AVX-512 is an instruction set that expands the CPU registers from 256-bit to 512-bit. It comes with a core specification, AVX-512 Foundation, and several extensions that can be added where it makes sense. For instance, AVX-512 Exponential and Reciprocal Instructions (ERI) help solve transcendental problems, which occur in geometry and are useful for GPU-style architectures. As such, it appears in Knights Landing but not anywhere else.
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Today's rumor is that Skylake, the successor to Broadwell, will not include any AVX-512 support in its consumer parts. According to the lineup, Xeons based on Skylake will support AVX-512 Foundation, Conflict Detection Instructions, Vector Length Extensions, Byte and Word Instructions, and Double and Quadword Instructions. Fused Multiply and Add for 52-bit Integers and Vector Byte Manipulation Instructions will not arrive until Cannonlake shrinks everything down to 10nm.
The main advantage of larger registers is speed. When you can fit 512 bits of data in a memory bank and operate upon it at once, you are able to do several, linked calculations together. AVX-512 has the capability to operate on sixteen 32-bit values at the same time, which is obviously sixteen times the compute performance compared with doing just one at a time… if all sixteen undergo the same operation. This is especially useful for games, media, and other, vector-based workloads (like science).
This also makes me question whether the entire Cannonlake product stack will support AVX-512. While vectorization is a cheap way to get performance for suitable workloads, it does take up a large amount of transistors (wider memory, extra instructions, etc.). Hopefully Intel will be able to afford the cost with the next die shrink.