[ibmibmibm]

• fma for decimal32_t, decimal64_t and the fast types added the exact product and z with add_impl. If z was far below the product, the nearest modes dropped z, and a tie in the product went to the even neighbor.
• fma now keeps the exact product of the significands in 64, 128 or 256 bits, and it does not move the product up. z moves to the product. A far z moves up p digits, and the product moves down to it.
• The operand that moves down records a dropped digit that is not zero. After the sum, this record goes into the last digit. Because of this, a carry or a borrow does not remove it.
• A sum of 2p - 1 or 2p digits rounds in one step with the finalizers of multiplication. Other sums round in the constructor. A zero operand leaves one operation, which rounds one time.
• For a finite x and y and an infinite z, fma returned x * y + z. A large product rounded to an infinity, and with z of the other sign the sum was NaN. fma now returns z in this case, for all six types.
• test_fma_tie tests ties with a far z, a dropped digit with a carry or a borrow, the bounds of the finalizers, and the five rounding modes. github_issue_1522 tests the infinite z.

Fixes #1522

Correctness

• 3000000 random triples for each of decimal32_t, decimal64_t and decimal128_t, in the five rounding modes, against Intel libbid 2.0 Update 4. Each value that is not the same is an underflow error of libbid. Python decimal gives our value in each of these cases.
• 4500000 triples with a far z, for all six types and the five modes, against Python decimal. All results are correct.
• In the tie_far set of the benchmark, develop gives a different value from libbid for 6720 of 16384 decimal32_t triples and 6193 of 16384 decimal64_t triples. The fix gives 0.
• test_fma, test_fma_tie, github_issue_1522 and test_cmath pass with GCC 16.2 and Clang 23.1 in C++14 and C++20 with -Wall -Wextra -Werror. They also pass without the builtin 128-bit integer, with BOOST_DECIMAL_FAST_MATH, and with BOOST_DECIMAL_NO_CONSTEVAL_DETECTION.

Speed

The table gives ns for each call, the best of 3 runs. Each set has 16384 triples: random significands and exponents (random), a tie with a far z (tie_far), a z near the product (near), a z near -(x * y) (cancel) and a zero z (zero_z). I used -O3 -march=native on an AMD Ryzen 9 3900X, on one core. libbid is built with GCC at -O2.

type	set	GCC develop	GCC fix	Clang develop	Clang fix	libbid
decimal32_t	random	40.3	18.9	39.8	17.6	42.4
decimal32_t	tie_far	28.0	15.6	30.6	15.0	42.3
decimal32_t	near	45.6	22.7	42.8	22.0	37.9
decimal32_t	cancel	27.5	15.3	28.3	14.4	16.8
decimal32_t	zero_z	26.6	14.3	29.8	12.2	37.6
decimal64_t	random	112.7	30.0	92.6	28.8	37.8
decimal64_t	tie_far	55.4	26.1	49.4	23.1	29.4
decimal64_t	near	159.9	30.3	124.6	30.2	46.0
decimal64_t	cancel	62.3	17.1	60.0	16.1	27.0
decimal64_t	zero_z	40.0	21.5	38.3	18.4	31.9
decimal128_t	random	135.2	58.1	115.6	56.4	154.1
decimal128_t	tie_far	118.6	55.0	99.9	49.3	129.3
decimal128_t	near	131.6	59.2	111.9	59.1	177.1
decimal128_t	cancel	44.9	41.7	46.4	49.1	91.6
decimal128_t	zero_z	102.6	37.1	83.6	32.1	113.1
decimal_fast32_t	random	37.1	13.5	36.6	15.9
decimal_fast32_t	tie_far	26.1	10.1	26.8	13.4
decimal_fast32_t	near	41.5	16.9	40.3	19.3
decimal_fast32_t	cancel	23.6	13.4	25.1	13.0
decimal_fast32_t	zero_z	24.9	6.8	25.8	9.1
decimal_fast64_t	random	110.6	25.8	89.5	25.1
decimal_fast64_t	tie_far	57.5	22.5	47.3	20.2
decimal_fast64_t	near	153.5	26.3	120.7	26.8
decimal_fast64_t	cancel	58.1	18.5	60.0	15.7
decimal_fast64_t	zero_z	43.0	16.7	36.7	13.8
decimal_fast128_t	random	135.3	58.4	118.8	56.5
decimal_fast128_t	tie_far	121.5	55.0	102.4	50.0
decimal_fast128_t	near	128.6	60.4	115.2	58.6
decimal_fast128_t	cancel	49.1	46.9	46.9	51.6
decimal_fast128_t	zero_z	102.5	35.8	86.1	33.5

With Clang, the 128-bit types are 3 to 5 ns slower in the cancel set.

Size

One function calls fma for each of the six types. The table gives the size of all .text sections of its object file, in bytes.

compiler	flags	develop	fix
GCC 16.2	-O2	120683	117880
GCC 16.2	-Os	68440	68563
Clang 23.1	-O2	87270	82073
Clang 23.1	-Os	70472	68224

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 98.4%. Comparing base (4a2cea3) to head (e55f42c).

Additional details and impacted files

@@            Coverage Diff            @@
##           develop   #1523     +/-   ##
=========================================
- Coverage     98.6%   98.4%   -0.2%     
=========================================
  Files          321     323      +2     
  Lines        27261   27444    +183     
  Branches      2285    2288      +3     
=========================================
+ Hits         26858   26981    +123     
- Misses         403     463     +60     

Files with missing lines	Coverage Δ
include/boost/decimal/detail/cmath/fma.hpp	100.0% <100.0%> (ø)
test/github_issue_1522.cpp	100.0% <100.0%> (ø)
test/test_fma_tie.cpp	100.0% <100.0%> (ø)

... and 7 files with indirect coverage changes

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[mborland]

There was a merge conflict from your other PRs so the only change I made was to rebase this PR on develop