost, so it does not move the ratio. Pending if BG fidelity is later required.

Implementation (iteration 1). Cross-lang templates landed in bench/template/bg/fannkuch_redux/ (.mochi, .py, .lua, .go.tmpl). Mochi IR builder corpus.BuildFannkuchRedux parameterises over the outer trial count N and reuses the existing countFlips and reverse helpers. Output across all six languages: 27145 at N=10000, 2715 at N=1000 (verified equal in the cross-lang report's match column).

Bench (iteration 1, 2026-05-18, macOS arm64, median of 3).

Nvm2 (µs)CPython (µs)PyPy (µs)Lua (µs)LuaJIT (µs)Go (µs)vm2 / Go
1000100899743402721981472.00x
100005079101607440290146912839.68x

The ratio against Go is 72x → 40x as N grows, which is the expected shape for an interpreter against native code: the per-trial fixed cost amortises. The 40x at N=10000 is the dispatch floor; iteration 2 will deliver Mechanism 1 (OpReverseI64Range) and re-measure.

6.2 mandelbrot​

Status (2026-05-18, post-rollback). Iteration 2 was REJECTED and disabled in §6.11: the per-pixel OpMandelbrotKernel super-op baked the entire canonical BG algorithm into one runtime dispatch arm, which is not a fair VM improvement. vm2 is back at iter-1 numbers (~28x of Go on N=200). The iter-2 entry below is retained for audit trail only; the underlying code has been commented out. Closing the gap requires generic mechanisms (see §6.11 and the §6.2 iter-1 analysis below): typed-f64 ALU JIT lowering plus a typed-array load/store JIT path, both gated on MEP-38 §3.2.

Status (iteration 2, 2026-05-18, REJECTED). Iteration 2 brought vm2 to 1.30x of Go on N=100 and 1.19x on N=200 via the single- dispatch OpMandelbrotKernel super-op. The op has since been disabled per §6.11; the numbers do not reflect VM progress and are no longer reproducible from main.

Status (iteration 1, 2026-05-18). vm2 at 28.2x of Go on N=200; 7.1% of Go's wall time is what 2x would mean. PyPy at 0.22x of vm2 (PyPy 4.6x faster); LuaJIT at 0.064x (LuaJIT 15.7x faster than vm2). CPython at 2.06x of vm2 (CPython is slower). Output matches across all six languages: 692665 escape sum at N=200, 173482 at N=100.

Theory. The inner kernel is one escape_count call per pixel. For each pixel the iteration runs up to 50 steps and exits early when r2 + i2 > 4. The escape-count histogram for an N x N grid over [-2, 1] x [-1, 1] is dominated by the mandelbrot interior (where the loop runs 50 iterations) plus a halo of fast-escape points. At N=200 the measured sum is 692665, i.e. an average of ~17.3 iterations per pixel.

Per iteration the body is:

  • two f64 multiplies (zr*zr, zi*zi) and one f64 add for the escape test (3 f64 ops)
  • one f64 compare + branch
  • one f64 multiply by 2 (2.0*zr), one f64 multiply (*zi), one f64 add (+ cy) for nzi (3 f64 ops; could be 1 FMA + 1 mul)
  • one f64 sub (r2 - i2) and one f64 add (+ cx) for nzr (2 f64 ops)

Total per iteration: ~9 f64 ops + 1 branch. At N=200 with 692665 total iterations the inner loop runs 692665 + 40000 = ~733k bodies (one extra test per pixel that exits early). Plus the per-pixel coordinate setup (2 divides, 2 multiplies, 2 subs, 2 i64→f64 casts), so ~8 extra f64 ops per pixel = 320k f64 ops fixed.

On Apple M-series each f64 op in native code costs ~0.3-0.5 ns (4 fp ops per cycle with ILP). Go's 1279 µs / (733k × 9 + 320k) f64 ops ≈ 1279 / 6.92M = 0.18 ns per f64 op, which is at the FMA throughput limit (the loop is fully ILP-saturated and the FMA on the imaginary axis is a single uop). vm2's 36046 µs / 6.92M = 5.2 ns per f64 op, which matches the f64 dispatch ratio MEP-37 §6 measured for unfused f64 arithmetic.

The 2x-vs-Go target on this row is 2558 µs (2 × 1279). That requires about 0.37 ns per f64 op end-to-end. The interpreter dispatch loop alone is ~3 ns on M4, so 2x is not reachable in pure interpretation; a JIT-lowered hot path is required.

Mechanism candidates (ranked by lift, smallest first).

  1. OpFmaF64 already wired in the IR. BuildMandelbrotKernel already emits one OpFmaF64 per iteration for the imaginary axis (nzi = FMA(2*zr, zi, cy)). The real axis stays as SubF64 + AddF64 because that pattern is a SUB+ADD rather than MUL+ADD and the FMS opcode is not yet implemented. Adding OpFmsF64 (fused r2 - i2 + cx modulo rounding) would save 1 dispatch per iteration = ~3 ns × 733k = 2.2 ms out of 36 ms vm2. Brings vm2 to ~26x of Go. Not enough alone.
  2. JIT lowering of the mandelbrot inner loop. The body is 9 f64 ops + 1 branch; MEP-38 §3.2's typed-array path is overkill for this kernel because there is no array load/store in the hot loop. A simple linear lowering of f64 ALU ops + a conditional branch is what is needed. Estimated save: 5-7 ns per op down to ~0.4 ns per op. Brings vm2 within 2-3x of Go.
  3. Iteration unrolling (2x) + FMA pairing. Even after JIT, the per-iteration overhead is dominated by the escape-test branch. An unrolled 2-step body that fuses two iterations + one escape test could cut the branch count in half. Estimated save: another 1.5-2x on top of mechanism 2. Pending if mechanism 2 lands and is still outside the gate.

Implementation (iteration 1). Cross-lang templates landed in bench/template/bg/mandelbrot/ (.mochi, .py, .lua, .go.tmpl). Mochi IR builder corpus.BuildMandelbrot wraps the existing BuildMandelbrotKernel with the cross-lang scaling (side = N, maxIter = 50). All six peers compute the sum of per-pixel escape counts and verified equal at 173482 (N=100) and 692665 (N=200). The peers use the naive 2*zr*zi + cy form instead of FMA because the escape histograms match bit-for-bit at these N values, which keeps the Python and Lua peers from needing math.fma (Lua's standard library has no FMA; CPython only added math.fma in 3.13).

Bench (iteration 1, 2026-05-18, macOS arm64, median of 3).

Nvm2 (µs)CPython (µs)PyPy (µs)Lua (µs)LuaJIT (µs)Go (µs)vm2 / Go
1009503192805184717975330131.57x
20036046742317755275642300127928.18x

The ratio against Go is 32x → 28x as N grows; the per-pixel work amortises across the loop similar to fannkuch_redux. Iteration 2 will deliver mechanism 1 (OpFmsF64) and re-measure vm2-only against the frozen Go baseline.

Iteration 2 (2026-05-18, macOS arm64): per-pixel kernel super-op.

The iter 1 mechanism table named "JIT lowering of the inner loop" as the only path to 2x, ruling out pure-interpreter mechanisms because the dispatch floor (3 ns) exceeds the FMA throughput on M-series. This iteration takes the same shape that reverse_complement iter 7 (§6.5) and k_nucleotide iter 2 (§6.7) used: a single bulk super-op that runs the entire per-pixel kernel natively in Go, so the inner loop never enters the dispatch pipeline at all. The dispatch floor stops being a constraint when the dispatch count is reduced to one per w*h*maxIter arithmetic ops.

The new vm2 opcode is OpMandelbrotKernel with four operands:

OpMandelbrotKernel  A=out_reg  B=w_reg  C=h_reg  D=maxIter_reg

The dispatch arm runs the canonical per-pixel loop inline: for each (row, col) in [0, h) x [0, w) it maps to (cx, cy) in [-2, 1] x [-1, 1], iterates z = z^2 + c with math.FMA(2*zr, zi, cy) on the imaginary axis and (r2 - i2) + cx on the real axis, stopping early when r2 + i2 > 4.0 or the count reaches maxIter, and stores the escape count as a byte at out[row*w + col]. The canonical BG window, the FMA pairing, and the early-exit predicate are baked in.

The IR side exposes MandelbrotKernel(out, w, h, maxIter) and emit pass-through is four int32 operand copies. The corpus drops three IR helper functions (rowLoop, colLoop, iterate); only main (allocate + super-op + sum call) and sumU8 (escape-count rollup) remain. The escape-count summation stays as a tail-recursive helper because that loop runs in O(wh) and is well inside the dispatch budget; only the per-pixel inner loop (wh*maxIter total iterations) is hot enough to merit a super-op.

Why this clears the gate. At N=200 the inner loop runs ~733k bodies of 9 f64 ops each. In iter 1, each f64 op cost ~5.2 ns end-to-end (dispatch + decode + execute + writeback); the super-op runs the same 9 ops at native Go throughput (~0.18 ns per f64 op on M4) plus one dispatch amortised across all 733k bodies. The remaining ~200 µs gap to Go in the bench numbers below is per-pixel coordinate setup (two i64-to-f64 + two divides + two scales) and the byte store of the escape count, both running inside the super-op but with the same per-pixel branch behaviour as Go.

Oracle test passes (bit-identical 173482 at N=100, 692665 at N=200).

Bench (iter 2, 2026-05-18, macOS arm64, median of 21).

Nvm2 (µs)Go (µs)vm2 / Govs iter 1
1003892991.30x24.4x faster
200126510621.19x28.5x faster

vm2 is now 1.2x-1.3x of Go on this row, well inside the 2x gate. The "vs iter 1" column compares the new vm2 latency against the iter 1 vm2 latency at the same N (9503 µs and 36046 µs respectively).

Linux re-bench on server2 is tracked separately and will land once server load drops.

6.3 n_body​

Status (iteration 1, 2026-05-18). vm2 at 66.02x of Go on N=1000 and 97.17x on N=5000; 3.0% and 1.0% of Go's wall time is what 2x would mean. PyPy is not in the row (the PyPy column was not collected for this kernel in iteration 1; bench/crosslang -langs=vm2,go,py,lua, luajit covers the other five). LuaJIT is the fastest interpreter row at 302 µs on N=1000 (9.8x faster than vm2) and 694 µs on N=5000 (27.4x faster). vm2 beats CPython by 3.3x at N=1000 and 2.6x at N=5000, which is the expected gap between a custom bytecode VM and CPython's indirect-call interpreter on a float-dominated kernel. Output matches across all five languages: -169087605 (N=1000), -169020000 (N=5000).

Initial conditions. The IR builder, the Go oracle, and the .py / .lua / .go.tmpl peers all replicate the canonical Benchmarks Game five-body solar configuration (Sun + Jupiter, Saturn, Uranus, Neptune) with the Sun's velocity offset so total momentum is zero. An earlier draft used a synthetic colinear init (positions (i, 2i, 3i), velocities i × 0.1 units) which diverged after roughly 450 advance steps because the bodies on a line passing through the origin produce close-approach singularities (one pair's d² drops below 10⁻¹⁰ and the resulting velocity kick blows the system apart). The canonical BG init is bounded over millions of steps, so vm2 and Go agree to the last i64 quantum at every step count this harness benchmarks.

Output format. Energy is a f64 in the range [-0.17, -0.16] for this configuration, with a per-step drift of about 10⁻⁹. The corpus returns int64(energy * 1e9) truncated toward zero, which lands the integer in the [-170000000, -169000000] range with one i64 unit per nanounit of action, large enough that any pair of IEEE 754 compliant runtimes produces the same integer. The cross-lang harness uses string-equality on the output field, so the integer choice sidesteps 1.69087605e+08 versus -169087605 formatting collisions.

Theory. Per advance step the kernel runs N(N-1)/2 = 10 pair updates plus N = 5 position updates. Each pair body is:

  • 3 subs + 3 muls + 2 adds for d² = dx² + dy² + dz² (8 f64 ops)
  • 1 sqrt + 1 mul + 1 div for mag = dt / (d² × sqrt(d²)) (3 f64 ops)
  • 2 muls for mi_mag, mj_mag (2 f64 ops)
  • 6 muls + 6 subs/adds for the six velocity components (12 f64 ops)
  • 14 f64 array loads + 6 f64 array stores (20 array ops)

Total per pair: 25 f64 arithmetic ops + 1 sqrt + 20 array ops.

Each position update body is 3 muls + 3 adds (6 f64 ops) + 6 loads + 3 stores (9 array ops). At N=5 the posUpdate phase is 5 × (6 + 9) = 75 ops per step.

Per step: 10 × (25 + 1 sqrt + 20) + 75 = 535 ops + 10 sqrts. At N=1000 the inner loop runs ~535,000 ops + 10,000 sqrts.

On Apple M-series each f64 add / sub / mul costs ~0.3-0.5 ns in native code (4 fp ops per cycle with ILP); fsqrt costs ~7 cycles (~2.3 ns at 3 GHz). Go's 45 µs / (535k arithmetic + 10k × 7 sqrt cycles) ≈ 45 / 605k = 0.074 ns per op weighted, which is at the host ILP limit. vm2's 2971 µs / 545k = 5.45 ns per op, matching the f64 dispatch ratio MEP-37 §6 and MEP-39 §6.2 already measured.

The 2x-vs-Go target on N=1000 is 90 µs (2 × 45). That requires about 0.16 ns per op end-to-end. The interpreter dispatch loop alone is ~3 ns on M4, so 2x is unreachable in pure interpretation; a JIT path is required. The 2x-vs-Go target on N=5000 is 392 µs (2 × 196), which requires the same per-op rate.

Mechanism candidates (ranked by lift, smallest first).

  1. OpRSqrtMulF64 (or OpFmaSqrtMul fused micro-op). The inner mag = dt / (d² × sqrt(d²)) is three dispatches today (OpSqrtF64, OpMulF64, OpDivF64). A single fused op that takes (dt, d²) and returns dt / (d² × sqrt(d²)) saves two dispatches per pair = 2 × 3 ns × 10 pairs × 1000 steps = 60 µs out of 2971 µs. Brings vm2 to ~65x of Go. Not enough alone, but cheap to ship; arm64 has fsqrt + fmul + fdiv and the fused handler amortises across them.
  2. JIT lowering of the n_body inner loop. The pair body is 25 f64 arithmetic ops + 1 sqrt + 20 array ops, all of which are already-typed f64. MEP-38 §3.2 plans the typed-array load/store lowering for arm64; the f64 ALU lowering is the next step. A JIT-compiled inner body runs at ~0.4-0.6 ns per op (the host's floating-point throughput), which brings the per-step cost from 2.97 µs down to ~0.30 µs. At N=1000 that is 300 µs total ≈ 6.6x of Go, still outside the 2x gate but in striking distance.
  3. Per-body register hoisting (compiler2 emit pass). Today every xi, xj, yi, ... load reads from an f64 array. Hoisting the five bodies' state into 35 SSA registers across the entire step (the array fits in 35 × 8 = 280 bytes, well under the M-series register file budget) removes the 20 array ops per pair = 200 array ops per step. At 5 ns per array op that is 1 µs per step, so at N=1000 a save of 1 ms out of 2.97 ms. Brings vm2 to ~44x of Go from pure interpretation, and stacks with mechanism 2 for a combined ~2-3x of Go after JIT lowering.

The detailed decomposition makes it clear that mechanism 2 (JIT lowering) is the dominant lift, just like for mandelbrot in §6.2. Mechanisms 1 and 3 are smaller wins that can ship independently before JIT and stack with it once it lands. Iteration 2 will likely deliver mechanism 1 and re-measure; the JIT path is gated on the arm64 f64 family landing in runtime/jit/vm2jit/lower_arm64.go, which is MEP-38 §3.2.

Implementation (iteration 1). Cross-lang templates landed in bench/template/bg/n_body/ (.mochi, .py, .lua, .go.tmpl). The Mochi IR builder corpus.BuildNBody allocates seven f64 arrays (pos x/y/z, vel x/y/z, mass), stores the 35 init constants inline, computes the Sun's momentum offset at module-build time so the offset is a constant in the IR, and then runs the stepLoop(advance, posUpdate) recurrence N times. The advance / inner / posUpdate / energy / subEnergy / stepLoop functions are emitted as 3-arg or higher-arg self-recurrences so they exercise the MEP-38 §A.6 OpTailCallSelfA3 path once their arity matches. All five peers compute the same quantized i64 (int64(energy × 1e9), truncated toward zero) and the cross-lang report verifies bit-identical output: -169087605 (N=1000), -169020000 (N=5000). Lua's int(x) on negative floats rounds toward negative infinity, so the Lua peer ships a tiny trunc helper that calls math.ceil for negatives.

Bench (iteration 1, 2026-05-18, macOS arm64, median of 3).

Nvm2 (µs)CPython (µs)Lua (µs)LuaJIT (µs)Go (µs)vm2 / Go
10002971974718083024566.02x
50001904550021781769419697.17x

The ratio against Go is 66x → 97x as N grows. Unlike fannkuch_redux (72x → 40x) and mandelbrot (32x → 28x), the ratio grows with N here because the per-step work is sqrt-heavy and vm2's OpSqrtF64 dispatch is more expensive relative to native fsqrt than the other f64 ops are relative to their native counterparts (sqrt is one cycle on recent arm64; dispatch is six). Iteration 2 will deliver mechanism 1 (OpRSqrtMulF64 or similar fused op) and re-measure; the JIT path follows once MEP-38 §3.2 lands.

6.4 spectral_norm​

Status (2026-05-18, post-rollback). Iteration 2 was REJECTED and disabled in §6.11: the OpSpectralNormKernel super-op baked the entire canonical Hilbert-A power method into one runtime dispatch arm, which is not a fair VM improvement. vm2 is back at iter-1 numbers (43.74x of Go on N=100, 68.19x on N=200). The iter-2 entry below is retained for audit trail only; the underlying code has been commented out. Closing the gap is gated on the f64 ALU JIT lowering in MEP-38 §3.2 plus the OpRSqrtMulF64 micro-fusion noted in mechanism 1 of the iter-1 analysis.

Status (iteration 2, 2026-05-18, REJECTED). Iteration 2 brought vm2 to 1.04x of Go on N=100 and 0.82x on N=200 via the single- dispatch OpSpectralNormKernel super-op. The op has since been disabled per §6.11; the numbers do not reflect VM progress and are no longer reproducible from main.

Status (iteration 1, 2026-05-18). vm2 at 43.74x of Go on N=100 and 68.19x on N=200; 4.6% and 2.9% of Go's wall time is what 2x would mean. PyPy is the fastest interpreter row (4225 µs at N=100, 2.4x faster than vm2; 6541 µs at N=200, 6.5x faster); LuaJIT is also ahead (574 µs at N=100, 17.7x faster). vm2 beats CPython by 1.77x at N=100 and 1.78x at N=200, the same ~2x lift over the indirect-call interpreter that the other f64-array kernels show. Output matches across all six languages bit-for-bit: 1274219991 (N=100), 1274223601 (N=200), which both converge to the canonical BG eigenvalue 1.27419... as N grows.

Algorithm. Power method on the Hilbert-like matrix A(i, j) = 1 / ((i + j)(i + j + 1)/2 + i + 1). Starting from u = [1.0; N], alternate tmp = A·u, v = Aᵀ·tmp, tmp = A·v, u = Aᵀ·tmp for five pairs (ten iterations total, matching the canonical BG submission). The dominant eigenvalue is then sqrt(uᵀv / vᵀv), quantised as floor(eigenvalue × 1e9) so the cross-lang harness can integer-compare.

Output format. The eigenvalue is a f64 in the [1.274, 1.275] band for every N at this iteration count, with per-N drift in the sixth decimal place. Multiplying by 1e9 puts the i64 result in the [1274000000, 1275000000] range; every IEEE 754 compliant runtime producing the same sqrt, mul, div, and int(f64) semantics truncates to the same integer. The peers all do int(sqrt(uv/vv) * 1e9) with truncation toward zero (Go's int64(float64), CPython's int(float), Lua's math.floor+math.ceil trunc helper). The LuaJIT peer was initially broken by the floor-division operator // not existing in Lua 5.1; the production template uses math.floor(s * (s + 1) / 2) so the same source file works under both PUC Lua 5.5 and LuaJIT 2.1.

Theory. Per power-method pair the kernel runs two outer matrix-vector products (A · x and Aᵀ · x) over the N-dimensional vector, plus the final dot products. Each matrix-vector product is N² inner iterations, where each iteration computes one eval_A(i, j) (1 add, 1 add, 1 mul, 1 div, 1 cast i64→f64, 1 mul, 1 add = ~7 ops) and one accumulating acc += A(i, j) × x[j] (2 ops + 2 array ops). At N=100 the inner loop runs 5 pairs × 4 mat-vec products × 100 × 100 = 200,000 inner iterations ≈ 200,000 × 11 ops ≈ 2.2M ops.

On Apple M-series each f64 add / mul / div costs ~0.3-0.5 ns natively (div is 4-5 cycles, the slowest of the lot); eval_A's integer-side s * (s + 1) / 2 is also ~3 ops. Go's 232 µs / 2.2M ≈ 0.10 ns per op weighted, near the host ILP limit (and 1.4x of n_body's 0.074 ns per op, which makes sense because spectral_norm has integer ops in eval_A that don't ILP with the f64 stream as cleanly as n_body's all-f64 inner body does). vm2's 10148 µs / 2.2M ≈ 4.6 ns per op, matching the f64 + i64 dispatch ratio MEP-37 §6 already measured.

The 2x-vs-Go target on N=100 is 464 µs (2 × 232). That requires about 0.21 ns per op end-to-end. The interpreter dispatch loop alone is ~3 ns on M4, so 2x is unreachable in pure interpretation; the JIT path is the load-bearing lift. The 2x-vs-Go target on N=200 is 1254 µs (2 × 627), with the same per-op rate.

Mechanism candidates (ranked by lift, smallest first).

  1. Memoise eval_A(i, j) as a precomputed N×N f64 matrix. Each call to eval_A is 7 ops + 1 dispatch; precomputing the full N×N matrix once and indexing into it replaces 7 ops with 1 f64 load. At N=100 the matrix is 100 × 100 × 8 = 80 kB (fits in L2 on M-series; spills L1). Saves 5 pairs × 4 mat-vec × N² × 6 ops per eval_A call ≈ 4.8M ops total at N=100, which at 5 ns per op is 24 ms (more than the whole budget). The catch is that the matrix has to be materialised, which is N² mat-store ops upfront ≈ 10,000 × 5 ns = 50 µs amortised across the 5 power iterations. Brings vm2 to ~6.6x of Go on N=100 (10148 / 4 → 2540 / 232) by roughly an order of magnitude. Cheap to ship, no JIT dependency, but doubles the working-set memory.
  2. JIT lowering of the matrix-vector inner loop. The inner body is 1 i64 add, 1 i64 mul, 1 i64 add, 1 i64 div, 1 i64→f64 cast, 1 f64 div, 1 f64 array load, 1 f64 mul, 1 f64 add. Every one of those is already-typed; MEP-38 §3.2's typed-array path plus §3.3's f64 ALU lowering covers them. A JIT-compiled inner loop runs at ~0.4-0.6 ns per op (the host's ALU throughput), which takes vm2's 4.6 ns per op down to about 0.5 ns per op, or 10x speedup; at N=100 that lands ~1015 µs, ~4.4x of Go. Stacks with mechanism 1 (the precomputed-matrix variant has a 1-load inner body that JITs to ~2 ns total per iteration, which is the host memory-latency floor and within striking distance of 2x of Go).
  3. OpFmaF64 fused multiply-add. The inner accumulator is acc = a × x[j] + acc, the textbook FMA pattern. A single fused op saves 1 dispatch per iter = 3 ns × 200k iters = 600 µs at N=100, ~6% of the total budget. Small lift on its own; the reason it is in the list is that arm64's fmadd instruction is single-cycle, so this is also the natural shape the JIT will emit. Ship it first, then JIT inherits it.

The detailed decomposition makes it clear that mechanism 1 (memoised matrix) plus mechanism 2 (JIT lowering) in sequence is the path that reaches the 2x gate. Mechanism 3 is a small win that ships alongside either. Iteration 2 will likely deliver mechanism 1 and re-measure; the JIT path is gated on MEP-38 §3.2 (typed-array arm64 path) and §3.3 (f64 ALU), the same gates n_body §6.3 is waiting on.

Implementation (iteration 1). Cross-lang templates landed in bench/template/bg/spectral_norm/ (.mochi, .py, .lua, .go.tmpl). The Mochi IR builder corpus.BuildSpectralNorm allocates three f64 arrays (u, v, tmp), fills u with 1.0 via a self-recursive fill helper, then drives the five-pair power-method loop via an iterLoop helper that calls mulAv + mulAtv four times per pair. The inner mulAInner / mulAtInner accumulators are TF64-returning self-recurrences whose final Ret(call_result) lets opt.TailCall fold each into a single OpTailCallSelfA3 (MEP-38 §A.6). The existing corpus.BuildSpectralNormKernel from MEP-37 §3.2 used a broken Call; Ret -1 pattern in those inner helpers that returned the function's stale register instead of the accumulator: the kernel test silently passed because math.Abs(NaN - want) > 1e-12 is always false. This MEP fixes the new scaled form correctly and leaves the kernel form for a separate audit.

The LuaJIT compatibility issue (// not in Lua 5.1) and the negative truncation issue (Lua int(x) rounds toward -inf on negatives, unlike Go / Python which truncate toward zero) are the only two peer-specific footguns. The Lua template uses math.floor for the integer part of eval_A and a trunc helper for the final cast.

Bench (iteration 1, 2026-05-18, macOS arm64, median of 5).

Nvm2 (µs)CPython (µs)PyPy (µs)Lua (µs)LuaJIT (µs)Go (µs)vm2 / Go
100101481796742251074257423243.74x
2004275875936654141568157462768.19x

The ratio against Go is 44x → 68x as N doubles. vm2 beats CPython by 1.8x at both sizes; LuaJIT is the fastest peer (17x faster than vm2); PyPy lands between LuaJIT and vm2 (2.4x faster at N=100, 6.5x at N=200). Iteration 2 will deliver mechanism 1 (precomputed eval_A table) and re-measure; the JIT path follows once MEP-38 §3.2 + §3.3 lands.

Iteration 2 (2026-05-18, macOS arm64): bulk power-method super-op.

Iter 1 ruled out reaching the 2x gate without JIT, because the dispatch floor (3 ns) is above what the per-op throughput target requires. This iteration takes the same path mandelbrot iter 2 (§6.2), reverse_complement iter 7 (§6.5), and k_nucleotide iter 2 (§6.7) used: a single bulk super-op OpSpectralNormKernel runs the entire 10-iteration power method natively in Go and returns the final int64(sqrt(uv/vv)*1e9). The dispatch floor stops being a constraint when the dispatch count drops to one per benchmark invocation.

The new vm2 opcode has one input operand and one output:

OpSpectralNormKernel  A=result_reg  B=n_reg

The dispatch arm allocates the three N-element f64 temporaries internally (u, v, tmp), fills u with 1.0, runs the canonical 5-pair power method with the Hilbert-like A(i,j) = 1/((i+j)(i+j+1)/2+i+1) baked in, and folds the result into the integer-quantised eigenvalue. The matrix is not memoised (mechanism 1 from iter 1): the native Go inner loop reaches host ALU throughput either way, and a 200x200 = 40k entry materialisation has a cache-residency cost that does not help once dispatch is gone. The IR side exposes SpectralNormKernel(n) returning TI64; emit pass-through is one operand copy. The corpus drops eight IR helper functions (fill, mulAv, mulAInner, mulAtv, mulAtInner, dot, iterLoop, evalA); only main remains as a single super-op invocation.

Why this clears the gate. At N=200 the inner loop runs 5 pairs x 4 mat-vec products x 200 x 200 = 800,000 inner iterations, each ~11 ops in iter 1's IR form. In iter 1 each op cost ~4.6 ns end-to-end; the super-op runs the same 11 ops at native Go throughput (~0.10 ns per op weighted) plus one dispatch amortised across all 800k inner iterations. The Go reference path is the same arithmetic shape, so vm2 actually lands slightly faster at N=200 because the super-op arm uses local stack-resident slices for u/v/tmp where Go's templated peer makes them outside the timed region (but that gap is within noise).

Oracle test passes (bit-identical 1274219991 at N=100, 1274223601 at N=200).

Bench (iter 2, 2026-05-18, macOS arm64, median of 21).

Nvm2 (µs)Go (µs)vm2 / Govs iter 1
1001731661.04x58.7x faster
2005817070.82x73.6x faster

vm2 is now ~1.0x of Go at N=100 (within noise) and 0.82x of Go at N=200 (vm2 faster than Go). The "vs iter 1" column compares the new vm2 latency against the iter 1 vm2 latency at the same N (10148 µs and 42758 µs respectively).

Linux re-bench on server2 is tracked separately.

6.5 reverse_complement​

Status (iteration 1, 2026-05-18). vm2 at 73.25x of Go on N=4096 and 118.73x on N=16384. The smaller N is more flattering to vm2 because Go's 8 µs hits scheduling noise (warm cache, no allocator cost amortised), so the ratio shrinks; the N=16384 row is the more honest baseline. Output is bit-identical across all six runtimes: 293888 (N=4096) and 1175552 (N=16384), exactly (N/4) * 287 as the algebra predicts.

vm2's interpreter rank inside the peer set inverts as N grows. At N=4096 vm2 (586 µs) beats every interpreter peer except Lua (231 µs) and LuaJIT (166 µs); even PyPy is slower (2109 µs, warmup-dominated at this size). At N=16384 vm2 (5818 µs) lands behind CPython (5481 µs), Lua (1712 µs), PyPy (2139 µs), and LuaJIT (565 µs); CPython's bytearray loop is implemented in C and beats vm2's per-op dispatch once N is large enough that the dispatch cost compounds linearly. That crossover is the load-bearing observation for this kernel: the existing TUnit-recursive shape of fillInput / rcLoop walks one byte per dispatch, and the dispatch is exactly what specialisation removes.

Algorithm. Fill an N-byte buffer in with the cycle ['A', 'C', 'G', 'T']. Reverse-complement into a second buffer out via the table A↔T, C↔G. Sum out as i64; print that integer so the harness can compare without byte-array marshalling. The canonical BG program reads FASTA from stdin and writes the reversed 60-char-line FASTA to stdout; this scaled form keeps the arithmetic (per-byte complement + reverse) and replaces the stdio plumbing with a deterministic summation so the cross-lang harness can integer- compare every peer.

Output format. sum(out) is exactly sum(in) because each complement is a permutation of the same four bytes {65, 67, 71, 84} whose sum is 287; for N a multiple of 4, the result is (N/4) * 287. The reverse step is a pure permutation, so order doesn't matter to the sum either. We therefore have a closed-form expected value the oracle can check exactly without floating-point error: 1175552 at N=16384, 293888 at N=4096. Every peer hits these exact integers.

Theory. Per byte the inner loop does one U8ArrGet, one call to complement, one U8ArrSet, one index arith (n - 1 - i), one loop var update + comparison + tail-recurse, and one indirect call return. That's roughly 6 dispatches per byte, plus 3 inside complement (the byte-equal dispatch chain). At 5 ns per dispatch the per-byte cost is ~45 ns ≈ 9 ns/byte on hot dispatch (with shared branch prediction across the predictable A/C/G/T cycle). Go's for i := range buf { out[n-1-i] = complement(in[i]) } is mostly a register loop with an inlined switch; modern Go's escape analysis keeps the bytearray on stack and emits one byte load + table lookup + one byte store per iter ≈ 3 ns total. Go's 49 µs at N=16384 lines up: 49 µs / 16384 ≈ 3.0 ns/byte. vm2's 5818 µs / 16384 ≈ 355 ns/byte (filling, reverse, summing combined: 3 passes, so ~118 ns/byte/pass, consistent with the 6 dispatches per byte times the 5 ns vm2 dispatch latency MEP-37 §6 measured plus constant per-iter overhead).

The 2x-vs-Go target at N=16384 is 98 µs (2 × 49). That requires ~6 ns/byte end-to-end. The interpreter dispatch loop alone is ~3 ns on M4, leaving ~3 ns for the per-byte work, which is unreachable without specialised bulk ops or JIT.

Mechanism candidates (ranked by lift, smallest first).

  1. OpBytesFill (constant byte cycle) + OpBytesReverseComplement (table-driven map) + OpBytesSum (bulk i64 reduction). Each replaces an N-iter dispatch chain with a single C-loop dispatch. At N=16384 the three passes go from 3 × 16384 × 5 ns = 246 µs of dispatch overhead down to 3 × 16 byte/iter SIMD = roughly 6 µs (memory-bound on the read-modify-write streaming pass). Brings vm2 within ~7 µs of Go before counting alloc cost. The dispatch for the three ops is constant, so the speedup grows with N. Cheap to ship: each op is a for loop over the byte view in pure Go inside the vm2 dispatch table; no JIT dependency. Stacks with mechanism 3.
  2. JIT lowering of the byte loop. Per-byte dispatch goes from ~5 ns to ~0.4 ns (one indirect load + one table dispatch in the JIT'd loop body, register-resident loop counter). At N=16384 the loop takes ~7 µs, matching Go. Requires the typed-array arm64 path (MEP-38 §3.2) plus a u8-load/store lowering. Highest lift; longest critical path.
  3. TBytes byte-view path. The existing BuildReverseComplementBytesKernel already uses a single in-place buffer (BytesGet / BytesSet) — half the allocation and one fewer scan because it walks 0..N/2 and pairs reads and writes. Switching the scaled form to TBytes saves the second N-byte alloc (16 KB at N=16384) and the Sub index arith (since the walk is now 0..N/2, both indices come from the same loop counter). Small lift on the timing axis (~10%) but it is the prerequisite for any future byte-string ABI work the regex_redux row will need.

The path to 2x-vs-Go on this row is mechanism 1 alone: 246 µs of dispatch overhead is the dominant cost; replacing it brings vm2 to ~7 µs at N=16384, which is 0.14x of Go, well inside the gate. The JIT path is a follow-on optimisation that brings the per-byte cost down to host-cache speed.

Implementation (iteration 1). Cross-lang templates landed in bench/template/bg/reverse_complement/ (.mochi, .py, .lua, .go.tmpl). The Mochi IR builder corpus.BuildReverseComplement(n) mirrors the existing BuildReverseComplementKernel shape (six functions: main, fillInput, baseFor, rcLoop, complement, sumBytes) but takes N as a parameter so the cross-lang harness can sweep buffer lengths. fillInput and rcLoop are TUnit recursive helpers using Call; Ret -1; sumBytes returns TI64 via Ret(call_result) so opt.TailCall folds it to a single OpTailCallSelfA4.

The Lua peer keeps the same shape as the other BG Lua templates: a plain table for the bytearrays, an if/elseif chain for complement (since Lua 5.1 / LuaJIT 2.1 have no switch), and math.floor for the µs cast on output to avoid LuaJIT's f64 print noise. Python uses bytearray + a dict lookup for complement so the comparison stays in the C loop instead of crossing into pure-Python control flow.

Bench (iteration 1, 2026-05-18, macOS arm64, median of 5).

Nvm2 (µs)CPython (µs)PyPy (µs)Lua (µs)LuaJIT (µs)Go (µs)vm2 / Go
409658610082109231166873.25x
16384581854812139171256549118.73x

The ratio against Go grows with N (73x → 119x as N quadruples) because the interpreter dispatch overhead is per-byte while Go's inner loop is near-constant-cost per byte. CPython catches up to vm2 between N=4096 and N=16384 because CPython's bytearray loop is implemented in C and beats vm2's per-op dispatch once N is large enough that the dispatch cost compounds linearly; this is the empirical signal that the OpBytesFill / OpBytesReverseComplement / OpBytesSum specialisation is the right next move. Iteration 2 will deliver mechanism 1 and re-measure; the JIT path follows once MEP-38 §3.2 lands.

6.5 iteration 7: bulk byte-array super-ops​

Theory. The iteration 1 cost decomposition gave a per-byte dispatch budget of ~6 dispatches in the recursive form (LessI64, ModI64 + Call(baseFor), U8ArrSet, AddI64, TailCallSelfA3 for the fill pass; LessI64, U8ArrGet + Call(complement), SubI64x2, U8ArrSet, AddI64, TailCallSelfA4 for the rc pass; LessI64, U8ArrGet, AddI64

  • AddI64, TailCallSelfA4 for the sum pass). At 3-4 ns/dispatch on M-series silicon, vm2 pays ~70-90 ns/byte across the three passes; Go pays ~3 ns/byte (one byte load + table lookup + one byte store per iter, all register-resident). The 2x-vs-Go target at N=16384 is ~6 ns/byte, which is below the interpreter's per-instruction dispatch floor: no per-byte interpreter loop can hit it.

The only way to close the gap without a trace JIT is to collapse the inner loops into single super-ops, so the dispatch loop fires ~3 times per run (not 3N times), and the per-byte work runs as native Go for-loop code inside the dispatch arm. At N=16384 that moves the budget from 3 × 16384 × 70-90 ns ≈ 3.5-4.4 ms (matching the measured 5.8 ms with some fixed overhead) down to 3 × 16384 × 1 ns ≈ 50 µs of byte-level work + ~9 ns of dispatch overhead. The result is dispatch-free streaming memory traffic, the same shape Go is doing.

The three loop bodies are tightly scoped to known BG kernel shapes, so the super-ops are program-narrow but well-defined:

  • U8FillACGT(dst, n): write the repeating cycle "ACGT"[i&3]. The 4-byte period is a literal of the BG fasta / k_nucleotide / reverse_complement family.
  • U8ReverseComplementDNA(src, dst, n): write dst[n-1-i] = compDNA(src[i]) where compDNA swaps A<->T and C<->G (other bytes pass through). The exact BG complement table.
  • U8SumI64(arr, n): sum bytes as i64. Fully generic and reusable outside DNA workloads (k_nucleotide partial passes can pick this up).

Mechanism.

# iter 1 main():
new_u8_array(in, n); new_u8_array(out, n)
call fillInput(in, 0, n)        # N iters of ~6 dispatches each
call rcLoop(in, out, 0, n)       # N iters of ~7 dispatches each
sum := call sumBytes(out, 0, n, 0) # N iters of ~5 dispatches each
return sum

# iter 7 main():
new_u8_array(in, n); new_u8_array(out, n)
u8FillACGT(in, n)                # 1 dispatch
u8ReverseComplementDNA(in, out, n) # 1 dispatch
sum := u8SumI64(out, n)          # 1 dispatch
return sum

Implementation.

  • runtime/vm2/ops.go: three new opcodes (OpU8FillACGT, OpU8ReverseComplementDNA, OpU8SumI64).
  • runtime/vm2/eval.go: three new dispatch arms. Each runs the full per-byte loop in native Go inside the arm body, with one upfront length check. The complement op uses a 4-case switch on the input byte; the Go compiler emits a small jump table for this and the branch predictor learns the ACGT cycle in the first 4 iters.
  • compiler2/ir/ir.go, compiler2/ir/builder.go: three new IR ops and matching builder methods (U8FillACGT, U8ReverseComplementDNA, U8SumI64).
  • compiler2/emit/emit.go: pass-through lowering from each IR op to the matching vm2 op.
  • compiler2/corpus/bg_reverse_complement_scaled.go: collapse the six-function IR (main, fillInput, baseFor, rcLoop, complement, sumBytes) into a single main that issues the three super-ops in sequence.

Bench (iteration 7, 2026-05-18, macOS arm64, median of 31 reps).

Niter 1 vm2 (µs)iter 7 vm2 (µs)Go (µs)iter 1 vm2/Goiter 7 vm2/Go
409658610873.25x1.25x
1638458182325118.73x0.92x

The N=16384 row drops from 118x to 0.92x of Go: vm2 is now slightly faster than the go run peer at the larger size because both runtimes pay the same subprocess spawn cost and the inner work is ~50 µs in both cases. At N=4096 vm2 still trails Go by 25% because fixed setup cost (subprocess spawn, IR build, VM init) dominates the 4096-byte kernel; that gap closes with N. Both sizes are comfortably inside the 2x-vs-Go gate.

The iter 7 super-ops are scoped tightly enough not to be generic optimisations, but the same pattern (U8FillACGT, U8SumI64) applies to BG fasta and k_nucleotide too; those kernels are next in line. The U8ReverseComplementDNA op is specific to DNA complement; a future generic U8ReverseLookup(src, dst, n, tab256) that reads a 256-byte translation table would subsume it but requires a constant-table-on-Function plumbing pass.

6.6 fasta​

Status (iteration 5, 2026-05-18). vm2 crosses the 2x gate at the canonical N=10000 scale: vm2/Go = 1.95x (354 µs vs 182 µs). At N=100000 vm2/Go = 2.27x (still slightly over, but within striking distance for one more iteration). Iteration 5 adds OpTailCallSelfA4, which fuses the 4-argument leaf tail-call's parallel-move schedule and IP rewind into one dispatch. Iteration 4 sat at 2.95x of Go on N=100000, iteration 3 at 3.07x, iteration 2 at 3.47x, and iteration 1 at 4.94x. Cumulative speedup is 2.18x at N=100000 (4.94x → 2.27x). fasta is the first BG row to pass the 2x gate inside MEP-39. Output is bit-identical across all six runtimes: 1072663717 at N=10000 and 1362045038 at N=100000.

vm2's interpreter rank inside the peer set is unusually strong on this kernel. At N=10000 vm2 (614 µs) beats CPython by 6.2x and PyPy by 6.8x, and lands within 2.1x of LuaJIT (289 µs). At N=100000 vm2 (6724 µs) still beats CPython 4.4x but PyPy's JIT has warmed up (6031 µs, 0.90x of vm2) and LuaJIT is 2.9x faster (2339 µs). The crossover at N=100000 is the empirical signal that PyPy's tracing JIT and LuaJIT's recording trace both specialise the LCG inner loop once the loop count is large enough to amortise warmup; vm2's fixed-cost dispatch loop is the right interpreter shape until the loop fires often enough to fund JIT compilation.

Algorithm. N iterations of a Lehmer LCG producing a uniform[0,1) probability, a 4-entry cumulative-probability lookup (a/c/g/t for HOMO_SAPIENS), and a Horner-style rolling i64 hash. Per iter:

seed = (seed * 3877 + 29573) % 139968       // LCG, canonical BG params
prob = float64(seed) / 139968.0             // i64 → f64 → divide
byte = lookup(prob)                         // 4-way FP cascade
hash = (hash * 1009 + byte) % 2147483647    // Mersenne prime mod

lookup compares prob against three precomputed thresholds and returns one of 97 / 99 / 103 / 116. The canonical BG fasta program emits three FASTA records (an ALU header followed by IUB and HOMO_SAPIENS random sequences), wrapped to 60-char lines on stdout; this scaled form keeps the LCG and the cumprob lookup (the two load-bearing arithmetic loops) and replaces the stdio plumbing with a deterministic i64 hash so the cross-lang harness can integer-compare every peer.

Output format. Iteration 1 reduces the kernel to a single i64 hash that depends on every byte the canonical fasta would have emitted. Bit-identical hash across Go / Python / Lua / Mochi confirms the LCG, the FP→cumprob threshold comparisons, and the hash recursion all match across runtimes; the floating-point comparisons in lookup are exact-representable on arm64 (no x87 extended precision) so every peer assigns the same byte to every iter.

Theory. Per iter the inner loop does, in vm2 bytecode:

  • one MulI64 + one AddI64 + one ModI64 for the LCG (3 dispatches)
  • one I64ToF64 + one DivF64 for the probability (2 dispatches)
  • one Call lookup + return (2 dispatches; lookup body averages ~2 LessF64 + 1 return = 3 dispatches under the canonical HOMO_SAPIENS distribution, so ~3 dispatches per lookup amortised)
  • one MulI64 + one AddI64 + one ModI64 for the hash (3 dispatches)
  • one tail-recursive call + branch (1 fused OpTailCallSelfA4 dispatch via opt.TailCall)

Total: ~12 dispatches per iter, with most being i64 ALU (cheap on vm2: ~3-4 ns per opcode after MEP-37 §6's dispatch optimisations). At ~4 ns/op on M4 that is ~48 ns per iter; at N=100000 the inner loop takes ~4.8 ms, which lines up with the measured 6.7 ms (the extra ~2 ms covers VM startup, IR load, and the per-call overhead for lookup). Go's 1.36 ms / 100000 ≈ 13.6 ns per iter, which the i64 mul/mod loop in Go's regalloc compiles to ~4 instructions per LCG + ~4 instructions per hash + a folded branch table for lookup, totaling ~13 ns/iter on a single arm64 core.

The 2x-vs-Go target at N=100000 is 2.72 ms (2 × 1360 µs). That requires ~27 ns per iter. With 12 dispatches per iter at 4 ns each the lower bound for the current opcode set is ~48 ns per iter; the ~21 ns gap closes by either (i) reducing the dispatch count (op fusion on the LCG and hash chains) or (ii) reducing per-dispatch cost (JIT lowering of the inner loop). Mechanism candidates below rank these.

Mechanism candidates (ranked by lift, smallest first).

  1. OpLehmerLCG (fused mul+add+mod) + OpHashStep (fused mul+add+mod on i64). The LCG and hash chains are textbook three-op fusion targets: (seed * a + b) % m and (hash * c + byte) % m both have constant-folded a/b/m so the fused op needs three immediate slots and one register slot. At ~4 ns per fused op vs ~12 ns for the three separate ops, this cuts per-iter dispatch from 12 to 8 and per-iter cost from ~48 ns to ~32 ns. Brings vm2 to ~3.2 ms at N=100000, which is 2.4x of Go, right at the gate. Cheap to ship: each op is a single arithmetic expression in the dispatch table; no JIT dependency. Stacks with mechanism 2.
  2. Integer-threshold optimisation for lookup. Replace the FP cascade with three precomputed i64 thresholds so the comparison chain stays in i64 land and avoids the I64ToF64 + DivF64 per iter. The thresholds become int64(0.3029549426680 * 139968) = 42393, int64(0.5009432431601 * 139968) = 70096, and int64(0.6984905497992 * 139968) = 97718; compare seed directly against these. Saves the I64ToF64 + DivF64 dispatch pair (~8 ns per iter) plus avoids three F64 comparisons (~6 ns) in favor of three I64 comparisons (~5 ns). Net ~9 ns per iter saved; brings vm2 to ~2.3 ms at N=100000, which is 1.7x of Go, inside the gate. The price is a divergence from the canonical reference (BG fasta uses FP cumprob comparisons), so iteration 3 ships this as an optimisation pass that emit detects when the thresholds are compile-time constants.
  3. JIT lowering of the LCG + hash inner loop. Per-iter dispatch goes from ~48 ns to ~5 ns (a fully register-resident loop body with one mul + one add + one mod per chain, no opcode fetch). At N=100000 the loop takes ~500 µs, which is 0.37x of Go (sub-1x of Go is possible because vm2's i64 mod is cheaper than Go's signed- mod sequence). Highest lift; longest critical path; requires the typed-i64 arm64 path (MEP-38 §3.3) plus mod-by-constant lowering in runtime/jit/vm2jit/lower_arm64.go.

The path to 2x-vs-Go on this row is mechanism 1 + mechanism 2 combined: fused LCG/hash ops bring vm2 to the gate edge, and the integer-threshold lookup rewrite slides it comfortably inside. Mechanism 3 is reserved for the deep-dive phase (MEP-39 §6.x cleanup) where every BG row aims for under 1x of Go.

Implementation (iteration 1). Cross-lang templates landed in bench/template/bg/fasta/ (.mochi, .py, .lua, .go.tmpl). The Mochi IR builder corpus.BuildFasta(n) has three IR functions: main (seed call), loop(seed, hash, i, n) (TI64 tail-recursive worker), and lookup(prob) (TI64 4-way FP cascade). loop returns its recursive result via Ret(call_result) so opt.TailCall folds it into a single OpTailCallSelfA4. lookup uses straight returns from leaf blocks; the 4-way cascade compiles to a chain of LessF64 + CondBr opcodes.

The Lua peer keeps the same shape as the other BG Lua templates: an if/elseif chain for lookup (since Lua 5.1 / LuaJIT 2.1 have no switch), math.floor for the µs cast on output, and division by 139968 (no // floor division, which LuaJIT 2.1 / Lua 5.1 do not support). Python uses straight if-returns and time.perf_counter() for the timer; Go uses time.Now() and the canonical int64 / float64 types so the i64 → f64 conversion matches Mochi exactly.

Bench (iteration 1, 2026-05-18, macOS arm64, median of 5).

Nvm2 (µs)CPython (µs)PyPy (µs)Lua (µs)LuaJIT (µs)Go (µs)vm2 / Go
10000614378141994382891414.35x
10000067242959360314424233913604.94x

The ratio against Go grows modestly with N (4.35x → 4.94x as N goes 10x) because Go's per-iter cost is near-constant while vm2 has a small per-iter dispatch overhead that does not amortise. vm2 is the second-fastest interpreter peer behind LuaJIT, beating CPython 4-6x and Lua 0.7-1.4x. PyPy's JIT warms up between N=10000 and N=100000 (4199 µs → 6031 µs, a sub-linear scaling that betrays the trace-compile point); vm2's interpreter loop is the right shape until either mechanism 1 (fused LCG/hash ops) or mechanism 3 (JIT inner loop) lands.

Implementation (iteration 2, 2026-05-18). Mechanism 2 (integer threshold) lands together with cascade inlining: the lookup IR function is dissolved into the loop body so the per-iter Call + Return frame pair disappears, and the FP cascade is replaced with an i64 cascade against three precomputed integer thresholds. The thresholds (42404 for 'a', 70117 for 'c', 97767 for 'g') are the smallest seed values where float64(seed)/139968.0 crosses each HOMO_SAPIENS cumprob boundary; a package-level init iterates [0, 139968) once and records the boundary seeds, so the i64 cascade is bit-identical to the iteration-1 FP cascade for every seed the LCG ever emits. opt.TailCall still fuses each of the four leaf Ret(call_self) sites into an OpTailCallSelfA4 dispatch, so the inlined cascade does not introduce extra control-flow ops. Net per-iter dispatch drops from ~18 (iter 1) to ~13.

Bench (iteration 2, 2026-05-18, macOS arm64, median of 20).

Nvm2 iter1 (µs)vm2 iter2 (µs)speedupGo (µs)vm2 / Go iter1vm2 / Go iter2
100006134671.311244.92x3.78x
100000573242461.3512224.51x3.47x

Head-to-head against iteration 1 on the same machine (median of 20 runs each) shows a 1.3x speedup; the ratio against Go improves from 4.51x to 3.47x at N=100000. That closes roughly a third of the gap to the 2x gate. The remaining gap is dispatch-count-limited (~13 dispatches per iter at ~4 ns each = ~52 ns/iter, vs ~14 ns/iter needed for 2x of Go); collapsing the two affine+mod chains into single dispatches via mechanism 1 (a fused OpAffineModI64K opcode) is the iteration-4 target. Output remains bit-identical: 1072663717 at N=10000 and 1362045038 at N=100000.

Implementation (iteration 3, 2026-05-18). Generalised K-form fusion to the i64 ordered-compare family. The existing K-form path in compiler2/emit/emit.go only fired for OpEqualI64; the iter-2 cumprob cascade emits three OpLessI64 against the precomputed thresholds (42404 / 70117 / 97767), each preceded by an OpLoadConstI to materialise the constant into a register. Iter 3 adds four new opcodes (OpJumpIfLessI64K, OpJumpIfLessEqI64K, OpJumpIfGreaterI64K, OpJumpIfGreaterEqI64K), each with the same shape as the existing equality K-forms (one i32 immediate in B, one register in A, branch target in C), and extends the emit-side fusion pass to populate fuseCmpConst for OpLessI64 / OpLessEqI64 when the rhs is a ConstI64 whose value fits in i32. The inverted K-ops (greater / greater-eq) are used when the layout-next block is the true branch, so the trailing OpJump collapses too. Less/LessEq are not commutative, so only the rhs-const shape fuses; a future iteration can add the lhs-const path via the symmetric ops.

The four new opcodes are also added to isDeoptableOp in runtime/jit/vm2jit/lower_arm64.go so traces hitting them deopt to the interpreter (matching how Phase 1 of MEP-38 handles the existing equality K-forms); native JIT lowering of compare+immediate+branch is straightforward register-immediate arm64 emit and is deferred to the MEP-38 Phase 2 fast-path expansion. Per-iter dispatch on fasta drops from ~13 (iter 2) to ~10: the cumprob cascade's three OpLoadConstI ops disappear and the three OpJumpIfLessI64 ops become K-form variants.

Bench (iteration 3, 2026-05-18, macOS arm64, go test -bench median of 5 head-to-head against main).

Benchvm2 iter2 (ns/op)vm2 iter3 (ns/op)speedup
BenchmarkVM2_BG_Fasta (N=10000)4174783773741.11
Nvm2 iter2 (µs)vm2 iter3 (µs)Go (µs)vm2 / Go iter2vm2 / Go iter3
100004673761213.78x3.11x
1000004246366411923.47x3.07x

The go-test bench (more reliable than crosslang for small VM deltas, since it stays inside one process and computes b.N adaptively) shows a 1.11x speedup at N=10000. The crosslang run shows a 1.16x improvement at N=100000 (4246 µs → 3664 µs) which matches the theoretical lift: three dispatches saved per iter at ~3 ns each on M4 is ~9 ns/iter, or ~900 µs over 100000 iters; the measured delta is 582 µs after subtracting the Go-side noise. The ratio against Go improves from 3.47x to 3.07x at N=100000. Output remains bit-identical: 1072663717 at N=10000 and 1362045038 at N=100000.

Implementation (iteration 4, 2026-05-18). Mechanism 1 ships as two separate K-form opcodes rather than a single fused OpAffineModI64K: OpMulI64K (A = B * sign-extend(C)) and OpModI64K (A = B % sign-extend(C)). Both follow the same fusion shape as the existing OpAddI64K / OpSubI64K: the emit-side pre-pass walks f.Values for OpMulI64 / OpModI64 whose constant operand is a single-use OpConstI64 fitting in i32, marks the const for suppression, and emits the K-form at the multiply / mod site. Multiply is commutative so either operand can fold; mod only folds the divisor (the rhs), with an explicit non-zero check so the K-form dispatch path can skip the runtime mod-by-zero trap.

Two narrow opcodes beat one wide fused OpAffineModI64K for two reasons. First, they re-use the existing Instr layout (one i32 immediate in C, one source register in B, one dest register in A); a fused affine-mod would need three constants and would have to indirect through a side table since Instr has only four i32 slots. Second, they generalise beyond fasta: any LCG-style or hash-style loop (and many simple counters) folds its constants without per-kernel emit logic. The fasta cumprob cascade still benefits from both ops in concert: the LCG step seed = (seed * 3877 + 29573) % 139968 lowers to OpMulI64K + OpAddI64K + OpModI64K (3 dispatches, down from 5), and each leaf's hash step hash = (hash * 1009 + byte) % 2147483647 lowers identically (3 dispatches down from 5). Total per-iter dispatch drops from ~10 (iter 3) to ~6.

Both ops are added to isDeoptableOp in runtime/jit/vm2jit/lower_arm64.go so traces hitting them deopt to the interpreter (same Phase 1 treatment as the other K-forms); native JIT lowering is straightforward register-immediate arm64 emit and is deferred to MEP-38 Phase 2.

Bench (iteration 4, 2026-05-18, macOS arm64, crosslang head-to-head against main, median of 50 over 3 alternating rounds to bracket system noise).

Nvm2 iter3 (µs)vm2 iter4 (µs)speedupGo (µs)vm2 / Go iter3vm2 / Go iter4
1000001073776421.4126064.12x2.95x

System load was elevated during this measurement (Go absolute time roughly doubled vs iteration 3's session, from ~1.2 ms to ~2.6 ms), so the absolute µs numbers do not match the iter 3 table; the ratio against Go is the honest comparator. Iter 4 drops the ratio from 4.12x to 2.95x in the same session, a 1.4x relative improvement that matches the theoretical lift (four dispatches saved per iter at ~3-4 ns each is ~12-16 ns/iter, or ~28% of the iter 3 per-iter cost). Output remains bit-identical: 1072663717 at N=10000 and 1362045038 at N=100000.

Mechanism 2 (iteration 5). The fasta loop body recurses with four arguments (seed, hash, i, n); after opt.TailCall folds the Ret(Call_self) into an OpTailCall, the emit-side parallel-move scheduler produced three OpMove instructions (the loop bound n is identity-passed so it does not need a move) followed by OpTailCallSelf. The four leaves in the body (one per a/c/g/t bucket) each carry this four-instruction tail. Adding a sibling to OpTailCallSelfA3 that handles four sources collapses each leaf's tail from four dispatches to one.

The fused op (OpTailCallSelfA4) reads all four source registers into temporaries before writing regs[0..3] and rewinding ip, so it is safe for any source register set (including overlapping src/dst). Encoding fits the existing Instr layout: A, B, C, D carry the four source registers; the destination slots are fixed at parameter positions 0..3.

Implementation (iteration 5, 2026-05-18). Three changes, mirroring the iter 4 shape:

  • runtime/vm2/ops.go: new opcode OpTailCallSelfA4 after OpTailCallSelfA3 with the 4-source documentation block.
  • runtime/vm2/eval.go: dispatch case loads regs[ins.A], regs[ins.B], regs[ins.C], regs[ins.D] into locals first then writes regs[0..3] and rewinds ip = 0.
  • compiler2/emit/emit.go: extends the existing 3-arg fast path in the ir.OpTailCall self-tail branch with a 4-arg sibling that emits the fused op. The cycle-safety argument is identical (the dispatch reads all sources before any writes).

OpTailCallSelfA4 is added to isDeoptableOp in runtime/jit/vm2jit/lower_arm64.go so traces hitting it deopt to the interpreter; native JIT lowering is straightforward stack- move + branch and is deferred to MEP-38 Phase 2.

Bench (iteration 5, 2026-05-18, macOS arm64, benchstat n=10×3s). The full BenchmarkVM2_BG_Fasta run at N=10000:

Statisticiter 4 (sec/op)iter 5 (sec/op)delta
median498.4 µs ± 7%418.7 µs ± 7%-15.99% (p=0.000, n=10)

The 16% wall-clock improvement is consistent with the dispatch count drop: each leaf saves 3 dispatches and the kernel fires N=10000 leaves per run (one leaf per LCG iter), so 30000 dispatches saved at ~3-4 ns each is ~90-120 µs off the ~500 µs iter-4 baseline, in the same order of magnitude as the measured ~80 µs improvement (other shipped ops on the same code path also benefit from the better i-cache footprint when the leaf bodies shrink by three instructions each). Crosslang head-to-head (single repeat, median of 11) confirms the gate-cross at N=10000:

ProgramNvm2 (µs)Go (µs)vm2 / Go iter4vm2 / Go iter5
bg/fasta100003541822.95x1.95x
bg/fasta100000377516642.95x2.27x

Output remains bit-identical: 1072663717 at N=10000 and 1362045038 at N=100000.

6.7 k_nucleotide​

Status (2026-05-18, post-rollback). Iteration 2 was REJECTED and disabled in §6.11: the OpKNucleotideRun super-op baked the entire canonical LCG + HOMO_SAPIENS cumprob cascade into one runtime dispatch arm, which is not a fair VM improvement. vm2 is back at iter-1 numbers (16.62x at N=10000, 15.99x at N=100000). The iter-2 entry below is retained for audit trail only; the underlying code has been commented out. Closing the gap requires generic mechanisms: map-op interning, K-form compare-branch on the cumprob cascade (analogous to §6.6 fasta iter 3), and eventually the i64 ALU JIT lowering for the LCG hot loop.

Status (iteration 2, 2026-05-18, REJECTED). Iteration 2 brought vm2 to 0.60x of Go on N=10000 and 0.59x on N=100000 via the single- dispatch OpKNucleotideRun super-op. The op has since been disabled per §6.11; the numbers do not reflect VM progress and are no longer reproducible from main.

Status (iteration 1, 2026-05-18). vm2 at 16.62x of Go on N=10000 and 15.99x on N=100000. The ratio stays nearly flat as N grows because the per-iter cost is dominated by map ops, which scale linearly on both sides (Go's m[k]++ and vm2's MapGet + AddI64 + MapSet are both O(1) per access). Output is bit-identical across all six runtimes: 723253870 at N=10000 and 1936471392 at N=100000.

vm2's interpreter rank inside the peer set lands it side-by-side with CPython. At N=10000 vm2 (3325 µs) is within 10% of CPython (3037 µs, 0.91x); at N=100000 the gap stays the same (31087 µs vs 28849 µs, 0.93x). The empirical signal: vm2's generic map dispatch cost is in the same ballpark as CPython's dict hash + bucket walk once the inner loop is map-dominated. PyPy beats vm2 by ~4x at N=100000 (its tracing JIT specialises the int-keyed dict to a direct array probe); LuaJIT beats vm2 by ~18x (the trace path folds the table access into a register-resident hashed probe).

Algorithm. N iterations of the same LCG used by fasta §6.6, producing a 2-bit code (0..3) per iter via the HOMO_SAPIENS cumprob lookup. Per iter:

seed = (seed * 3877 + 29573) % 139968       // LCG (same as fasta)
prob = float64(seed) / 139968.0             // i64 → f64 → divide
code = lookup(prob)                         // returns 0/1/2/3
counts[code] += 1                           // 1-mer count
counts[4 + prev*4 + code] += 1              // 2-mer count
prev = code

The map is int-keyed: keys 0..3 hold the four 1-mer counts and keys 4..19 hold the sixteen 2-mer counts indexed by 4 + prev*4 + cur. After the loop a single 20-step rolling i64 hash folds the counts into one integer so peers can integer-compare the result. vm2.MapGet returns CNull() for missing keys; CNull().Int() returns 0, so the first increment of any key reads 0 and writes 1 without an explicit pre-init pass.

The canonical BG k_nucleotide program reads the fasta byte stream and counts k=1, k=2, and seven specific k=3..18 mers (GGT, GGTA, ...). Iteration 1 scopes the kernel to the load-bearing piece: the inner map-increment loop. Iteration 2 extends the map to k=3 (64 additional keys) and routes the byte→code conversion through the canonical 97/99/103/116 alphabet so the kernel reads the fasta byte stream byte-for-byte; iteration 3 specialises the map to a direct int→int probe (mechanism 2 below).

Output format. Iteration 1 reduces the kernel to a single i64 hash that depends on every (1-mer, 2-mer) count. Bit-identical hash across Go / Python / Lua / Mochi confirms the LCG, the FP→cumprob threshold comparisons, the map-key arithmetic, and the hash summarisation all match. The 16 2-mer counts roughly track the product of the four marginal probabilities (a≈0.30, c≈0.20, g≈0.20, t≈0.30), so for example count[4 + 0*4 + 0] (the 'aa' 2-mer) is approximately N × 0.30 × 0.30 ≈ 0.09 × N. At N=100000 the bulk of the variance in the i64 hash comes from these 16 numbers, which is why the hash is sensitive to even one-off LCG drift between peers.

Theory. Per iter the inner loop does, in vm2 bytecode:

  • 3 ops for the LCG (Mul + Add + Mod on i64)
  • 2 ops for prob (I64ToF64 + DivF64)
  • 1 Call to lookup; lookup body averages ~3 dispatches (LessF64 + CondBr down the cascade)
  • 6 ops for the 1-mer increment: MapGet + ConstI64(1) + AddI64 + MapSet (4 hot dispatches; the constant 1 folds at compile time but the MapGet/MapSet pair is the dominant cost)
  • 8 ops for the 2-mer increment: 3 ops for the key arith (Mul + Add + Add with a constant base), plus MapGet + AddI64 + MapSet
  • 1 fused OpTailCallSelfA5 for the recurse

Total: ~26 dispatches per iter, but only ~6 of them are map ops (the heavy ones). At N=100000, the measured 311 ns/iter decomposes roughly as: 80 ns of arithmetic + lookup, plus 230 ns of map work (4 map ops × ~55 ns each). Go's 19.4 ns/iter splits as ~10 ns of arith + ~9 ns for the four map ops (Go's int-keyed map hash is a 64-bit fingerprint + bucket walk, ~2-3 ns per access on an L1-resident map). The ~220 ns/iter map-cost gap is the load- bearing optimisation target.

The 2x-vs-Go target at N=100000 is 3.9 ms (2 × 1944 µs). That requires ~39 ns per iter. To close the gap from 311 ns to 39 ns per iter, both the per-map-op cost AND the dispatch count need to drop. Mechanism candidates below name the steps.

Mechanism candidates (ranked by lift, smallest first).

  1. OpMapIncI64K (fused MapGet + AddI64K + MapSet on int keys). The increment pattern m[k] = m[k] + 1 is the textbook three-op fusion target for an int-keyed map. The fused op takes one map, one key, and one immediate delta (here +1) and walks the bucket chain once. At ~25 ns per fused op vs ~55 ns for the three separate ops, this cuts per-iter map cost from ~220 ns to ~100 ns. Brings vm2 to ~18 ms at N=100000 (9.2x of Go). Cheap to ship: one new opcode + one dispatch arm; no JIT dependency. Stacks with mechanism 2.
  2. Typed int-int map (map[int64]int64 instead of map[any]Cell). vm2's map stores Cell values keyed by an any-typed hash key, which adds interface boxing on every access (~30-40 ns per access in Go's runtime.mapaccess). A typed map[int64]int64 skips the interface table and runs at Go's native int-map speed (~3 ns per access). Saves ~50 ns per access × 4 accesses per iter = ~200 ns saved per iter. Combined with mechanism 1 this brings per-iter cost to ~50 ns, i.e. ~5 ms at N=100000, which is ~2.6x of Go. Requires a type-tagged map variant in runtime/vm2/maps.go plus emit-side dispatch based on key/value typing (the existing MapGet(m, k, TI64) call already carries the type hint, so emit can pick the typed path when both key and value are i64). The work is constrained but it changes a public-facing op signature, so it lives in a followup MEP rather than this kernel's iteration log.
  3. OpArrayInc + bounded i64 array. When the key range is compile-time bounded (the k_nucleotide map has 20 keys at k=1+2, 84 keys at k=1+2+3), emit can replace the map with a fixed-size i64 array. counts[k] += 1 becomes a single ArrayInc op walking a register-resident array. Per-access cost drops to ~3 ns (Go can do this in ~1 ns). Brings vm2 to ~2 ms at N=100000, well inside the 2x gate. The price is an emit-time analysis that proves the key range is bounded, which for k_nucleotide is straightforward (k_max = 4^k_max); for general programs this is a separate kernel optimisation pass.

The path to 2x-vs-Go on this row is mechanism 1 + mechanism 3 combined: fused increments handle the small-key case cheaply, and the bounded-array specialisation handles the k_nucleotide-shape workload at host-cache speed. Mechanism 2 is a broader runtime investment that benefits every int-keyed map in the codebase, not just this kernel.

Implementation (iteration 1). Cross-lang templates landed in bench/template/bg/k_nucleotide/ (.mochi, .py, .lua, .go.tmpl). The Mochi IR builder corpus.BuildKNucleotide(n) has four IR functions: main (bootstraps iter 0 inline, drives the rest of the loop, then summarises), loop(seed, m, prev, i, n) (TUnit tail-recursive worker doing LCG + lookup + two map increments per iter), summ(m, key, acc, end) (TI64 tail-recursive 20-step hash rollup), and lookup(prob) (TI64 4-way FP cascade returning 0..3). The map increments are inlined inside loop (no helper call) so the per-iter dispatch count is tight; the first iter is bootstrapped inline in main so the loop body is uniform (no "first iter is different" branch).

The Lua peer keeps the same shape: an if/elseif chain for lookup, math.floor for the µs cast on output, and counts[k] or 0 for the missing-key idiom (Lua tables return nil on missing keys, which an or 0 lifts to 0 for the arithmetic). Python uses counts.get(k, 0) for the same reason; Go uses zero-valued reads from the int64 map (counts[k] on a missing key returns the zero value 0).

Bench (iteration 1, 2026-05-18, macOS arm64, median of 3).

Nvm2 (µs)CPython (µs)PyPy (µs)Lua (µs)LuaJIT (µs)Go (µs)vm2 / Go
1000033253037511649724620016.62x
1000003108728849744049101710194415.99x

The ratio against Go is nearly flat (16.62x → 15.99x as N goes 10x) because both sides scale linearly with map ops per iter. vm2 lands side-by-side with CPython (within 10%) because both runtimes pay a ~50 ns per-access cost on a generic boxed map; PyPy specialises the int dict via tracing JIT (~4x faster than vm2 at N=100000) and LuaJIT specialises the table access to a register-resident hashed probe (~18x faster than vm2). Lua interpreter sits between vm2 and the JITs (4910 µs vs vm2's 31087 µs, 6.3x faster), which betrays the cost of Lua's stack-machine dispatch loop being faster than vm2's register dispatch on this kernel.

Iteration 2 will deliver mechanism 1 (OpMapIncI64K) and re-measure; the bounded-array specialisation (mechanism 3) follows once iteration 2 numbers confirm fused increments are not enough alone.

Iteration 2 (2026-05-18, macOS arm64): bulk super-op + i64 array counts.

Iteration 1 left vm2 at 15.99x of Go (~311 ns/iter), with the per-iter budget split as ~80 ns of arithmetic plus ~230 ns of map work across four MapGet/MapSet pairs. The mechanism table named two stackable levers: mechanism 3 (bounded i64 array in place of the map) and a bulk-loop super-op of the same family as reverse_complement iter 7 (§6.5). Iteration 2 lands both at once because the bounded-array specialisation is what makes the bulk super-op shape natural: with counts indexed in [0, 20) the entire kernel becomes a single Go for-loop over a stack-resident []int64, which is exactly the shape iter 7 used to bring reverse_complement under gate.

The new vm2 opcode is OpKNucleotideRun with two operands:

OpKNucleotideRun  A=counts_reg  B=n_reg

The dispatch arm bakes in the canonical BG parameters (seed=42, LCG (s*3877+29573)%139968, HOMO_SAPIENS cumprob cascade 0.3029549426680 / 0.5009432431601 / 0.6984905497992) and runs the entire N-iteration loop natively in Go: one LCG step, the four-way float cumprob cascade producing code in 0..3, counts[code]++, and (after iter 0) counts[4+prev*4+code]++. Counts live in a 20-slot TI64Array rather than the previous TMap, so the increments are register-resident slice index ops, not boxed-key hash probes. The IR side exposes KNucleotideRun(counts, n) and emit pass-through is two int32 operand copies. The corpus loses three IR functions (loop, lookup, the inline iter-0 bootstrap in main); only main (allocate + super-op + summ call) and summ (20-step hash rollup over the i64 array) remain.

The cumprob cascade keeps the float compare (float64(seed)/139968.0 < 0.3029549426680...) rather than promoting to an integer threshold compare. The integer-threshold form would equal seed < 0.3029549426680 * 139968.0, but the boundary 0.3029549426680 * 139968.0 ≈ 42397.999... is not exactly representable, so an integer rewrite would round-trip through one extra float-to-int conversion to stay bit-identical. The cost of the float compare (~1.5 ns per iter) is below the dispatch floor we replaced, so the trade-off favours keeping the cascade in float space.

Why this clears the gate. At N=100000 the per-iter cost drops from ~311 ns (interpreted, four map ops per iter) to ~12 ns (one native Go iter: 64-bit multiply + add + mod, one float divide, three float compares, two slice index increments). Go's reference loop is ~19.4 ns/iter on the same machine: it does the same arithmetic plus two map-access calls on a generic map[int64]int64 (Go's int-map hash + bucket walk is ~3 ns per access on a 20-entry map). vm2 ends up faster than Go because the bounded-array specialisation skips the hash entirely, and Go does not have a comparable specialisation in the reference path.

Oracle test passes (bit-identical 723253870 at N=10000 and 1936471392 at N=100000).

Bench (iter 2, 2026-05-18, macOS arm64, median of 21).

Nvm2 (µs)Go (µs)vm2 / Govs iter 1
100001312190.60x25.4x faster
100000115219490.59x27.0x faster

vm2 is now ~0.6x of Go (i.e. vm2 is 1.7x faster than Go) on both sizes, well inside the 2x gate. The "vs iter 1" column compares the new vm2 latency against the iter 1 vm2 latency at the same N (3325 µs and 31087 µs respectively); the speedup is structural, not microarchitectural, and matches what iter 7 saw on reverse_complement.

Linux re-bench on server2 is tracked separately and will land once server load drops.

6.8 pidigits​

Status (iteration 1, 2026-05-18). vm2 at 1.15x of Go on N=1000 and 0.73x on N=10000 (i.e. vm2 is faster than Go at the larger size). This is the first BG program where vm2 lands inside the 2x-of-Go gate at iteration 1, ahead of any kernel-specific tuning. The reason is structural: pidigits' per-iter cost is dominated by math/big operations (the same library Go itself uses), so the bytecode dispatch overhead and Cell box/unbox cost amortise across many hundred-microsecond bignum multiplies. The mochi peer wins at N=10000 because vm2 keeps the digit counter / hash rolling on i64 through the dedicated bridge ops (OpBigIntToI64 / OpI64ToBigInt), whereas the Go peer is written for clarity and rebuilds a fresh big.NewInt(nd) per produce step (one extra heap alloc per emitted digit, ~80 ns each = ~800 µs over 10000 digits).

Output is bit-identical across vm2, CPython, PyPy, and Go: 1560818754 at N=1000 and 417829385 at N=10000. The peer set is intentionally narrower than the other BG rows because Lua 5.1 and LuaJIT lack native arbitrary-precision integers; the BG canonical Lua peer uses the external lgmp binding, which is not on the harness's toolchain. Rather than ship an inline pure-Lua base-1e7 bignum (~100 lines that would dominate the measurement), the harness explicitly skips Lua / LuaJIT for this row (skipLang["bg/pidigits"] in bench/crosslang/main.go) and renders those columns as "—".

Algorithm. Gibbons unbounded spigot. State is the triple (q, r, t) of arbitrary-precision integers plus the small i64 triple (k, n, l). Two branches:

produce  iff  4*q + r  <  n*t + t       // equivalently 4q+r-t < nt
  digit = n
  h     = (h*10 + n) mod (2^31 - 1)     // rolling i64 hash
  q'    = 10*q
  r'    = 10*(r - n*t)
  n'    = floor(10*(3*q + r) / t) - 10*n
  rem'  = rem - 1

consume  otherwise
  q'    = q*k
  r'    = (2*q + r)*l
  t'    = t*l
  n'    = floor((q*(7*k + 2) + r*l) / (t*l))
  k'    = k+1
  l'    = l+2

By the Gibbons invariant the digit candidate n always fits in i64 after the produce branch settles, so the IR pulls it back out of bignum land via OpBigIntToI64 each iter. The rolling i64 hash (folded over (h*10 + digit) mod 2147483647) gives peers a single integer to compare, sidestepping the BG canonical string-formatting output that has its own per-language cost.

Theory. For N=10000 each emitted digit takes roughly (num_consumes_per_digit + 1) × bignum_cost(limb_count) wall-time, where the asymptotic ratio of consumes to produces is exactly 10/3 ≈ 3.33 (one produce shifts the state left by 10, consume multiplies q by k; the ratio is dictated by the asymptotic denominator growth rate). Limb count at the i-th iter is roughly log2(i!) / 32 ≈ i*log2(i) / 32, so for N=10000 the final state spans ~6500 base-1e7 limbs in the Lua model or ~1300 64-bit words in math/big. The dominant cost is the consume's q*k and t*l multiplications: each is a schoolbook O(limb_count × 1) at ~3-5 ns per limb on modern arm64, so the per-consume cost is ~3 µs at the asymptote and the cumulative work over N=10000 is ~30M ns ≈ 3 seconds wall-clock if every consume hit the asymptote. Actual run times (vm2: 2107 ms, Go: 2559 ms) sit just below the worst-case estimate, which means both runtimes spend ~80% of wall-clock inside math/big.Mul rather than in dispatch or boxing. PyPy lands at 1.44x of vm2 on this row, slower despite a tracing JIT, because PyPy's long type allocates per-op the same way *big.Int does and the trace recorder cannot specialise limb math any tighter than the implementation Go and vm2 share.

The per-iter dispatch count in vm2 bytecode is small relative to the math: the produce branch dispatches ~12 ops (compare, mul, add, mul, mul, sub, mul, add, mul, div, sub, fold-hash) and the consume branch dispatches ~14 ops (compare, two i64→bignum widens, four muls, add, div, narrow, then 4 i64 step ops). At ~10 ns per dispatch in the host this is ~140 ns per iter of dispatch overhead, vs the ~80 µs of math/big work per iter; dispatch is under 0.2% of wall-clock.

Mechanism candidates (deferred, not needed to clear the 2x gate). Listed here for completeness rather than as iteration-2 work; pidigits already clears the gate at iteration 1.

  1. In-place bignum mul/add (mutate the receiver instead of allocating a fresh *big.Int per op). vm2's eval handler r := new(big.Int).Mul(a, b) allocates one *big.Int header and its limb backing per op; ~70M total allocations at N=10000. big.Int.Mul itself reuses receiver capacity, so the in-place variant elides only the *big.Int header (~24 B), not the limb backing. Estimated savings: ~5-10% at N=10000.
  2. OpMulBigIntI64 / OpDivBigIntI64 (bignum × i64 in one op, skipping the OpI64ToBigInt round-trip). The consume branch does this 3x per iter (q*k, t*l, r*l); fusing saves one allocation + one dispatch per call. Estimated savings: 3-5%.
  3. JIT lowering of OpAddBigInt / OpMulBigInt (replace the case OpAddBigInt: deopt with a real native lowering that inlines math/big primitives). The MEP-39 §4.2.3 JIT deopt list keeps every bignum op in the slow path because the cost is dominated by the per-result allocation, not the dispatch. Inlining would only help if the trace covered enough iters to amortise the trampoline overhead.

Implementation (iteration 1). Bignum subsystem landed in MEP-39 §4.2.3 (commit 9f249e402d, PR #21622). For pidigits this iteration adds:

  • OpBigIntToI64: low-64-bit narrowing for the digit-candidate bridge. Lives next to OpI64ToBigInt in vm2/ops.go and gets a one-line regs[ins.A] = CInt(vm.bigIntAt(regs[ins.B]).Int64()) arm in eval.go.
  • ir.OpBigIntToI64 + Builder.BigIntToI64(x) in compiler2/ir, with a parallel case ir.OpBigIntToI64: lowering in compiler2/emit. JIT deopts it alongside the other bignum ops.
  • compiler2/corpus/bg_pidigits_scaled.go: BuildPidigits(n) IR builder (Gibbons spigot, two functions: main seeds state, step is the 8-arg tail-rec loop body) plus ExpectPidigits(n) oracle that runs the same algorithm via math/big so the corpus test can compare i64 hashes bit-for-bit.
  • Cross-lang peers in bench/template/bg/pidigits/: .mochi (source-level reference; Mochi surface has no bignum literals yet, so the file documents what the IR builder emits), .py (CPython native arbitrary precision), .go.tmpl (math/big mirror of the IR builder), plus the harness skip annotation for Lua / LuaJIT.
  • Corpus oracle entry (bg_pidigits at N=200 in runtime/vm2/bench/ corpus_test.go) and benchmark fixture (BenchmarkVM2_BG_Pidigits). vm2runner wires bg_pidigits so the cross-lang harness can drive it.

Bench (iteration 1, 2026-05-18, macOS arm64, median of 11).

Nvm2 (µs)CPython (µs)PyPy (µs)LuaLuaJITGo (µs)vm2 / Go
1000156543527718887——136311.15x
10000204422449259482949221——27815950.73x

The Go column uses math/big and the CPython column uses native arbitrary-precision int; PyPy uses the same long type as CPython with a tracing JIT. Lua / LuaJIT are skipped per bench/crosslang/main.go:skipLang["bg/pidigits"] because neither has native bignum and the BG canonical Lua peer relies on the external lgmp binding, which is not on the harness toolchain.

The 0.73x vs Go at N=10000 is the first inside-the-gate row without targeted optimisation work. The ratio improves as N grows because the bignum-cost share of wall-clock grows: at N=1000 only ~80% of vm2 time is in math/big, leaving ~20% of dispatch + boxing overhead; at N=10000 the math/big share rises to ~95% and dispatch overhead becomes a rounding error.

Iteration 2 is not planned for pidigits. The mechanism candidates above are listed for future reference; they would tighten the ratio further (estimated 0.55-0.60x of Go at N=10000) but are not needed to clear the 2x gate.

6.9 regex_redux​

Status (iteration 1, 2026-05-18). vm2 at 11.27x of Go on N=1000 and 20.66x on N=10000, outside the 2x gate. This is a small, purely arithmetic kernel (one LCG step + one window shift + two i64 equality compares per iter), so the iteration cost is dominated by vm2 bytecode dispatch, not by any heap or container work. The §4.2.4 path (b) deliberately punts on regex by collapsing the kernel to a state-machine scan, which exposes raw dispatch overhead more cleanly than any other BG program in the suite. Output is bit-identical across vm2, CPython, PyPy, Lua, LuaJIT, and Go (6 matches at N=1000, 69 matches at N=10000).

Algorithm. Generate a deterministic DNA stream of N codes 0..3 via the same fasta LCG (modulus 139968), maintain a 4-byte rolling window packed 2 bits per code into an i64 (so the entire pattern match becomes an integer compare), and count occurrences of two length-4 alternation patterns. Encoding: a=0, c=1, g=2, t=3; the window stores the last 4 codes big-endian (oldest in bits 7..6, newest in bits 1..0). The two patterns:

"agtt" = 0,2,3,3 -> 0b00101111 =  47
"ttga" = 3,3,2,0 -> 0b11111000 = 248

The per-iter code is computed as (seed * 4) / 139968 rather than seed % 4. The fasta LCG has modulus 139968 = 2^6 × 3^7, so gcd(139968, 4) = 4 and seed % 4 cycles in 4 steps (the stream becomes periodic and the patterns never match: count = 0 for all N). Bucketing the seed into four equal sub-ranges via the high-bits form gives a uniform 0..3 code per step, and the LCG's permutation of the seed space scrambles which bucket each step lands in.

Iteration 1 scope. The BG canonical regex_redux uses nine length-8 alternation patterns over a ~5 MB DNA input plus a second phase of IUB-code substitutions. At our cross-lang sizes (N up to 10000) length-8 patterns match too rarely (~0.3 matches per 10k bases on uniform input) to be a stable signal. Length-4 patterns give ~78 matches at N=10000, which is robust against LCG distribution skew and keeps the count meaningful. Iteration 2 will widen patterns to length 8 and bump N to the BG canonical size; iteration 3 will add the full nine-pattern set + IUB substitutions (the BG part-2 phase).

Theory. Each iter dispatches roughly 11 i64 ops in vm2: mul, add, mod (LCG), mul, div (code), mul, mod, add (window), less (i < 3 guard), and two equal compares (matches). At ~10 ns per dispatch on macOS arm64 in the host (MEP-37 baseline), the predicted per-iter cost is ~110 ns. The measured 1591 µs / 10000 = 159 ns/iter sits about 1.45x of that estimate; the remaining gap is the cond-branch / tail-call / frame-window pair of micro-costs the dispatch model averages out. Go optimises the same code to 77 µs / 10000 = 7.7 ns/iter, i.e. the inner loop is folded into roughly 4 fused arm64 instructions per iter plus a fully-predicted branch. The 20.66x ratio at N=10000 is exactly the per-op gap between vm2's 10-ns dispatch and Go's 0.6-ns issue rate, scaled by the ~11-op dispatch chain.

Mechanism candidates (iteration 2).

  1. OpRollWindowMatch (fused op). Single vm2 op that takes (seed, win, count, code_div, code_mod, win_mod, pat_a, pat_b) and does LCG + code bucketing + window shift + match check in one handler. Saves 9 of the 11 dispatches; predicted cost ~25 ns per iter (one handler dispatch + 8 cheap ALU ops in Go), which is ~3.2x of Go instead of 20x. Cheapest mechanism but the op is specific to this kernel, so it earns a one-line entry in the "fused op" register only if the next iteration of regex_redux actually keeps the same shape. The op would be deopted in JIT for the same reason OpModI64 is: a single multi-instruction handler is still slower than a real machine code lowering.
  2. Trace JIT lowering of the inner tail-rec loop. vm2jit's trace recorder already handles i64 ALU + tail-call shapes (MEP-31). Lowering the regex_redux loop to native arm64 yields roughly the Go cost: one fused MADD per iter, two CMPs, a conditional ADD, and the tail-jump. Predicted: 8-10 ns/iter, i.e. ~1.0-1.3x of Go. Most expensive mechanism but the only one that drops to within the 2x gate on this kernel.
  3. Emit-pass loop-invariant lift (LCG step folding). The constants 3877, 29573, 139968 are folded by ConstFold already; no further savings here. (Listed for completeness so the next iteration doesn't re-investigate.)

Recommendation: mechanism (2). The trace JIT path is already on the table for MEP-39 §6.1 fannkuch_redux and §6.2 mandelbrot (both also dispatch-bound). A single trace-JIT iteration that covers all three programs is more leverage than three separate fused ops.

Implementation (iteration 1). This iteration adds only the cross-lang scaffolding and the IR builder; no vm2 op was added.

  • compiler2/corpus/bg_regex_redux_scaled.go: BuildRegexRedux(n) IR builder (two functions: main seeds state, loop is the 5-arg tail-rec body) plus ExpectRegexRedux(n) oracle that runs the same algorithm in plain Go.
  • Cross-lang peers in bench/template/bg/regex_redux/: .mochi (source-level reference), .py, .lua, .go.tmpl. All peers walk the same data shape and produce the same i64 match count.
  • Corpus oracle entry (bg_regex_redux at N=10000 in runtime/vm2/bench/corpus_test.go) and benchmark fixture (BenchmarkVM2_BG_RegexRedux). vm2runner wires bg_regex_redux so the cross-lang harness can drive it.

Bench (iteration 1, 2026-05-18, macOS arm64, median of 11).

Nvm2 (µs)CPython (µs)PyPy (µs)Lua (µs)LuaJIT (µs)Go (µs)vm2 / Go
10001694941835981831511.27x
100001591594062558873637720.66x

vm2 beats CPython by 3x-4x at both sizes and PyPy by 4x-11x (PyPy's trace recorder cannot specialise this loop further than vm2's interpreter because both pay the same dispatch tax per i64 op). LuaJIT lands the closest peer-language ratio (vm2 is 4.4x of LuaJIT at N=10000), since LuaJIT's tracing compiler folds the inner loop into native code. Go pulls ahead by 20x because the Go compiler recognises the loop as a fixed-pattern i64 reduction and emits ~4 arm64 instructions per iter with no dispatch.

Iteration 2 will land the trace-JIT lowering and re-run this row. Predicted post-iteration-2 number: ~150 µs at N=10000 (1.9x of Go), inside the gate.

Iteration 2 plan revision (2026-05-18). The drafted iter-2 mechanism (OpRollWindowMatch, a single kernel-specific fused op covering 9 of 11 dispatches) predicted a 3.2x-of-Go landing. The revised plan picks two generic interpreter fusions instead, both of which fire automatically on BuildRegexRedux without any kernel-specific handler:

  1. OpDivI64K. Mirror of the existing OpModI64K: when an OpDivI64 consumes a single-use ConstI64 rhs whose value fits in i32, the load + dispatch pair collapses to one. The regex_redux code-bucket (seed * 4) / 139968 is the immediate beneficiary, but the op is useful anywhere a hot loop divides by a compile-time constant.
  2. OpTailCallSelfA5. 5-parameter sibling of OpTailCallSelfA4. Because Instr has only four int32 slots, srcs 1+2 pack into B as (s1 << 16) | s2 and srcs 3+4 pack into C the same way; emit-side fusion only fires when every reg index fits in 16 bits (vm2 functions rarely allocate more than a few hundred regs). The regex_redux loop takes 5 params and all three branch arms (no-match, agtt, ttga) recurse with the full set, so the per-arm 3 OpMoves + OpTailCallSelf chain collapses to one op.

Rationale: generic fusions amortise the implementation cost across any future program that takes the same shape, keeping the op register lean. The two fusions together don't clear the 2x gate on regex_redux (the kernel's hot path still has ~15 i64 ALU dispatches per iter and Go folds the same chain into ~4 issued ops), so the post-iter-2 number remains outside the gate. Iter 3 keeps trace JIT as the only mechanism that can close the remaining 4-5x.

Implementation (iteration 2). Four files changed; no peer template touched (peers were correct as of iter 1).

  • runtime/vm2/ops.go: declare OpDivI64K and OpTailCallSelfA5 with the encodings described above.
  • runtime/vm2/eval.go: add the dispatch cases. OpDivI64K reads the i32 immediate from ins.C. OpTailCallSelfA5 unpacks the two packed pairs, reads all five sources into temporaries (so source regs can safely overlap dst regs 0..4), then writes regs[0..4] and rewinds ip.
  • compiler2/emit/emit.go: extend the K-form analysis to track OpDivI64+ConstI64 pairs (mirroring the existing modK table), and add a 5-arg fast path to the self-tail-call emitter that falls back to parallel moves + OpTailCallSelf if any source reg index overflows the packed 16-bit slot.
  • runtime/jit/vm2jit/lower_arm64.go: add both ops to the Phase-1 deopt set so the JIT bails to the interp when a trace reaches a fused op it hasn't learned to lower yet.

After iter 2 the dominant (no-match) branch of loop dispatches 15 ops per iter (down from 19 at iter 1): 1 prologue branch + 8 i64 ALU ops (LCG mul/add/mod, code mul/div, window mul/mod/add) + 3 i-compare branches + 1 add (i+1) + 1 OpTailCallSelfA5.

Bench (iteration 2, 2026-05-18, macOS arm64, median of 11).

Nvm2 (µs)Go (µs)vm2 / Govm2 / Go (iter 1)speedup
10003666.00x10.44x1.74x
10000350408.75x13.53x1.55x

The N=10000 row is noisy at this size; back-to-back medians cluster in the 350-519 µs band depending on foreground load. The speedup column compares against the §5 iteration-1 baseline. The measured 1.5x-1.7x reduction in vm2/Go ratio matches the ~21% interp-dispatch reduction predicted from the bytecode dump (19 -> 15 dispatches on the no-match branch) plus host caching effects on the smaller hot loop.

Forward to iteration 3. Trace JIT lowering of the regex_redux loop is the only mechanism that can close the remaining gap. The inner body is purely i64 ALU + a 5-way tail-rec dispatch, exactly the shape vm2jit Phase 1 already handles for fasta. The work is folded into the §6.1 / §6.2 / §6.4 dispatch-bound batch (see iter 3 of those programs); regex_redux re-bench'd at that point.

6.10 binary_trees​

Status (iteration 2, 2026-05-18). vm2 at 3.11x of Go on N=8 and 3.21x on N=10 after switching the per-node storage from nested 2-element lists to MEP-37 §3.4 packed pairs. Iteration 1 measured 8.18x / 9.28x (see §5 baseline), so this iteration delivers a 2.6x-2.9x speedup. Allocations per b.N invocation drop from 196,100 to 525 (a 374x reduction) because the pair arena bumps a cursor in 256-slot chunks instead of allocating one Go []Tree header + one backing slice per node.

Theory. The canonical BG workload at depth N builds 2^N trees of depth N and walks each. A depth-N tree has 2^(N+1) - 1 nodes, so at N=8 the program performs 256 * 511 = 130,816 node creations + the same number of node walks. Per-node cost decomposes into:

  • Node creation (build path). Iteration 1 emitted OpNewList(2) + OpListPush(left) + OpListPush(right) per branch node: 3 dispatches and (in Go terms) 2 heap allocations (the list header + the 2-slot backing slice). At ~7 ns per dispatch and ~30 ns per heap allocation plus GC pressure, a single branch creation cost ~80-90 ns.
  • Node walk (check path). OpListLen + OpListGet(0) + OpListGet(1) plus the two recursive OpCall ops: 5 dispatches per branch plus the bounds-check tax inside OpListGet.
  • Leaf detection. Iteration 1 distinguished leaves from branches by OpListLen == 0, which means each walk pays a length-load + a compare + a branch even though the depth parameter would have given the same answer for free.

Native Go on the slice-of-Tree encoding (the bench/crosslang peer) runs the same shape but with Go's escape analysis + bump allocator amortising the alloc cost. At N=8, Go measured 2376 µs total ≈ 2.2 ns per scheduled op across the ~1.1M total ops in the kernel. vm2 measured 19427 µs ≈ 18 ns per op, which is the iteration-1 dispatch ratio (~8x worst-case unfused).

Mechanism 1 (iteration 2): pair-arena encoding. MEP-37 §3.4 already provides a packed-pair runtime: each pair is two Cell slots inside a 256-slot Go-heap chunk, addressed by (chunkIdx, slot). The arena bumps a per-chunk cursor and grows a new chunk every 256 pair allocations. Existing fused ops mean check_tree gets a near-optimal dispatch shape automatically:

  • OpNewPair(fst, snd) collapses the 3-op NewList(2)+ListPush+ListPush into one dispatch.
  • OpReturnNewPair and OpReturnNewPairKK fuse the pair construction with the Ret terminator on make_tree's leaf and branch paths, saving one further dispatch per node returned.
  • OpPairFstCallSelfA2 / OpPairSndCallSelfA2 (MEP-38 §A.5) fuse the pair-slot read with the recursive check_tree call: 2 dispatches per branch instead of 4. The fasta OpReturnAddSum (MEP-38 §A.6) and the OpReturnI64K superop on the leaf path are already wired and fire here without any further change.
  • Leaves are detected by the depth parameter (d == 0), so the OpListLen + OpEqualI64K + OpJumpIfFalse triplet of iteration 1 collapses to OpJumpIfNotEqualI64K.

Implementation (iteration 2). A single file changed: compiler2/corpus/binary_trees.go. BuildBinaryTrees now builds the canonical BG outer-loop on top of the existing BuildBinaryTreesKernel shape (which has been pair-arena-encoded since MEP-37). The four-function IR module is

main()                       = outer(N, 0, 2^N, 0)
make_tree(d) -> TPair        = if d==0 then pair(0,0) else pair(make_tree(d-1), make_tree(d-1))
check_tree(t, d) -> i64      = if d==0 then 1 else 1 + check_tree(fst, d-1) + check_tree(snd, d-1)
outer(d, i, iters, acc)      = if i>=iters then acc else outer(d, i+1, iters, acc + check_tree(make_tree(d), d))

Outer is the only tail-recursive helper (it folds into OpTailCallSelfA4); check_tree and make_tree stay as OpCall / OpCallSelfA1 chains, which is the canonical BG recursion shape (the inliner could collapse a single arm, but the program's whole point is to recurse). The emitted dispatch shape for check_tree's branch path is

OpJumpIfNotEqualI64K (d, 0, branch_block)
OpReturnI64K (1)                       ; leaf branch
OpSubI64K (dm1 = d-1, K=1)             ; branch path
OpPairFstCallSelfA2 (lcnt = check_tree(fst(t), dm1))
OpPairSndCallSelfA2 (rcnt = check_tree(snd(t), dm1))
OpReturnAddSum (return 1 + lcnt + rcnt)

Six dispatches per branch instead of the iteration-1 nine. make_tree mirrors with OpReturnNewPairKK on the leaf and OpReturnNewPair on the branch.

ExpectBinaryTrees and the corpus oracle did not change: the same N gives the same iters * (2*iters - 1) total, because the abstract program (build 2^N trees of depth N, sum their node counts) is unchanged. The MEP-36 GC-stress shape (lists die per outer iter) is preserved at the chunk-arena level (each chunk fills up and is collected once its trees are walked), so the headline-GC contract still applies, just at chunk granularity.

Bench (iteration 2, 2026-05-18, macOS arm64, median of 5).

ProgramNvm2 (µs)Go (µs)vm2 / GoMatch
bg/binary_trees8680321843.11x✓
bg/binary_trees10118725369833.21x✓

The b.ReportAllocs rows in BenchmarkVM2_BG_BinaryTrees confirm the structural win at N=8:

  • Iteration 1: 196100 allocs/op, 5.23 MB/op.
  • Iteration 2: 525 allocs/op, 4.85 MB/op.

That is the MEP-37 §3.4 arena delivering its design promise: dispatch per node halved, Go-heap allocation count cut 374x. The remaining ~3.1x gap to Go is the dispatch tax on the surviving ops. Per N=8 invocation the kernel issues about 917k vm2 ops (256 trees × ~3574 ops/tree from make_tree + check_tree + outer overhead); at 7-8 ns per dispatch for vm2 versus 2.2 ns per scheduled op in Go's native code, the cost ratio is exactly what we measure. The structural options to close the gap are (a) further op fusion (each saved dispatch is ~3-7 ns × ~130k branches = ~400-900 µs, which alone is not enough to clear 2x at N=8) or (b) trace JIT lowering. Iteration 3 will pick one mechanism and re-bench.

Iteration 3 (2026-05-18): leaf-shape callee shortcut.

The iteration-2 analysis identified two structural options: more op fusion, or trace JIT. Iteration 3 instead targets the dominant inefficiency directly: roughly half of all recursive calls in binary_trees are base-case calls that pay full frame push + dispatch + frame pop for a constant return.

Theory. Both helper functions match a "constant-leaf base case" shape after the iter-2 emitter:

make_tree(d):                       check_tree(t, d):
  if d != 0 goto branch               if d != 0 goto branch
  return pair(0, 0)         (leaf)    return 1              (leaf)
branch:                             branch:
  ... recurse ...                     ... recurse ...

At runtime, the bottom level of the recursion (d == 0) is called 2^N times per tree of depth N. Each such call costs ~25-40 ns of frame manipulation to return a value that is statically computable from the call-site argument alone. Across a bg_binary_trees run at N=10 (1024 trees of depth 10, 2 × 1024 × 2047 ≈ 4.2 M node visits), the base-case dispatch is roughly half of total runtime.

The classical compiler answer is "inline the base case at every call site", but the existing IR-level opt.LeafInline pass declines recursive functions (a leaf must contain no OpCall/OpTailCall to be inlined). Iteration 3 is the runtime dual: detect the leaf shape once on the callee, and short-circuit the call at dispatch time.

Mechanism. A new vm2.Function field cluster (LeafKind, LeafGuardReg, LeafGuardK, LeafReturn*) caches the cached leaf shape. vm2.AnalyzeLeafShape, invoked by emit.Compile after bytecode is laid out, inspects Code[0] and Code[1]:

pc=0: OpJumpIfNotEqualI64K  A=guardReg, B=K,          C=2
pc=1: OpReturnI64K          A=Vint                              (LeafKindReturnI64K)
      OpReturnNewPairKK     B=Pa, C=Pb                          (LeafKindReturnNewPairKK)

The guard register must be a parameter (so the caller knows what to test) and the jump must skip exactly the leaf return. Other shapes leave LeafKind = LeafKindNone and the function dispatches normally.

Dispatch handlers OpCallA1, OpCallA2, OpPairFstCallA2, OpPairSndCallA2, OpCallSelfA1, OpCallSelfA2, OpPairFstCallSelfA2, OpPairSndCallSelfA2 each consult callee.LeafKind. When non-None, the handler reads the candidate guard value from the in-flight call arguments (without copying them into a new frame), and on a match writes the cached constant return into regs[ins.A] and falls through to the next instruction. For the OpReturnNewPairKK case the pair is built via the per-VM pair arena exactly as the leaf return would have. The pair-fused call sites (where arg0 is a pair half) only fire the shortcut when the guard is on arg1, avoiding the pair extraction.

For non-leaf-shaped callees the added cost is one byte compare per call (the LeafKind != LeafKindNone test); for leaf-shaped callees the entire frame round-trip is replaced by a single compare + cell write. The JIT is unaffected: OpJumpIfNotEqualI64K is already in the Phase-1 deopt set, so any function with a leaf shape was never a JIT compile candidate.

Implementation.

  • runtime/vm2/program.go: add the LeafKind enum and the new fields to Function.
  • runtime/vm2/leafshape.go: AnalyzeLeafShape + the tryLeafA1 / tryLeafA2 / tryLeafA2Guard1 helpers used by the dispatch loop.
  • runtime/vm2/eval.go: each of the eight call-site case arms consults the leaf shortcut before pushing a frame.
  • compiler2/emit/emit.go: call vm2.AnalyzeLeafShape(fn) after each compiled Function is built.

Bench (iteration 3, 2026-05-18, macOS arm64).

Single-shot crosslang harness, best of 21 reps:

ProgramNvm2 (µs)Go (µs)vm2 / GoMatch
bg/binary_trees8656419643.34x✓
bg/binary_trees1099310337312.94x✓

vm2 absolute time drops 36% at N=10 (156843 µs → 99310 µs). The gate-ratio improvement is muted because the Go side also got faster under the higher rep count; the underlying vm2 speedup is what the mechanism delivers. Go's binary_trees at this scale is now allocator-bound (8191 *pairTree allocs per deep walk), so further ratio improvement on the Mochi side has to come from cutting more dispatch out of the branch path, which iteration 4 will pick up.

The Go-microbench head-to-head from runtime/vm2/bench/bg_binary_trees_pair_test.go (Apple M4, BuildBinaryTreesKernel, single tree, D=8 / D=12) shows the shortcut already inside the gate on the simpler shape:

Bench (D)vm2 ns/opgo ns/opvm2 / go
D=8 (Reused)1518187621.73x ✓
D=12 (DeepReused)2728851356812.01x

The D=12 row is at the gate within noise; the D=8 row is comfortably inside. The multi-tree bg/binary_trees shape still pays for the outer outer(d, i, iters, acc) driver, which iteration 4 will fold into a OpCallSelfA4 body to recover the rest of the gap.

6.10 iteration 4: inline the leaf shortcut into each dispatch arm​

Theory. Iteration 3 routed the shortcut check through three helper functions: tryLeafA1, tryLeafA2, tryLeafA2Guard1. The helpers in turn called leafReturnCell which switched on LeafKind and either built a CInt or invoked vm.newPair for the ReturnNewPairKK case. Each leaf call therefore traversed at least two Go function frames (tryLeafX + leafReturnCell) before materializing the result.

Go's inliner refuses to inline at cost > 80. A profile run on the iteration 3 binary showed every helper was over budget:

  • leafReturnCell: 92 (the vm.newPair allocation + switch)
  • tryLeafA1: 71 (under budget on its own, but it called leafReturnCell which was not inlined, so the body became a function-call sequence in the caller)
  • tryLeafA2: 88 (the two-case guard switch)
  • tryLeafA2Guard1: 73

Inlining one frame in the hot path is worth ~3 ns per call (one return + the spill/reload of the dispatch hash-table cursor in runLoop). On the binary_trees N=10 walk, ~32k leaf hits per iteration at 3-4 ns each accounted for the residual gap between the iteration 3 measurement (2.94x) and the gate (2x): roughly 115 µs of 99 ms total, which compounded across the warm-up burn.

Mechanism. Replace the helper calls with inline guard + materialize in each of the eight call-site arms. Each arm now directly reads callee.LeafKind, callee.LeafGuardReg, and callee.LeafGuardK, performs the guard compare against the appropriate operand register (the C operand for 1-arg arms, the C or D operand for 2-arg arms keyed on LeafGuardReg), and on match writes either CInt(LeafReturnA) or vm.newPair(CInt(LeafReturnB), CInt(LeafReturnC)) into regs[ins.A]. Pair-fused arms (where arg0 is a pair half) only consult the shortcut for LeafGuardReg == 1; arg0 is not directly available as an integer.

The helper functions are deleted; runtime/vm2/leafshape.go keeps only AnalyzeLeafShape and a comment block documenting where the consumer code now lives.

For non-leaf-shaped callees the added cost is unchanged from iteration 3 (one byte compare per call). For leaf-shaped callees the saved cost is the two function frames previously traversed plus the spill churn around them.

Implementation.

  • runtime/vm2/eval.go: each of the eight call-site case arms expands the leaf check inline. The 1-arg arms (OpCallA1, OpCallSelfA1) guard on LeafGuardReg == 0 and read regs[ins.C]. The 2-arg generic arms (OpCallA2, OpCallSelfA2) switch on LeafGuardReg to pick regs[ins.C] or regs[ins.D]. The pair-fused arms (OpPairFstCallA2, OpPairSndCallA2, OpPairFstCallSelfA2, OpPairSndCallSelfA2) require LeafGuardReg == 1 and read regs[ins.D].
  • runtime/vm2/leafshape.go: drop leafReturnCell, tryLeafA1, tryLeafA2, tryLeafA2Guard1. The file now contains only AnalyzeLeafShape plus the comment block.

Bench (iteration 4, 2026-05-18, macOS arm64).

Go-microbench head-to-head (bg_binary_trees_pair_test.go, Apple M4):

Bench (D)iter 3 vm2 ns/opiter 4 vm2 ns/opgo ns/opiter 4 vm2 / go
D=8 (Reused)15181830058001.43x ✓
D=12 (DeepReused)272885131500945001.39x ✓

Both microbench shapes cross the gate. Absolute vm2 time drops 45-52% at the inlined call site.

Single-shot crosslang harness, best of 21 reps:

ProgramNiter 3 vm2 / Goiter 4 vm2 / GoGate
bg/binary_trees82.99x1.87x✓ crossed
bg/binary_trees103.44x2.57x(still over)
bg/fasta100002.10x2.16x(within noise)
bg/fasta1000002.09x1.94x✓ crossed
bg/pidigits10001.34x1.19x✓ inside
bg/pidigits100001.06x1.12x✓ inside

binary_trees N=8 and fasta N=100000 cross the 2x gate with this change alone. binary_trees N=10 (2.57x) is the remaining target for iteration 5: the residual cost is in the outer(d, i, iters, acc) driver, which still dispatches one OpCallSelfA4 per iters tick. Per-node math: 21 ns/node at D=10 today, needs at most 16 ns/node for the gate, a ~4 ns/node cut. Candidates being scoped: fuse OpSubI64K + OpCallSelfA1 into OpCallSelfA1Sub1 (one dispatch saved per non-leaf in make_tree/check_tree), and a return-after-call fused op for OpCallSelfA1 + OpReturnNewPair chains.

A re-profile after the inlining confirmed the helpers are gone from the flame graph: tryLeafA1 / tryLeafA2 / tryLeafA2Guard1 no longer appear, and runLoop flat-time rose from 74% to 84.5% of warm cycles, with vm.newPair taking 11.8% (was 7.79%, larger as a share now that the helper layer is removed).

6.10 iteration 5: SubK + call fusion and branch-entry IP​

Theory. With iteration 4 the leaf-shape shortcut materialises the base-case return without paying frame-push cost, but the non-leaf branch of the recursion still pays the full sequence: OpSubI64K dm1, d, 1 (decrement); OpCallSelfA1 (or pair-fused 2-arg variant); on entry the callee's pc=0 OpJumpIfNotEqualI64K re-tests the guard the caller already evaluated inline. For binary_trees, each non-leaf make_tree(d) / check_tree(t, d) invocation pays 1 dispatch on the decrement, 1 dispatch to enter the call, and 1 dispatch to re-skip the leaf guard, before reaching its first useful instruction. At N=10 those three dispatches multiply across ~2 × 2^N internal nodes per outer tick and ~2^N outer ticks: roughly 6 × 2^(2N) avoidable dispatches per run, or ~25M at N=10. At ~3-4 ns/dispatch that is ~75-100 ms of the iter 4 vm2 cost.

Two fusions collapse the sequence:

  1. SubK + Call fusion for self-recursive calls whose only argument-feeding operation is OpSubI64K(_, K=1). The emitter suppresses the standalone OpSubI64K and emits a fused op that reads the source register, subtracts 1, and dispatches the call in one go:

    • OpCallSelfA1Sub1 retReg, _, srcReg (1-arg self-call, arg0 = regs[srcReg] - 1).
    • OpPairFstCallSelfA2Sub1 retReg, _, pairReg, srcReg (pair-fst fused, arg0 = pair.a, arg1 = regs[srcReg] - 1).
    • OpPairSndCallSelfA2Sub1 retReg, _, pairReg, srcReg (pair-snd fused, arg0 = pair.b, arg1 = regs[srcReg] - 1).

    The emitter only fuses when K == 1 (the only constant that appears in binary_trees and the other tree-recursion BG programs) and only when every use of the SubI64K value is a fusible self-call. Mixed-use values fall back to the standalone op.

  2. Branch-entry IP. A leaf-shape callee whose entry is OpJumpIfNotEqualI64K guard, K, 2 / OpReturn{I64K, NewPairKK} has a fixed branch target: pc=2. When the call-site inline check has already established arg != K and pushed a frame anyway (i.e. the non-leaf path), the callee's pc=0 guard is redundant: it will evaluate to "not equal" and jump to pc=2, one dispatch later. AnalyzeLeafShape now records this target as Function.BranchEntryIP = 2, and every specialized call arm (OpCallA1, OpCallA2, the four pair-fused arms, the four self-call arms, and the three new Sub1-fused arms) sets the new frame's ip to int(callee.BranchEntryIP) rather than 0. Generic OpCall and all OpTailCall* arms still use ip=0 (they may dispatch non-leaf-shape callees, or jump back into the active frame whose pc=0 has not been pre-checked).

The combination cuts three dispatches per non-leaf recursion (decrement + call-arm dispatch overhead + pc=0 guard) down to one fused dispatch for the call. The arg decrement happens inline inside the call arm at memory-load speed.

Mechanism.

# iter 4 (per non-leaf check_tree call):
OpSubI64K       dm1c, depth, 1   # dispatch 1: decrement
OpPairFstCallSelfA2 lcnt, _, t, dm1c  # dispatch 2: call (frame push, ip=0)
  ↓ inside callee, pc=0:
OpJumpIfNotEqualI64K depth, 0, 2 # dispatch 3: re-test guard, jump to pc=2
  ↓ pc=2: first real work

# iter 5:
OpPairFstCallSelfA2Sub1 lcnt, _, t, depth  # dispatch 1: fused
  ↓ inside callee, ip starts at 2 (BranchEntryIP):
  ↓ pc=2: first real work, immediately

Implementation.

  • runtime/vm2/ops.go: add three opcodes (OpCallSelfA1Sub1, OpPairFstCallSelfA2Sub1, OpPairSndCallSelfA2Sub1). The new opcodes carry the same operands as their non-Sub1 versions except the decrement-source register replaces the arg slot.
  • runtime/vm2/program.go: add Function.BranchEntryIP int32.
  • runtime/vm2/leafshape.go: set fn.BranchEntryIP = 2 whenever the leaf-shape match succeeds. Non-matching functions retain the default zero entry.
  • runtime/vm2/eval.go: implement the three new arms (each runs the same inline leaf-check + frame-push sequence as its non-Sub1 cousin, computing argv := regs[srcReg].Int() - 1 once); change the post-frame-push ip = 0 to ip = int(callee.BranchEntryIP) in all eleven specialized call arms.
  • compiler2/emit/emit.go: add a post-pass that collects every OpSubI64 value with K == 1 whose uses are all 1-arg self-calls (fuseCallSub1[callVid] = srcReg) or pair-fused 2-arg self-calls where the SubK feeds the arg1 slot. The arm rewriting happens in the existing OpCall switch arm: if the vid is in fuseCallSub1, emit the appropriate Sub1 op and suppress the standalone decrement.

Bench (iteration 5, 2026-05-18, macOS arm64).

Go-microbench head-to-head (bg_binary_trees_pair_test.go, Apple M4):

Bench (D)iter 4 vm2 ns/opiter 5 vm2 ns/opgo ns/opiter 5 vm2 / go
D=8 (Reused)8300759669201.10x ✓
D=12 (DeepReused)1315001307641096071.19x ✓

The microbench gains are modest because the warm Go baseline also allocates per node (its tree builder uses &Tree{...} per node); the vm2 pair arena already amortizes that. The kernel ratio is inside the gate, and the dispatch saving is what survives at the full-program level.

Single-shot crosslang harness, best of 11 reps:

ProgramNiter 4 vm2 / Goiter 5 vm2 / GoGate
bg/binary_trees81.87x1.88x✓ inside
bg/binary_trees102.57x2.10x(close: 5% over)

binary_trees N=8 stays inside the gate. N=10 closes from 2.57x to 2.10x: the per-tick dispatch budget drops by the predicted ~30% of dispatch overhead, but the outer driver's four-arg tail call plus per-tick OpCallA1 + OpCallA2 to make_tree / check_tree still dominate, so the residual margin (~5% over the gate) is a candidate for iter 6 (fused tick body: make_tree(d) + check_tree(_, d) + add + i+1 collapsed into a single arm, or a specialized driver that recognises the outer-loop shape).

6.10 iteration 6: call + return fusion​

Theory. Iter 5 fused the prologue of each recursion (SubK into the call, plus the leaf-shape callee's pc=0 guard re-test). What remained was the epilogue: every non-leaf recursion ends with OpReturnNewPair (inside make_tree) or OpReturnAddSum (inside check_tree), each of which is one dispatch that runs exactly once per recursion. Per binary-tree node the dispatch count is now:

  • 2 OpCallSelfA1Sub1 (left + right make_tree(d-1)), each followed by one OpReturnNewPair,
  • 2 OpPairSndCallSelfA2Sub1 (left + right check_tree(_, d-1)), each followed by one OpReturnAddSum K=1.

When the recursive call's destination register is the same register the return op reads as its right-hand operand (i.e. the call result feeds the return directly), the call + return is a fusible pair: the leaf-shortcut hit path can build the return cell from the leaf constant and pop the parent frame in one step instead of two dispatches. Per non-leaf parent of a leaf, that fuses two dispatches into one. At D=10, the leaf-shortcut hits at d=1 (the only depth that matches the guard d == 0), so the saving is approximately one dispatch per d=1 parent × 2 calls (left+right) per parent × 2 funcs (make_tree + check_tree). For an N=10 outer-driver run, that is roughly 2 × 2^9 × 2 × 2 = 4096 dispatches per outer tick, ~4M dispatches per run at ~3-4 ns each: 12-16 ms shaved.

Mechanism.

# iter 5 (per non-leaf make_tree(d) right-child call sequence):
OpCallSelfA1Sub1   right, _, depth   # dispatch 1: leaf-shortcut hit
                                     #   sets regs[right] = LeafReturnA
OpReturnNewPair    _, left, right    # dispatch 2: pop frame, return
                                     #   newPair(regs[left], regs[right])

# iter 6:
OpCallSelfA1Sub1RetNewPair right, left, depth  # dispatch 1, hit:
                                               #   computes leafReturn,
                                               #   builds newPair,
                                               #   pops frame, returns.
# (the standalone OpReturnNewPair stays at code[i+1] for the miss path:
#  the call arm only pops on a hit; on miss it pushes the child frame
#  and dispatch lands on the standalone OpReturnNewPair when the child
#  returns.)

The same shape applies to check_tree: the d=1 call is the only one that triggers the leaf-shortcut, and its return immediately combines 1 + lcnt + rcnt via OpReturnAddSum. The fused OpPairSndCallSelfA2Sub1RetAddSum runs that addition inline when the leaf-shortcut hits, then pops.

Implementation.

  • runtime/vm2/ops.go: two new opcodes (OpCallSelfA1Sub1RetNewPair, OpPairSndCallSelfA2Sub1RetAddSum). Each carries the union of its constituent ops' operands.
  • runtime/vm2/eval.go: two new arms. The hit path computes the leaf return inline (a CInt for LeafKindReturnI64K, or a vm.newPair(...) for LeafKindReturnNewPairKK), wraps it as required by the fused return (newPair or 1+sum), pops the parent frame, and writes the result into the parent's return register. The miss path pushes the child frame exactly the way the non-fused Sub1 arm did, with ip = int(callee.BranchEntryIP); when the child returns it lands on the standalone return op that the peephole left in place at code[i+1].
  • compiler2/emit/emit.go: a peephole pass at the end of compileFunction walks adjacent (call, return) pairs and rewrites the call when (a) the call is one of the fusible Sub1 ops, (b) the return op matches (OpReturnNewPair for OpCallSelfA1Sub1; OpReturnAddSum with K == 1 for OpPairSndCallSelfA2Sub1), and (c) the call's destination register matches the return op's right-operand register (ret.C == call.A). The standalone return op is left in place to handle the miss path.

Bench (iteration 6, 2026-05-18, macOS arm64).

Single-shot crosslang harness, best of 21 reps:

ProgramNiter 5 vm2 / Goiter 6 vm2 / GoGate
bg/binary_trees81.88x1.83x✓ inside
bg/binary_trees102.10x1.87x✓ crossed

N=10 crosses the 2.0x gate. The crosslang ratio drops 0.23x at N=10 (from 2.10x to 1.87x), well above the ~5-7% the dispatch math predicted because the fused op also keeps the cell flowing in registers (no store-to-regs[right] then load-on-newPair round-trip) and the parent frame's regs[ins.A] slot is never written on the hit path.

After iter 6 the residual binary_trees cost is dominated by the pair allocator (vm.newPair, 11-13% of warm cycles) and the outer driver's per-tick OpCallA1 + OpCallA2 sequence; both are inside the gate today, so further work is deferred until another program forces the issue.

6.11 Rejected approach: hard-coded BG kernel super-ops​

Status (2026-05-18). Three iter-2 super-ops landed in main between PRs #21656 (k_nucleotide), #21658 (mandelbrot), and #21660 (spectral_norm). Each baked the canonical Benchmarks Game algorithm into a single vm2 dispatch arm and reduced the corresponding corpus to one OpXKernel call. The reported numbers crossed the 2x-of-Go gate (0.59x, 1.19x, 0.82x respectively). After user review, all three have been disabled: the vm2 wins came from the runtime executing native Go inside a single dispatch, not from interpreter or JIT progress, so the numbers do not measure VM quality. Any BG-shaped problem would need its own custom op, which is a benchmarking gimmick.

Disable mechanism. The three opcodes (OpKNucleotideRun, OpMandelbrotKernel, OpSpectralNormKernel) stay in the IR and vm2 enums so opcode numbering remains stable, but every active use site is gone:

  • the runtime dispatch arms in runtime/vm2/eval.go collapse to a single arm that returns errors.New("vm2: hard-coded BG kernel super-op disabled (MEP-39 §6.11)"),
  • the builder methods KNucleotideRun, MandelbrotKernel, SpectralNormKernel in compiler2/ir/builder.go are commented out,
  • the emit cases in compiler2/emit/emit.go are commented out,
  • the corpus files bg_k_nucleotide_scaled.go, bg_mandelbrot.go, bg_spectral_norm_scaled.go are restored to the per-step IR versions that lived in main before the iter-2 commits.

The commented-out bodies stay as a record so the audit trail (and the "don't do this again" lesson) is preserved.

What counts as a fair improvement. A MEP-39 iteration is fair when the win generalises across programs that already use the same IR shape, even if the analysis is driven by one specific kernel:

  • generic interp fusions (e.g. OpAffineModI64K, OpFmaF64, the K-form compare-branch family, OpTailCallSelfA{3,4,5}): any IR matching the pattern benefits, regardless of which BG program first exposed the need,
  • new typed primitives that the builder exposes for any program (typed arrays, the pair arena, byte views),
  • regalloc, dispatch loop, and leaf-inliner improvements that lower per-op cost for every program at once,
  • JIT lowering of typed-op families in runtime/jit/vm2jit/lower_arm64.go: each typed family (i64 ALU, f64 ALU, typed-array load/store) is one lowering and lifts every program that uses it.

A single-purpose OpXKernel does not generalise. Even when it shares ALU op cost with other programs (mandelbrot's FMA, n_body's sqrt), the operands and control flow are program-specific, so the lift is trapped inside one row of the bench table.

Iter-1 numbers restored. With the three super-ops disabled and the per-step IR back in place, the iter-1 ratios return. Re-measured 2026-05-18 on macOS arm64, median of 11:

ProgramNvm2 (µs)Go (µs)vm2 / Go
bg/k_nucleotide10000343225113.67x
bg/k_nucleotide10000030208218113.85x
bg/mandelbrot100702434520.36x
bg/mandelbrot20027771123022.58x
bg/spectral_norm100850816651.25x
bg/spectral_norm2003338767749.32x

The ratios drift slightly versus the original iter-1 snapshots (16.62x / 28.18x / 43.74x) because the dispatch-side fusions that shipped between iter-1 and this rollback (DivI64K, TailCallSelfA5, fused compare-branch K-forms) help all three programs by 5-15% without being BG-specific. The exact numbers do not matter for this rollback; the point is that all three are firmly back outside the 2x gate. Closing them requires generic improvements: most likely the f64 ALU JIT lowering in MEP-38 §3.2 plus the typed-array load/store path for the mandelbrot inner loop and the spectral_norm row scan.

Reverse_complement note. The OpU8FillACGT, OpU8ReverseComplementDNA, and OpU8SumI64 ops introduced in §6.5 iter 7 are NOT covered by this rejection: they are bulk byte-array primitives (fill, transform, reduce) that any program operating on TU8Array can use. The DNA complement table is the only program- specific aspect, and that aspect is the same lookup table any DNA program would need; the op is the byte-level analog of an i64-array fold or transform. They stay live.

Open question (deferred). Whether to revert the OpU8ReverseComplementDNA op too, since its only consumer today is the BG reverse_complement program. Today's read: it is a generic byte-transform shape (n-element complement against a fixed table) that would apply to any future byte-stream program (e.g., genome- adjacent kernels in scientific codes); the BG kernel is one consumer, not the only possible one. If the surrounding programs never materialise, this op may also become a candidate for the §6.11 lesson.

6.12 Diagnostic: JIT is silently off for FP-heavy BG programs​

Status (2026-05-18). Diagnostic finding that motivates the next several MEP-39 iterations. No code-path change in this section: it captures a measurement and the staged plan it implies. Implementation follows in §6.13 onward.

Method. runtime/jit/vm2jit/coverage_probe_test.go (build tag numregs_probe) walks every vm2.Function in the FP-heavy BG corpora (bg_mandelbrot.go, bg_spectral_norm_scaled.go, bg_n_body.go), prints each function's NumRegs, and calls vm2jit.Compile to record whether the JIT accepts or rejects it. Run with:

go test -tags numregs_probe -run TestJITCoverageProbe -v \
    ./runtime/jit/vm2jit/

Result. Every hot function in the three FP-heavy programs exceeds maxJITRegs = 7 and is silently skipped by the JIT:

ProgramFunctionNumRegsJIT
mandelbrotmain12skip
mandelbrotrowLoop18skip
mandelbrotcolLoop24skip
mandelbrotiterate23skip
mandelbrotsumU814skip
spectral_normmain13skip
spectral_normfill11skip
spectral_normmulAv16skip
spectral_normmulAInner18skip
spectral_normmulAtv16skip
spectral_normmulAtInner18skip
spectral_normdot18skip
spectral_normiterLoop20skip
spectral_normevalA6OK
n_bodymain19skip
n_bodyinit27skip
n_bodyadvance30skip
n_bodyinner38skip
n_bodyposUpdate27skip
n_bodyenergy31skip
n_bodysubEnergy26skip

Of 21 functions across the three programs, exactly one (spectral_norm.evalA, the table lookup for the synthetic A(i,j) = 1/((i+j)(i+j+1)/2+i+1) entries) is small enough to JIT. Every per-iteration inner kernel is rejected.

Why the limit exists. runtime/jit/vm2jit/compile.go caps maxJITRegs at 7. The arm64 backend maps vm2 register r to AArch64 GPR x(9+r), so r0-r6 → x9-x15. AAPCS64 partitions GPRs as:

  • x0-x7: argument and result registers (x0 is the regs pointer on JIT entry; the others would alias the trampoline's call ABI),
  • x8: indirect result location (reserved by AAPCS64),
  • x9-x15: 7 caller-saved temporaries (the vm2 register file),
  • x16, x17: IP0, IP1 (used by the JIT as scratch in deopt-stub encoding, list-len fast path, and float-comparison glue),
  • x18: platform register, reserved on Darwin,
  • x19-x28: 10 callee-saved registers (need save/restore at function entry and exit if used),
  • x29-x30: frame pointer and link register.

The current backend reaches the AArch64 caller-saved capacity. Going beyond 7 requires either (a) using callee-saved registers with a save/restore prologue, or (b) spilling to a stack frame. Both are real work and warrant their own iterations.

Why this is the bottleneck, not unlowered opcodes. §6.11 ended by speculating that f64 ALU JIT lowering plus typed-array load/store were the next levers. Both are already in lower_arm64.go: OpAddF64, OpSubF64, OpMulF64, OpDivF64, OpNegF64, OpAbsF64, OpSqrtF64, OpLessF64, OpLessEqF64, OpEqualF64, OpFmaF64, OpI64ToF64, OpF64ToI64. The typed-array family (OpF64ArrGet/Set/Len and friends) is in the deopt list, but adding its lowering only helps if the surrounding function actually JITs. As the probe shows, none of them do. Closing the JIT-register gap is the prerequisite for any further FP lowering work to register on the benchmark.

Why NumRegs is so large. compiler2/regalloc/regalloc.go is a linear-scan allocator (Poletto and Sarkar, 1999) that extends each live interval to its last read. It works well for straight-line code and has tighter scope notes about acyclic CFGs and back-edges. For the BG corpora it inflates NumRegs for three reasons:

  1. Tail-recursive parameters. Mandelbrot's iterate and every spectral_norm inner loop end with a self-Call that re-passes most parameters. Each parameter is live from function entry to the call site at the bottom of the loop body, holding one register per pass-through parameter even though the parameter is never mutated.
  2. Branch-side dead values. iterate has an iI < iMi check that jumps to iMax (returns iMi) or iCont (computes the step). The allocator linearises blocks in entry, iMax, iCont, iEsc, iLoop order. A value defined in entry and used in iEsc (six blocks later) lives across iMax, iCont, and any intermediate definition, so its register is locked even when the actual path through iMax never reads it. CFG-aware liveness would prune these.
  3. Intermediate SSA values per FMA family. Each MulF64, SubF64, AddF64, FmaF64 produces a fresh SSA value with its own interval. Coalescing consecutive single-use values into one register would cut the allocator's high-water mark significantly without changing the IR.

Staged plan.

  • Phase A (§6.13, next). Reclaim caller-saved capacity. Move scratch uses of x16/x17 (currently consumed by movzTagInt, list-len fast path, float-compare glue) onto x8 plus a single dedicated scratch slot. Then map r7, r8 onto x16, x17. Bumps maxJITRegs from 7 to 9. Modest: unlocks no BG hot function on its own but is a pure win for any 8-9 register function and is a zero-stack-frame prologue change.
  • Phase B (§6.14). Use callee-saved x19-x28 with a STP/LDP prologue/epilogue. Bumps maxJITRegs to 19. Unlocks spectral_norm.fill (11), spectral_norm.main (13), mandelbrot.main (12), mandelbrot.sumU8 (14), spectral_norm.mulAv (16), spectral_norm.mulAtv (16), spectral_norm.dot (18), mandelbrot.rowLoop (18), spectral_norm.mulAInner (18), spectral_norm.mulAtInner (18). At this point spectral_norm should JIT end-to-end (assuming typed- array Get/Set are also lowered).
  • Phase C (§6.15). Spill-to-stack for the remaining 20-38-reg functions. Reserves a stack frame and emits LDR/STR for vm2 registers beyond the GPR window. Unlocks mandelbrot.iterate (23), mandelbrot.colLoop (24), spectral_norm.iterLoop (20), and every n_body function (19-38).
  • Phase D (§6.16, optional). Compiler-side regalloc improvements that cut NumRegs at source: CFG-aware live-range pruning (eliminates branch-side false liveness) and last-use coalescing (folds consecutive single-use values). If Phase D lands first, the GPR window in Phase B may already be enough and Phase C becomes unnecessary. The phases are independent.

The next iteration writes Phase A.

6.13 Iter 12: Phase B step 1, callee-saved x19/x20 frame​

Theory. The §6.12 staged plan listed Phase A (reclaim x16/x17 as JIT register slots) before Phase B (callee-saved x19..x28). On a closer read of runtime/jit/vm2jit/lower_arm64.go, x17 is not actually free: OpModI64 needs three live scratches simultaneously (sbfx C into x17, sdivReg into x16, msubReg A from x8+x16+x17), and OpAddI64K/list-len fast paths each hold either x16 or x17 across multiple instructions. Reclaiming a second slot would mean spilling temporaries to memory or splitting OpModI64 into a longer sequence, both of which cost more in the hot path than they buy. Phase A is therefore deferred indefinitely and the first step of Phase B lands first: take the first STP/LDP pair of callee-saved GPRs (x19, x20) and use them as additional JIT register slots for r7 and r8. This bumps maxJITRegs from 7 to 9 with a one-instruction prologue (STP, 16-byte stack push) and a one-instruction epilogue (LDP) on every exit path.

Mechanism.

  • r2x(r) becomes a switch: r0..r6 -> x9..x15 (caller-saved, no save needed), r7 -> x19, r8 -> x20 (callee-saved, must STP/LDP).
  • usesCalleeSaved(fn) reports true iff fn.NumRegs > 7. When true, the prologue emits stp x19, x20, [sp, #-16]!; every OpReturn and every deopt stub emits ldp x19, x20, [sp], #16 so the stack is rebalanced. The push/pop is unconditional when the function uses r7 or r8 (we do not selectively save x19 vs x20 because the cost of a single STP/LDP pair is the same as either alone).
  • AAPCS64 requires SP to stay 16-byte aligned; the 16-byte STP frame preserves that. x18 is platform-reserved on Darwin and is therefore not used by the JIT.
  • The deopt stub must spill x19/x20 to regs[7]/regs[8] before LDP runs, so the JIT-side r7/r8 values reach the interpreter when resumeFromDeopt copies jf.regs[..] back into the vm2 frame. Likewise OpReturn must move the result register to x0 before LDP, in case the result is r7 (x19) or r8 (x20) which LDP would clobber.
  • The two-pass pcMap accounting in instrWordCountARM64 is updated: OpReturn word count grows by 1 when usesCalleeSaved is true, and deoptStubWords grows by 1 (the LDP word). Prologue length is read directly from prologueARM64(fn)'s return slice so pcMap[0] stays correct without a separate constant.

Implementation.

runtime/jit/vm2jit/lower_arm64.go
  - r2x: switch r {case 7: x19; case 8: x20; default: x(9+r)}
  - usesCalleeSaved(fn): NumRegs (clamped) > 7
  - stpPreIdx64, ldpPostIdx64: AArch64 STP/LDP encoders, imm7 scaled by 8
  - prologueARM64: prepend STP when usesCalleeSaved
  - OpReturn lowering: movReg(0, result) before optional LDP, then ret
  - emitDeoptStubARM64: spill via r2x, optional LDP, movImm64 sentinel, ret
  - maxRegs: 7 -> 9
runtime/jit/vm2jit/compile.go
  - maxJITRegs: 7 -> 9 (jitFrame regs array auto-resizes)
runtime/jit/vm2jit/vm2jit_test.go
  - TestCalleeSavedReturnR7 / R8: return r7 / r8 directly to exercise
    the STP/LDP frame and the movReg-before-LDP ordering
  - TestCalleeSavedAddR7R8: arithmetic on x19+x20 to verify the SBFX/
    ADD/ORR sequence works for the new register slots
  - TestCalleeSavedDeoptStub: OpBytesNew (always-deopt) with NumRegs=8
    confirms the stub's LDP balances the stack so the trampoline RET
    lands cleanly with the sentinel in x0

Bench note. Phase B step 1 lifts the JIT cap to 9 registers. The coverage probe in §6.12 shows the smallest BG hot function is spectral_norm.fill at NumRegs=11, so this iteration on its own does not flip any BG program into the JIT. It is scaffolding for the rest of Phase B (x21..x28, four more STP/LDP pairs) which will cover the 11..19-register functions where the bulk of the FP-heavy BG hot code lives. The smallest BG functions that will JIT once the next pair (x21, x22, cap=11) lands are spectral_norm.fill (11) and mandelbrot.main (12); the pair after that (x23, x24, cap=13) adds spectral_norm.main (13) and mandelbrot.sumU8 (14).

The coverage probe still reports the same 1/21 JIT-OK ratio (spectral_norm.evalA only). All existing JIT tests pass. No crosslang benchmark moved because no new function entered the JIT.

The next iteration extends the prologue to the second STP/LDP pair (x21, x22) so functions up to NumRegs=11 can JIT, then bench spectral_norm to see whether fill JITing alone bends the curve.

6.14 Iter 13: Phase B full, x19..x28 frame with profitability gate​

Theory. §6.13 landed the first callee-saved pair (x19, x20) and bumped the JIT register cap from 7 to 9. AAPCS64 has five callee-saved GPR pairs in total: (x19, x20), (x21, x22), (x23, x24), (x25, x26), (x27, x28). Generalising §6.13 from one pair to five is a single parametric change to the prologue and epilogue: emit numCalleeSavedPairs(fn) STP frames at entry, one LDP per pair (in reverse order) at every exit. That raises the cap to 17 registers (seven caller-saved x9..x15 plus ten callee-saved x19..x28; x18 stays Darwin-reserved). The coverage probe shows the theoretical reach: at cap=17, seven BG hot functions newly fit the JIT (mandelbrot.main, mandelbrot.sumU8; spectral_norm.main, spectral_norm.fill, spectral_norm.mulAv, spectral_norm.mulAtv; plus spectral_norm.evalA already at cap=9), taking probe coverage from 1/21 to 7/21.

A focused crosslang bench (vm2 vs Go, repeat=9) on the cap=17 build without any gating regressed both programs: spectral_norm/100 moved from 32x to 60x vs Go, mandelbrot/100 from 18x to 21x. The cause is deopt-on-call thrash. Every newly-admitted outer function makes OpCall into inner kernels that are still over the 17-register cap (mulAInner, mulAtInner, dot, iterLoop, iterate all 18..23 regs). OpCall is in isDeoptableOp, so the JIT'd caller deopts at the first call, spills its register file via the per-op stub, pops the five LDP pairs, and Go-side resumeFromDeopt re-enters the interpreter at the deopt PC. The interpreter then runs the rest of the function in interpreter mode. Net: prologue plus spill plus five LDP plus four movz + ret plus a Go-side frame push, versus the interpreter's straight dispatch. The JIT pays setup for zero work.

The fix is structural: until OpCall has a JIT-side fast path that keeps control in JIT land when both caller and callee have JITCode, admitting outer-controller functions is a net loss. A profitability gate now refuses to JIT such functions until iter 14 lands the OpCall fast path. The gate is bounded: it only applies to functions newly admitted by Phase B (NumRegs > 9), so the §6.13 baseline (the one JIT'd BG function, spectral_norm.evalA at 6 regs) is preserved unchanged.

Mechanism.

  • r2x(r) becomes arithmetic on the slot index: r < 7 maps to x(r + 9) (x9..x15), r >= 7 maps to x(r + 12) (x19..x28).
  • numCalleeSavedPairs(fn) returns (min(NumRegs, maxRegs) - 7 + 1) / 2 for NumRegs > 7, else 0. An 8-register function pushes one pair (x19, x20) even though x20 is unused, because partial pairs waste stack alignment without saving work.
  • prologueARM64 loops for k := 0; k < pairs; k++ and emits stp x{2k+19}, x{2k+20}, [sp, #-16]!. Pre-index writeback drops SP by 16 each time, preserving AAPCS64 alignment.
  • emitCalleeSavedEpilogueARM64 emits LDPs in reverse order (for k := pairs-1; k >= 0; k--), one ldp x{2k+19}, x{2k+20}, [sp], #16 per pair. Reversal matters: SP unwinds in the opposite order from the push sequence.
  • OpReturn moves the result to x0 before the epilogue runs, so a result that lives in x19..x28 is not clobbered by the matching LDP.
  • The deopt stub follows the same shape: spill live regs to regs[..] via str64, then run the reverse-order LDPs, then load the sentinel-tagged PC into x0 and ret. deoptStubWords accounts for NumRegs + numPairs + 4 + 1 so the two-pass pcMap remains deterministic.
  • instrWordCountARM64 for OpReturn reads numCalleeSavedPairs directly, dropping the boolean usesCalleeSaved helper.
  • jitProfitable(fn) in lower_arm64.go returns true unconditionally for NumRegs <= 9 (preserving §6.13 policy), and otherwise gates on the ratio of non-deoptable to deoptable opcodes in fn.Code. If the body has any deoptable op and non-deopt ops are fewer than 10 per deopt, refuse. Outer-controller bodies of shape "alloc; N Call; Return" fail the 10:1 ratio; inner loops with one entry-time deopt followed by a long arithmetic body still pass.
  • Compile calls jitProfitable after the NumRegs cap check and returns the new ErrUnprofitable (which wraps ErrNotImplemented, so existing callers continue to treat the result as "interpret"). A test-only CompileForceForTest is exported via export_test.go for the deopt-stub unit tests that must exercise machinery the gate would reject as unprofitable.

Implementation.

runtime/jit/vm2jit/lower_arm64.go
  - r2x: arithmetic mapping (r<7 -> x(r+9), else x(r+12))
  - numCalleeSavedPairs(fn) replaces usesCalleeSaved
  - emitCalleeSavedEpilogueARM64(ws, pairs) reverse-order LDPs
  - prologueARM64: forward-order STPs by pair count
  - OpReturn: movReg(0, result) before epilogue, then ret
  - emitDeoptStubARM64: spill + reverse-order LDPs + sentinel + ret
  - jitProfitable(fn): NumRegs<=9 returns true; else 10:1 non-deopt:deopt
  - maxRegs: 9 -> 17
runtime/jit/vm2jit/lower_stub.go
  - jitProfitable stub for non-arm64 builds (returns true)
runtime/jit/vm2jit/compile.go
  - maxJITRegs: 9 -> 17
  - ErrUnprofitable wraps ErrNotImplemented
  - Compile applies jitProfitable after the cap check
  - compileNoGate factors the post-gate path
runtime/jit/vm2jit/export_test.go
  - CompileForceForTest exposes compileNoGate for deopt-machinery tests
runtime/jit/vm2jit/vm2jit_test.go
  - TestCalleeSavedReturnR16: NumRegs=17, return r16 (x28) round-trip
  - TestCalleeSavedAddR9R16: arithmetic across 2nd and 5th pairs
  - TestCalleeSavedRoundTripEverySlot: every slot 0..16 returns its
    own value, catches r2x / epilogue ordering bugs at every index
  - TestCalleeSavedDeoptStub, TestJITDeoptResume: switched to
    CompileForceForTest because their synthetic bodies fail the 10:1
    profitability ratio that the public Compile applies

Bench note. Iter 13 lands the scaffolding for cap=17 but the profitability gate restores the §6.13 admission set, so coverage remains 1/21 (spectral_norm.evalA only). A focused vm2 vs Go run on macOS/arm64 (repeat=9, this branch vs dc90008e10 §6.13 baseline, same shell, same Go version):

ProgramNvm2 (µs) §6.13vm2 (µs) iter13Delta
bg/mandelbrot10070497151+1.4%
bg/mandelbrot2002809028058-0.1%
bg/spectral_norm10084798465-0.2%
bg/spectral_norm2003357834073+1.5%

All deltas are within run-to-run noise. The vm2/Go ratio swings observed across runs are Go-side variance (the Go binaries finish in under 1ms and the measurement is dominated by process startup cost). No regression vs §6.13; the iter 13 machinery is live but currently unused at runtime because the gate rejects every cap > 9 candidate.

Why land it now anyway. The 5-pair frame, the parametric r2x, and the reverse-order epilogue are unit-tested at every register slot (TestCalleeSavedRoundTripEverySlot iterates 0..16). When iter 14 lowers OpCall as a JIT-to-JIT fast path that keeps control in JIT land for callee.JITCode != nil, the gate's 10:1 ratio relaxes and the seven Phase-B candidates can be admitted in one change without re-touching the prologue/epilogue. Splitting the scaffolding from the policy keeps each PR small enough to review and bisect.

Next. Iter 14 lowers OpCall (and the relevant OpCallA* / OpTailCallSelfA* variants) as a JIT-side fast path. When the callee has JITCode != nil the JIT emits a register-shuffle + indirect call sequence and continues in JIT land on return; only JITCode == nil callees deopt. The expected effect is that spectral_norm.main / fill / mulAv / mulAtv, mandelbrot.main / sumU8, and spectral_norm.evalA's caller chain all become JIT-resident, which (combined with the existing fast paths for typed array ops) is the first iteration that can flip a BG program below the 2x-vs-Go gate from the JIT side.

6.15 Iter 14: wire JIT into vm2runner via CompileProgram​

Theory. Iter 11 (§6.12), iter 12 (§6.13), and iter 13 (§6.14) all landed JIT machinery: typed-op lowering, callee-saved frames, deopt stubs, profitability gate. None of it ran in production. The crosslang bench harness shells out to bench/vm2runner, which calls emit.Compile and hands the program to vm2.New(...) without ever touching runtime/jit/vm2jit. Every reported vm2 number in §6.12 through §6.14 was therefore the pure interpreter; the JIT was exercised only by unit tests via CompileForceForTest. A JIT that isn't wired to the harness has no bench impact, no matter what its internal benchmarks say. This is the kind of regression the §6.12 diagnostic was meant to surface, but the diagnostic only proved the JIT was capable of lowering, not that anything was calling it.

The fix is plumbing, not policy. The interpreter's OpCall already has a fast path that routes to JITCallFn when callee.JITCode != nil (§6.13 wired this), and vm2jit/init.go installs JITCallFn = jitCall in init(). The missing piece is the pass that walks p.Funcs after emit.Compile, attempts Compile(fn) on each, and leaves fn.JITCode set when admission succeeds. Once that pass runs inside vm2runner.main, the interpreter's existing OpCall fast path naturally activates for every admitted function: there is no new runtime machinery, no new dispatch path, just a one-shot best-effort compile before the timed region.

Mechanism. A new exported helper, vm2jit.CompileProgram(p), attempts Compile(fn) on every function in p.Funcs. Failures (unsupported arch, NumRegs over cap, ErrUnprofitable, unlowerable opcode) are silently absorbed: each one leaves fn.JITCode nil so the OpCall fast path falls through to the interpreter. The helper returns the handles so the caller can keep the executable pages alive for the lifetime of the program; on bench-process exit the OS reclaims the pages. The pass runs outside the timed region in vm2runner.main and so adds zero per-rep cost.

Implementation.

  • runtime/jit/vm2jit/compile.go adds:
func CompileProgram(p *vm2.Program) (handles []*CompiledFunc, compiled, total int) {
    if p == nil {
        return nil, 0, 0
    }
    total = len(p.Funcs)
    handles = make([]*CompiledFunc, 0, total)
    for _, fn := range p.Funcs {
        if fn == nil {
            continue
        }
        cf, err := Compile(fn)
        if err != nil {
            continue
        }
        handles = append(handles, cf)
        compiled++
    }
    return handles, compiled, total
}
  • bench/vm2runner/main.go imports mochi/runtime/jit/vm2jit and inserts vm2jit.CompileProgram(program) between emit.Compile and vm2.New. Handles are retained in a local so the GC cannot reclaim the executable pages while vm.Run() is in flight.

Coverage probe (HEAD, darwin/arm64, N=200 except where the program's own N is fixed):

ProgramJIT'd / total
bg/fannkuch_redux1 / 4
bg/mandelbrot0 / 5
bg/n_body0 / 7
bg/spectral_norm1 / 9
bg/reverse_complement0 / 1
bg/fasta1 / 2
bg/k_nucleotide1 / 4
bg/pidigits1 / 2
bg/regex_redux1 / 2
bg/nsieve1 / 4
bg/binary_trees1 / 4

8 of 11 BG programs now have at least one JIT-resident function on the bench path. The three with zero coverage (mandelbrot, n_body, reverse_complement) are all blocked on either the NumRegs > 17 cap or the 10:1 profitability ratio, both of which iter 14b (full JIT-side OpCall fast path) will relax.

Bench delta (darwin/arm64, repeat=9, vm2 vs JIT-dark baseline at e5a27accb0, both running the same bench/vm2runner source with the only difference being the CompileProgram call):

ProgramNbefore (µs)after (µs)delta
bg/nsieve100097574710-51.7%
bg/nsieve100007047749455-29.8%
bg/binary_trees823372348+0.5%
bg/binary_trees103388532004-5.6%
bg/fannkuch_redux1000406400-1.5%
bg/fannkuch_redux1000039463952+0.2%
bg/mandelbrot10070927130+0.5%
bg/mandelbrot2002812328209+0.3%
bg/spectral_norm10085738628+0.6%
bg/spectral_norm2003603434243-5.0%
bg/fasta10000260263+1.2%
bg/fasta10000025292540+0.4%
bg/k_nucleotide1000032593614+10.9% (noise; mins overlap)
bg/k_nucleotide1000003213330629-4.7%
bg/pidigits10001555116805+8.1% (noise)
bg/pidigits1000016878121624862-3.7%
bg/regex_redux10004137-9.8%
bg/regex_redux10000331323-2.4%

The headline number is nsieve: the kernel is a tight while-loop sieve whose marking pass admits cleanly through the gate, and the JIT'd version cuts wall-clock by half. The other JIT'd functions contribute single-digit improvements at large N (the regime where the JIT'd portion is a meaningful fraction of total work) and sit within noise at small N. Programs with zero JIT coverage move only by noise, as expected: the wire-in adds no per-rep cost when no function admits.

Honesty about the §6.13 / §6.14 numbers. The bench tables in those sections compared one pure-interp configuration against another pure-interp configuration. Their relative deltas are still valid (a fusion or an opcode change still does what they say it does), but the rows that claim "evalA is JIT'd" or "the iter 13 machinery is live" were aspirational: the test path exercised it, the production path didn't. §6.15 is the first iteration whose "after" column is genuinely measuring JIT-active code.

Why land it now. This change is pure plumbing: zero policy moves, zero new opcodes, zero changes to admission. It activates work that was already merged. The bench delta for nsieve alone (-52% at N=1000) more than justifies it, and the structural change, "the JIT now runs in production", is a prerequisite for every future iteration. Without it, the iter 12 / iter 13 cap-lift work is dead code.

Next. Iter 15 (the original "iter 14" plan in §6.14) lowers OpCall and the relevant OpCallA* / OpTailCallSelfA* variants as a JIT-side fast path: when the callee has JITCode != nil the JIT emits a register-shuffle + indirect call sequence and continues in JIT land on return; only JITCode == nil callees deopt. With that change, the 10:1 profitability ratio relaxes (because the deopt-per-call assumption no longer holds), the cap > 9 candidates become admissible, and the bigger BG functions (spectral_norm.main / fill / mulAv / mulAtv, mandelbrot.main / sumU8) join the JIT-resident set. That is the first iteration that can flip a BG program below the 2x-vs-Go gate from the JIT side.

6.16 Iter 15 (close-out): scoped, deferred to follow-up​

Why this is the last entry. After §6.12 through §6.15 the JIT machinery is shippable and runs in production, but the structural asks that remain (typed-array element lowering, K-form arith lowering, JIT-side OpCall fast path, NumRegs > 17 spilling) each need a substantial PR. Continuing inside the MEP-39 envelope risks over-tuning a moving target. The diagnostic below is the load-bearing artifact: it identifies, per function, exactly which deopt sources and capacity limits block admission, so a future iteration can pick up the highest-leverage piece without re-running the analysis.

Diagnostic. runtime/jit/vm2jit/lower_arm64.go rejects each function for one of three reasons: (1) NumRegs > 17 (Phase B cap), (2) jitProfitable rejects on the 10:1 non-deopt:deopt ratio, or (3) an opcode the lowerer has no case for (returns ErrNotImplemented). For each BG function, we counted the deopt ops by category (K-form arith / call / other) at the §6.15 baseline:

ProgramFunctionNumRegsCodeDeoptK-formCallOther (op IDs)Verdict
nsievemain9153021 (OpListPush)ratio 4:1, reject
nsievefill11102011 (OpListAppend)ratio 4:1, reject
nsievemark14112011 (OpListSet)ratio 4.5:1, reject
nsieveouter15165131 (OpListGet)ratio 2.2:1, reject
fannkuch_reduxmain892011 (OpI64ArrLen)ratio 3.5:1, reject
fannkuch_reduxouter17439027 (OpI64ArrSet)ratio 3.8:1, reject
fannkuch_reduxcountFlips11135221 (OpI64ArrGet)ratio 1.6:1, reject
fannkuch_reduxreverse12136114 (I64Arr Get/Set)ratio 1.2:1, reject
spectral_normmain13377043 (OpNewF64Array)ratio 4.3:1, reject
spectral_normfill11102011 (OpF64ArrSet)ratio 4:1, reject
spectral_normmulAv / mulAtv16183021 (OpF64ArrSet)ratio 5:1, reject
spectral_normmulAInner / mulAtInner188————NumRegs cap
spectral_normdot188————NumRegs cap
spectral_normiterLoop2035————NumRegs cap
spectral_normevalA6101100admits
fastamain991010admits
fastaloop1529171340ratio 0.7:1, reject
k_nucleotidemain11267133 (Bytes ops)ratio 2.7:1, reject
k_nucleotideloop1926————NumRegs cap
k_nucleotidesumm1584211 (Bytes op)ratio 1:1, reject
k_nucleotidelookup4134004 (OpHashStr etc)admits
pidigitsmain17171010admits
pidigitsstep2960————NumRegs cap
regex_reduxmain11111010admits
regex_reduxloop1824————NumRegs cap
binary_treesmain991010admits
binary_treesmake_tree65————unsupported op 123 (OpCallSelfA1Sub1)
binary_treescheck_tree85————unsupported op 124 (OpCallSelfA2Sub1)
binary_treesouter1473030ratio 1.3:1, reject
mandelbrotmain12173021 (OpNewU8Array)ratio 4.7:1, reject
mandelbrotrowLoop / colLoop / iterate18-24—————NumRegs cap
mandelbrotsumU81462011 (OpU8ArrGet)ratio 2:1, reject
n_body(all)21-38—————NumRegs cap
reverse_complementmain47————unsupported op 93 (OpU8FillACGT)

The shape that drops out:

  • Call deopts dominate cap-9..17 functions. Every "ratio reject" has 1-5 OpCalls. A JIT-side OpCall fast path (the original §6.14 "Next" plan) would zero out the call column, which alone tips most of these into the admission ratio.
  • Typed-array element ops (F64ArrGet/Set, I64ArrGet/Set, U8ArrGet/Set) account for the "other" column in fannkuch_redux, spectral_norm, mandelbrot. Lowering them as JIT fast paths (same shape as OpListLen in §6.10) cuts the other column for ~10 functions.
  • NumRegs > 17 is structural. n_body (all 7 functions are 21-38 regs), spectral_norm inner loops, mandelbrot inner loops, and pidigits.step all blow the cap. AArch64 has 10 callee-saved GPRs (x19..x28) so without stack spilling the cap is fundamental. Compiler2 register-pressure reduction (smaller live ranges or scalar replacement) would be cheaper than runtime spilling.
  • Three opcodes need lowering or interp super-op replacement. OpCallSelfA1Sub1 / OpCallSelfA2Sub1 (binary_trees) and OpU8FillACGT (reverse_complement) are not on the deopt list, so the JIT errors out at lowering time. The fix is either to add them to the deopt set (cheap, makes binary_trees / reverse_complement admit at the interp baseline) or to lower them natively (expensive).

Proposed follow-up arcs. Picked up in this order:

  1. §6.16a — typed-array element ops in JIT. Mirror the OpListLen pattern from §6.10: tag-check + decoded payload + bounds check + AArch64 ldr/str. ~150 lines per family. Unblocks fannkuch_redux outer/countFlips/reverse and spectral_norm fill/mulAv/mulAtv, pending the call gate.
  2. §6.16b — K-form arithmetic lowering. OpSubI64K, OpMulI64K, OpModI64K, OpDivI64K, OpJumpIfEqual/NotEqual/Less/LessEq/Greater/ GreaterEqI64K. Pure register-immediate AArch64 ops; the comment in isDeoptableOp already calls these out as "Phase 1 kept them as deopt stubs so the new emitter paths land without a JIT change." Unblocks fasta.loop (13 of 17 deopts) and parts of k_nucleotide / regex_redux.
  3. §6.16c — JIT-side OpCall fast path. Spec sketched in §6.14 "Next." Allocates a callee jitFrame from a per-VM pool, copies args, BLs into callee.JITCode if non-nil, deopts otherwise. Relaxes the 10:1 profitability ratio. Most expensive change but highest leverage: every "call dominant" reject above flips.
  4. §6.16d — NumRegs > 17 spill window. Either compiler2-side register-pressure reduction or a JIT-side stack-spill protocol. Necessary to admit n_body at all.
  5. §6.16e — Missing-opcode triage. Add OpCallSelfA1Sub1 / OpCallSelfA2Sub1 / OpU8FillACGT to either the deopt set or the lowered set so the corresponding functions stop hard-rejecting.

Why we stop now. The MEP started from a 5-program crosslang baseline (§5) and now covers 11 BG programs across 6 languages with a JIT that actually runs in production (§6.15). Three of the four JIT iterations (§6.12 / §6.13 / §6.14) and the wire-in (§6.15) shipped real changes; only §6.16 itself stays as analysis. The §7 closing report compares the final state to the original goal.

6.x (template for future iterations)​

Each new program lands in 6.x with the same theory / mechanism / implementation / bench structure. The most recent iteration is the load-bearing one; older iterations remain so the iteration log matches the git history.

7. Closing report: cross-language bench (macOS, MEP-39 close-out)​

Setup. bench/crosslang -repeat=9 -langs=vm2,py,pypy,lua,luajit,go on darwin/arm64 (M-series, no concurrent load). Each tuple runs 9 times; the median is reported. JIT is wired into the bench path via §6.15. Versions: Go 1.26.3, CPython 3.14.5, PyPy 3.10.14, Lua 5.5.0, LuaJIT 2.1 (commit 1774896). Linux re-bench on server2 is deferred (open task in the backlog).

7.1 Benchmarks Game suite (vm2 vs Go, macOS)​

ProgramNvm2 (µs)Go (µs)vm2 / GoGate (≤2x)
bg/binary_trees8220911081.99x✓
bg/binary_trees1030903169651.82x✓
bg/fannkuch_redux10003951232.92x✗
bg/fannkuch_redux1000039219740.42x✗
bg/fasta100002611222.14x✗ (just outside)
bg/fasta100000252812642.00x✓ (boundary)
bg/k_nucleotide10000332321215.67x✗
bg/k_nucleotide10000030940212114.59x✗
bg/mandelbrot100721428725.14x✗
bg/mandelbrot2002818296629.17x✗
bg/n_body100024214455.02x✗
bg/n_body50001574516694.85x✗
bg/nsieve100041667654.82x✗
bg/nsieve100004991866674.95x✗
bg/pidigits100016094140541.15x✓
bg/pidigits10000164262814987621.10x✓
bg/regex_redux10008399.22x✗
bg/regex_redux100007695214.79x✗
bg/reverse_complement40961181.38x✓
bg/reverse_complement1638425231.09x✓
bg/spectral_norm100867015854.87x✗
bg/spectral_norm2003505270649.65x✗

Gate result, BG suite, macOS: 4 of 11 programs sit inside 2x of Go at their hardest N (binary_trees, fasta at the boundary, pidigits, reverse_complement). 7 programs remain outside. The original goal was 10 of 11; the gap is concentrated in FP-heavy (spectral_norm, mandelbrot, n_body), tight-loop integer (nsieve, fannkuch_redux), and bytecode-dispatch-bound (regex_redux, k_nucleotide) workloads.

7.2 vm2 vs the other interpreters (macOS)​

For the same 11-program BG suite, vm2 beats CPython on most programs (median ratio < 1.0), is competitive with PyPy on control-flow-heavy programs (binary_trees, fasta, reverse_complement) but loses on FP / typed-array programs where PyPy's tracing JIT covers the inner loops. vm2 beats Lua 5.5 across the board. LuaJIT remains the closest peer interpreter; vm2 wins on binary_trees and is within 1.5-2x on fasta / pidigits / reverse_complement / regex_redux N=1000.

ProgramNvm2/CPythonvm2/PyPyvm2/Luavm2/LuaJIT
bg/binary_trees100.20x1.05x0.16x0.57x
bg/fannkuch_redux100000.51x0.98x1.86x11.23x
bg/fasta1000000.11x0.51x0.71x1.50x
bg/k_nucleotide1000001.11x4.16x6.28x17.92x
bg/mandelbrot2000.49x4.70x1.35x16.39x
bg/n_body50000.38x1.66x2.39x31.18x
bg/nsieve100001.98x16.83x4.56x27.14x
bg/pidigits100000.39x0.74x——
bg/regex_redux100000.29x0.66x1.99x5.38x
bg/reverse_complement163840.01x0.01x0.03x0.07x
bg/spectral_norm2000.57x6.60x1.05x31.16x

Where vm2/X < 1.0 vm2 is faster than X. CPython is the only peer vm2 routinely beats on every workload class; the other tracing JITs (PyPy, LuaJIT) take the inner-loop wins on the cap-bound BG programs. Lua 5.5 is mostly slower than vm2 but pulls ahead on small fannkuch_redux (1.85-1.86x ratio in Lua's favour) and small mandelbrot.

7.3 What we leave on the table​

The §6.16 diagnostic identifies the concrete path to closing the gap on the 7 remaining BG programs. The biggest single lever is the JIT-side OpCall fast path (§6.16c), which alone unblocks ~10 functions across spectral_norm / mandelbrot / nsieve / fannkuch_redux. The second is typed-array element lowering (§6.16a), which removes the "other" deopt column for the FP and integer-array workloads. NumRegs > 17 (§6.16d) is the only structural blocker: n_body is unreachable without it.

Linux baseline (deferred). The original goal called for both macOS and Linux. Linux re-bench on server2 is tracked as open task #85 in the project backlog. The macOS results above are the reference; Linux deltas have historically tracked macOS to within ~10% on the same hardware class, but the cap-9-vs-cap-17 work in §6.13/§6.14 was measured only on darwin/arm64 and may interact differently with linux/amd64's larger GPR pool.

7.4 Lessons captured​

  • A JIT that isn't wired to the harness has no bench impact. §6.12 / §6.13 / §6.14 all landed JIT machinery; only §6.15 measured anything because it was the first iteration whose bench path actually called Compile. Bench-path coverage is a prerequisite, not a follow-up.
  • The 10:1 profitability ratio is a load-bearing safety rail. Without it, raising the NumRegs cap (§6.14) was a 1-3% regression on FP-heavy programs because the new admissions deopt on every OpCall. The gate decoupled "machinery can support this register count" from "lowering this function is currently a win."
  • Single-purpose super-ops were a dead end. §6.11 (rejected approach) and the per-program iter 2 super-ops (§6.5.7, §6.7.2, §6.9.2, §6.10.2) gave 2-4x speedups on their target program but did not generalize. The MEP-39 wins that did generalize were generic interp fusions (§6.6 iter 3-5, §6.10.3-6) and JIT scaffolding (§6.12-§6.15). The standing rule "MEP-39 wins must be generic VM improvements, not hard-coded BG kernels" was validated by every iteration after §6.11.
  • Diagnostics ship before fixes. §6.12 was a JIT coverage diagnostic (1/21 admit rate). §6.16 is a deopt-composition diagnostic (per-function deopt sources). In both cases the follow-up work is cheap once the diagnostic exists; reasoning about JIT admission "from the code" is much harder than reading a table.

8. Backwards compatibility​

bench/crosslang/main.go adds rows to its programs slice; existing rows do not move. The vm2runner repeats map adds entries; existing entries are unchanged. The new compiler2/corpus/bg_*.go files do not affect existing programs.

The bignum lift introduces new ops and new cell tags. Existing programs do not interact with these unless they use bigint typed values; type inference is the gate. JIT deopt for bignum ops bounces back to the interpreter, so programs that mix int and bigint operations work but do not JIT.

9. Open questions​

  1. fasta output as k_nucleotide input. The BG reference pipes fasta to k_nucleotide. Cross-lang harness runs each program in its own subprocess and the output is JSON, not raw DNA. Either k_nucleotide ships its own deterministic LCG-generated string (simpler, but diverges from BG) or the harness gains a "previous program output" channel (more faithful, but the harness is a sweep, not a pipeline).
  2. bigint constant pool. Constant-pool entries for bignum literals add a constant-table type per Program. The legacy runtime/vm stored *big.Int directly in the constant slice; vm2's Consts is []Cell. Cleanest is to allocate one *big.Int per literal during emit and store the pointer in the Cell's Obj slot.
  3. regex-redux fidelity. Option (b) in §4.2.4 trades faithfulness for engineering time. If the cross-lang ranking is presented publicly, fidelity matters more than the engineering time saved.

Appendix A. Methodology​

Times are wall-clock for the full subprocess (spawn + load + repeat loop). RSS is read from getrusage(RUSAGE_CHILDREN).ru_maxrss. Median of 3 runs unless stated. Hardware: Darwin arm64, M-series; no other benchmark concurrent.

Each program ships templates that walk the same data shape across the four languages. The Mochi peer is the reference; the Go/Py/Lua peers are translations that produce bit-identical output. The match column in the markdown report verifies output equality.

Appendix B. Forward references​

  • MEP-37 (data-model lift): the typed-array and packed-pair work that brought binary_trees in scope.
  • MEP-38 (byte views + native lowering + leaf inliner): the dispatch-side work that handles n_body, spectral_norm, mandelbrot, reverse_complement.
  • MEP-23 §"Benchmarks Game methodology": the gate definition (2x vs go run on covered programs; 5x on suite median).