Linus Baumberger
96 lines
3.8 KiB
Markdown
96 lines
3.8 KiB
Markdown
# Optimization: SIMD polynomial exp inside the j loop
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- **Author**: Linus
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- **Predecessor**: variant_inlining_7_SoA_AVX2.cpp
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- **File(s)**: variants/variant_inlining_7_SoA_AVX2_polyexp.cpp
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- **Status**: TBD, positive locally (2 times)
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## What this changes
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The predecessor still called libm `exp()` 4 times into a scratch array
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inside the j loop. A profile pointed at those calls as ~54% of
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inner-loop cycles. This variant replaces them with one SIMD polynomial
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approximation, `exp256_pd`, that operates on a `__m256d` directly.
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The approximation is the textbook 2^n bit trick combined with a
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degree-7 Taylor polynomial. We write `exp(x) = 2^n * exp(r)` with
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`n = round(x / ln2)` and `r = x - n*ln2`, so after range reduction
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`r` lives in roughly `[-0.347, 0.347]`. On that range a degree-7
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Taylor evaluated with Horner is accurate to ~1e-9 absolute, way under
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the 1e-3 tolerance verify.py uses. The `2^n` part is free in IEEE 754
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doubles, you just pack `(n + 1023)` into the exponent slot via a left
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shift by 52.
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## Code (relevant snippet)
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See `variants/variant_inlining_7_SoA_AVX2_polyexp.cpp` for the full
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`exp256_pd` helper. Quick walkthrough: a round-to-nearest of `x / ln2`
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gives `n`, then an `fnmadd` strips `n*ln2` out of `x` to get the
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reduced argument `r`, which lives in roughly `[-0.347, 0.347]`. Seven
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`fmadd`s evaluate a degree-7 Horner polynomial for `exp(r)` on that
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range. In parallel we build `2^n` straight into the IEEE 754 exponent
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field by converting `n` to int32, adding the bias 1023, left shifting
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by 52, then reinterpreting as double. No actual `pow` call needed.
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Final multiply of polynomial value with `2^n` gives `exp(x)`.
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In the j loop the old scratch-array path (4 libm calls into `tmp_arg`
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/ `tmp_val`) becomes a single `__m256d e_val = exp256_pd(e_arg);`.
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## Benchmark
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Conditions: min of N runs (TBD), N_ITER=1000, -O3 -march=native -flto.
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Diff is vs `variant_inlining_7_SoA_AVX2`.
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local runs show a sizeable speedup over the AVX2-with-scalar-exp
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predecessor (approx 2times)
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### x86
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**Machine**: TBD
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**Frequency**: TBD
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| N | W(n) [Flops] | T(n) [Cycles] | Performance [Flops/Cycle] | Diff vs Predecessor |
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| ---- | ------------ | ------------- | ------------------------- | ------------------- |
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| 50 | | | | |
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| 100 | | | | |
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| 200 | | | | |
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| 500 | | | | |
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| 1000 | | | | |
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### ARM (Apple)
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Not applicable, AVX2-only. A NEON port would need its own vectorized
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exp.
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## Analysis
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`exp` was the single biggest cost in the predecessor's inner loop, so
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the expectation was that killing it would lift the whole loop by a
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large fraction. The SIMD polynomial is roughly 9 fmadds +
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a round + a couple of converts + a shift, no library call, no scratch
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array, no scalar fallback. Should pipeline well.
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Accuracy: degree-7 Taylor on the reduced range is good to ~1e-9
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absolute. verify.py's tolerance is 1e-3, so there is a lot of headroom.
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If we ever need tighter accuracy we could bump the degree or move to an
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Estrin schedule, but right now there is no point.
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## Takeaway
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Biggest single step on top of the first AVX2 pass. The inner loop is
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now essentially fma / sqrt / div bound with no libm calls left in it.
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## References
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- Schraudolph, N. N. (1999). "A Fast, Compact Approximation of the
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Exponential Function." Neural Computation 11(4), 853-862.
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- IEEE 754 double-precision format. Wikipedia.
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https://en.wikipedia.org/wiki/Double-precision_floating-point_format
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- Horner's method. Wikipedia.
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https://en.wikipedia.org/wiki/Horner%27s_method
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