# Should @FastPow not make this faster?

**URL:** <https://discourse.julialang.org/t/should-fastpow-not-make-this-faster/93296>\
**Category:** Performance\
**Created:** [January 21, 2023, 7:34am UTC](https://discourse.julialang.org/t/should-fastpow-not-make-this-faster/93296 "2023-01-21T07:34:48Z")\
**Posts on this page:** 8\
**Page:** 1

<div class="post-metadata">

**Author:** ![Ahmed\_Salih](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ahmed_salih/32/206579_2.png) [@Ahmed\_Salih](https://discourse.julialang.org/u/Ahmed_Salih)\
**Post date:** [January 21, 2023, 7:34am UTC](https://discourse.julialang.org/t/should-fastpow-not-make-this-faster/93296/1 "2023-01-21T07:34:48Z")

</div>

Hello everyone!

Suppose I have a set of parameters as:

```julia
ρ₀ = 1000; c₀ = 20; γ = 7; ρ = ρ₀ .+ ρ₀*rand(10^6)*0.1;

```

And I wish to evaluate the function:

```julia
function Pressure(ρ,c₀,γ,ρ₀)
    return ((c₀^2*ρ₀)/γ) * ((ρ/ρ₀)^γ - 1)
end

```

Benchmarking the code as follows gives:

```julia
@benchmark Pressure.($ρ,$Cb,$γ,$ρ₀)
BenchmarkTools.Trial: 707 samples with 1 evaluation.
 Range (min … max): 6.260 ms … 12.147 ms ┊ GC (min … max): 0.00% … 0.00%
 Time (median): 6.604 ms ┊ GC (median): 0.00%
 Time (mean ± σ): 7.070 ms ± 1.258 ms ┊ GC (mean ± σ): 5.35% ± 11.09%

  ▄▆██▇▅▄▃▁▁▁
  █████████████▇▅▄▆▆▆▆▅▆▁▄▁▄▄▁▁▁▁▁▁▁▄▁▁▁▁▁▁▄▁▄▁▆▇███▇▇▅▇▁▆▁▆ █
  6.26 ms Histogram: log(frequency) by time 11.5 ms <

```

And using @fastpow:

```julia
@benchmark @fastpow Pressure.($ρ,$Cb,$γ,$ρ₀)
BenchmarkTools.Trial: 706 samples with 1 evaluation.
 Range (min … max): 6.279 ms … 13.528 ms ┊ GC (min … max): 0.00% … 45.26%
 Time (median): 6.621 ms ┊ GC (median): 0.00%
 Time (mean ± σ): 7.083 ms ± 1.266 ms ┊ GC (mean ± σ): 5.34% ± 11.02%

  ▁▆▇█▇▅▄▃▁▁ ▁
  ██████████▇▅▆█▆▇▆▄▆▆▄▄▄▄▁▁▁▁▄▁▁▁▁▁▄▁▁▁▁▁▁▄▁▁▁▁▄▇▇██▇█▇▆▄▆▄ █
  6.28 ms Histogram: log(frequency) by time 11.4 ms <

 Memory estimate: 7.63 MiB, allocs estimate: 2.

```

Which seems to give no difference in results. The exponentation happens since `γ = 7`.

Any apparent way to speed up this calculation?

Kind regards

---

<div class="post-metadata">

**Author:** ![giordano](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/giordano/32/2166_2.png) [@giordano](https://discourse.julialang.org/u/giordano)\
**Post date:** [January 21, 2023, 9:00am UTC](https://discourse.julialang.org/t/should-fastpow-not-make-this-faster/93296/2 "2023-01-21T09:00:26Z")

</div>

That’s not how macros work. Macros rewrite _ **unevaluated** _ expressions into some other expressions. In particular, the `@fastpow` macro would rewrite expressions which contain calls to `^` into something else. But the raw expression `Pressure.($ρ,$Cb,$γ,$ρ₀)` doesn’t have any such calls, so there is no transformation to perform to start with. You can use the [`@macroexpand`](https://docs.julialang.org/en/v1/base/base/#Base.macroexpand) macro to see the effect of applying another macro.

However the [README.MD](https://github.com/JuliaMath/FastPow.jl/blob/f84a99a9bed96a3bfb06c14ed77c81729b0126c7/README.md) file of the package shows that you can apply the macro to function definitions which contain calls to the power operator, which is likely what you want to do here.

---

<div class="post-metadata">

**Author:** ![Ahmed\_Salih](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ahmed_salih/32/206579_2.png) [@Ahmed\_Salih](https://discourse.julialang.org/u/Ahmed_Salih)\
**Post date:** [January 21, 2023, 9:08am UTC](https://discourse.julialang.org/t/should-fastpow-not-make-this-faster/93296/3 "2023-01-21T09:08:01Z")

</div>

Thank you for explanation

I do as they state in documentation:

```julia
using FastPow
@fastpow function Pressure(ρ,c₀,γ,ρ₀)
    return ((c₀^2*ρ₀)/γ) * ((ρ/ρ₀)^γ - 1)
end

```

Then benchmark the code:

```julia
@benchmark Pressure.($ρ,$c₀,$γ,$ρ₀)
BenchmarkTools.Trial: 649 samples with 1 evaluation.
 Range (min … max): 6.287 ms … 16.706 ms ┊ GC (min … max): 0.00% … 31.21%
 Time (median): 7.064 ms ┊ GC (median): 0.00%
 Time (mean ± σ): 7.696 ms ± 1.789 ms ┊ GC (mean ± σ): 5.97% ± 11.83%

    █▄▄▁▄▁   
  ▅███████▇▆▄▃▂▃▂▃▂▂▃▂▂▁▂▂▂▂▁▂▂▂▁▂▂▂▂▃▃▃▃▃▃▃▃▂▃▂▂▂▂▃▂▂▁▂▁▂▁▂ ▃
  6.29 ms Histogram: frequency by time 14.1 ms <

 Memory estimate: 7.63 MiB, allocs estimate: 2.

```

Which still gives around 6 ms and no improvement. Am I missing something else I need to do for it to work?

Kind regards

---

<div class="post-metadata">

**Author:** ![giordano](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/giordano/32/2166_2.png) [@giordano](https://discourse.julialang.org/u/giordano)\
**Post date:** [January 21, 2023, 9:15am UTC](https://discourse.julialang.org/t/should-fastpow-not-make-this-faster/93296/4 "2023-01-21T09:15:54Z")

</div>

I’m away from keyboard so I can’t test myself, but the README of the [`FastPow.jl`](https://github.com/JuliaMath/FastPow.jl) package talks about _ **literal** _ integer powers, your exponent is a variable, not a literal integer. Again, I encourage you to use the `@macroexpand` macro to see the effect of applying macros.

---

<div class="post-metadata">

**Author:** ![Ahmed\_Salih](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ahmed_salih/32/206579_2.png) [@Ahmed\_Salih](https://discourse.julialang.org/u/Ahmed_Salih)\
**Post date:** [January 21, 2023, 9:20am UTC](https://discourse.julialang.org/t/should-fastpow-not-make-this-faster/93296/5 "2023-01-21T09:20:27Z")

</div>

Thank you!

That was the step I had misunderstood, I thought one just had to ensure it was integer input. By putting in “7” directly instead of gamma I get:

```julia
@benchmark Pressure.($ρ,$c₀,$γ,$ρ₀)
BenchmarkTools.Trial: 2510 samples with 1 evaluation.
 Range (min … max): 1.128 ms … 10.429 ms ┊ GC (min … max): 0.00% … 0.00%
 Time (median): 1.516 ms ┊ GC (median): 0.00%
 Time (mean ± σ): 1.987 ms ± 1.516 ms ┊ GC (mean ± σ): 21.78% ± 20.62%

  ▁█▄▁▁▃   
  ██████▆▅▄▃▃▂▂▂▂▁▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▂▁▁▁▁▁▁▁▁▂▁▁▁▁▁▁▁▂▃▃▃▃▃▃▂▂▂ ▃
  1.13 ms Histogram: frequency by time 7.1 ms <

 Memory estimate: 7.63 MiB, allocs estimate: 2.

```

Which is 6 times faster and what I wanted 🙂 Sometimes gamma can vary (but still be Int) so I will see if I can get that to work some way

Kind regards

---

<div class="post-metadata">

**Author:** ![lmiq](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/lmiq/32/18314_2.png) [@lmiq](https://discourse.julialang.org/u/lmiq)\
**Post date:** [January 21, 2023, 6:00pm UTC](https://discourse.julialang.org/t/should-fastpow-not-make-this-faster/93296/6 "2023-01-21T18:00:52Z")

</div>

It is conceivable that `@fastpow` could work if the exponent was passed as a value-type? Or that could be combined with generated functions perhaps?

It does not work as it is:

```julia
julia> using FastPow

julia> f(x) = 1/x^13 - 1/x^7
f (generic function with 2 methods)

julia> f2(x) = @fastpow 1/x^13 - 1/x^7
f2 (generic function with 1 method)

julia> f3(x,::Val{N}) where N = @fastpow 1/x^(2N - 1) - 1/x^N
f3 (generic function with 1 method)

julia> v = rand(1000);

julia> f.(v) ≈ f2.(v) ≈ f3.(v,Val(7))
true

julia> @btime f.($rand(1000));
  13.027 μs (2 allocations: 15.88 KiB)

julia> @btime f2.($rand(1000)); # @fastpow doing its job
  2.223 μs (2 allocations: 15.88 KiB)

julia> @btime f3.($rand(1000),$Val(7));
  13.593 μs (5 allocations: 15.95 KiB)

```

---

<div class="post-metadata">

**Author:** ![giordano](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/giordano/32/2166_2.png) [@giordano](https://discourse.julialang.org/u/giordano)\
**Post date:** [January 21, 2023, 6:17pm UTC](https://discourse.julialang.org/t/should-fastpow-not-make-this-faster/93296/7 "2023-01-21T18:17:18Z")

</div>

Again, macros work on _ **unevaluated** _ expressions, it doesn’t make any difference if `N` is a variable with an integer value or a value type: to a macro that’s always the symbol `N`, not a literal integer:

```julia
julia> Meta.@dump 1/x^(2N - 1) - 1/x^N
Expr
  head: Symbol call
  args: Array{Any}((3,))
    1: Symbol -
    2: Expr
      head: Symbol call
      args: Array{Any}((3,))
        1: Symbol /
        2: Int64 1
        3: Expr
          head: Symbol call
          args: Array{Any}((3,))
            1: Symbol ^
            2: Symbol x
            3: Expr
              head: Symbol call
              args: Array{Any}((3,))
                1: Symbol -
                2: Expr
                3: Int64 1
    3: Expr
      head: Symbol call
      args: Array{Any}((3,))
        1: Symbol /
        2: Int64 1
        3: Expr
          head: Symbol call
          args: Array{Any}((3,))
            1: Symbol ^
            2: Symbol x
            3: Symbol N

```

You _can_ make `@fastpow` work with variables by complicating its internals which would probably slow it down, which I presume is the reason why it wasn’t done.

---

<div class="post-metadata">

**Author:** ![ademonts](https://avatars.discourse-cdn.com/v4/letter/a/ba8739/32.png) [@ademonts](https://discourse.julialang.org/u/ademonts)\
**Post date:** [January 21, 2023, 6:25pm UTC](https://discourse.julialang.org/t/should-fastpow-not-make-this-faster/93296/8 "2023-01-21T18:25:08Z")

</div>

It works if you are willing to do a @generated function:

```julia
julia> @generated function f4(x,::Val{N}) where N 
           return quote
               @fastpow 1/x^(2*$N -1) - 1/x^$N
           end
       end

julia> @generated function f5(x,::Val{N1}, ::Val{N2}) where {N1,N2}
           return quote
               @fastpow 1/x^($N1) - 1/x^$N2
           end
       end

julia> v7 = Val(7)

julia> v13 = Val(13)

julia> @btime f4.($rand(1000),$v7);
  7.394 μs (2 allocations: 15.88 KiB)

julia> @btime f5.($rand(1000),$v13, $v7);
  1.898 μs (2 allocations: 15.88 KiB)

```