Reduce sqrt128 multiply cost: mul128By64, precompute target, 128-bit sig_z²

[justinzhuguangwen]

Summary

Reduce redundant and oversized multiplications in sqrt implementations.

Key changes:

• Precompute target in sqrt32/64/128: sig_gx × scale computed once, was recomputed 2–4× per call
• mul128By64 (new in u256.hpp): 128×64→256 using 2 muls; replaces umul256 (4 muls) for initial sig_z
• 128-bit sig_z²: sig_z < √10 × 10³³ fits in 128 bits → umul256(u128, u128) (4 muls) replaces u256 × u256 (16 muls via Knuth), called 4× per sqrt128

Performance delta (this PR only, sqrt_bench.py, WSL2 x86_64, g++ -O3)

Type	Before	After	Delta
decimal128_t	0.80M ops/s	0.93M ops/s	+16%
decimal_fast128_t	0.71M ops/s	0.81M ops/s	+14%

decimal32/64 within noise — optimization targets decimal128 multiply path.

Overall speedup vs original baseline

Type	Baseline	Current	Speedup
decimal32_t	1.49M	11.36M	7.60×
decimal64_t	0.74M	4.64M	6.23×
decimal128_t	0.23M	0.93M	3.97×
decimal_fast32_t	1.71M	9.34M	5.46×
decimal_fast64_t	0.78M	3.69M	4.71×
decimal_fast128_t	0.22M	0.81M	3.66×

Test

All existing sqrt tests pass (test_sqrt, test_cmath, test_constants, github_issue_1107, github_issue_1110).

[justinzhuguangwen]

Hi @ckormanyos @mborland,

Follow-up micro-optimization on sqrt — reviewed each step for redundant work:

• Precompute target once (was recomputed 2–4× per call)
• Add mul128By64 for 128×64 case (2 muls vs 4)
• Use umul256(u128, u128) for sig_z² (4 muls vs 16, called 4× in sqrt128)

decimal128 throughput +16%, all tests pass.

[cppalliance-bot]

An automated preview of the documentation is available at https://1337.decimal.prtest3.cppalliance.org/libs/decimal/doc/html/index.html

If more commits are pushed to the pull request, the docs will rebuild at the same URL.

2026-02-09 02:02:55 UTC

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 98.9%. Comparing base (fc669ff) to head (f566d56).

Additional details and impacted files

@@            Coverage Diff            @@
##           develop   #1337     +/-   ##
=========================================
+ Coverage     98.9%   98.9%   +0.1%     
=========================================
  Files          283     283             
  Lines        18327   18337     +10     
  Branches      1939    1939             
=========================================
+ Hits         18109   18120     +11     
+ Misses         218     217      -1     

Files with missing lines	Coverage Δ
...e/boost/decimal/detail/cmath/impl/sqrt128_impl.hpp	100.0% <100.0%> (ø)
...de/boost/decimal/detail/cmath/impl/sqrt32_impl.hpp	100.0% <100.0%> (ø)
...de/boost/decimal/detail/cmath/impl/sqrt64_impl.hpp	100.0% <ø> (ø)
include/boost/decimal/detail/u256.hpp	96.7% <100.0%> (+0.3%)	⬆️

... and 1 file with indirect coverage changes

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