# Taking power with float exponent x^y is slower than exp(y\*log(x))?

**URL:** <https://discourse.julialang.org/t/taking-power-with-float-exponent-x-y-is-slower-than-exp-y-log-x/132270>\
**Category:** Performance\
**Created:** [September 11, 2025, 2:46am UTC](https://discourse.julialang.org/t/taking-power-with-float-exponent-x-y-is-slower-than-exp-y-log-x/132270 "2025-09-11T02:46:18Z")\
**Posts on this page:** 9\
**Page:** 1

<div class="post-metadata">

**Author:** ![Norman](https://avatars.discourse-cdn.com/v4/letter/n/97f17d/32.png) [@Norman](https://discourse.julialang.org/u/Norman)\
**Post date:** [September 11, 2025, 2:46am UTC](https://discourse.julialang.org/t/taking-power-with-float-exponent-x-y-is-slower-than-exp-y-log-x/132270/1 "2025-09-11T02:46:19Z")

</div>

Updated results: There was some sloppiness on my benchmark approach. The discrepancy isn’t that large.

```julia-auto
julia> @benchmark x^1.5 setup=(x=rand()) samples=100000
BenchmarkTools.Trial: 100000 samples with 998 evaluations per sample.
 Range (min … max): 14.586 ns … 49.423 ns ┊ GC (min … max): 0.00% … 0.00%
 Time (median): 14.788 ns ┊ GC (median): 0.00%
 Time (mean ± σ): 14.805 ns ± 0.229 ns ┊ GC (mean ± σ): 0.00% ± 0.00%

          ▅██▅ ▁▂▁ ▂ ▃
  ▄▃▃▃▃▁▃▇████▇▅▅▄▄▇█▇▇▅▃▄▆███▇▄▅▄▃▁▃▁▁▃▄▃▁▁▁▃▁▃▃▁▁▁▁▁▁▁▃▅▆██ █
  14.6 ns Histogram: log(frequency) by time 15.8 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

julia> @benchmark exp(1.5*log(x)) setup=(x=rand()) samples=100000
BenchmarkTools.Trial: 100000 samples with 1000 evaluations per sample.
 Range (min … max): 4.629 ns … 65.165 ns ┊ GC (min … max): 0.00% … 0.00%
 Time (median): 11.923 ns ┊ GC (median): 0.00%
 Time (mean ± σ): 11.649 ns ± 1.867 ns ┊ GC (mean ± σ): 0.00% ± 0.00%

  ▅ █▃▁▂ ▂▂▅ ▃
  █▆▆▃▃▁▅▆▃▃▄▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▃▁▁▁▄▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁█████▄▄▅███ █
  4.63 ns Histogram: log(frequency) by time 13.3 ns <

 Memory estimate: 0 bytes, allocs estimate: 0.

```

I don’t understand why the latter has a cluster of very fast results though. But it’s probably a different issue.

* * *

Original post:

I know that `x^y` with `y` being Float64 is slower than `y` being integers. But, I just realized that `x^y` is also noticeably slower than `exp(y*log(x))`. Is this expected? I imagine that `x^y` is slightly more accurate?

Here are my results on both a Macbook and an X86 desktop.

```julia-auto
julia> versioninfo()
Julia Version 1.12.0-rc2
Commit 72cbf019d04 (2025-09-06 12:00 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: macOS (arm64-apple-darwin24.0.0)
  CPU: 10 × Apple M1 Pro
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, apple-m1)
  GC: Built with stock GC
Threads: 8 default, 1 interactive, 8 GC (on 8 virtual cores)
Environment:
  JULIA_NUM_THREADS = 8

julia> using BenchmarkTools

julia> @btime x^1.5 setup=(x=rand()) # Slow float exponent
  14.320 ns (0 allocations: 0 bytes)
0.1302126529886922

julia> @btime exp(1.5*log(x)) setup=(x=rand()) # Faster but mathematically equivalent
  8.466 ns (0 allocations: 0 bytes)
0.05577688891274848

```

```julia-auto
julia> versioninfo()
Julia Version 1.12.0-rc2
Commit 72cbf019d04 (2025-09-06 12:00 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 32 × AMD Ryzen 9 9950X 16-Core Processor
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver5)
  GC: Built with stock GC
Threads: 16 default, 1 interactive, 16 GC (on 32 virtual cores)
Environment:
  JULIA_NUM_THREADS = 16

julia> using BenchmarkTools

julia> @btime x^1.5 setup=(x=rand()) # Slow float exponent
  14.576 ns (0 allocations: 0 bytes)
0.7008843249519519

julia> @btime exp(1.5*log(x)) setup=(x=rand()) # Faster but mathematically equivalent
  4.849 ns (0 allocations: 0 bytes)
0.962818987187762

```

Update: The `x` should be randomized now. The gap is even larger between the two. These comparisons are motivated by something observed in my actual usage where lots of `^` are involved.

---

<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:** [September 11, 2025, 5:17am UTC](https://discourse.julialang.org/t/taking-power-with-float-exponent-x-y-is-slower-than-exp-y-log-x/132270/2 "2025-09-11T05:17:08Z")

</div>

I highly suspect that you’re not benchmarking what you think you’re benchmarking. By interpolating the global variable, you’re effectively benchmarking the literal expression `0.09969088930420678 ^ 1.5` which is surely not representative of a generic benchmark (I’d even be surprised if this was not constant folded in the function generated by the `@btime` macro internally). Use the `setup` keyword for initializing `x` instead.

---

<div class="post-metadata">

**Author:** ![Norman](https://avatars.discourse-cdn.com/v4/letter/n/97f17d/32.png) [@Norman](https://discourse.julialang.org/u/Norman)\
**Post date:** [September 11, 2025, 5:36am UTC](https://discourse.julialang.org/t/taking-power-with-float-exponent-x-y-is-slower-than-exp-y-log-x/132270/3 "2025-09-11T05:36:13Z")

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Thank you! The revised version should look right now. The gap is even larger now (but consistent with what I observed in actual usage environment).

---

<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:** [September 11, 2025, 5:40am UTC](https://discourse.julialang.org/t/taking-power-with-float-exponent-x-y-is-slower-than-exp-y-log-x/132270/4 "2025-09-11T05:40:44Z")

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Keep in mind that `@btime` shows you the fastest of all executions, which may also not be representative. What does the graph shown by the full `@benchmark` look like? The random numbers given by `rand()` are also always in the half-open interval `[0, 1)`, so this may also be a factor (though won’t influence the optimizer anymore).

---

<div class="post-metadata">

**Author:** ![Norman](https://avatars.discourse-cdn.com/v4/letter/n/97f17d/32.png) [@Norman](https://discourse.julialang.org/u/Norman)\
**Post date:** [September 11, 2025, 5:52am UTC](https://discourse.julialang.org/t/taking-power-with-float-exponent-x-y-is-slower-than-exp-y-log-x/132270/5 "2025-09-11T05:52:20Z")

</div>

That’s a good catch. I should have tried `@benchmark` for such things. The median does seem much closer.

---

<div class="post-metadata">

**Author:** ![Oscar\_Smith](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/oscar_smith/32/25343_2.png) [@Oscar\_Smith](https://discourse.julialang.org/u/Oscar_Smith)\
**Post date:** [September 11, 2025, 12:39pm UTC](https://discourse.julialang.org/t/taking-power-with-float-exponent-x-y-is-slower-than-exp-y-log-x/132270/6 "2025-09-11T12:39:28Z")

</div>

x^y requires some heroics to be accurate. specifically you need about 16 bits of extra accuracy out of your log

---

<div class="post-metadata">

**Author:** ![mikmoore](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mikmoore/32/31109_2.png) [@mikmoore](https://discourse.julialang.org/u/mikmoore)\
**Post date:** [September 11, 2025, 2:43pm UTC](https://discourse.julialang.org/t/taking-power-with-float-exponent-x-y-is-slower-than-exp-y-log-x/132270/7 "2025-09-11T14:43:27Z")

</div>

As stated above, we take pains to ensure greater accuracy in `x^y` than `exp(y*log(x))` would give (just as [IEEE-754 says we should](https://en.wikipedia.org/wiki/IEEE_754#Recommended_operations)). But if you don’t need full precision, then the exp+log approach is fine if it benchmarks noticeably faster.

For the specific case of `x^1.5`, I measure `x*sqrt(x)` as 8x faster than either and it should only a bit or two less accurate than `^`.

---

<div class="post-metadata">

**Author:** ![Oscar\_Smith](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/oscar_smith/32/25343_2.png) [@Oscar\_Smith](https://discourse.julialang.org/u/Oscar_Smith)\
**Post date:** [September 11, 2025, 3:12pm UTC](https://discourse.julialang.org/t/taking-power-with-float-exponent-x-y-is-slower-than-exp-y-log-x/132270/8 "2025-09-11T15:12:03Z")

</div>

`x*sqrt(x)` should be as accurate as `^`

---

<div class="post-metadata">

**Author:** ![stevengj](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/stevengj/32/71_2.png) [@stevengj](https://discourse.julialang.org/u/stevengj)\
**Post date:** [September 11, 2025, 4:27pm UTC](https://discourse.julialang.org/t/taking-power-with-float-exponent-x-y-is-slower-than-exp-y-log-x/132270/9 "2025-09-11T16:27:28Z")

</div>

> [@Oscar\_Smith](#):
>
> `x*sqrt(x)` should be as accurate as `^`

This doesn’t seem to be quite true. Here is the error difference in ulps for random x \in [0,1):

```julia-auto
julia> err_diff = [let x = rand()
                        e = big(x)^(3//2) # "exact" result
                        Float64(abs(x * sqrt(x) - e) - abs(x^1.5 - e)) / eps(Float64(e))
                    end for _ in 1:10^6]

```

The result shows that the error of `x * sqrt(x)` is worse much more often than the error is better, up to 1ulp worse, though on average it is only 0.08ulps worse.

```julia-auto
julia> count(err_diff .< 0) # x * sqrt(x) is better
5169

julia> count(err_diff .> 0) # x * sqrt(x) is worse
238882

julia> mean(err_diff)
0.082866970466614

julia> extrema(err_diff)
(-0.0425258183554384, 1.0)

```

(and it is never more than slightly better than `x^1.5`).