Intel64 Multiprecision Arithmetic

Eric Bainville - Dec 2006

These pages relate my experiments on 64-bit multiprecision arithmetic running on Core 2 Duo processors. Sep 2009: I updated some of these pages with Core i7 experiments, and the latest versions of glibc and gmp.

I describe a set of basic memory, arithmetic, and logic operations on (long) vectors of 64-bit words. The vector (A,N) is a vector of N consecutive 64-bit words A0, A1,..., AN-1. Such a vector represents an unsigned integer A0+A1.264+...+AN-1.264(N-1). All integers in the interval 0..264N-1 are representable, and have an unique representation.

To simplify the functions, I impose N to be a multiple of 8: the size of a vector is a multiple of 64 bytes (cache line). Vectors must be 8-byte aligned on the AMD64, and 16-byte aligned on the Intel64, where 128-bit SSE instructions are used.

I used information from the following sources:

The GMP library already provides a very fast and comprehensive set of low-level multiprecision functions (mpn). I will often use it as a starting point for further optimizations, along with some optimized memory function of the GNU C Library.

On the CPU, execution time for each function is evaluated for 8KB vectors (N=1024 words), using the rdtsc instruction surrounded by two cpuid instructions to ensure all instructions are counted (because rdtsc may be executed out-of-order, but cpuid can't be). With this size we can get a very precise measure of the number of required cycles per word. Each function is called a large number of times, and we take the average number of cycles.

Here are the best timings (in CPU cycles per 64-bit word) I could get on the Core 2 Duo (Conroe):

 Function glibc gmp this page Z ← 0 1.00 - 0.50 Z ← op Z (not) - - 0.50 Z ← op Z (neg) - - 1.00 Z ← X 1.75 - 1.50 Z ← Z op X (and or xor) - - 1.63 Z ← Z op X (+ -) - 2.63 2.53 Z ← Z op not X (and or xor) - - - Z ← Z * k - 4.00 - Z ← Z * k op X (+ -) - 4.54 - Z ← Z << k - 2.40 - Z ← Z / k - 33.29 -

glibc refers to glibc 2.4, and gmp refers to gmp 4.2.1 with Jason W. Martin's patch.

Sep 2009: Here are the timings on a Core i7 920 (Bloomfield):

 Function glibc gmp this page Z ← 0 0.50 - 0.50 Z ← op Z (not) - - - Z ← op Z (neg) - - - Z ← X 1.00 - 1.00 Z ← Z op X (and or xor) - - - Z ← Z op X (+ -) - 2.25 - Z ← Z op not X (and or xor) - - - Z ← Z * k - 3.90 - Z ← Z * k op X (+ -) - 5.00 - Z ← Z << k - 1.78 - Z ← Z / k - 20.94 -

glibc refers to glibc 2.9, and gmp refers to gmp 4.3.1. Hyperthreading is enabled in the processor (4 cores, 8 threads).