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sym-diff.pl
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sym-diff.pl
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% diff(x^(x*log(cos(x))), x, N).
% N = x^(exp(log(1/x))-1)*(exp(log(1/x))+ - (1)/(x*x)/(1/x)*exp(log(1/x))*x*log(x))*cos(x^exp(log(1/x)))
% diff(expr, var, div)
%%%%%%%%%%%%%%%%%%%%%%%%%%
opt_sum(X, X, 0).
opt_sum(Y, 0, Y).
opt_sum(W, X, Y) :-
number(X),
number(Y),
W is X + Y.
opt_sum(2*X, X, Y) :-
X == Y.
opt_sum(X+Y, X, Y).
%%%%%%%%%%%%%%%%%%%%%%%%%%
opt_sub(X, X, 0).
opt_sub(-Y, 0, Y).
opt_sub(W, X, Y) :-
number(X),
number(Y),
W is X - Y.
opt_sub(0, X, Y) :-
X == Y.
opt_sub(X-Y, X, Y).
%%%%%%%%%%%%%%%%%%%%%%%%%%
opt_mul(0, 0, _).
opt_mul(0, _, 0).
opt_mul(Y, 1, Y).
opt_mul(X, X, 1).
opt_mul(Y, X, Y) :-
X == 1.
opt_mul(W, X, Y) :-
number(X),
number(Y),
W is X * Y.
opt_mul(X*Y, X, Y).
%%%%%%%%%%%%%%%%%%%%%%%%%%
opt_div(_,_,0) :-
throw(error(evaluation_error(zero_divisor),(is)/2)).
opt_div(0, 0, N).
opt_div(X, X, 1).
opt_div(1, X, Y) :-
X == Y.
opt_div(W, X, Y) :-
number(X),
number(Y),
W is X / Y.
opt_div(X/Y, X, Y).
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%
% diff
%%%%%%%%%%%%%%%%%%%%%%%%%%
% constant
diff(E, _, 0) :- number(E).
% the same variable
diff(E, V, 0) :-
atom(E),
E \== V.
% other variables
diff(E, V, 1) :-
atom(E),
E == V.
% sum
diff(E1 + E2, V, W) :-
diff(E1, V, Ed1),
diff(E2, V, Ed2),
opt_sum(W, Ed1, Ed2).
% subtraction
diff(E1 - E2, V, W) :-
diff(E1, V, Ed1),
diff(E2, V, Ed2),
opt_sub(W, Ed1, Ed2).
% (fg)' = f'g + fg'
diff(E1 * E2, V, W) :-
diff(E1, V, Ed1),
diff(E2, V, Ed2),
opt_mul(L, Ed1, E2),
opt_mul(P, E1, Ed2),
opt_sum(W, L, P).
% (f/g)' = (f'g - fg') / (g*g)
diff(E1 / E2, V, E) :-
diff(E1, V, Ed1),
diff(E2, V, Ed2),
opt_mul(A, Ed1,E2),
opt_mul(B, E1,Ed2),
opt_mul(C, E2,E2),
opt_sub(D, A, B),
opt_div(E, D,C).
% functions
% (h(g))' = h'(g) * g'
diff(sin(E), V, M) :-
diff(E,V, Ed),
opt_mul(M, Ed, cos(E)).
diff(cos(E), V, K) :-
diff(E,V, Ed),
opt_mul(M, Ed,sin(E)),
opt_sub(K, 0, M).
diff(tan(E), V, F) :-
diff(E,V, Ed),
opt_mul(A, Ed, 2),
opt_mul(B, 2, E),
opt_mul(C, 2, cos(B)),
opt_sum(D, C, 1),
opt_div(F, A, D).
diff(exp(E), V, F) :-
diff(E,V, Ed),
opt_mul(F, Ed, exp(E)).
diff(log(E), V, F) :-
diff(E,V, Ed),
opt_div(F, Ed, E).
% (f^g)' = f^(g-1)*(gf' + g'f logf)
diff(F^G, V, FF) :-
diff(F, V, Fd),
diff(G, V, Gd),
opt_sub(AA, G, 1),
opt_mul(BB, G,Fd),
opt_mul(CC, Gd, F),
opt_mul(DD, CC, log(F)),
opt_sum(EE, BB, DD),
opt_mul(FF, F^(AA), EE).