Non algebraic equations

Let consider the following system

 
eq1:=x^2-x*cos(y)-1:
eq2:=y^2-cos(x)-1:

for which we want to determine if real roots exist in the range [-5,5], [-5,5]. With a depth level 0 (no bisection is done) the analyzer proposes the following ranges as possibly containing a real root:

 
Range 1( x y)
[-0.86812856880248989722,-0.86812856880248989722]
[-1.28306500467802298,-1.28306500467802298]
Range 2( x y)
[1.219833908062562422,1.219833908062562422]
[-1.1592247392497665448,-1.1592247392497665448]
Range 3( x y)
[1.2198350997243703198,1.2198367316341245381]
[-1.159223547587958647,-1.1592195879408853099]
Range 4( x y)
[-0.86812856880248978619,-0.86812856880248978619]
[1.28306500467802298,1.28306500467802298]
Range 5( x y)
[1.219833908062562422,1.219833908062562422]
[1.1592247392497667668,1.1592247392497667668]
Range 6( x y)
[1.2198352388536839452,1.2198367316341249822]
[1.1592195879408850878,1.1592236867169092296]

Among this 6 ranges, 4 are reduced to point and are therefore a result of the application of the Newton scheme. The other 4 ranges are very close to one of the solution, but the algorithm has not been able to eliminate them. With a depth level of 1 the algorithm proposes directly the four solutions of this system:

 
Range 1( x y)
[-0.86812741994331155126,-0.86812741994331155126]
[-1.2830653469203250339,-1.2830653469203250339]
Range 2( x y)
[1.2198339926936523359,1.2198339926936523359]
[-1.1592245852349318813,-1.1592245852349318813]
Range 3( x y)
[-0.86812741994331166229,-0.86812741994331166229]
[1.2830653469203250339,1.2830653469203250339]
Range 4( x y)
[1.219833901899016082,1.219833901899016082]
[1.159224777165462017,1.159224777165462017]

Note that the initial range [-5,5] may be largely expanded without modifying the result and with only a low amount of additional computation time. For example for the range [-1000,1000] for both variables the computation time remains the same.

The previous system may be considered as a special occurrence of the system:

 
eq1:=x^2-x*cos(y)-x1:
eq2:=y^2-cos(x)-1:

where the parameter x1 has value 1. This special case of the generic system may be analyzed by indicating in a parameter file:

 
x1 1

Now just by changing the value of x1 in this file to

 
x1 2

and modifying the configuration file to include the sentence:

 
Directory /tmp

we will get the solutions:

 
Range 1( x y)
[-1.2271026453420417202,-1.2271026453420417202]
[-1.1562745250557344701,-1.1562745250557344701]
Range 2( x y)
[1.757647484313144215,1.757647484313144215]
[-0.90234927430355182931,-0.90234927430355182931]
Range 3( x y)
[-1.2271026453420417202,-1.2271026453420417202]
[1.1562722309470763182,1.1562722309470763182]
Range 4( x y)
[1.757647484313144215,1.757647484313144215]
[0.90234927430355205136,0.90234927430355205136]