Performance of Distributions¶

The performance is measured on an MacBook Pro with an Intel Core i7-4960HQ CPU running macOS Sierra (version 10.12.4).

Three compilers are tested, LLVM clang (version Apple 8.1.0), GNU GCC (version 6.3.0), and Intel C++ compiler (version 17.0.2). Results of the LLVM clang compiler is shown here. The results with other compilers are similar.

Two usage cases of RNGs are considered. The first is generating random integers within a loop, each iteration generate a single random integer,

Four usage cases of the distributions are considered. First, if the distribution is available in the standard library, we measure the following case,

std::normal_distribution<double> rnorm_std(0, 1);
for (size_t i = 0; i = n; ++i)
    r[i] = rand(rng, rnorm_std);

Second, we measure the performance of the library’s implementation,

::mckl::NormalDistribution<double> rnorm_mckl(0, 1);
for (size_t i = 0; i = n; ++i)
    r[i] = ::mckl::rand(rng, rnorm_mckl);

The third and the fourth are the vectorized performance,

::mckl::rand(rng, rnorm_mckl, n, r.data());

See Vectorized Random Number Generating. The difference is that we first test the performance without the Intel MKL library and then with it. When Intel MKL is not used, the assembly implementation of selected functions are used. For all the four above, the ARS RNG is used (see AES Round Function Based Random Number Generators). The last is when there are mkl routines for generating random numbers from the distribution, either directly or indirectly. In this case, we use the MKL_ARS5 RNG (see MKL Random Number Generators),

::mckl::MKL_ARS5 rng_mkl;
::mckl::rand(rng_mkl, rnorm_mckl, n, r.data());

In all cases, we repeat the simulations 100 times, each time with \(n\) chosen randomly between 5,000 and 10,000. The total number of cycles of the 100 simulations are recorded, and then divided by the total number of elements generated. This gives the performance measurement in cpE. This experiment is repeated ten times, and the best results are shown. The five cases are labeled “STD”, “MCKL”, “VMF”, “VML” and “MKL”, respectively, in tables below.

Performance of Standard Uniform Distributions¶
Distribution	STD	MCKL	VMF	VML	MKL
`U01Canonical`	16.6	16.1	4.02	4.02	—
`U01CC`	—	10.9	2.54	2.53	—
`U01CO`	16.6	7.86	2.53	2.54	3.46
`U01OC`	—	8.59	2.54	2.54	—
`U01OO`	—	8.58	2.54	2.51	—

Performance of Distributions using the Inverse Method¶
Distribution	STD	MCKL	VMF	VML	MKL
`Arcsine(0,1)`	—	64.5	12.9	12.8	—
`Cauchy(0,1)`	97.2	71.9	15.9	19.6	11.7
`Exponential(1)`	68.6	48.4	12.2	9.94	9.34
`ExtremeValue(0,1)`	130	103	22.0	17.8	17.2
`Laplace(0,1)`	—	72.5	15.3	13.2	15.3
`Logistic(0,1)`	—	63.1	20.6	18.7	19.7
`Pareto(1,1)`	—	107	18.3	16.1	—
`Rayleigh(1)`	—	51.1	19.9	14.7	14.3
`UniformReal(-1.3,1.3)`	17.6	9.85	3.00	3.00	2.86
`Weibull(1,1)`	187	47.7	12.2	10.2	10.3

Performance of Beta Distribution¶
Distribution	STD	MCKL	VMF	VML	MKL
`Beta(0.3,0.3)`	—	244	167	71.7	51.6
`Beta(0.5,0.5)`	—	77.8	12.8	13.8	57.0
`Beta(0.5,1)`	—	98.7	18.1	15.8	14.7
`Beta(0.5,1.5)`	—	291	273	271	86.8
`Beta(0.9,0.9)`	—	249	231	231	74.0
`Beta(1,0.5)`	—	104	18.3	16.1	14.7
`Beta(1,1)`	—	21.4	2.59	2.59	14.7
`Beta(1,1.5)`	—	104	18.3	16.1	14.7
`Beta(1.5,0.5)`	—	292	273	273	87.0
`Beta(1.5,1)`	—	98.7	18.1	15.8	14.7
`Beta(1.5,1.5)`	—	257	67.7	62.8	60.3

Performance of \(\chi^2\)-Distribution¶
Distribution	STD	MCKL	VMF	VML	MKL
`ChiSquared(0.2)`	198	180	45.8	42.2	47.3
`ChiSquared(1)`	258	235	81.5	77.8	68.5
`ChiSquared(1.4)`	271	263	67.4	60.2	50.7
`ChiSquared(1.8)`	266	220	51.0	45.6	39.1
`ChiSquared(2)`	89.6	66.6	12.3	10.1	10.4
`ChiSquared(3)`	315	235	44.2	45.0	39.4
`ChiSquared(30)`	286	239	39.5	41.8	36.3

Performance of Gamma Distribution¶
Distribution	STD	MCKL	VMF	VML	MKL
`Gamma(0.1,1)`	198	179	45.7	42.3	47.2
`Gamma(0.5,1)`	257	234	81.0	77.7	68.4
`Gamma(0.7,1)`	270	261	67.4	60.1	50.7
`Gamma(0.9,1)`	265	218	50.8	44.8	39.2
`Gamma(1,1)`	88.2	65.2	12.2	10.2	10.4
`Gamma(1.5,1)`	310	234	43.9	45.2	39.5
`Gamma(15,1)`	285	236	40.4	41.8	36.3

Performance of Fisher’s F-Distribution¶
Distribution	STD	MCKL	VMF	VML	MKL
`FisherF(0.5,0.5)`	443	402	168	155	140
`FisherF(0.5,1)`	482	431	193	179	153
`FisherF(0.5,1.5)`	493	454	187	173	130
`FisherF(0.5,3)`	543	454	140	134	118
`FisherF(0.5,30)`	512	455	131	125	114
`FisherF(1,0.5)`	482	432	195	180	153
`FisherF(1,1)`	520	461	217	201	164
`FisherF(1,1.5)`	530	483	211	195	142
`FisherF(1,3)`	577	485	163	156	130
`FisherF(1,30)`	544	486	154	147	126
`FisherF(1.5,0.5)`	495	445	187	173	130
`FisherF(1.5,1)`	529	471	211	196	142
`FisherF(1.5,1.5)`	540	492	204	190	117
`FisherF(1.5,3)`	594	491	156	149	108
`FisherF(1.5,30)`	555	493	148	142	103
`FisherF(3,0.5)`	551	438	139	134	119
`FisherF(3,1)`	564	464	163	155	130
`FisherF(3,1.5)`	579	481	157	150	107
`FisherF(3,3)`	615	477	107	109	93.7
`FisherF(3,30)`	601	478	99.2	101	89.4
`FisherF(30,0.5)`	508	437	131	124	114
`FisherF(30,1)`	549	465	155	148	126
`FisherF(30,1.5)`	561	480	149	144	103
`FisherF(30,3)`	595	475	99.3	101	89.6
`FisherF(30,30)`	555	474	91.3	92.5	84.9

Performance of Normal and Related Distributions¶
Distribution	STD	MCKL	VMF	VML	MKL
`Normal(0,1)`	87.8	77.3	17.6	18.4	15.4
`Lognormal(0,1)`	143	119	23.6	24.2	19.9
`Levy(0,1)`	—	83.3	29.2	30.9	27.6

Performance of Stable Distribution¶
Distribution	STD	MCKL	VMF	VML	MKL
`Stable(0.5,1,0,1)`	—	387	202	67.0	—
`Stable(1,0,0,1)`	—	194	49.0	51.5	—
`Stable(2,0,0,1)`	—	388	202	75.2	—

Performance of Student’s t-Distribution¶
Distribution	STD	MCKL	VMF	VML	MKL
`StudentT(0.2)`	298	283	91.4	87.4	87.4
`StudentT(1)`	362	335	137	134	112
`StudentT(1.4)`	374	353	146	139	91.7
`StudentT(1.8)`	369	315	104	97.2	76.8
`StudentT(2)`	187	178	49.1	45.1	44.7
`StudentT(3)`	413	329	87.3	86.8	77.9
`StudentT(30)`	374	330	78.5	78.5	72.3

Performance of Discrete Distributions¶
Distribution	STD	MCKL	VMF	VML	MKL
`Geometric(0.5)`	119	50.8	15.7	14.0	14.7
`UniformInt(-10,10)`	142	26.8	6.40	6.83	7.34