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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
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).
Here are timing results with GMP 4.2.2 (sorted after processor speed):
| Number of digits | Time (s) Athlon64 2.6 GHz | Time (s) "Core 2" 3.0 GHz | Time (s) PowerPC 970 2.7 GHz | Time (s) Itanium 2 1.6 GHz | Time (s) Pentium 4 3.2 GHz |
|---|---|---|---|---|---|
| 100,000 | 0.05 | 0.08 | 0.13 | 0.13 | 0.15 |
| 1,000,000 | 1.23 | 1.68 | 1.92 | 2.33 | 2.9 |
| 10,000,000 | 22.5 | 30.9 | 33.3 | 42.2 | 52.3 |
| 100,000,000 | 393 | 549 | 565 | 740 | 902 |
| 1,000,000,000 | 6860 | 9278 | 14400 |
How do these numbers compare to other π computing programs out there?
It seems gmp-chudnovsky.c with GMP 4.2 is faster than all
specialized π programs on Athlon, but a tad bit slower on Pentium 4.
Many π programs proclaim themselves as "the fastest", but then they
are actually slower than gmp-chudnovsky.c with the current GMP
release. Compare the numbers!
The next version of GMP will make gmp-chudnovsky.c beat
'em also on Pentium 4 and crush them on Athlon. E.g., linking
gmp-chudnovsky.c with the current development version of GMP,
computing 100,000,000 digits needs about 200 seconds on a high end Athlon64
system.
Using GMP 4.2, a fast 64-bit computer, and sufficient memory, it should be possible to compute up to 41 billion (41e9) 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 4.2'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 (1e15) digits, and unlike GMP 4.x, GMP 5
will operate fine with operands on disk. You'll need around 10000 high-end
swap disks in order to compute 1 quintillion digits, but surely that will
qualify you for a good discount ("buy 10000, pay for 9999").