In hardware this would be called a carry save adder [1].
One interesting fact about carry save adders is that the carry part of the register can be extended in order to avoid any carrying between words for very long periods of time. Instead of using 53 bits to represent 52 bits, use 60 bits, and now you can perform 256 sequential additions with no carrying between words before the carry part saturates and needs to be handled.
Somewhat surprisingly, it's even possible to use this construction in the context of a quantum computation [2][3]. The reason it's surprising is because tacking on additional registers like this can act as an information leakage mechanism (which is an error in a quantum computation). For analogy, you can imagine that if you computed a public key using this trick that you would not be comfortable handing an attacker the registers storing your public key before you removed the carry padding that made the computation faster. But it turns out that if you just initialize the carry padding randomly, subtracting out of the next word at the start to ensure everything sums to the correct result at the end, then you can show that the information leakage and chance-of-bad-overflow is exponentially suppressed in the length of the carry padding. Currently the most efficient known way to implement Shor's algorithm uses this technique [4].
One interesting fact about carry save adders is that the carry part of the register can be extended in order to avoid any carrying between words for very long periods of time. Instead of using 53 bits to represent 52 bits, use 60 bits, and now you can perform 256 sequential additions with no carrying between words before the carry part saturates and needs to be handled.
Somewhat surprisingly, it's even possible to use this construction in the context of a quantum computation [2][3]. The reason it's surprising is because tacking on additional registers like this can act as an information leakage mechanism (which is an error in a quantum computation). For analogy, you can imagine that if you computed a public key using this trick that you would not be comfortable handing an attacker the registers storing your public key before you removed the carry padding that made the computation faster. But it turns out that if you just initialize the carry padding randomly, subtracting out of the next word at the start to ensure everything sums to the correct result at the end, then you can show that the information leakage and chance-of-bad-overflow is exponentially suppressed in the length of the carry padding. Currently the most efficient known way to implement Shor's algorithm uses this technique [4].
1: https://en.wikipedia.org/wiki/Carry-save_adder
2: https://arxiv.org/abs/1905.08488
3: https://youtu.be/upTipX9yXNg?t=777
4: https://arxiv.org/abs/1905.09749