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Top1. Introduction
The applications of fuzzy mathematics and fuzzy logic play major rule in various areas of mathematics, physics and engineering (Wei et al., 2016; Wei et al., 2016; Kumar et al.; 2016). One of the important topics in fuzzy mathematics is to evaluate fuzzy integrals which was investigated by many authors such as (Goetschel & Voxman, 1986; Kaleva, 1987; Nanda, 1989; Ralescu & Adams, 1980; Wang, 1984; Zimmerman, 1987; Wu, 2000; Bede & Gal, 2005; Fariborzi Araghi & Khadem, 2011; Pasrija 2013; Fariborzi Araghi & Khadem, 2014). Also, Allahviranloo (2005a) used the Romberg integration rule (Atkinson, 1993; Dutka, 1984; Evans & Megson, 1986; Havie, 1986; Iyengar & Jain, 2009; Stoer & Bulirsch, 2002; Tseng & Lee, 2008) in fuzzy case as an accurate and efficient scheme to approximate the fuzzy integrals.
In these works, a fixed number of points and step size have been considered to apply the given numerical integration rule. Also, the exact solution of all examples must be accessible to compare the approximate and the exact solutions. The packages like Matlab, Mathematica or Maple were applied to implement the given algorithm which are worked based on the floating-point arithmetic. In this case, the accuracy of results cannot be estimated and the results are not optimized. Also, for the problems which the exact solution is not accessible the given algorithms cannot be applied, since the distance between the exact and approximate solutions is considered as the obtained error. Furthermore, by changing the step size, the results are changed. So, the best approximation in computer point of view cannot be computed and the optimal number of points cannot be found. This means the results may not valid informatically.
The CESTAC method is based on a probabilistic approach of the round-off error propagation which replaces the floating-point arithmetic by the stochastic arithmetic which was developed by Vignes and Laporte (Vignes; 1993). Also, the CADNA library, which was designated by Chesneaux (Chesneaux,1998; Chesneaux & Vignes, 1992; Jezequel & Chesneaux, 2004; Vignes, 1993) in order to automatic implementation of the CESTAC method was applied to validate the results of different problems such as (Chesneaux, 1990; Chesneaux, 1992; Chesneaux & Vignes, 1992; Chesneaux, 1994; Toutounian,1997; Chesneaux,1998; Chesneaux & Jezequel, 1998; Khojasteh Salkuyeh & Toutounian, 2009; Khojasteh Salkuyeh & Toutounian, 2006; Abbasbandy & Fariborzi Araghi, 2002a; Abbasbandy & Fariborzi Araghi, 2002b; Abbasbandy & Fariborzi Araghi, 2002c; Jezequel & Chesneaux, 2004; Abbasbandy & Fariborzi Araghi, 2004; Khojasteh Salkuyeh & Toutounian, 2006; Khojasteh Salkuyeh & Toutounian, 2009; Fariborzi Araghi & Noeighdam, 2016).