| Computing billions of π digits using GMP |
While GMP is a general-purpose library for arithmetic on large numbers, it also works very well for such special tasks as computing a silly number of digits of π. This program, written by Hanhong Xue, is all that's needed.
How do these numbers compare to other π computing programs out there?
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, 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'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. 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. One needs around 250 top-capacity rust swap disks in order to compute 1 quadrillion digits, but hopefully that will qualify for a discount ("buy 250, pay for 249").
Timing results are in seconds.
Note: All Intel processors below are affected by the security bugs Meltdown, Spectre, Foreshadow, and MDS. Skylake is further affected by the security bug Jcc. With proper workarounds, the performance of the Intel processors would be considerably lower!
| Number of digits | AMD K10 Thuban Phenom II 1090T | AMD Piledriver FX-8350 | AMD Zen1 Ryzen 2700X | AMD Zen2 Ryzen 3700X | Intel Penryn Xeon E3110 | Intel Nehalem Xeon X3470 | Intel Sandybridge Xeon E3-1270 | Intel Haswell Xeon E3-1271v3 | Intel Broadwell Xeon E3-1285Lv4 | Intel Skylake Xeon E3-1270v5 | IBM POWER7 | IBM POWER8 | IBM POWER9 3800 MHz |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 100,000 | 0.024 | 0.024 | 0.015 | 0.012 | 0.034 | 0.037 | 0.026 | 0.016 | 0.016 | 0.016 | 0.028 | 0.020 | 0.014 |
| 1,000,000 | 0.46 | 0.48 | 0.25 | 0.21 | 0.61 | 0.61 | 0.37 | 0.29 | 0.27 | 0.27 | 0.51 | 0.53 | 0.31 |
| 10,000,000 | 8.24 | 8.36 | 4.33 | 3.70 | 11.0 | 10.8 | 6.25 | 4.94 | 4.61 | 4.38 | 8.96 | 9.06 | 5.17 |
| 100,000,000 | 135 | 134 | 69 | 59 | 177 | 170 | 99 | 78 | 72 | 70 | 144 | 143 | 80 |
| 1,000,000,000 | 2107 | 2033 | 1057 | 876 | 2540 | 1489 | 1179 | 1107 | 1079 | 2197 | 2143 | 1228 |
| Number of digits | AMD Athlon (K8) 2.2 GHz | AMD K10 Thuban 3.2 GHz | Intel Pentium 4 3.4 GHz | Intel Conroe 2.13 GHz | Intel Nehalem 2.67 GHz | Intel Sandybridge 3.3 GHz | IBM PowerPC 970 1.8 GHz | Intel Itanium 2 0.9 GHz |
|---|---|---|---|---|---|---|---|---|
| 100,000 | 0.03 | 0.08 | 0.04 | 0.03 | 0.02 | 0.13 | 0.09 | |
| 1,000,000 | 0.48 | 1.49 | 0.89 | 0.69 | 0.49 | 1.73 | 1.67 | |
| 10,000,000 | 8.2 | 26.3 | 16.4 | 12.0 | 8.22 | 30.8 | 29.3 | |
| 100,000,000 | 134 | 430 | 269 | 191 | 131 | 497 | 494 | |
| 1,000,000,000 | 2097 | 6656 | 2896 |
| Number of digits | AMD Athlon (K8) 2.2 GHz | AMD K10 Thuban 3.2 GHz | Intel Pentium 4 3.2 GHz | Intel Conroe 2.13 GHz | Intel Nehalem 2.67 GHz | IBM 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 |
| Number of digits | AMD Athlon (K8) 2.2 GHz | AMD K10 Thuban) 3.2 GHz | Intel Pentium 4 3.2 GHz | Intel Conroe 2.13 GHz | Intel Nehalem 2.67 GHz | IBM 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 |