[maleadt]

With Julia built against LLVM 23 (JuliaLang/julia#63626), the compensated summation and dot product tests fail for Float32: 38 of 636 tests in the summation/compensated and dot product/compensated sets. Over 2200 random ill-conditioned sums (generate_sum(N, 1f5), N = 100:110), sum_oro gives:

	results ≠ exact reference	worst error
LLVM 22	8-11 / 2200	1 ulp
LLVM 23	~1390 / 2200	262144 ulp
LLVM 23, with this PR	6-9 / 2200	1 ulp

The cause is the @fastmath on the first line of two_sum. It marks that addition fast, which allows reassociation. In the horizontal reduction of Base.sum(acc::CompSumAcc{<:Vec}), the hi values form a chain s = ((s + x1) + x2) + .... LLVM 23's SLP vectorizer turns that chain into a vector reduction. Because the adds are fast, it computes the final s with a tree-shaped llvm.vector.reduce.fadd, while the error terms are still derived from the sequential partial sums:

%foldExtExtBinop    = fadd fast <4 x float> %shift, %res.i211          ; sequential partial sums,
%foldExtExtBinop357 = fadd fast <4 x float> %shift356, %foldExtExtBinop ; used for the error terms
...
%30 = call fast float @llvm.vector.reduce.fadd.v4f32(float 0.0, <4 x float> %res.i211)  ; final hi, tree order

The final hi then no longer matches the lo terms. The fast flag allows this, so it isn't a compiler bug; LLVM 22 just didn't vectorize this chain. An error-free transformation needs hi to be exactly fl(a + b).

The @fastmath came in with a39fd86 as the replacement for SIMDops.@fusible, which only meant "may contract into an FMA". @fastmath allows much more than that (reassociation, no NaN/Inf, no signed zeros), and none of it is safe inside an error-free transformation.

With this PR, the test suite passes on LLVM 23 (636/636) and still passes on LLVM 22, both on assertion builds of Julia master and of the LLVM 23 PR. I left the @fastmath on the error accumulators (acc.e += e). It only reorders the small correction terms, though dropping it would make results independent of vectorization choices.

Investigated with the help of Claude Code; reproduced and tested locally.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 47.77%. Comparing base (3427289) to head (e18c2a3).

Additional details and impacted files

@@            Coverage Diff             @@
##           master      #43      +/-   ##
==========================================
- Coverage   50.11%   47.77%   -2.34%     
==========================================
  Files          13       13              
  Lines         447      427      -20     
==========================================
- Hits          224      204      -20     
  Misses        223      223              

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