Improving the efficiency of the code

It may happen that the evaluation of an expression involves many time the evaluation of a sub-expression. Clearly evaluating only once these sub-expressions will speed up the code. This may be done through a user-provided Maple procedure that must be called ALIAS_FSIMPLIFY. This procedure takes as input a file descriptor and an expression expr. It will be first called right after the creation of the evaluation file with a string: detecting that expr is a string allow to write some initialization. Then it will be called before writing any equation to allow for simplification. For example assume that an expression that will be treated by the Make[FJH] procedure involves numerous time the evaluation of the sine and cosine of the first variable x (whose name in ALIAS-C++ is v_IS(1)).The ALIAS_FSIMPLIFY procedure may be written as:

 
 ALIAS_FSIMPLIFY:=proc(fid,expr)
 local aux:
 if type(expr,string) then
     fprintf(fid,"INTERVAL SS,CC;\n"):
     fprintf(fid,"SS=Sin(v_IS(1));\n"):
     fprintf(fid,"CC=Cos(v_IS(1));\n"):
     RETURN(0):
 fi:
 aux:=expr:
 aux:=subs(sin(x)=SS,cos(x)=CC,aux):
 RETURN(aux):
 end:

This procedure will be first called with a string for expr and a consequence is that at the beginning of the evaluation file fid the interval variable SS,CC will be defined and then assigned to the value of $\sin(x), \cos(x)$. Then the procedure will be called for each expression that will be assigned to expr: each occurrence of the sine and cosine of $x$ in the expression will be substituted by SS, CC.

The procedure Math_Func, see section 9.4, may be used to identify mathematical functions occurring in an expression and the list provided by this function may be used to write a generic ALIAS_FSIMPLIFY procedure that will automatically compute only once the more complex components of an expression. The procedure Auto_Diff, see section 2.3.2, may also be used to speed up the interval evaluation of an expression. Note also that a similar mechanism exists for expression involving determinants of matrices.