# AMD64 Multiprecision Arithmetic

Eric Bainville - Dec 2006These pages relate my experiments on 64-bit multiprecision arithmetic running on Athlon 64 processors.

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 A_{0}, A_{1},..., A_{N-1}.
Such a vector represents an unsigned integer A_{0}+A_{1}.2^{64}+...+A_{N-1}.2^{64(N-1)}. All integers in
the interval 0..2^{64N}-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:

- AMD Manuals,
- Intel Manuals,
**Agner Fog**'s Software Optimization Resources page, containing a lot of precious and easy to find information.

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.

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 I could get so far:

Function | glibc | gmp | this page |

Z ← 0 | 0.50 | - | 0.50 |

Z ← op Z (not neg) | - | - | 1.13 |

Z ← X | 1.50 | - | 1.50 |

Z ← Z op X (+ - and or xor) | - | 2.00 | 1.50 |

Z ← Z op not X (and or xor) | - | - | 1.50 |

Z ← Z * k | - | 3.00 | 3.00 |

Z ← Z * k op X (+ -) | - | 3.13 | 3.13 |

Z ← Z << k | - | 2.50 | 2.25 |

Z ← Z / k | - | 22.00 | 17.00 |

glibc refers to glibc 2.4, and gmp to gmp 4.2.1 with P. Gaudry's patch.

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