EigenRand : The Fastest C++11-compatible random distribution generator for Eigen
EigenRand is a header-only library for Eigen, providing vectorized random number engines and vectorized random distribution generators.
Since the classic Random functions of Eigen relies on an old C function rand(),
there is no way to control random numbers and no guarantee for quality of generated numbers.
In addition, Eigen's Random is slow because rand() is hard to vectorize.
EigenRand provides a variety of random distribution functions similar to C++11 standard's random functions,
which can be vectorized and easily integrated into Eigen's expressions of Matrix and Array.
You can get 5~10 times speed by just replacing old Eigen's Random or unvectorizable c++11 random number generators with EigenRand.
Features
C++11-compatible Random Number Generator
5~10 times faster than non-vectorized functions
Header-only (like Eigen)
Can be easily integrated with Eigen's expressions
Currently supports only x86 and x86-64 architecture
a vectorized version of Mersenne Twister algorithm. It generates two 64bit random integers simultaneously with SSE2 and four integers with AVX2.
std::mt19937_64
Performance
The following charts show the relative speed-up of EigenRand compared to references(equivalent functions of C++ std or Eigen).
The following charts are about multivariate distributions.
The following result is a measure of the time in seconds it takes to generate 1M random numbers.
It shows the average of 20 times.
Intel(R) Xeon(R) Platinum 8171M CPU @ 2.60GHz (Ubuntu 16.04, gcc5.4)
C++ std (or Eigen)
EigenRand (No Vect.)
EigenRand (SSE2)
EigenRand (SSSE3)
EigenRand (AVX)
EigenRand (AVX2)
balanced*
9.0
5.9
1.5
1.4
1.3
0.9
balanced(double)*
8.7
6.4
3.3
2.9
1.7
1.7
binomial(20, 0.5)
400.8
118.5
32.7
36.6
30.0
22.7
binomial(50, 0.01)
71.7
22.5
7.7
8.3
7.9
6.6
binomial(100, 0.75)
340.5
454.5
91.7
111.5
106.3
86.4
cauchy
36.1
54.4
6.1
7.1
4.7
3.9
chiSquared
80.5
249.5
64.6
58.0
29.4
28.8
discrete(int32)
-
14.0
2.9
2.6
2.4
1.7
discrete(fp32)
-
21.9
4.3
4.0
3.6
3.0
discrete(fp64)
72.4
21.4
6.9
6.5
4.9
3.7
exponential
31.0
25.3
5.5
5.3
3.3
2.9
extremeValue
66.0
60.1
11.9
10.7
6.5
5.8
fisherF(1, 1)
178.1
35.1
33.2
39.3
22.9
18.7
fisherF(5, 5)
141.8
415.2
136.47
172.4
92.4
74.9
gamma(0.2, 1)
207.8
211.4
54.6
51.2
26.9
27.0
gamma(5, 3)
80.9
60.0
14.3
13.3
11.4
8.0
gamma(10.5, 1)
81.1
248.6
63.3
58.5
29.2
28.4
geometric
43.0
22.4
6.7
7.4
5.8
lognormal
66.3
55.4
12.8
11.8
6.2
6.2
negativeBinomial(10, 0.5)
312.0
301.4
82.9
100.6
95.3
77.9
negativeBinomial(20, 0.25)
483.4
575.9
125.0
158.2
148.4
119.5
normal(0, 1)
38.1
28.5
6.8
6.2
3.8
3.7
normal(2, 3)
37.6
29.0
7.3
6.6
4.0
3.9
poisson(1)
31.8
25.2
9.8
10.8
9.7
8.2
poisson(16)
231.8
274.1
66.2
80.7
74.4
64.2
randBits
5.2
5.4
1.4
1.3
1.1
1.0
studentT(1)
122.7
120.1
15.3
19.2
12.6
9.4
studentT(20)
102.2
111.1
15.4
19.2
12.2
9.4
uniformInt(0~63)
22.4
4.7
1.7
1.6
1.4
1.1
uniformInt(0~100k)
21.8
10.1
6.2
6.7
6.6
5.4
uniformReal
12.9
5.7
1.4
1.2
1.4
0.7
weibull
41.0
35.8
17.7
15.5
8.5
8.5
Since there is no equivalent class to balanced in C++11 std, we used Eigen::DenseBase::Random instead.
C++ std
EigenRand (No Vect.)
EigenRand (SSE2)
EigenRand (SSSE3)
EigenRand (AVX)
EigenRand (AVX2)
Mersenne Twister(int32)
4.7
5.6
4.0
3.7
3.5
3.6
Mersenne Twister(int64)
5.4
5.3
4.0
3.9
3.4
2.6
Python 3.6 + scipy 1.5.2 + numpy 1.19.2
EigenRand (No Vect.)
EigenRand (SSE2)
EigenRand (SSSE3)
EigenRand (AVX)
EigenRand (AVX2)
Dirichlet(4)
6.47
6.60
2.39
2.49
1.34
1.67
Dirichlet(100)
75.95
189.97
66.60
72.11
38.86
34.98
InvWishart(4)
140.18
7.62
4.21
4.54
3.58
3.39
InvWishart(50)
1510.47
1737.4
697.39
733.69
604.59
554.006
Multinomial(4, t=20)
3.32
4.12
0.95
1.06
1.00
1.03
Multinomial(4, t=1000)
3.51
192.51
35.99
39.58
27.84
35.45
Multinomial(100, t=20)
69.19
4.80
2.00
2.20
2.28
2.09
Multinomial(100, t=1000)
139.74
179.43
49.48
56.19
40.78
43.18
MvNormal(4)
2.32
0.96
0.36
0.37
0.25
0.30
MvNormal(100)
49.09
57.18
17.17
18.51
10.82
11.03
Wishart(4)
71.19
5.28
2.70
2.93
2.04
1.94
Wishart(50)
1185.26
1360.49
492.91
517.44
359.03
324.60
Intel(R) Xeon(R) CPU E5-1650 v2 @ 3.50GHz (macOS 10.15, clang-1103)
C++ std (or Eigen)
EigenRand (No Vect.)
EigenRand (SSE2)
EigenRand (SSSE3)
EigenRand (AVX)
balanced*
6.5
7.3
1.1
1.4
1.1
balanced(double)*
6.6
7.5
2.6
3.3
2.4
binomial(20, 0.5)
38.8
164.9
27.7
29.3
24.9
binomial(50, 0.01)
21.9
27.6
6.6
7.0
6.3
binomial(100, 0.75)
52.2
421.9
93.6
94.8
89.1
cauchy
36.0
30.4
5.6
5.8
4.0
chiSquared
84.4
152.2
44.1
48.7
26.2
discrete(int32)
-
12.4
2.1
2.6
2.2
discrete(fp32)
-
23.2
3.4
3.7
3.4
discrete(fp64)
48.6
22.9
4.2
5.0
4.6
exponential
22.0
18.0
4.1
4.9
3.2
extremeValue
36.2
32.0
8.7
9.5
5.1
fisherF(1, 1)
158.2
73.1
32.3
32.1
18.1
fisherF(5, 5)
177.3
310.1
127.0
121.8
74.3
gamma(0.2, 1)
69.8
80.4
28.5
33.8
19.2
gamma(5, 3)
83.9
53.3
10.6
12.4
8.6
gamma(10.5, 1)
83.2
150.4
43.3
48.4
26.2
geometric
39.6
19.0
4.3
4.4
4.1
lognormal
43.8
40.7
9.0
10.8
5.7
negativeBinomial(10, 0.5)
217.4
274.8
71.6
73.7
68.2
negativeBinomial(20, 0.25)
192.9
464.9
112.0
111.5
105.7
normal(0, 1)
32.6
28.6
5.5
6.5
3.8
normal(2, 3)
32.9
30.5
5.7
6.7
3.9
poisson(1)
37.9
31.0
7.5
7.8
7.1
poisson(16)
92.4
243.3
55.6
57.7
53.7
randBits
6.5
6.5
1.1
1.3
1.1
studentT(1)
115.0
54.1
15.5
15.7
8.3
studentT(20)
121.2
53.8
15.8
16.0
8.2
uniformInt(0~63)
20.2
9.8
1.8
1.8
1.6
uniformInt(0~100k)
25.7
16.1
8.1
8.5
7.2
uniformReal
12.7
7.0
1.0
1.2
1.1
weibull
23.1
19.2
11.6
13.6
7.6
Since there is no equivalent class to balanced in C++11 std, we used Eigen::DenseBase::Random instead.
C++ std
EigenRand (No Vect.)
EigenRand (SSE2)
EigenRand (SSSE3)
EigenRand (AVX)
Mersenne Twister(int32)
6.2
6.4
1.7
2.0
1.8
Mersenne Twister(int64)
6.4
6.3
2.5
3.1
2.4
Python 3.6 + scipy 1.5.2 + numpy 1.19.2
EigenRand (No Vect.)
EigenRand (SSE2)
EigenRand (SSSE3)
EigenRand (AVX)
Dirichlet(4)
3.54
3.29
1.25
1.25
0.83
Dirichlet(100)
57.63
145.32
49.71
49.50
29.13
InvWishart(4)
210.92
7.53
3.72
3.66
3.10
InvWishart(50)
1980.73
1446.40
560.40
559.73
457.07
Multinomial(4, t=20)
2.60
5.22
1.48
1.50
1.42
Multinomial(4, t=1000)
3.90
208.75
29.19
29.50
27.70
Multinomial(100, t=20)
47.71
7.09
3.71
3.63
3.60
Multinomial(100, t=1000)
128.69
215.19
44.48
44.63
43.76
MvNormal(4)
2.04
1.05
0.35
0.34
0.19
MvNormal(100)
48.69
47.10
16.25
16.12
11.41
Wishart(4)
81.11
13.24
9.87
9.81
5.90
Wishart(50)
1419.02
1087.40
448.06
442.97
328.20
Intel(R) Xeon(R) Platinum 8171M CPU @ 2.60GHz (Windows Server 2019, MSVC2019)
C++ std (or Eigen)
EigenRand (No Vect.)
EigenRand (SSE2)
EigenRand (AVX)
EigenRand (AVX2)
balanced*
20.7
7.2
3.3
4.0
2.2
balanced(double)*
21.9
8.8
6.7
4.3
4.3
binomial(20, 0.5)
718.3
141.0
38.1
30.2
32.7
binomial(50, 0.01)
61.5
21.4
7.5
6.5
8.0
binomial(100, 0.75)
495.9
1042.5
100.6
95.2
93.0
cauchy
71.6
30.0
6.8
6.4
3.0
chiSquared
243.0
147.3
63.5
34.1
24.0
discrete(int32)
-
12.4
3.5
2.7
2.2
discrete(fp32)
-
19.2
5.1
3.6
3.7
discrete(fp64)
83.9
19.0
6.7
7.4
4.6
exponential
58.7
16.0
6.8
6.4
3.0
extremeValue
64.6
27.7
13.5
9.8
5.5
fisherF(1, 1)
178.7
75.2
35.3
28.4
17.5
fisherF(5, 5)
491.0
298.4
125.8
87.4
60.5
gamma(0.2, 1)
211.7
69.3
43.7
24.7
18.7
gamma(5, 3)
272.5
42.3
17.6
17.2
8.5
gamma(10.5, 1)
237.8
146.2
63.7
33.8
23.5
geometric
49.3
17.0
7.0
5.8
5.4
lognormal
169.8
37.6
12.7
7.2
5.0
negativeBinomial(10, 0.5)
752.7
462.3
87.0
83.0
81.6
negativeBinomial(20, 0.25)
611.4
855.3
123.7
125.3
116.6
normal(0, 1)
78.4
21.1
6.9
4.6
2.9
normal(2, 3)
77.2
22.3
6.8
4.8
3.1
poisson(1)
77.4
28.9
10.0
8.1
10.1
poisson(16)
312.9
485.5
63.6
61.5
60.5
randBits
6.0
6.2
3.1
2.7
2.7
studentT(1)
175.8
53.9
17.3
12.5
7.7
studentT(20)
173.2
55.5
17.9
12.7
7.6
uniformInt(0~63)
39.1
5.2
2.0
1.4
1.6
uniformInt(0~100k)
38.5
12.3
7.6
6.0
7.7
uniformReal
53.4
5.7
1.9
2.3
1.0
weibull
75.1
44.3
18.5
14.3
7.9
Since there is no equivalent class to balanced in C++11 std, we used Eigen::DenseBase::Random instead.
C++ std
EigenRand (No Vect.)
EigenRand (SSE2)
EigenRand (AVX)
EigenRand (AVX2)
Mersenne Twister(int32)
6.5
6.4
5.6
5.1
4.5
Mersenne Twister(int64)
6.6
6.5
6.9
5.9
5.1
Python 3.6 + scipy 1.5.2 + numpy 1.19.2
EigenRand (No Vect.)
EigenRand (SSE2)
EigenRand (AVX)
EigenRand (AVX2)
Dirichlet(4)
4.27
3.20
2.31
1.43
1.25
Dirichlet(100)
69.61
150.33
67.01
47.34
32.47
InvWishart(4)
482.87
14.52
8.88
13.17
11.28
InvWishart(50)
2222.72
2211.66
902.34
775.36
610.60
Multinomial(4, t=20)
2.99
5.41
1.99
1.92
1.78
Multinomial(4, t=1000)
4.23
235.84
49.73
42.41
40.76
Multinomial(100, t=20)
58.20
9.12
5.84
6.02
5.98
Multinomial(100, t=1000)
130.54
234.40
72.99
66.36
55.28
MvNormal(4)
2.25
1.89
0.35
0.32
0.25
MvNormal(100)
57.71
68.80
24.40
18.28
13.05
Wishart(4)
70.18
16.25
4.49
3.97
3.07
Wishart(50)
1471.29
1641.73
628.58
485.68
349.81
AMD Ryzen 7 3700x CPU @ 3.60GHz (Windows 10, MSVC2017)
C++ std (or Eigen)
EigenRand (SSE2)
EigenRand (AVX)
EigenRand (AVX2)
balanced*
20.8
1.9
2.0
1.4
balanced(double)*
21.7
4.1
2.7
3.0
binomial(20, 0.5)
416.0
27.7
28.9
29.1
binomial(50, 0.01)
37.8
6.3
6.0
6.6
binomial(100, 0.75)
309.1
72.4
66.0
67.0
cauchy
42.2
4.8
5.1
2.7
chiSquared
153.8
33.5
21.2
17.0
discrete(int32)
-
2.4
2.3
2.5
discrete(fp32)
-
2.6
2.3
3.5
discrete(fp64)
55.8
5.1
4.7
4.3
exponential
33.4
6.4
2.8
2.2
extremeValue
39.4
7.8
4.6
4.0
fisherF(1, 1)
103.9
25.3
14.9
11.7
fisherF(5, 5)
295.7
85.5
58.3
44.8
gamma(0.2, 1)
128.8
31.9
18.3
15.8
gamma(5, 3)
156.1
9.7
8.0
5.0
gamma(10.5, 1)
148.5
33.1
21.1
17.2
geometric
27.1
6.6
4.3
4.1
lognormal
104.0
6.6
4.7
3.5
negativeBinomial(10, 0.5)
462.1
60.0
56.4
58.6
negativeBinomial(20, 0.25)
357.6
84.5
80.6
78.4
normal(0, 1)
48.8
4.2
3.7
2.3
normal(2, 3)
48.8
4.5
3.8
2.4
poisson(1)
46.4
7.9
7.4
8.2
poisson(16)
192.4
43.2
40.4
40.9
randBits
4.2
1.7
1.5
1.8
studentT(1)
107.0
12.3
6.8
5.7
studentT(20)
107.1
12.3
6.8
5.8
uniformInt(0~63)
31.2
1.1
1.0
1.2
uniformInt(0~100k)
27.7
5.6
5.6
5.4
uniformReal
30.7
1.1
1.0
0.6
weibull
46.5
10.6
6.4
5.2
Since there is no equivalent class to balanced in C++11 std, we used Eigen::DenseBase::Random instead.
C++ std
EigenRand (SSE2)
EigenRand (AVX)
EigenRand (AVX2)
Mersenne Twister(int32)
5.0
3.4
3.4
3.3
Mersenne Twister(int64)
5.1
3.9
3.9
3.3
Accuracy
Since vectorized mathematical functions may have a loss of precision, I measured how well the generated random number fits its actual distribution.
32768 samples were generated and Earth Mover's Distance between samples and its actual distribution was calculated for each distribution.
Following table shows the average distance (and stdev.) of results performed 50 times for different seeds.
C++ std
EigenRand
balanced*
.0034(.0015)
.0034(.0015)
chiSquared(7)
.0260(.0091)
.0242(.0079)
exponential(1)
.0065(.0025)
.0072(.0022)
extremeValue(1, 1)
.0097(.0029)
.0088(.0025)
gamma(0.2, 1)
.0380(.0021)
.0377(.0025)
gamma(1, 1)
.0070(.0020)
.0065(.0023)
gamma(5, 1)
.0169(.0065)
.0170(.0051)
lognormal(0, 1)
.0072(.0029)
.0067(.0022)
normal(0, 1)
.0070(.0024)
.0073(.0020)
uniformReal
.0018(.0008)
.0017(.0007)
weibull(2, 1)
.0032(.0013)
.0031(.0010)
(* Result of balanced were from Eigen::Random, not C++ std)
The smaller value means that the sample result fits its distribution better.
The results of EigenRand and C++ std appear to be equivalent within the margin of error.
License
MIT License
History
0.3.5 (2021-07-16)
Now UniformRealGen generates accurate double values.
Fixed a bug where non-vectorized double-type NormalGen would get stuck in an infinite loop.
New overloading functions balanced and balancedLike which generate values over [a, b] were added.
0.3.4 (2021-04-25)
Now Eigen 3.3.4 - 3.3.6 versions are additionally supported.
0.3.3 (2021-03-30)
A compilation failure with some RNGs in double type was fixed.
An internal function name plgamma conflict with one of SpecialFunctionsPacketMath.h was fixed.
0.3.2 (2021-03-26)
A default constructor for DiscreteGen was added.
0.3.1 (2020-11-15)
Compiling errors in the environment EIGEN_COMP_MINGW && __GXX_ABI_VERSION < 1004 was fixed.
0.3.0 (2020-10-17)
Potential cache conflict in generator was solved.
Generator classes were added for efficient reusability.
Multivariate distributions including Multinomial, Dirichlet, MvNormal, Wishart, InvWishart were added.
0.2.2 (2020-08-02)
Now ParallelRandomEngineAdaptor and MersenneTwister use aligned array on heap.
0.2.1 (2020-07-11)
A new template class ParallelRandomEngineAdaptor yielding the same random sequence regardless of SIMD ISA was added.
0.2.0 (2020-07-04)
New distributions including cauchy, studentT, fisherF, uniformInt, binomial, negativeBinomial, poisson and geometric were added.
A new member function uniform_real for PacketRandomEngine was added.
