Simplify symbolic expression for optimized code generation

I'm trying to generate optimized c code (reduce the number of operations) from symbolic MATLAB expressions. However I MATLAB doesn't seem to want to simplify even the simplest equations.

lets take the following expression as an example:

X_eq = [
q0 + (dt*q1*(gyrobx - omegax))/2 + (dt*q2*(gyroby - omegay))/2 + (dt*q3*(gyrobz - omegaz))/2;
q1 - (dt*q0*(gyrobx - omegax))/2 + (dt*q3*(gyroby - omegay))/2 - (dt*q2*(gyrobz - omegaz))/2;
q2 - (dt*q0*(gyroby - omegay))/2 - (dt*q3*(gyrobx - omegax))/2 + (dt*q1*(gyrobz - omegaz))/2;
q3 + (dt*q2*(gyrobx - omegax))/2 - (dt*q1*(gyroby - omegay))/2 - (dt*q0*(gyrobz - omegaz))/2];

What i'm looking for is (basically factor out dt and (1/2)):

X_eq = [
 q0 + (((q1*(gyrobx - omegax)) + (q2*(gyroby - omegay)) + (q3*(gyrobz - omegaz)))*dt)/2;
 q1 - (((q0*(gyrobx - omegax)) + (q3*(gyroby - omegay)) - (q2*(gyrobz - omegaz)))*dt)/2;
 q2 - (((q0*(gyroby - omegay)) - (q3*(gyrobx - omegax)) + (q1*(gyrobz - omegaz)))*dt)/2;
 q3 + (((q2*(gyrobx - omegax)) - (q1*(gyroby - omegay)) - (q0*(gyrobz - omegaz)))*dt)/2];

simplify(X_eq, 'steps', 100) returns the original expression so it's useless:

ans =
 q0 + (dt*q1*(gyrobx - omegax))/2 + (dt*q2*(gyroby - omegay))/2 + (dt*q3*(gyrobz - omegaz))/2
 q1 - (dt*q0*(gyrobx - omegax))/2 + (dt*q3*(gyroby - omegay))/2 - (dt*q2*(gyrobz - omegaz))/2
 q2 - (dt*q0*(gyroby - omegay))/2 - (dt*q3*(gyrobx - omegax))/2 + (dt*q1*(gyrobz - omegaz))/2
 q3 + (dt*q2*(gyrobx - omegax))/2 - (dt*q1*(gyroby - omegay))/2 - (dt*q0*(gyrobz - omegaz))/2

collect(X_eq, [dt]) returns the following which is a step forward but doesn't factorize (1/2):

ans =
 ((q1*(gyrobx - omegax))/2 + (q2*(gyroby - omegay))/2 + (q3*(gyrobz - omegaz))/2)*dt + q0
 ((q3*(gyroby - omegay))/2 - (q0*(gyrobx - omegax))/2 - (q2*(gyrobz - omegaz))/2)*dt + q1
 ((q1*(gyrobz - omegaz))/2 - (q3*(gyrobx - omegax))/2 - (q0*(gyroby - omegay))/2)*dt + q2
 ((q2*(gyrobx - omegax))/2 - (q1*(gyroby - omegay))/2 - (q0*(gyrobz - omegaz))/2)*dt + q3

Finally, collect(X_eq, [1/2, -1/2, dt, -dt]) does the following which comes close but is still not what I expect:

ans =
          ((q1*(gyrobx - omegax) + q2*(gyroby - omegay) + q3*(gyrobz - omegaz))*dt)/2 + q0
 ((q3*(gyroby - omegay))*dt)/2 - ((q0*(gyrobx - omegax) + q2*(gyrobz - omegaz))*dt)/2 + q1
 ((q1*(gyrobz - omegaz))*dt)/2 - ((q0*(gyroby - omegay) + q3*(gyrobx - omegax))*dt)/2 + q2
 ((q2*(gyrobx - omegax))*dt)/2 - ((q1*(gyroby - omegay) + q0*(gyrobz - omegaz))*dt)/2 + q3

I've tried subexpr() as well which has it's used by doesn't help me in this situation. The number of equations is high so finding an automated way of reducing the number of calculations is important.

Answer 1

[Andrew Newell]

I don't know how particular you are about the exact form of the simplified expression, but this certainly produces a simpler expression:

Q = [q0; q1; q2; q3];
X_eq = Q + simplify(expand(X_eq-Q));

[Andrew Newell]

It's also worth pointing out that you could substitute

x = gyrobx-omegax

etc. to greatly simplify the expressions.

[Ehsan Tofigh]

Yes i've already used subexpr() to extract "gyrobx-omegax" and similar terms. The problem is I need to find a more general way since not all of my equations have that form. The fact that

Q = [q0; q1; q2; q3];
X_eq = Q + simplify(expand(X_eq-Q));

works suggest to me that the simplify function isn't working as hard as it should. It needs a lot of help even in simple cases.

[Andrew Newell]

Yes, simplify often doesn't do what we expect it to do. Hard to know what to suggest since I don't know how your equations depart from this form.

[Ehsan Tofigh]

I was afraid of that. Thanks for the response none the less! can you tell me why collect() doesn't want to factor out all of the (1/2) factors? it does it for some terms but not all of them.

[Andrew Newell]

Maybe because 1/2 is not a variable. How about this?

X_eq = collect(2*expand(X_eq),dt)/2;