Is it possible to use SSE and SSE2 to make a 128-bit wide integer?

SIMD is meant to work on multiple small values at the same time, hence there won't be any carry over to the higher unit and you must do that manually. In SSE2 there's no carry flag but you can easily calculate the carry as carry = sum < a or carry = sum < b like this. Worse yet, SSE2 doesn't have 64-bit comparisons either, so you must use some workaround like the one here

Here is an untested, unoptimized C code based on the idea above:

inline bool lessthan(__m128i a, __m128i b){
    a = _mm_xor_si128(a, _mm_set1_epi32(0x80000000));
    b = _mm_xor_si128(b, _mm_set1_epi32(0x80000000));
    __m128i t = _mm_cmplt_epi32(a, b);
    __m128i u = _mm_cmpgt_epi32(a, b);
    __m128i z = _mm_or_si128(t, _mm_shuffle_epi32(t, 177));
    z = _mm_andnot_si128(_mm_shuffle_epi32(u, 245),z);
    return _mm_cvtsi128_si32(z) & 1;
}

inline __m128i addi128(__m128i a, __m128i b)
{
    __m128i sum = _mm_add_epi64(a, b);
    __m128i mask = _mm_set1_epi64(0x8000000000000000);    
    if (lessthan(_mm_xor_si128(mask, sum), _mm_xor_si128(mask, a)))
    {
        __m128i ONE = _mm_setr_epi64(0, 1);
        sum = _mm_add_epi64(sum, ONE);
    }

    return sum;
}

As you can see, the code requires many more instructions and even after optimizing it may still be much longer than a simple 2 ADD/ADC pair in x86_64 (or 4 instructions in x86)

SSE2 will help though, if you have multiple 128-bit integers to add in parallel. However you need to arrange the high and low parts of the values properly so that we can add all the low parts at once, and all the high parts at once

See also

• practical BigNum AVX/SSE possible?
• Can long integer routines benefit from SSE?