fix pow_fast(x, y) for large integer y

[KlausC]

Fixes #53857

For x::Union{Float32,Float64} and y::Integer with y % Int32 != y a stack trace appears when calling @fastmath x^y.

This fix calls x^y in those cases, while leaving the other cases as is.

[KlausC]

@oscardssmith , can you have a look on this?

[oscardssmith]

it looks good to me, but I do wonder whether with that extra branch it is faster than just calling pow.

[KlausC]

You are partially right:
The following examples show, that on my machine, both 64-bit and 32-bit versions,

1. Float64: @fastmath a^n is faster than a^n if n = 2^30
2. Float32: @fastmath a^n is slower than a^n if n = 2^22
   which suggests to use the pow_fast as is only for Float64, and let pow_fast(a::Float32, n::Integer) = a^n

Julia Version 1.12.0-DEV.564
Commit c641bf67d1 (2024-05-20 07:24 UTC)
Platform Info:
  OS: Linux (x86_64-redhat-linux)
  CPU: 12 × Intel(R) Core(TM) i7-10710U CPU @ 1.10GHz
  WORD_SIZE: 64
  LLVM: libLLVM-17.0.6 (ORCJIT, skylake)
Threads: 1 default, 0 interactive, 1 GC (on 1 virtual cores)
Environment:
  JULIA_CPU_THREADS = 1
  LD_LIBRARY_PATH = /usr/local/intel/compilers_and_libraries_2020.1.217/linux/mkl/lib/intel64:/usr/local/lib:
  JULIA_NUM_THREADS = 1
  JULIA_EDITOR = /usr/bin/vim -R -c"set nonumber"

julia> n = 2^30
1073741824

julia> a = 1.0000000001
1.0000000001

julia> using BenchmarkTools
Precompiling BenchmarkTools...
  1 dependency successfully precompiled in 2 seconds. 13 already precompiled.

julia> @btime $b^$n
ERROR: UndefVarError: `b` not defined in `Main`
Suggestion: check for spelling errors or missing imports.
Stacktrace:
 [1] top-level scope
   @ ~/.julia/packages/BenchmarkTools/QNsku/src/execution.jl:496

julia> @btime $a^$n
  23.736 ns (0 allocations: 0 bytes)
1.1133507826969948

julia> @btime @fastmath $a^$n
  21.254 ns (0 allocations: 0 bytes)
1.113350782267788

julia> @btime @fastmath $a^$(Int32(n))
  19.284 ns (0 allocations: 0 bytes)
1.113350782267788

julia> n = 2^22
4194304

julia> b = Float32(1 + 1/n)
1.0000002f0

julia> @btime $b^$n
  9.662 ns (0 allocations: 0 bytes)
3.7182815f0

julia> @btime @fastmath $b^$n
  13.764 ns (0 allocations: 0 bytes)
2.7175977f0

julia> @btime @fastmath $b^$(Int32(n))
  12.772 ns (0 allocations: 0 bytes)
2.7175977f0

Julia Version 1.12.0-DEV.564
Commit c641bf67d1 (2024-05-20 07:24 UTC)
Platform Info:
  OS: Linux (i686-redhat-linux)
  CPU: 12 × Intel(R) Core(TM) i7-10710U CPU @ 1.10GHz
  WORD_SIZE: 32

julia> a = 1.000001
1.000001

julia> n = 2^30
1073741824

julia> @btime $a^$n
  96.309 ns (0 allocations: 0 bytes)
1.1133507826969948

julia> @btime @fastmath $a^$n
  28.971 ns (0 allocations: 0 bytes)
1.1133507826730735

julia> n = 2^22
4194304

julia> b = Float32(1 + 1/n)
1.0000002f0

julia> @btime $b^$n
  14.665 ns (0 allocations: 0 bytes)
2.7182815f0

julia> @btime @fastmath $b^$n
  21.108 ns (0 allocations: 0 bytes)
2.7182815f0

julia> @btime @fastmath $b^$(Int32(n))
  21.111 ns (0 allocations: 0 bytes)
2.7182815f0

[KlausC]

ping