 kill(all)$
 decimales:3$
 prec:10^decimales$
 x0:1$ x1:4$ x2:6$ x3:5$ x4:3$ 
 f0:0$ f1:1.38629$ f2:1.79176$ f3:1.60944$ f4:1.09861$  
 f01:(f1-f0)/(x1-x0)$
 f12:(f2-f1)/(x2-x1)$
 f23:(f3-f2)/(x3-x2)$
 f34:(f4-f3)/(x4-x3)$
 f012:(f12-f01)/(x2-x0)$ 
 f123:(f23-f12)/(x3-x1)$ 
 f234:(f34-f23)/(x4-x2)$ 
 f0123:(f123-f012)/(x3-x0)$ 
 f1234:(f234-f123)/(x4-x1)$ 
 f01234:(f1234-f0123)/(x4-x0)$
 p1(x):=f0+f01*(x-x0);
 p2(x):=f0+f01*(x-x0)+f012*(x-x0)*(x-x1);
 p3(x):=f0+f01*(x-x0)+f012*(x-x0)*(x-x1)
        +f0123*(x-x0)*(x-x1)*(x-x2);
 p4(x):=f0+f01*(x-x0)+f012*(x-x0)*(x-x1)
        +f0123*(x-x0)*(x-x1)*(x-x2)
        +f01234*(x-x0)*(x-x1)*(x-x2)*(x-x3);
 poli_1:expand(p1(x));
 poli_2:expand(p2(x));
 poli_3:expand(p3(x));
 poli_4:expand(p4(x));