Example.

The Gaussian integral

was computed with 14 accurate digits.

The results are given below.

SINGLE PRECISION
Tol	Trapezoidal		Simpson
	error	n	error	n
0.1E+01	-.2E-01		0.5E-03
0.1E+00	-.2E-01		0.4E-04
0.1E-01	-.1E-02		0.4E-04
0.1E-02	-.3E-03		0.4E-04
0.1E-03	-.2E-04		0.3E-05
0.1E-04	-.1E-05		0.2E-06
0.1E-05	-.2E-06		-.8E-07
0.1E-06	-.2E-06		0.0E+00
0.1E-07	-.1E-04		0.8E-07
0.1E-08	-.1E-04		0.8E-07
0.1E-09	-.1E-04		0.8E-07
0.1E-10	-.1E-04		0.8E-07
0.1E-11	-.1E-04		0.8E-07

DOUBLE PRECISION
TOL	Trapezoidal		Simpson
	error	n	error	n
0.1E+01	-.2E-01		0.5E-03
0.1E+00	-.2E-01		0.4E-04
0.1E-01	-.1E-02		0.4E-04
0.1E-02	-.3E-03		0.4E-04
0.1E-03	-.2E-04		0.3E-05
0.1E-04	-.1E-05		0.2E-06
0.1E-05	-.3E-06		0.1E-07
0.1E-06	-.2E-07		0.7E-09
0.1E-07	-.1E-08		0.7E-09
0.1E-08	-.3E-09		0.4E-10
0.1E-09	-.2E-10		0.3E-11
0.1E-10	-.1E-11		0.2E-12
0.1E-11	-.3E-12		0.6E-14

Aside from this presentation, we compare the results above with the results obtained via Gaussian Quadrature integration formula. This method gives excellent results with only several node points; it is therefore much more efficient than both trapezoidal and Simpson rules.

Gauss Quadrature
error	n
-3.0E-4	2
-1.2E-5	3
4.4E-7	4
-8.0E-9	5