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Needs review. On a 30-collision Pb-Pb time frame, 7 of 4,416,506 clusters differed between the CPU and the Metal backend, each by one step of a packed width, sigmaTime or sigmaPad. Those widths were all within one float ulp below a quantisation step: sigma * 32 was 11.4999994, 14.4999992, 29.4999996 or 31.4999988, where the packing rounds at n + 0.5, so any difference in the last bit decides between n and n + 1. The CPU build compiles this with -ffast-math. Its arm64 code for finalize() computes 1 / Q once, the means as products with it, the variance as one fused multiply-add, fma(SS, 1 / Q, -mean * mean), and an IEEE sqrt. Metal's fast-math division and sqrt are not correctly rounded. Do the same operations on Metal explicitly, with precise::divide, fma and precise::sqrt: the pbpb clusters then match the CPU bit for bit, with no measurable cost; the rest of the Metal code keeps fast math. This matches the CPU build that the compiler produces on arm64 with clang. An x86 build, or another compiler, may choose a different sequence under -ffast-math. Writing the sequence explicitly for all backends would make the widths independent of those choices, but can change the CPU results on some platforms.
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Needs review. On a 30-collision Pb-Pb time frame, 7 of 4,416,506 clusters
differed between the CPU and the Metal backend, each by one step of a
packed width, sigmaTime or sigmaPad. Those widths were all within one
float ulp below a quantisation step: sigma * 32 was 11.4999994,
14.4999992, 29.4999996 or 31.4999988, where the packing rounds at n + 0.5,
so any difference in the last bit decides between n and n + 1.
The CPU build compiles this with -ffast-math. Its arm64 code for
finalize() computes 1 / Q once, the means as products with it, the
variance as one fused multiply-add, fma(SS, 1 / Q, -mean * mean), and an
IEEE sqrt. Metal's fast-math division and sqrt are not correctly rounded.
Do the same operations on Metal explicitly, with precise::divide, fma and
precise::sqrt: the pbpb clusters then match the CPU bit for bit, with no
measurable cost; the rest of the Metal code keeps fast math.
This matches the CPU build that the compiler produces on arm64 with
clang. An x86 build, or another compiler, may choose a different sequence
under -ffast-math. Writing the sequence explicitly for all backends would
make the widths independent of those choices, but can change the CPU
results on some platforms.