[agourakis82]

For complex inputs with a sufficiently large negative imaginary part, xsimd::tan returns an imaginary component of +1 instead of -1. For example, tan(complex<float>(1, -45)) and tan(complex<double>(1, -356)) both produce (0, +1) on the unchanged base, while std::tan produces (0, -1).

The overflow handling detects isinf(sinh(2 * imag(z))) but selects +1 for either sign. Use copysign to retain the overflowing value's sign. The denominator-zero handling and the fast-math path are unchanged.

Add float/double regression tests with mixed SIMD lanes, both imaginary signs, zero/nonzero real parts, maximum finite imaginary magnitudes and ordinary finite controls. The tests check finite results and both component values against a C++23 scalar reference.

Validation on base ebe9991 plus this patch:

• Native Apple clang 21, arm64/NEON64, C++23: the same new regression fails on the unchanged headers for both types, then passes with the correction. All four cases in the complex trigonometric unit pass (96 assertions).
• The same unit passes AddressSanitizer and UndefinedBehaviorSanitizer.
• Full native CMake/CTest suite passes on arm64: 432 doctest cases, 77,716 assertions; CTest 1/1.
• Native GCC 13.3, Linux x86-64, C++23: the before/after comparison and complete complex trigonometric unit pass in SSE2 and AVX2 builds (Slurm 17825, exit 0:0).
• The original issue reproducer was compiled unchanged with C++23 before and after the fix. Its negative-imaginary tan examples acquire the correct imaginary sign; other reported defects remain visible.

This addresses only the imaginary saturation-sign cases in #1432 and should not close the issue. The original reproducer also exposes real-component signed-zero differences, which remain outside this patch. The remaining tan pole cases, other complex functions and fast-math behavior are not addressed here. This is independent of the finite-real-axis sqrt correction in #1433.

AI assistance supported investigation, implementation and validation. A local model review was attempted but did not return within the bounded review window; the results above are executable checks, not model approval.