What are the special arithmetic operations for 128-bit results in x86-64?

Full multiply and divide treat %rdx:%rax as a single 128-bit value: mulq/imulq produce a 128-bit product there, and divq/idivq divide that 128-bit value, leaving quotient in %rax and remainder in %rdx.

When a 64×64 multiply could overflow 64 bits, or a division needs a 128-bit dividend, the architecture pairs two registers — %rdx holding the high half, %rax the low half.

The dividend setup is the part you must not skip:

# signed x / y
movq x, %rax
cqto              # sign-extend %rax across %rdx:%rax
idivq y           # %rax = quotient, %rdx = remainder

# unsigned x / y
movq x, %rax
movq $0, %rdx     # clear the high half
divq y

Warning: if %rdx holds leftover bits, the division uses a bogus 128-bit dividend; dividing by zero faults outright.

Go deeper:

• Felix Cloutier — IMUL — signed multiply, including the full-width RDX:RAX form.
• Felix Cloutier — DIV — the DIV side of the 128-bit RDX:RAX dividend setup.