pgsql: Optimise numeric multiplication using base-NBASE^2 arithmetic.

Optimise numeric multiplication using base-NBASE^2 arithmetic.

Currently mul_var() uses the schoolbook multiplication algorithm,
which is O(n^2) in the number of NBASE digits. To improve performance
for large inputs, convert the inputs to base NBASE^2 before
multiplying, which effectively halves the number of digits in each
input, theoretically speeding up the computation by a factor of 4. In
practice, the actual speedup for large inputs varies between around 3
and 6 times, depending on the system and compiler used. In turn, this
significantly reduces the runtime of the numeric_big regression test.

For this to work, 64-bit integers are required for the products of
base-NBASE^2 digits, so this works best on 64-bit machines, on which
it is faster whenever the shorter input has more than 4 or 5 NBASE
digits. On 32-bit machines, the additional overheads, especially
during carry propagation and the final conversion back to base-NBASE,
are significantly higher, and it is only faster when the shorter input
has more than around 50 NBASE digits. When the shorter input has more
than 6 NBASE digits (so that mul_var_short() cannot be used), but
fewer than around 50 NBASE digits, there may be a noticeable slowdown
on 32-bit machines. That seems to be an acceptable tradeoff, given the
performance gains for other inputs, and the effort that would be
required to maintain code specifically targeting 32-bit machines.

Joel Jacobson and Dean Rasheed.

Discussion: https://postgr.es/m/9d8a4a42-c354-41f3-bbf3-199e1957db97%40app.fastmail.com

Branch
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master

Details
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https://git.postgresql.org/pg/commitdiff/8dc28d7eb868b6ce5a51614628bf46fc63c7e90c

Modified Files
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src/backend/utils/adt/numeric.c | 224 +++++++++++++++++++++++++++-------------
1 file changed, 150 insertions(+), 74 deletions(-)