[Reuben006]

Description of Change

Fix floating-point nontermination in math/sqrt_double.cpp.

The previous implementation could stop making progress before reaching its fixed absolute error tolerance. Input 10000000001 timed out during local testing. The smallest positive subnormal also timed out because the reciprocal workaround overflows.

This change:

• Normalizes the input using std::frexp before bisection.
• Stops when rounding makes the midpoint equal an endpoint.
• Restores the result's scale using std::ldexp.
• Defines behavior for zero, negative inputs, NaN, and infinity.
• Adds Doxygen documentation, a worked example, and noninteractive assertion tests.

The implementation still uses bisection. std::sqrt is used only as a test reference; correct rounding is not guaranteed.

Related to #2456.

Validation

• Debug and optimized C++17 builds passed with strict compiler warnings.
• Built-in regression and boundary tests passed.
• Local CMake target build, clang-format, and clang-tidy checks passed.
• Fork CI passed: Code Formatter and compilation on Ubuntu, Windows, and macOS.
• An additional local verification harness, outside this PR, checked 99,957 finite positive inputs against std::sqrt, within relative error 2 * std::numeric_limits<double>::epsilon().

Related Work

• fix: sqrt_double hangs on x < 1 (#760) #768 introduced the existing reciprocal workaround.
• [fix/docs]: fit sqrt_double.cpp to guidelines #2932 proposed documentation and tests but closed without merging.
• Fix for https://github.com/TheAlgorithms/C-Plus-Plus/issues/3234 #3263 overlaps in Doxygen tag syntax only; this PR addresses numerical correctness and adds regression tests.

Checklist

• Added description of change
• Existing filename follows repository naming guidelines
• Added tests and examples; tests pass
• Added educational Doxygen documentation
• Updated relevant documentation/comments
• PR title follows semantic commit guidelines
• Searched previous suggestions for duplicates
• I acknowledge that all my contributions will be made under the project's license.