Little-endian vs. big-endian

Additional cache coherency/lock-free posts are still in the pipe, I just haven’t gotten around to writing much lately.

In the meantime, here’s a quick post on something else: little-endian (LE) vs. big-endian (BE) and some of the trade-offs involved. The whole debate comes up periodically by proponents of LE or BE with missionary zeal, and then I get annoyed, because usually what makes either endianness superior for some applications makes it inferior for others. So for what it’s worth, here’s the trade-offs I’m aware of:

Doing math vs. indexing/sorting/searching

• LE stores bytes in the order you do most math operations on them (if you were to do it byte by byte and not in larger chunks, that is). Additions and subtractions proceed from least-significant bit (LSB) to most-significant bit (MSB), always, because that’s the order carries (and borrows) are generated. Multiplications form partial products from smaller terms (at the limit, individual bits, though for hardware you’re more likely to use radix-4 booth recoding or similar) and add them, and the final addition likewise is LSB to MSB. Long division is the exception and works its way downwards from the most significant bits, but divisions are generally much less frequent than additions, subtractions and multiplication.

  Arbitrary-precision arithmetic (“bignum arithmetic”) thus typically chops up numbers into segments (“legs”) matching the word size of the underlying machine, and stores these words in memory ordered from least significant to most significant – on both LE and BE architectures.

  All 8-bit ISAs I’m personally familiar with (Intel 8080, Zilog Z80, MOS 6502) use LE, presumably for that reason; it’s the more natural byte order for 16-bit numbers if you only have an 8-bit ALU. (That said, Motorola’s 8-bit 6800 apparently used BE). And consequently, if you’re designing a new architecture with the explicit goal of being source-code compatible with the 8080 (yes, x86 was already constrained by backwards-compatibility considerations even for the original 8086!) it’s going to be little-endian.

• BE stores bytes in the order you compare them (assuming a lexicographic compare).

  So if you want to do a lexicographic sort, memcmp does the right thing on BE but not on LE: encoding numbers in BE is an order-preserving bijection (if the ordering predicate is lexicographic comparison). This is a very useful property if you’re in the business of selling your customers machines that spend a large chunk of their time sorting, searching and retrieving records, and likely one of the reasons why IBM’s architectures dating back as far as the mainframe era are big-endian. (It doesn’t make sense to speak of “endianness” before the IBM 7030, since that was the machine that introduced byte-addressable memory in the first place; machines before that point had word-based memory). It’s still common to encode numbers in BE for databases and key-value stores, even on LE architectures.

Byte order vs. bit order

• All LE architectures I’m aware of have the LSB as bit 0 and number both bits and bytes in order of increasing significance. Thus, bit and byte order agree: byte 0 of a number on a LE machine stores bits 0-7, byte 1 stores bits 8-15 and so forth. (Assuming 8-bit bytes, that is.)

• BE has two schools. First, there’s “Motorola style”, which is bits numbered with LSB=0 and from then on in increasing order of significance; but at the same time, byte 0 is the most significant byte, and following bytes decrease in significance. So by these conventions, a 16-bit number would store bits 8-15 in byte 0, and bits 0-7 in byte 1. As you can see, there’s a mismatch between byte order and bit order.

• Finally, there’s BE “IBM style”, which instead labels the MSB as bit 0. As the bit number increases, they decrease in significance. In this scheme, same as in LE, byte 0 stores bits 0-7 of a number, byte 1 stores bits 8-15, and so forth; these bytes are exactly reversed compared to the LE variant, but bit and byte ordering are in agreement again.

  That said, referring to the MSB as bit 0 is confusing in other ways; people normally expect bit 0 to have mask 1 << 0, and with MSB-first bit numbering that’s not the case.

Memory access

• For LE, the 8/16/32-bit prefixes of a 64-bit number all start at the same address as the number itself. This can be viewed as either an advantage (“it’s convenient!”) or a disadvantage (“it hides bugs!”).

  A LE load of 8/16/32/64 bits will always put all source bits at the same position in the destination register; as you make the load wider, it will just zero-clear (or sign-extend) less of them. Flow of data through the load/store circuitry is thus essentially the same regardless of operand size; different AND masks corresponding to the load size, but that’s it.

• For BE, prefixes start at different offsets. Again, can be viewed as either an advantage (“it prevents bugs!”) or a disadvantage (“I can’t transparently widen fields after the fact!”).

  A BE load of 8/16/32/64 bits puts the source bits in different locations in the destination register; instead of a width-dependent mask, we get a width-dependent shift. In a circuit, this is a Mux of differently-shifted versions of the source operand, which is (very slightly) more complicated than the masking for LE. (Not that I actually think anyone cares about HW complexity at that level today, or has in over a decade for that matter.)

That said, the difference can be slightly interesting if you don’t have a full complement of differently-sized loads; the Cell SPUs are an example. If you don’t have narrow loads, BE is hit a bit more than LE is. A synthesized LE narrow load is wide_unaligned_load(addr) & mask (where the wide unaligned load might itself consist of multiple steps, like it does on the SPUs); synthesized BE narrow load is wide_unaligned_load(addr + offs) & mask. Note the extra add of a non-zero offset, which means one more instruction. You can get rid of it in principle by just having all addresses for e.g. byte-aligned data be pre-incremented by offs, but that’s obnoxious too.

And that’s it for now, off the top of my head.