rewrite

Rewrite expression in terms of another function

Syntax

R = rewrite(expr,target)

Description

R = rewrite(expr,target) rewrites the symbolic expression expr in terms of another function as specified by target. The rewritten expression is mathematically equivalent to the original expression. If expr is a vector or matrix, rewrite acts element-wise on expr.

Examples

Rewrite Between Trigonometric and Exponential Functions

Rewrite any trigonometric function in terms of the exponential function by specifying the target "exp".

syms x
sin2exp = rewrite(sin(x),"exp")

sin2exp = 
$$\frac{{\mathrm{e}}^{-x\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}}{2}-\frac{{\mathrm{e}}^{x\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}}{2}$$

tan2exp = rewrite(tan(x),"exp")

tan2exp = 
$$-\frac{{\mathrm{e}}^{2\hspace{0.17em}x\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}-\mathrm{i}}{{\mathrm{e}}^{2\hspace{0.17em}x\hspace{0.17em}\mathrm{i}}+1}$$

Rewrite the exponential function in terms of any trigonometric function by specifying the trigonometric function as the target. For a full list of target options, see target.

syms x
exp2sin = rewrite(exp(x),"sin")

exp2sin = 
$$-2\hspace{0.17em}{\sin \left(\frac{x\hspace{0.17em}\mathrm{i}}{2}\right)}^2-\sin \left(x\hspace{0.17em}\mathrm{i}\right)\hspace{0.17em}\mathrm{i}+1$$

exp2tan = rewrite(-(exp(x*2i)*1i - 1i)/(exp(x*2i) + 1),"tan")

exp2tan = 
$$-\frac{\frac{\left(\tan \left(x\right)-\mathrm{i}\right)\hspace{0.17em}\mathrm{i}}{\tan \left(x\right)+\mathrm{i}}+\mathrm{i}}{\frac{\tan \left(x\right)-\mathrm{i}}{\tan \left(x\right)+\mathrm{i}}-1}$$

Simplify exp2tan to the expected form by using simplify.

exp2tan = simplify(exp2tan)

exp2tan = $$\tan \left(x\right)$$

Rewrite Between Trigonometric Functions

Rewrite any trigonometric function in terms of any other trigonometric function by specifying the target. For a full list of target options, see target.

Rewrite tan(x) in terms of the sine function by specifying the target "sin".

syms x
tan2sin = rewrite(tan(x),"sin")

tan2sin = 
$$-\frac{\sin \left(x\right)}{2\hspace{0.17em}{\sin \left(\frac{x}{2}\right)}^2-1}$$

Rewrite Between Inverse Trigonometric and Logarithm Functions

Rewrite any inverse trigonometric function in terms of the logarithm function by specifying the target "log". For a full list of target options, see target.

Rewrite acos(x) and acot(x) in terms of the log function.

syms x
acos2log = rewrite(acos(x),"log")

acos2log = $$-\log \left(x+\sqrt{1-x^2}\hspace{0.17em}\mathrm{i}\right)\hspace{0.17em}\mathrm{i}$$

acot2log = rewrite(acot(x),"log")

acot2log = 
$$\frac{\log \left(1-\frac{\mathrm{i}}{x}\right)\hspace{0.17em}\mathrm{i}}{2}-\frac{\log \left(1+\frac{\mathrm{i}}{x}\right)\hspace{0.17em}\mathrm{i}}{2}$$

Similarly, rewrite the logarithm function in terms of any inverse trigonometric function by specifying the inverse trigonometric function as the target.

Rewrite Elements of Matrix

Rewrite each element of a matrix by calling rewrite on the matrix.

Rewrite all elements of a matrix in terms of the exp function.

syms x
matrix = [sin(x) cos(x); sinh(x) cosh(x)];
R = rewrite(matrix,"exp")

R = 
$$\left(\begin{array}{cc}\frac{{\mathrm{e}}^{-x\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}}{2}-\frac{{\mathrm{e}}^{x\hspace{0.17em}\mathrm{i}}\hspace{0.17em}\mathrm{i}}{2}& \frac{{\mathrm{e}}^{-x\hspace{0.17em}\mathrm{i}}}{2}+\frac{{\mathrm{e}}^{x\hspace{0.17em}\mathrm{i}}}{2}\\ \frac{{\mathrm{e}}^x}{2}-\frac{{\mathrm{e}}^{-x}}{2}& \frac{{\mathrm{e}}^{-x}}{2}+\frac{{\mathrm{e}}^x}{2}\end{array}\right)$$

Rewrite Between Sine and Cosine Functions

Rewrite the cosine function in terms of the sine function. Here, the rewrite function rewrites the cosine function using the identity $\cos \left(\mathit{x}\right)=1-2{\sin \left(\frac{\mathit{x}}{2}\right)}^2$, which is valid for any $\mathit{x}$.

syms x
R = rewrite(cos(x),"sin")

R = 
$$1-2\hspace{0.17em}{\sin \left(\frac{x}{2}\right)}^2$$

rewrite does not rewrite sin(x) as either $-\sqrt{1-{\cos }^2\left(\mathit{x}\right)}$ or $\sqrt{1-{\cos }^2\left(\mathit{x}\right)}$ because these expressions are not valid for all x. However, using the square of these expressions to represent sin(x)^2 is valid for all x. Thus, rewrite can rewrite sin(x)^2.

syms x
R1 = rewrite(sin(x),"cos")

R1 = $$\sin \left(x\right)$$

R2 = rewrite(sin(x)^2,"cos")

R2 = $$1-{\cos \left(x\right)}^2$$

Rewrite root Function

Since R2023a

Find the root of a polynomial by using root. The result is a column vector in terms of the root function with k = 1, 2, 3, 4, or 5 for the kth root of the polynomial.

syms x
sols = root(x^5 - x^4 - 1,x)

sols = 
$$\left(\begin{array}{c}\mathrm{root}\left(x^5-x^4-1,x,1\right)\\ \mathrm{root}\left(x^5-x^4-1,x,2\right)\\ \mathrm{root}\left(x^5-x^4-1,x,3\right)\\ \mathrm{root}\left(x^5-x^4-1,x,4\right)\\ \mathrm{root}\left(x^5-x^4-1,x,5\right)\end{array}\right)$$

Expand the root function in sols by using rewrite with the "expandroot" option. The result is in terms of arithmetic operations such as ^, *, /, +, and – that operate on exact symbolic numbers. Because the expanded result can involve many terms that operate arithmetically, numerical approximation of this result can be inaccurate (due to accumulation of round-off errors).

R = rewrite(sols,"expandroot")

R = 
$$\begin{array}{l}\left(\begin{array}{c}\frac{1}{2}-\frac{\sqrt{3}\hspace{0.17em}\mathrm{i}}{2}\\ \frac{1}{2}+\frac{\sqrt{3}\hspace{0.17em}\mathrm{i}}{2}\\ \frac{1}{3\hspace{0.17em}{\sigma }_1}+{\sigma }_1\\ -\frac{1}{6\hspace{0.17em}{\sigma }_1}-\frac{{\sigma }_1}{2}-\frac{\sqrt{3}\hspace{0.17em}\left(\frac{1}{3\hspace{0.17em}{\sigma }_1}-{\sigma }_1\right)\hspace{0.17em}\mathrm{i}}{2}\\ -\frac{1}{6\hspace{0.17em}{\sigma }_1}-\frac{{\sigma }_1}{2}+\frac{\sqrt{3}\hspace{0.17em}\left(\frac{1}{3\hspace{0.17em}{\sigma }_1}-{\sigma }_1\right)\hspace{0.17em}\mathrm{i}}{2}\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1={\left(\frac{\sqrt{23}\hspace{0.17em}\sqrt{108}}{108}+\frac{1}{2}\right)}^{1/3}\end{array}$$

As an alternative, you can numerically approximate sols directly by using vpa to return variable-precision symbolic numbers. The resulting numeric values have the default 32 significant digits, which are more accurate.

solsVpa = vpa(sols)

solsVpa = 
$$\left(\begin{array}{c}-0.66235897862237301298045442723905-0.56227951206230124389918214490937\hspace{0.17em}\mathrm{i}\\ -0.66235897862237301298045442723905+0.56227951206230124389918214490937\hspace{0.17em}\mathrm{i}\\ 0.5-0.86602540378443864676372317075294\hspace{0.17em}\mathrm{i}\\ 0.5+0.86602540378443864676372317075294\hspace{0.17em}\mathrm{i}\\ 1.3247179572447460259609088544781\end{array}\right)$$

To use sols without Symbolic Math Toolbox™, you can generate code and convert sols to a MATLAB® function by using matlabFunction. The generated file uses the roots function that operates on the numeric double data type.

matlabFunction(sols,"File","myfile");
type myfile

function sols = myfile
%MYFILE
%    SOLS = MYFILE

%    This function was generated by the Symbolic Math Toolbox version 9.3.
%    03-Mar-2023 07:29:21

t0 = roots([1.0,-1.0,0.0,0.0,0.0,-1.0]);
t2 = t0(1);
t0 = roots([1.0,-1.0,0.0,0.0,0.0,-1.0]);
t3 = t0(2);
t0 = roots([1.0,-1.0,0.0,0.0,0.0,-1.0]);
t4 = t0(3);
t0 = roots([1.0,-1.0,0.0,0.0,0.0,-1.0]);
t5 = t0(4);
t0 = roots([1.0,-1.0,0.0,0.0,0.0,-1.0]);
t6 = t0(5);
sols = [t2;t3;t4;t5;t6];
end

Rewrite symsum and root Functions

Since R2023a

Find the indefinite integral of a polynomial fraction. The result is in terms of the symsum and root functions as represented by the $\sum$ and $\mathrm{root}$ symbols.

syms x
F = int(1/(x^3 + x - 1),x)

F = 
$${\sum }_{k=1}^3\log \left(\mathrm{root}\left(z^3-\frac{3\hspace{0.17em}z}{31}-\frac{1}{31},z,k\right)\hspace{0.17em}\left(3\hspace{0.17em}x+\mathrm{root}\left(z^3-\frac{3\hspace{0.17em}z}{31}-\frac{1}{31},z,k\right)\hspace{0.17em}\left(6\hspace{0.17em}x-9\right)\right)\right)\hspace{0.17em}\mathrm{root}\left(z^3-\frac{3\hspace{0.17em}z}{31}-\frac{1}{31},z,k\right)$$

Rewrite the result in terms of arithmetic operations. Because the symbolic summation is the outermost operation, first apply the rewrite function with the "expandsum" option to expand symsum. Then, apply rewrite again with the "expandroot" option to rewrite the root function. The resulting symbolic expression is in terms of arithmetic operations such as ^, *, /, +, and –.

R = rewrite(rewrite(F,"expandsum"),"expandroot")

R = 
$$\begin{array}{l}\log \left(\left(3\hspace{0.17em}x+\left(6\hspace{0.17em}x-9\right)\hspace{0.17em}{\sigma }_3\right)\hspace{0.17em}{\sigma }_3\right)\hspace{0.17em}{\sigma }_3-\log \left(-\left(3\hspace{0.17em}x-\left(6\hspace{0.17em}x-9\right)\hspace{0.17em}{\sigma }_2\right)\hspace{0.17em}{\sigma }_2\right)\hspace{0.17em}{\sigma }_2-\log \left(-\left(3\hspace{0.17em}x-\left(6\hspace{0.17em}x-9\right)\hspace{0.17em}{\sigma }_1\right)\hspace{0.17em}{\sigma }_1\right)\hspace{0.17em}{\sigma }_1\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1=\frac{1}{62\hspace{0.17em}{\sigma }_4}+\frac{{\sigma }_4}{2}-\frac{\sqrt{3}\hspace{0.17em}\left(\frac{1}{31\hspace{0.17em}{\sigma }_4}-{\sigma }_4\right)\hspace{0.17em}\mathrm{i}}{2}\\ \\ \mathrm{ }{\sigma }_2=\frac{1}{62\hspace{0.17em}{\sigma }_4}+\frac{{\sigma }_4}{2}+\frac{\sqrt{3}\hspace{0.17em}\left(\frac{1}{31\hspace{0.17em}{\sigma }_4}-{\sigma }_4\right)\hspace{0.17em}\mathrm{i}}{2}\\ \\ \mathrm{ }{\sigma }_3=\frac{1}{31\hspace{0.17em}{\sigma }_4}+{\sigma }_4\\ \\ \mathrm{ }{\sigma }_4={\left(\frac{\sqrt{27}\hspace{0.17em}\sqrt{119164}}{119164}+\frac{1}{62}\right)}^{1/3}\end{array}$$

Input Arguments

expr — Input to rewrite

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix | symbolic multidimensional array

Input to rewrite or replace, specified as a symbolic number, variable, expression, function, vector, matrix, or multidimensional array.

target — Target function or function to expand

string scalar | character vector

Target function or function to expand, specified as a string scalar or character vector. This table summarizes the rewriting rules for all allowed target options.

Target	Rewrite or Replace These Functions	In Terms of These Functions
"exp"	All trigonometric and hyperbolic functions including inverse functions	exp, log
"log"	All inverse trigonometric and hyperbolic functions	log
"sincos"	tan, cot, exp, sinh, cosh, tanh, coth	sin, cos
"sin", "cos", "tan", or "cot"	sin, cos, exp, tan, cot, sinh, cosh, tanh, coth except the target	Target trigonometric function
"sinhcosh"	tan, cot, exp, sin, cos, tanh, coth	sinh, cosh
"sinh", "cosh", "tanh", "coth"	tan, cot, exp, sin, cos, sinh, cosh, tanh, coth except the target	Target hyperbolic function
"asin", "acos", "atan", "acot"	log, and all inverse trigonometric and inverse hyperbolic functions	Target inverse trigonometric function
"asinh", "acosh", "atanh", "acoth"	log, and all inverse trigonometric and inverse hyperbolic functions	Target inverse hyperbolic function
"sqrt"	abs(x + 1i*y)	sqrt(x^2 + y^2)
"heaviside"	sign, triangularPulse, rectangularPulse	heaviside
"piecewise"	abs, heaviside, sign, triangularPulse, rectangularPulse	piecewise
"expandroot"	root	Arithmetic operations such as ^, *, /, +, and –
"expandsum"	symsum	Arithmetic operations such as + and –

Tips

• rewrite replaces symbolic function calls in expr with another function as specified by target only if the replacement is mathematically valid. Otherwise, it keeps the original function calls. For example, see Rewrite Between Sine and Cosine Functions.

Version History

Introduced in R2012a

R2023a: Expand root and symsum functions

You can rewrite or expand symbolic expressions that use the root and symsum functions in terms of arithmetic operations. For examples, see Rewrite root Function and Rewrite symsum and root Functions.