/**
         
 * Prolog code for the symbolic derivation benchmark.
         
 *
         
 * This is the benchmark of page 222 of:
         
 * Warren, D.H.D. (1983): Applied Logic - Its Use and
         
 * Implementation as a Programming Tool,
         
 * Technical Note 290, SRI International, 1983
         
 *
         
 * Warranty & Liability
         
 * To the extent permitted by applicable law and unless explicitly
         
 * otherwise agreed upon, XLOG Technologies AG makes no warranties
         
 * regarding the provided information. XLOG Technologies AG assumes
         
 * no liability that any problems might be solved with the information
         
 * provided by XLOG Technologies AG.
         
 *
         
 * Rights & License
         
 * All industrial property rights regarding the information - copyright
         
 * and patent rights in particular - are the sole property of XLOG
         
 * Technologies AG. If the company was not the originator of some
         
 * excerpts, XLOG Technologies AG has at least obtained the right to
         
 * reproduce, change and translate the information.
         
 *
         
 * Reproduction is restricted to the whole unaltered document. Reproduction
         
 * of the information is only allowed for non-commercial uses. Selling,
         
 * giving away or letting of the execution of the library is prohibited.
         
 * The library can be distributed as part of your applications and libraries
         
 * for execution provided this comment remains unchanged.
         
 *
         
 * Restrictions
         
 * Only to be distributed with programs that add significant and primary
         
 * functionality to the library. Not to be distributed with additional
         
 * software intended to replace any components of the library.
         
 *
         
 * Trademarks
         
 * Jekejeke is a registered trademark of XLOG Technologies AG.
         
 */
         

         
/*****************************************************************/
         
/* Normal Test Cases                                             */
         
/*****************************************************************/
         

         
% ops8(-Expr)
         
ops8(E) :-
         
	d((x + 1) * ((x ^ 2 + 2) * (x ^ 3 + 3)), x, E).
         

         
% divide10(-Expr)
         
divide10(E) :-
         
	d(((((((((x / x) / x) / x) / x) / x) / x) / x) / x) / x, x, E).
         

         
% log10(-Expr)
         
log10(E) :-
         
	d(log(log(log(log(log(log(log(log(log(log(x)))))))))), x, E).
         

         
% times10(-Expr)
         
times10(E) :-
         
	d(((((((((x * x) * x) * x) * x) * x) * x) * x) * x) * x, x, E).
         

         
% deriv
         
deriv :-
         
	ops8(_), divide10(_), log10(_), times10(_).
         

         
/*****************************************************************/
         
/* The Derivator                                                 */
         
/*****************************************************************/
         

         
% d(+Expr, +Var, -Expr)
         
d(U + V, X, DU + DV) :- !,
         
	d(U, X, DU),
         
	d(V, X, DV).
         
d(U - V, X, DU - DV) :- !,
         
	d(U, X, DU),
         
	d(V, X, DV).
         
d(U * V, X, DU * V + U * DV) :- !,
         
	d(U, X, DU),
         
	d(V, X, DV).
         
d(U / V, X, (DU * V - U * DV) / V ^ 2) :- !,
         
	d(U, X, DU),
         
	d(V, X, DV).
         
d(U ^ N, X, DU * N * U ^ N1) :- !,
         
	N1 is N - 1,
         
	d(U, X, DU).
         
d(-U, X, -DU) :- !,
         
	d(U, X, DU).
         
d(exp(U), X, exp(U) * DU) :- !,
         
	d(U, X, DU).
         
d(log(U), X, DU / U) :- !,
         
	d(U, X, DU).
         
d(X, X, 1) :- !.
         
d(_, _, 0).