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| Computing billions of π digits using GMP |
This program, written by Hanhong Xue, can be used to compute tens of billions of digits of π, provided you have enough memory in your system: gmp-chudnovsky.c (last updated 2008-12-05).
For the program to work properly for more than 231 bits (about 600 million
digits) you need to to use GMP 4.2 (or later).
Timing results with GMP 4.3:
| Number of digits | Time (s) AMD Opteron (K8) 2.2 GHz | Time (s) AMD Opteron (K10) 2.3 GHz | Time (s) Intel "Core 2" 2.13 GHz | Time (s) PowerPC 970 1.6 GHz | Time (s) Pentium 4 3.2 GHz |
|---|---|---|---|---|---|
| 100,000 | 0.04 | 0.04 | 0.05 | 0.15 | 0.10 |
| 1,000,000 | 0.90 | 0.77 | 1.08 | 2.3 | 1.77 |
| 10,000,000 | 16.8 | 13.9 | 19.7 | 40.4 | 31.0 |
| 100,000,000 | 291 | 246 | 349 | 692 | 542 |
| 1,000,000,000 | - | 4520 | - |
Timing results with GMP 4.2.4:
| Number of digits | Time (s) AMD Opteron (K8) 2.2 GHz | Time (s) AMD Opteron (K10) 2.3 GHz | Time (s) Intel "Core 2" 2.13 GHz | Time (s) PowerPC 970 1.6 GHz | Time (s) Pentium 4 3.2 GHz |
|---|---|---|---|---|---|
| 100,000 | 0.06 | - | 0.12 | 0.17 | 0.15 |
| 1,000,000 | 1.48 | - | 2.35 | 2.92 | 2.9 |
| 10,000,000 | 26.8 | - | 42.6 | 52.5 | 52.3 |
| 100,000,000 | 467 | - | 756 | 902 | 902 |
| 1,000,000,000 | - | - | - |
How do these numbers compare to other π computing programs out there?
It seems gmp-chudnovsky.c with GMP 4.3 is faster than all
specialized π programs on Athlon, Core 2 and 64-bit Pentium 4, but a tad
bit slower on 32-bit Pentium 4.
Many π programs proclaim themselves as "the fastest", but then they
are actually several times slower than gmp-chudnovsky.c with the
current GMP release. Compare the numbers!
Using GMP 4.3, a fast 64-bit computer, and sufficient memory, it should be possible to compute up to 41 billion digits. Unfortunately, the memory requirements are about 8n bytes for computing n digits, which will make most desktop computers unfit for 41 billion digit computations. Memory locality in GMP 4.x's FFT multiply code is not good enough for efficient computation with operands on disk.
Attempting computations of more than 41 billion digits will cause overflow
in the mpz type. The fabled GMP 5 will allow the patient and wealthy to
compute up to at least 1 quintillion (1015) digits, and unlike GMP
4.x, GMP 5 will operate fine with operands on disk. You'll need around 4000
high-end swap disks in order to compute 1 quintillion digits, but surely that
will qualify you for a discount ("buy 4000, pay for 3999").