simplify

Simplify these symbolic expressions:

syms x a b c
S = simplify(sin(x)^2 + cos(x)^2)

S = $$1$$

S = simplify(exp(c*log(sqrt(a+b))))

S = $${\left(a+b\right)}^{c/2}$$

Call simplify for this symbolic matrix. When the input argument is a vector or matrix, simplify tries to find a simpler form of each element of the vector or matrix.

syms x
M = [(x^2 + 5*x + 6)/(x + 2), sin(x)*sin(2*x) + cos(x)*cos(2*x);
		(exp(-x*1i)*1i)/2 - (exp(x*1i)*1i)/2, sqrt(16)];
S = simplify(M)

S = 
$$\left(\begin{array}{cc}x+3& \cos \left(x\right)\\ \sin \left(x\right)& 4\end{array}\right)$$

Simplify a symbolic expression that contains logarithms and powers. By default, simplify does not combine powers and logarithms because combining them is not valid for generic complex values.

syms x
expr = (log(x^2 + 2*x + 1) - log(x + 1))*sqrt(x^2);
S = simplify(expr)

S = $$-\left(\log \left(x+1\right)-\log \left({\left(x+1\right)}^2\right)\right)\hspace{0.17em}\sqrt{x^2}$$

To apply the simplification rules that allow the simplify function to combine powers and logarithms, set 'IgnoreAnalyticConstraints' to true:

S = simplify(expr,'IgnoreAnalyticConstraints',true)

S = $$x\hspace{0.17em}\log \left(x+1\right)$$

Simplify this expression:

syms x
expr = ((exp(-x*1i)*1i) - (exp(x*1i)*1i))/(exp(-x*1i) + exp(x*1i));
S = simplify(expr)

S = 
$$-\frac{{\mathrm{e}}^{2\hspace{0.17em}x\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}-\mathrm{i}}{{\mathrm{e}}^{2\hspace{0.17em}x\hspace{0.17em}\mathrm{i}}+1}$$

By default, simplify uses one internal simplification step. You can get different, often shorter, simplification results by increasing the number of simplification steps:

S10 = simplify(expr,'Steps',10)

S10 = 
$$\frac{2\hspace{0.17em}\mathrm{i}}{{\mathrm{e}}^{2\hspace{0.17em}x\hspace{0.17em}\mathrm{i}}+1}-\mathrm{i}$$

S30 = simplify(expr,'Steps',30)

S30 = 
$$\frac{\left(\cos \left(x\right)-\sin \left(x\right)\hspace{0.17em}\mathrm{i}\right)\hspace{0.17em}\mathrm{i}}{\cos \left(x\right)}-\mathrm{i}$$

S50 = simplify(expr,'Steps',50)

S50 = $$\tan \left(x\right)$$

If you are unable to return the desired result, try alternate simplification functions. See Choose Function to Rearrange Expression.

Get equivalent results for a symbolic expression by setting the value of 'All' to true.

syms x
expr = cos(x)^2 - sin(x)^2;
S = simplify(expr,'All',true)

S = 
$$\left(\begin{array}{c}\cos \left(2\hspace{0.17em}x\right)\\ {\cos \left(x\right)}^2-{\sin \left(x\right)}^2\end{array}\right)$$

Increase the number of simplification steps to 10. Find the other equivalent results for the same expression.

S = simplify(expr,'Steps',10,'All',true)

S = 
$$\left(\begin{array}{c}\cos \left(2\hspace{0.17em}x\right)\\ 1-2\hspace{0.17em}{\sin \left(x\right)}^2\\ 2\hspace{0.17em}{\cos \left(x\right)}^2-1\\ {\cos \left(x\right)}^2-{\sin \left(x\right)}^2\\ \cot \left(2\hspace{0.17em}x\right)\hspace{0.17em}\sin \left(2\hspace{0.17em}x\right)\\ \frac{{\mathrm{e}}^{-2\hspace{0.17em}x\hspace{0.17em}\mathrm{i}}}{2}+\frac{{\mathrm{e}}^{2\hspace{0.17em}x\hspace{0.17em}\mathrm{i}}}{2}\end{array}\right)$$

Attempt to separate real and imaginary parts of an expression by setting the value of 'Criterion' to 'preferReal'.

syms x
f = (exp(x + exp(-x*1i)/2 - exp(x*1i)/2)*1i)/2 -...
    (exp(-x - exp(-x*1i)/2 + exp(x*1i)/2)*1i)/2;
S = simplify(f,'Criterion','preferReal','Steps',100)

S = $$\sin \left(\sin \left(x\right)\right)\hspace{0.17em}\cosh \left(x\right)+\cos \left(\sin \left(x\right)\right)\hspace{0.17em}\sinh \left(x\right)\hspace{0.17em}\mathrm{i}$$

If 'Criterion' is not set to 'preferReal', then simplify returns a shorter result but the real and imaginary parts are not separated.

S = simplify(f,'Steps',100)

S = $$\sin \left(\sin \left(x\right)+x\hspace{0.17em}\mathrm{i}\right)$$

When you set 'Criterion' to 'preferReal', the simplifier disfavors expression forms where complex values appear inside subexpressions. In nested subexpressions, the deeper the complex value appears inside an expression, the least preference this form of an expression gets.

Attempt to avoid imaginary terms in exponents by setting 'Criterion' to 'preferReal'.

Show this behavior by simplifying a complex symbolic expression with and without setting 'Criterion' to 'preferReal'. When 'Criterion' is set to 'preferReal', then simplify places the imaginary term outside the exponent.

expr = sym(1i)^(1i+1);
withoutPreferReal = simplify(expr,'Steps',100)

withoutPreferReal = 
$${\left(-1\right)}^{\frac{1}{2}+\frac{1}{2}\hspace{0.17em}\mathrm{i}}$$

withPreferReal = simplify(expr,'Criterion','preferReal','Steps',100)

withPreferReal = 
$${\mathrm{e}}^{-\frac{\pi }{2}}\hspace{0.17em}\mathrm{i}$$

Simplify expressions containing symbolic units of the same dimension by using simplify.

u = symunit;
expr = 300*u.cm + 40*u.inch + 2*u.m;
S = simplify(expr)

S = 
$$\frac{752}{125}\hspace{0.17em}\mathrm{m}\mathrm{"meter\; -\; a\; physical\; unit\; of\; length."}$$

simplify automatically chooses the unit to rewrite into. To choose a specific unit, use rewrite.

In most cases, to simplify a symbolic expression using Symbolic Math Toolbox™, you only need to use the simplify function. But for some large and complex expressions, you can obtain a faster and simpler result by using the expand function before applying simplify.

For instance, this workflow gives better results when finding the determinant of a matrix that represents the Kerr metric [1]. Declare the parameters of the Kerr metric.

syms theta real;
syms r rs a real positive;

Define the matrix that represents the Kerr metric.

rho = sqrt(r^2 + a^2*cos(theta)^2);
delta  = r^2 + a^2 - r*rs;
g(1,1) = - (1 - r*rs/rho^2);
g(1,4) = - (rs*a*r*sin(theta)^2)/rho^2;
g(4,1) = - (rs*a*r*sin(theta)^2)/rho^2;
g(2,2) = rho^2/delta;
g(3,3) = rho^2;
g(4,4) = (r^2 + a^2 + rs*a^2*r*sin(theta)^2/rho^2)*sin(theta)^2;

Evaluate the determinant of the Kerr metric.

det_g = det(g)

det_g = 
$$-\frac{{\sin \left(\theta \right)}^2\hspace{0.17em}\left(a^6\hspace{0.17em}{\cos \left(\theta \right)}^4+a^4\hspace{0.17em}{r}^2\hspace{0.17em}{\cos \left(\theta \right)}^4+2\hspace{0.17em}{a}^4\hspace{0.17em}{r}^2\hspace{0.17em}{\cos \left(\theta \right)}^2+\mathrm{rs}\hspace{0.17em}{a}^4\hspace{0.17em}r\hspace{0.17em}{\cos \left(\theta \right)}^2\hspace{0.17em}{\sin \left(\theta \right)}^2-\mathrm{rs}\hspace{0.17em}{a}^4\hspace{0.17em}r\hspace{0.17em}{\cos \left(\theta \right)}^2+2\hspace{0.17em}{a}^2\hspace{0.17em}{r}^4\hspace{0.17em}{\cos \left(\theta \right)}^2+a^2\hspace{0.17em}{r}^4-\mathrm{rs}\hspace{0.17em}{a}^2\hspace{0.17em}{r}^3\hspace{0.17em}{\cos \left(\theta \right)}^2+\mathrm{rs}\hspace{0.17em}{a}^2\hspace{0.17em}{r}^3\hspace{0.17em}{\sin \left(\theta \right)}^2-\mathrm{rs}\hspace{0.17em}{a}^2\hspace{0.17em}{r}^3+r^6-\mathrm{rs}\hspace{0.17em}{r}^5\right)}{a^2+r^2-\mathrm{rs}\hspace{0.17em}r}$$

Simplify the determinant using the simplify function.

D = simplify(det_g)

D = $$-{\sin \left(\theta \right)}^2\hspace{0.17em}\left(a^2\hspace{0.17em}{\cos \left(\theta \right)}^2+r^2\right)\hspace{0.17em}\left(-a^2\hspace{0.17em}{\sin \left(\theta \right)}^2+a^2+r^2\right)$$

Instead, flatten the expression using the expand function, and then apply the simplify function. The result is simpler with this extra step.

D = simplify(expand(det_g))

D = $$-{\sin \left(\theta \right)}^2\hspace{0.17em}{\left(-a^2\hspace{0.17em}{\sin \left(\theta \right)}^2+a^2+r^2\right)}^2$$