# Optimization: SIMD polynomial exp inside the j loop

- **Author**: Linus
- **Predecessor**: variant_inlining_7_SoA_AVX2.cpp
- **File(s)**: variants/variant_inlining_7_SoA_AVX2_polyexp.cpp
- **Status**: TBD, positive locally (2 times)

## What this changes

The predecessor still called libm `exp()` 4 times into a scratch array
inside the j loop. A profile pointed at those calls as ~54% of
inner-loop cycles. This variant replaces them with one SIMD polynomial
approximation, `exp256_pd`, that operates on a `__m256d` directly.

The approximation is the textbook 2^n bit trick combined with a
degree-7 Taylor polynomial. We write `exp(x) = 2^n * exp(r)` with
`n = round(x / ln2)` and `r = x - n*ln2`, so after range reduction
`r` lives in roughly `[-0.347, 0.347]`. On that range a degree-7
Taylor evaluated with Horner is accurate to ~1e-9 absolute, way under
the 1e-3 tolerance verify.py uses. The `2^n` part is free in IEEE 754
doubles, you just pack `(n + 1023)` into the exponent slot via a left
shift by 52.



## Code (relevant snippet)

See `variants/variant_inlining_7_SoA_AVX2_polyexp.cpp` for the full
`exp256_pd` helper. Quick walkthrough: a round-to-nearest of `x / ln2`
gives `n`, then an `fnmadd` strips `n*ln2` out of `x` to get the
reduced argument `r`, which lives in roughly `[-0.347, 0.347]`. Seven
`fmadd`s evaluate a degree-7 Horner polynomial for `exp(r)` on that
range. In parallel we build `2^n` straight into the IEEE 754 exponent
field by converting `n` to int32, adding the bias 1023, left shifting
by 52, then reinterpreting as double. No actual `pow` call needed.
Final multiply of polynomial value with `2^n` gives `exp(x)`.

In the j loop the old scratch-array path (4 libm calls into `tmp_arg`
/ `tmp_val`) becomes a single `__m256d e_val = exp256_pd(e_arg);`.


## Benchmark

Conditions: min of N runs (TBD), N_ITER=1000, -O3 -march=native -flto.
Diff is vs `variant_inlining_7_SoA_AVX2`.

local runs show a sizeable speedup over the AVX2-with-scalar-exp
predecessor (approx 2times)

### x86

**Machine**: TBD

**Frequency**: TBD

| N    | W(n) [Flops] | T(n) [Cycles] | Performance [Flops/Cycle] | Diff vs Predecessor |
| ---- | ------------ | ------------- | ------------------------- | ------------------- |
| 50   |              |               |                           |                     |
| 100  |              |               |                           |                     |
| 200  |              |               |                           |                     |
| 500  |              |               |                           |                     |
| 1000 |              |               |                           |                     |

### ARM (Apple)

Not applicable, AVX2-only. A NEON port would need its own vectorized
exp.

## Analysis

`exp` was the single biggest cost in the predecessor's inner loop, so
the expectation was that killing it would lift the whole loop by a
large fraction.  The SIMD polynomial is roughly 9 fmadds +
a round + a couple of converts + a shift, no library call, no scratch
array, no scalar fallback. Should pipeline well.

Accuracy: degree-7 Taylor on the reduced range is good to ~1e-9
absolute. verify.py's tolerance is 1e-3, so there is a lot of headroom.
If we ever need tighter accuracy we could bump the degree or move to an
Estrin schedule, but right now there is no point.


## Takeaway

Biggest single step on top of the first AVX2 pass. The inner loop is
now essentially fma / sqrt / div bound with no libm calls left in it.

## References

- Schraudolph, N. N. (1999). "A Fast, Compact Approximation of the
  Exponential Function." Neural Computation 11(4), 853-862.
- IEEE 754 double-precision format. Wikipedia.
  https://en.wikipedia.org/wiki/Double-precision_floating-point_format
- Horner's method. Wikipedia.
  https://en.wikipedia.org/wiki/Horner%27s_method