# FLINT: Fast Library for Number Theory

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## Benchmarks

This page shows how FLINT compares to other software on a selection of artificial benchmarks.

### Polynomial factorisation over Z/pZ

Time (s) to factor the degree-n polynomial (x + 2x + 3x3 + ... + nxn) modulo 17. CPU: AMD Opteron 6174 2.20 GHz, except Mathematica which was timed on a Xeon E5-2650, 2.00 GHz (>20% faster). FLINT, NTL and presumably also Magma use similar implementations of the Cantor-Zassenhaus algorithm with the Kaltofen-Shoup improvement, so this benchmark largely tests the speed of polynomial arithmetic.

n Mathematica 9.0 Pari/GP 2.5.4 NTL 6.0.0 Magma 2.19-8 FLINT 2.4
10 0.000097 0.000046 0.000076 0.000085 0.000032
30 0.000415 0.000266 0.000493 0.000480 0.000303
100 0.00534 0.00229 0.00721 0.00461 0.00166
300 0.105 0.0303 0.0670 0.0404 0.0158
1000 3.88 2.46 0.601 0.559 0.228
3000 65.8 107 5.15 4.67 2.09
10000 1229 - 166 84.8 59.8
30000 - - 825 664 376

Time (s) to compute a p-adic logarithm to precision 17n. CPU: Xeon X5670, 2.93 GHz. FLINT uses rectangular splitting series evaluation for small n and an asymptotically optimal algorithm (based on balanced argument reduction and binary splitting series evaluation) for large n.

n Magma 2.18-6 FLINT 2.3
32 0.000131 0.000018
512 0.011 0.00047
8192 12 0.035
131072 13388 2.4

### Factorisation of word-size integers

Time (s) to factor 10000 consecutive n-bit integers (2n-1 + 1), ..., (2n-1 + 10000), where n goes up to the word size (64 bits). CPU: Xeon E5-2650, 2.00 GHz. FLINT uses a combination of trial division, Hart's One Line Factor algorithm, SQUFOF, and quick tests for primes and perfect powers. The slowdown very close to 64 bits is possibly due to the fact that FLINT does not currently use the p+1 algorithm.

n Mathematica 9.0 Pari/GP 2.5.4 FLINT 2.4
16 0.0720 0.0160 0.00476
24 0.108 0.0560 0.0192
32 0.272 0.233 0.0812
40 0.960 0.608 0.199
48 1.87 1.17 0.430
56 2.76 2.28 1.19
64 4.60 4.21 3.86

### Power series over the real numbers

Time (s) to compute the Taylor series exp(exp(1+x)) to order xn at n-digit precision. CPU: Xeon E5-2650, 2.00 GHz. For large n, this example illustrates FLINT's efficient arithmetic with polynomials that have both high degree and large coefficients. It should be noted that the timings are not perfectly comparable since the results have different accuracy. Only FLINT 2.4/Arb computes rigorous error bounds.

n Mathematica 9.0 Pari/GP 2.5.4 FLINT 2.4/Arb
10 0.000113 0.000014 0.000021
30 0.000490 0.000066 0.000141
100 0.00473 0.00106 0.00149
300 0.0575 0.0276 0.0129
1000 2.17 1.66 0.215
3000 77.0 68.7 2.51
10000 - - 34.1
30000 - - 371

### Generating Bernoulli numbers

Time (s) to generate a table of the Bernoulli numbers B0, ..., Bn as exact fractions. CPU: Xeon E5-2650, 2.00 GHz. Pari/GP and FLINT 2.4/Arb compute the Bernoulli numbers by numerical approximation of the Riemann zeta function.

n Mathematica 9.0 Pari/GP 2.5.4 FLINT 2.4/Arb
1000 0.124 0.0360 0.0100
3000 4.01 0.913 0.103
10000 88.8 52.4 1.92
30000 1817 1653 33.2
100000 - - 669

### Integer partitions

Time (s) to compute the exact value of the partition function p(n). CPU: Xeon X5675, 3.07 GHz. All implementations (except Maple?) use the numerical Hardy-Ramanujan-Rademacher formula. Out of the two FLINT implementations, the newer "FLINT 2.4/Arb" version implements a provably correct algorithm (the older "FLINT 2.4" version depends on heuristics, with most of the speedup for very small n coming from use of hardware floating-point arithmetic).

Last updated: 2022-05-20 09:07:08 GMT

Contact: Fredrik Johansson, flint-devel mailing list.