32-bit versus 64-bit floating-point performance

Question

I have ran into a curious problem. An algorithm I am working on consists of lots of computations like this

q = x(0)*y(0)*z(0) + x(1)*y(1)*z(1) + ...

where the length of summation is between 4 and 7.

The original computations are all done using 64-bit precision. For experimentation, I tried using 32-bit precision for x,y,z input values (so that computations are performed using 32-bit), and storing final result as 64-bit value (straightforward cast).

I expected 32-bit performance to be better (cache size, SIMD size, etc.), but to my surprise there was no difference in performance, maybe even decrease.

The architecture in question is Intel 64, Linux, and GCC. Both codes do seem to use SSE and arrays in both cases are aligned to 16 byte boundary.

Why would it be so? My guess so far is that 32-bit precision can use SSE only on the first four elements, with the rest being done serially compounded by cast overhead.

Answer 1

On x87 at least, everything is really done in 80-bit precision internally. The precision really just determines how many of those bits are stored in memory. This is part of the reason why different optimization settings can change results slightly: They change the amount of rounding from 80-bit to 32- or 64-bit.

In practice, using 80-bit floating point (long double in C and C++, real in D) is usually slow because there's no efficient way to load and store 80 bits from memory. 32- and 64-bit are usually equally fast provided that memory bandwidth isn't the bottleneck, i.e. if everything is in cache anyhow. 64-bit can be slower if either of the following happens:

1. Memory bandwidth is the bottleneck.
2. The 64-bit numbers aren't properly aligned on 8-byte boundaries. 32-bit numbers only require 4-byte alignment for optimal efficiency, so they're less finicky. Some compilers (the Digital Mars D compiler comes to mind) don't always get this right for 64-bit doubles stored on the stack. This causes twice the amount of memory operations to be necessary to load one, in practice resulting in about a 2x performance hit compared to properly aligned 64-bit floats or 32-bit floats.

As far as SIMD optimizations go, it should be noted that most compilers are horrible at auto-vectorizing code. If you don't want to write directly in assembly language, the best way to take advantage of these instructions is to use things like array-wise operations, which are available, for example, in D, and implemented in terms of SSE instructions. Similarly, in C or C++, you would probably want to use a high level library of functions that are SSE-optimized, though I don't know of a good one off the top of my head because I mostly program in D.