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| Computing billions of π digits using GMP |
GMP is a general-purpose library for arithmetic on large numbers, but it works very well also for such special tasks as computing billions of digits of π. This program, written by Hanhong Xue, is all that's needed.
Timing results:
GMP 5.0
| Number of digits | AMD Athlon (K8) 2.2 GHz | AMD Phenom II (K10) 3.2 GHz | Intel Pentium 4 3.4 GHz | Intel Core 2 2.13 GHz | Intel Core i7 2.67 GHz | PowerPC 970 1.6 GHz |
|---|---|---|---|---|---|---|
| 100,000 | 0.03 | 0.08 | 0.04 | 0.03 | 0.13 | |
| 1,000,000 | 0.48 | 1.49 | 0.89 | 0.69 | 2.01 | |
| 10,000,000 | 8.2 | 26.3 | 16.4 | 12.0 | 35.3 | |
| 100,000,000 | 134 | 430 | 269 | 191 | 576 | |
| 1,000,000,000 | 6656 | 2896 |
GMP 4.3
| Number of digits | AMD Athlon (K8) 2.2 GHz | AMD Phenom II (K10) 3.2 GHz | Intel Pentium 4 3.2 GHz | Intel Core 2 2.13 GHz | Intel Core i7 2.67 GHz | PowerPC 970 1.6 GHz |
|---|---|---|---|---|---|---|
| 100,000 | 0.04 | 0.03 | 0.10 | 0.05 | 0.04 | 0.15 |
| 1,000,000 | 0.90 | 0.56 | 1.77 | 1.08 | 0.81 | 2.3 |
| 10,000,000 | 16.8 | 9.7 | 31.0 | 19.7 | 14.5 | 40.4 |
| 100,000,000 | 291 | 166 | 542 | 349 | 247 | 692 |
| 1,000,000,000 | 4069 |
GMP 4.2
| Number of digits | AMD Athlon (K8) 2.2 GHz | AMD Phenom II (K10) 3.2 GHz | Intel Pentium 4 3.2 GHz | Intel Core 2 2.13 GHz | Intel Core i7 2.67 GHz | PowerPC 970 1.6 GHz |
|---|---|---|---|---|---|---|
| 100,000 | 0.06 | 0.15 | 0.12 | 0.17 | ||
| 1,000,000 | 1.48 | 2.9 | 2.35 | 2.92 | ||
| 10,000,000 | 26.8 | 52.3 | 42.6 | 52.5 | ||
| 100,000,000 | 467 | 902 | 756 | 902 | ||
| 1,000,000,000 |
How do these numbers compare to other π computing programs out there?
It seems gmp-chudnovsky.c with GMP 5.0 is faster than all
specialised 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 5.0, 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 the FFT multiply code of GMP 5.0's 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. A planned future version of GMP will allow the patient and
wealthy to compute up to at least 1 quadrillion (1015) digits, and
unlike current GMP, this future GMP 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").