feat: add C language binding

[adeshkumar1]

Description

Add C language binding (ddog_trace_*) wrapping the C++ tracing library for integration from C-based projects.
Follows Datadog's ddog_ naming convention from libdatadog. Gated behind BUILD_C_BINDING=OFF.

Motivation

Enable C-based projects to use dd-trace-cpp without a C++ dependency at the call site.

Building and Testing

cmake -B build -DBUILD_C_BINDING=ON -DDD_TRACE_BUILD_TESTING=ON
cmake --build build -j
./build/binding/c/test_c_binding

[pr-commenter]

Benchmarks

Benchmark execution time: 2026-03-12 20:33:34

Comparing candidate commit 1f83a79 in PR branch adeshkumar1/c-binding-on-main with baseline commit f8c3913 in branch main.

Found 0 performance improvements and 0 performance regressions! Performance is the same for 1 metrics, 0 unstable metrics.

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 87.70%. Comparing base (f8c3913) to head (1f83a79).
⚠️ Report is 1 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #286      +/-   ##
==========================================
- Coverage   87.76%   87.70%   -0.07%     
==========================================
  Files          84       84              
  Lines        5658     5668      +10     
==========================================
+ Hits         4966     4971       +5     
- Misses        692      697       +5     

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[datadog-official]

🎯 Code Coverage (details)
• Patch Coverage: 100.00%
• Overall Coverage: 87.70% (-0.07%)

_{This comment will be updated automatically if new data arrives.
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[adeshkumar1]

All format changes/decisions are done wit the clang-format script

[xlamorlette-datadog]

Great work!
(Please address very latest comments, and we're good to go!)

[xlamorlette-datadog]

The order of the fields matters for ABI stability. So, what should be the order of these fields:

• Same order as in SpanConfig?
• Logical order? (But what is it? The one in SpanConfig does not fit since, in this C API, name is the only one mandatory and so should come first.)
• Alphabetical order? (When in doubt, this is what I would do. But if we ever add a field, it will no more be sorted.)

[xlamorlette-datadog]

nit: The comment about the caller and the callee (”The caller must allocate this struct […] Pass NULL to ignore errors.”) should not be here. I think the comment for dd_tracer_new() is sufficient.