Add with carry

For $n$-bit operands, an add with carry operation forms $X+Y+C_{\mathrm{in}}$, retains its residue modulo $2^n$, and outputs its arithmetic carry. Adding low limbs first and feeding each arithmetic carry into the next limb implements multiword unsigned addition. With $M=2^n$, the identities $X_0+Y_0=Z_0+Mc$ and $X_1+Y_1+c=Z_1+Md$ combine into $(M{X}_1+X_0)+(M{Y}_1+Y_0)=M{Z}_1+Z_0+M^{2}d$.^[ref]