Simplify Symbolic Expressions

Simplification of a mathematical expression is not a clearly defined subject. There is no universal idea as to which form of an expression is simplest. The form of a mathematical expression that is simplest for one problem turns out to be complicated or even unsuitable for another problem. For example, the following two mathematical expressions present the same polynomial in different forms:

(x + 1)(x - 2)(x + 3)(x - 4),

  x4 - 2x3 - 13x2 + 14x + 24.

The first form clearly shows the roots of this polynomial. This form is simpler for working with the roots. The second form serves best when you want to see the coefficients of the polynomial. For example, this form is convenient when you differentiate or integrate polynomials.

If the problem you want to solve requires a particular form of an expression, the best approach is to choose the appropriate simplification function. See Choose Function to Rearrange Expression.

Besides specific simplifiers, Symbolic Math Toolbox™ offers a general simplifier, simplify.

If you do not need a particular form of expressions (expanded, factored, or expressed in particular terms), use simplify to shorten mathematical expressions. For example, use this simplifier to find a shorter form for a final result of your computations.

simplify works on various types of symbolic expressions, such as polynomials, expressions with trigonometric, logarithmic, and special functions. For example, simplify these polynomials.

syms x y
simplify((1 - x^2)/(1 - x))
simplify((x - 1)*(x + 1)*(x^2 + x + 1)*(x^2 + 1)*(x^2 - x + 1)*(x^4 - x^2 + 1))
ans =
x + 1
 
ans =
x^12 - 1

Simplify expressions involving trigonometric functions.

simplify(cos(x)^(-2) - tan(x)^2)
simplify(cos(x)^2 - sin(x)^2)
ans =
1
 
ans =
cos(2*x)

Simplify expressions involving exponents and logarithms. In the third expression, use log(sym(3)) instead of log(3). If you use log(3), then MATLAB® calculates log(3) with the double precision, and then converts the result to a symbolic number.

simplify(exp(x)*exp(y))
simplify(exp(x) - exp(x/2)^2)
simplify(log(x) + log(sym(3)) - log(3*x) + (exp(x) - 1)/(exp(x/2) + 1))
ans =
exp(x + y)
 
ans =
0
 
ans =
exp(x/2) - 1

Simplify expressions involving special functions.

simplify(gamma(x + 1) - x*gamma(x))
simplify(besselj(2, x) + besselj(0, x))
ans =
0
 
ans =
(2*besselj(1, x))/x

You also can simplify symbolic functions by using simplify.

syms f(x,y)
f(x,y) = exp(x)*exp(y)
f = simplify(f)
f(x, y) =
exp(x)*exp(y)
 
f(x, y) =
exp(x + y)

Simplify Using Options

By default, simplify uses strict simplification rules and ensures that simplified expressions are always mathematically equivalent to initial expressions. For example, it does not combine logarithms.

syms x
simplify(log(x^2) + log(x))
ans =
log(x^2) + log(x)

You can apply additional simplification rules which are not correct for all values of parameters and all cases, but using which simplify can return shorter results. For this approach, use IgnoreAnalyticConstraints. For example, simplifying the same expression with IgnoreAnalyticConstraints, you get the result with combined logarithms.

simplify(log(x^2) + log(x),'IgnoreAnalyticConstraints',true)
ans =
3*log(x)

IgnoreAnalyticConstraints provides a shortcut allowing you to simplify expressions under commonly used assumptions about values of the variables. Alternatively, you can set appropriate assumptions on variables explicitly. For example, combining logarithms is not valid for complex values in general. If you assume that x is a real value, simplify combines logarithms without IgnoreAnalyticConstraints.

assume(x,'real')
simplify(log(x^2) + log(x))
ans =
log(x^3)

For further computations, clear the assumption on x by recreating it using syms.

syms x

Another approach that can improve simplification of an expression or function is the syntax simplify(f,'Steps',n), where n is a positive integer that controls how many steps simplify takes. Specifying more simplification steps can help you simplify the expression better, but it takes more time. By default, n = 1. For example, create and simplify this expression. The result is shorter than the original expression, but it can be simplified further.

syms x
y = (cos(x)^2 - sin(x)^2)*sin(2*x)*(exp(2*x) - 2*exp(x) + 1)/...
    ((cos(2*x)^2 - sin(2*x)^2)*(exp(2*x) - 1));
simplify(y)
ans =
(sin(4*x)*(exp(x) - 1))/(2*cos(4*x)*(exp(x) + 1))

Specify the number of simplification steps for the same expression. First, use 25 steps.

simplify(y,'Steps',25)
ans =
(tan(4*x)*(exp(x) - 1))/(2*(exp(x) + 1))

Use 50 steps to simplify the expression even further.

simplify(y,'Steps',50)
ans =
(tan(4*x)*tanh(x/2))/2

Suppose, you already simplified an expression or function, but want to simplify it further. The more efficient approach is to simplify the result instead of simplifying the original expression.

syms x
y = (cos(x)^2 - sin(x)^2)*sin(2*x)*(exp(2*x) - 2*exp(x) + 1)/...
    ((cos(2*x)^2 - sin(2*x)^2)*(exp(2*x) - 1));
 y = simplify(y)
y =
(sin(4*x)*(exp(x) - 1))/(2*cos(4*x)*(exp(x) + 1))
y = simplify(y,'Steps',25)
y =
(tan(4*x)*(exp(x) - 1))/(2*(exp(x) + 1))
y = simplify(y,'Steps',50)
y =
(tan(4*x)*tanh(x/2))/2

Simplify Using Assumptions

Some expressions cannot be simplified in general, but become much shorter under particular assumptions. For example, simplifying this trigonometric expression without additional assumptions returns the original expression.

syms n
simplify(sin(2*n*pi))
ans =
sin(2*pi*n)

However, if you assume that variable n represents an integer, the same trigonometric expression simplifies to 0.

assume(n,'integer')
simplify(sin(2*n*pi))
ans =
0

For further computations, clear the assumption.

syms n

Simplify Fractions

You can use the general simplification function, simplify, to simplify fractions. However, Symbolic Math Toolbox offers a more efficient function specifically for this task: simplifyFraction. The statement simplifyFraction(f) represents the expression f as a fraction, where both the numerator and denominator are polynomials whose greatest common divisor is 1. For example, simplify these expressions.

syms x y
simplifyFraction((x^3 - 1)/(x - 1))
ans =
x^2 + x + 1
simplifyFraction((x^3 - x^2*y - x*y^2 + y^3)/(x^3 + y^3))
ans =
(x^2 - 2*x*y + y^2)/(x^2 - x*y + y^2)

By default, simplifyFraction does not expand expressions in the numerator and denominator of the returned result. To expand the numerator and denominator in the resulting expression, use the Expand option. For comparison, first simplify this fraction without Expand.

simplifyFraction((1 - exp(x)^4)/(1 + exp(x))^4)
ans =
(exp(2*x) - exp(3*x) - exp(x) + 1)/(exp(x) + 1)^3

Now, simplify the same expressions with Expand.

simplifyFraction((1 - exp(x)^4)/(1 + exp(x))^4,'Expand',true)
ans =
(exp(2*x) - exp(3*x) - exp(x) + 1)/(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1)