Benchmarks

Comparison of FastTanhSinhQuadrature.jl vs FastGaussQuadrature.jl.

System:

CPU: Intel(R) Core(TM) Ultra 7 155U
Threads: 1 (Single-threaded execution)

Results

Legend:

TS: FastTanhSinhQuadrature.integrate1D
TS SIMD: FastTanhSinhQuadrature.integrate1D_avx (using LoopVectorization)
GQ: FastGaussQuadrature.gausslegendre

Function	Domain	Points	TS (ns)	TS SIMD (ns)	GQ (ns)	Ratio (TS/GQ)	Ratio (TS SIMD/GQ)
exp(x)	[-1, 1]	5	33.09	14.27	33.03	1.00	0.43
exp(x)	[-1, 1]	50	322.87	81.99	176.68	1.83	0.46
exp(x)	[-1, 1]	500	3374.75	782.08	1665.50	2.03	0.47
sin(x)^2	[-1, 1]	5	40.39	23.38	32.63	1.24	0.72
sin(x)^2	[-1, 1]	50	404.32	145.48	176.75	2.29	0.82
sin(x)^2	[-1, 1]	500	4361.71	1409.10	1768.40	2.47	0.80
1/(1+25x^2)	[-1, 1]	5	3.00	3.75	17.35	0.17	0.22
1/(1+25x^2)	[-1, 1]	50	41.82	39.17	34.77	1.20	1.13
1/(1+25x^2)	[-1, 1]	500	477.21	213.24	320.39	1.49	0.67
sqrt(1-x^2)	[-1, 1]	5	4.06	4.69	20.31	0.20	0.23
sqrt(1-x^2)	[-1, 1]	50	68.01	45.31	69.68	0.98	0.65
sqrt(1-x^2)	[-1, 1]	500	636.53	313.13	743.43	0.86	0.42
x^2	[-1, 1]	5	2.01	3.04	19.86	0.10	0.15
x^2	[-1, 1]	50	17.72	21.68	38.07	0.47	0.57
x^2	[-1, 1]	500	213.31	54.97	273.19	0.78	0.20
log(1-x)	[-1, 1]	5	38.06	30.29	34.32	1.11	0.88
log(1-x)	[-1, 1]	50	371.88	187.29	214.70	1.73	0.87
log(1-x)	[-1, 1]	500	4117.38	1978.80	2117.80	1.94	0.93
x^3	[-1, 1]	5	2.23	3.14	18.65	0.12	0.17
x^3	[-1, 1]	50	21.86	22.09	35.69	0.61	0.62
x^3	[-1, 1]	500	239.28	60.84	267.62	0.89	0.23
x^3+x^2+x+1	[-1, 1]	5	3.98	4.00	20.12	0.20	0.20
x^3+x^2+x+1	[-1, 1]	50	41.75	26.31	36.58	1.14	0.72
x^3+x^2+x+1	[-1, 1]	500	450.22	80.03	263.54	1.71	0.30

Analysis

Polynomials: FastTanhSinhQuadrature with SIMD acceleration (x^2, x^3) is significantly faster (up to ~3-5x) than Gauss-Legendre quadrature.
Singularities: Functions like sqrt(1-x^2) and log(1-x) are handled very efficiently, often matching or outperforming Gaussian quadrature due to the double exponential clustering of nodes.
Runge Function: 1/(1+25x^2) also shows competitive performance, especially with SIMD.

In summary, for smooth analytic functions, Gaussian quadrature (standard non-SIMD integrate1D vs GQ) is faster due to fewer nodes required for exactness using polynomials. However, FastTanhSinhQuadrature's SIMD implementation (integrate1D_avx) often bridges or exceeds this gap, and it is the superior choice for singular integrands.

Running Benchmarks

To run the benchmarks yourself:

julia --project=benchmark -e 'include("benchmark/benchmarks.jl"); run_benchmark()'