# Fma's and poly eval

**URL:** <https://discourse.julialang.org/t/fmas-and-poly-eval/61421>\
**Category:** Performance\
**Tags:** performance, polynomials\
**Created:** [May 19, 2021, 8:15am UTC](https://discourse.julialang.org/t/fmas-and-poly-eval/61421 "2021-05-19T08:15:39Z")\
**Posts on this page:** 5\
**Page:** 1

<div class="post-metadata">

**Author:** ![jarl](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jarl/32/4365_2.png) [@jarl](https://discourse.julialang.org/u/jarl)\
**Post date:** [May 19, 2021, 8:15am UTC](https://discourse.julialang.org/t/fmas-and-poly-eval/61421/1 "2021-05-19T08:15:39Z")

</div>

I want to evaluate a polynomial of degree 8. Two ways of doing this:

```julia
@inline function standard_eval(x)
    p = evalpoly(x, (0.16666666666666632, 0.04166666666666556, 0.008333333333401227,
                     0.001388888889068783, 0.00019841269447671544, 2.480157691845342e-5,
                     2.7558212415361945e-6, 2.758218402815439e-7, 2.4360682937111612e-8))
end
@inline function eval_ps(x)
    xx=x * x
    xxx=xx * x
    X3_1=muladd(2.4360682937111612e-8,xx,muladd(2.758218402815439e-7,x,2.7558212415361945e-6))
    X4_1=muladd(xx,2.480157691845342e-5,muladd(0.00019841269447671544,x,0.001388888889068783));
    Y_1=muladd(0.008333333333401227,xx,muladd(0.04166666666666556,x,0.16666666666666632))
    Y=muladd(xxx,muladd(xxx,X3_1,X4_1),Y_1)
    return Y
end

```

The `evalpoly` is based on Horner scheme and `eval_ps` is a hardcoded version of the Paterson-Stockmayer approach. They correspond to the evaluatation of the same polynomial

```julia
julia> standard_eval(0.1)-eval_ps(0.1)
-2.7755575615628914e-17

```

By using `@code_native` we see that `standard_eval` contains 8 fma-operations and `eval_ps` contains 6 fma-operations.

However, I don’t see a difference in performance:

```julia
julia> @btime eval_ps(0.3);
  0.036 ns (0 allocations: 0 bytes)
julia> @btime standard_eval(0.3);
  0.036 ns (0 allocations: 0 bytes)

```

Why is this? Can this be improved?

```julia
julia> @code_native eval_ps(22.0)
        .text
; ┌ @ REPL[94]:2 within `eval_ps'
; │┌ @ float.jl:356 within `*'
        vmulsd %xmm0, %xmm0, %xmm1
; │└
; │ @ REPL[94]:5 within `eval_ps'
; │┌ @ float.jl:363 within `muladd'
        vmovddup %xmm0, %xmm2 # xmm2 = xmm0[0,0]
        movabsq $.rodata.cst16, %rax
        vmovapd (%rax), %xmm3
        movabsq $140349345069936, %rax # imm = 0x7FA5A0DB7B70
        vfmadd213pd (%rax), %xmm2, %xmm3 # xmm3 = (xmm2 * xmm3) + mem
        vmovddup %xmm1, %xmm2 # xmm2 = xmm1[0,0]
        movabsq $140349345069952, %rax # imm = 0x7FA5A0DB7B80
        vfmadd132pd (%rax), %xmm3, %xmm2 # xmm2 = (xmm2 * mem) + xmm3
; │└
; │ @ REPL[94]:3 within `eval_ps'
; │┌ @ float.jl:356 within `*'
        vmulsd %xmm0, %xmm1, %xmm3
        movabsq $.rodata.cst8, %rax
        vmovsd (%rax), %xmm4 # xmm4 = mem[0],zero
        movabsq $140349345069976, %rax # imm = 0x7FA5A0DB7B98
; │└
; │ @ REPL[94]:7 within `eval_ps'
; │┌ @ float.jl:363 within `muladd'
        vfmadd213sd (%rax), %xmm0, %xmm4 # xmm4 = (xmm0 * xmm4) + mem
        movabsq $140349345069984, %rax # imm = 0x7FA5A0DB7BA0
        vfmadd231sd (%rax), %xmm1, %xmm4 # xmm4 = (xmm1 * mem) + xmm4
; │└
; │ @ REPL[94]:8 within `eval_ps'
; │┌ @ float.jl:363 within `muladd'
        vpermilpd $1, %xmm2, %xmm0 # xmm0 = xmm2[1,0]
        vfmadd213sd %xmm2, %xmm3, %xmm0 # xmm0 = (xmm3 * xmm0) + xmm2
        vfmadd213sd %xmm4, %xmm3, %xmm0 # xmm0 = (xmm3 * xmm0) + xmm4
; │└
; │ @ REPL[94]:9 within `eval_ps'
        retq
        nopl (%rax)
; └

```

Stems from [https://github.com/JuliaLang/julia/pull/40867](https://github.com/JuliaLang/julia/pull/40867)

---

<div class="post-metadata">

**Author:** ![fredrikekre](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/fredrikekre/32/1688_2.png) [@fredrikekre](https://discourse.julialang.org/u/fredrikekre)\
**Post date:** [May 19, 2021, 8:29am UTC](https://discourse.julialang.org/t/fmas-and-poly-eval/61421/2 "2021-05-19T08:29:03Z")

</div>

0.036 ns looks like the compiler figured out the answer and didn’t do any work. This is what I get when using the Ref trick to hide the value from the compiler:

```julia
julia> @btime eval_ps($(Ref(0.3))[]);
  2.163 ns (0 allocations: 0 bytes)

julia> @btime standard_eval($(Ref(0.3))[]);
  2.872 ns (0 allocations: 0 bytes)

```

---

<div class="post-metadata">

**Author:** ![jarl](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jarl/32/4365_2.png) [@jarl](https://discourse.julialang.org/u/jarl)\
**Post date:** [May 19, 2021, 8:43am UTC](https://discourse.julialang.org/t/fmas-and-poly-eval/61421/3 "2021-05-19T08:43:02Z")

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> [@fredrikekre](#):
>
> 0.036 ns looks like the compiler figured out the answer and didn’t do any work

Interesting. So is this a consequence of how the benchmarking is done? In a setting like it is used in `expm1` in the PR I referenced, do you think the compiler would also “figure out the answer”?

---

<div class="post-metadata">

**Author:** ![Sukera](https://avatars.discourse-cdn.com/v4/letter/s/ce7236/32.png) [@Sukera](https://discourse.julialang.org/u/Sukera)\
**Post date:** [May 19, 2021, 8:55am UTC](https://discourse.julialang.org/t/fmas-and-poly-eval/61421/4 "2021-05-19T08:55:18Z")

</div>

It’s a benchmarking artifact because the compiler can use constant propagation to propagate the literal `0.1`/`0.3` into the function, seeing that there’s no dynamic change, and computing the result at compile time.

The `Ref` trick referenced by @fredrikekre makes the “constant” invisible to the compiler, forcing it to treat the value as dynamic and able to change.

---

<div class="post-metadata">

**Author:** ![jarl](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jarl/32/4365_2.png) [@jarl](https://discourse.julialang.org/u/jarl)\
**Post date:** [May 19, 2021, 9:42am UTC](https://discourse.julialang.org/t/fmas-and-poly-eval/61421/5 "2021-05-19T09:42:33Z")

</div>

Thanks for the explanations. It turned into a PR: [https://github.com/JuliaLang/julia/pull/40870](https://github.com/JuliaLang/julia/pull/40870)