GH-44903: [C++] Add the Expm1(exponent) scalar arithmetic function

[felipecrv]

Expm1(exponent) is a more accurate way of computing Exp(exponent) - 1.0 for small values of exponent.

Rationale for this change

expm1(x) is specifically designed to compute exp(x)−1 more accurately, particularly for small x. It uses numerical techniques and approximations that minimize the loss of precision.

When x is very small (close to 0), exp(x) is approximately 1+x. Subtracting 1 from exp(x) (i.e., (exp(x)−1) can result in significant cancellation of significant digits due to floating-point arithmetic, leading to a loss of precision).

For example:

When x = 10^−8, exp(x) is close to 1 + 10^−8, so subtracting 1 leaves only the small 10^−8, which may lose accuracy due to floating-point limitations.

Are these changes tested?

Yes.

Are there any user-facing changes?

Yes and documentation was updated to list the new function.

• GitHub Issue: [C++] Add missing scalar arithmetic function: expm1() #44903

[pitrou]

LGTM in general. Just a couple minor comments.

[pitrou]

Nit: since the main point of expm1 is to be more precise when the input is close to zero, shouldn't we focus on this use case here? Though of course this is all dependent on the stdlib's implementation.

[felipecrv]

I added 0.0 but you're right about the need for other tiny numbers.

[felipecrv]

Done.

[zanmato1984]

+1. Only one nit.

[zanmato1984]

Shall we mention that "this is more accurate than directly computing exp(value) - 1" in the function doc?

[felipecrv]

I considered it, but I think it's better to keep this succinct. expm1 is very googlable and explaining why it's more accurate in a short phrase isn't easy.

[felipecrv]

R test failures seem unrelated. I will merge soon if no one is against it.

[conbench-apache-arrow]

After merging your PR, Conbench analyzed the 3 benchmarking runs that have been run so far on merge-commit 5e476b3.

There were 132 benchmark results with an error:

• Commit Run on arm64-t4g-2xlarge-linux at 2024-12-02 23:05:07Z
  • tpch (R) with engine=arrow, format=parquet, language=R, memory_map=False, query_id=TPCH-07, scale_factor=1
  • tpch (R) with engine=arrow, format=native, language=R, memory_map=False, query_id=TPCH-09, scale_factor=1
• and 130 more (see the report linked below)

There were no benchmark performance regressions. 🎉

The full Conbench report has more details. It also includes information about 32 possible false positives for unstable benchmarks that are known to sometimes produce them.