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If you have a large number of limbs, you can improve efficiencs by computing only half the triad of coefficients and doubling them - or something like that. I believe this reduces the computational complexity.
Every time that i try for it i miss something and multiprecision fails. but I#m really prone to typos and mistakes.
I would almost want to take this on in a GSoC 24 with some other optimizations unless it gets done sooner.
On a practical note, it is a bit harder to see the clean C++ interface to catch many use cases. You could provide a squaring functoin and clients knowing they need a square could use it. internally i think you could see if &left == &right in left * right = result, but I'm not sure if that is robust enough.
I am wondering if there is a function for computing the square of an integer. I ask because GMP has
mpn_sqr()
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