square() for arithmetic types: multiply instead of calling std::pow(x, 2)

[sims1253]

With default toolchain settings (gcc/clang default to -fmath-errno), std::pow must set errno on domain/range/overflow errors, so the optimizer cannot rewrite it to a multiply in general. glibc's pow is a branchy multi-path implementation (~105 instructions per call measured: 3,473,268 Ir over 33,078 calls) where x * x is one instruction.

On the measured GP regression model, square() accounts for 57 pow calls per gradient and 8.9% of gradient instructions (callgrind): 32,889 of 33,078 executed pow calls come from gp_exp_quad_cov (55 kernel pairs for N = 11, plus square(sigma) and square(l)).

Widen to double first, then multiply. The template is enabled for all arithmetic T; a naive x * x would compute an int product for integral x (overflow for |x| > 46,341 where the promoted path does not) and round twice for float x. Widening first reproduces the previous promote-then-round semantics exactly, minus the libm call

Eval

GP regression model (gp_exp_quad_cov, n = 11 → 55 kernel pairs), matched binaries, identical inputs, gcc 16.2.1, glibc, Zen 3. Medians of 3 interleaved reps; callgrind on a fixed seeded run (warmup 50 / samples 50, 577 gradient calls in both arms):

metric	stock	patched	delta
Ir / gradient (callgrind, deterministic)	66,950	60,864	−9.1%
pow instructions in run	3,473,268	19,923	−99.4% (residual is sampler-side Adam)
µs / logp_grad call (warmup stanza)	6.681	5.820	−12.9%
µs / logp_grad call (sampling stanza)	6.655	5.640	−15.2%

Checklist

• Copyright holder: Maximilian Scholz
  The copyright holder is typically you or your assignee, such as a university or company. By submitting this pull request, the copyright holder is agreeing to the license the submitted work under the following licenses:
  - Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
  - Documentation: CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

• the basic tests are passing

  • unit tests pass (to run, use: ./runTests.py test/unit)
  • header checks pass, (make test-headers)
  • dependencies checks pass, (make test-math-dependencies)
  • docs build, (make doxygen)
  • code passes the built in C++ standards checks (make cpplint)

• the code is written in idiomatic C++ and changes are documented in the doxygen

• the new changes are tested

[WardBrian]

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