Basic Operations

Here we discuss some of the most basic operations needed for expression manipulation in SymPy. Some more advanced operations will be discussed later in the advanced expression manipulation section.

>>> from sympy import *
>>> x, y, z = symbols("x y z")

Substitution

One of the most common things you might want to do with a mathematical expression is substitution. Substitution replaces all instances of something in an expression with something else. It is done using the subs method. For example

>>> expr = cos(x) + 1
>>> expr.subs(x, y)
cos(y) + 1

Substitution is usually done for one of two reasons:

1. Evaluating an expression at a point. For example, if our expression is cos(x) + 1 and we want to evaluate it at the point x = 0, so that we get cos(0) + 1, which is 2.

   >>> expr.subs(x, 0)
   2
   

2. Replacing a subexpression with another subexpression. There are two reasons we might want to do this. The first is if we are trying to build an expression that has some symmetry, such as \(x^{x^{x^x}}\). To build this, we might start with x**y, and replace y with x**y. We would then get x**(x**y). If we replaced y in this new expression with x**x, we would get x**(x**(x**x)), the desired expression.

   >>> expr = x**y
   >>> expr
   x**y
   >>> expr = expr.subs(y, x**y)
   >>> expr
   x**(x**y)
   >>> expr = expr.subs(y, x**x)
   >>> expr
   x**(x**(x**x))
   

   The second is if we want to perform a very controlled simplification, or perhaps a simplification that SymPy is otherwise unable to do. For example, say we have \(\sin(2x) + \cos(2x)\), and we want to replace \(\sin(2x)\) with \(2\sin(x)\cos(x)\). As we will learn later, the function expand_trig does this. However, this function will also expand \(\cos(2x)\), which we may not want. While there are ways to perform such precise simplification, and we will learn some of them in the advanced expression manipulation section, an easy way is to just replace \(\sin(2x)\) with \(2\sin(x)\cos(x)\).

   >>> expr = sin(2*x) + cos(2*x)
   >>> expand_trig(expr)
   2*sin(x)*cos(x) + 2*cos(x)**2 - 1
   >>> expr.subs(sin(2*x), 2*sin(x)*cos(x))
   2*sin(x)*cos(x) + cos(2*x)
   

There are two important things to note about subs. First, it returns a new expression. SymPy objects are immutable. That means that subs does modify it in-place. For example

>>> expr = cos(x)
>>> expr.subs(x, 0)
1
>>> expr
cos(x)
>>> x
x

Quick Tip

SymPy expressions are immutable. No function will change them in-place.

Here, we see that performing expr.subs(x, 0) leaves expr unchanged. In fact, since SymPy expressions are immutable, no function will change them in-place. All functions will return new expressions.

To perform multiple substitutions at once, pass a list of (old, new) pairs to subs.

>>> expr = x**3 + 4*x*y - z
>>> expr.subs([(x, 2), (y, 4), (z, 0)])
40

It is often useful to combine this with a list comprehension to do a large set of similar replacements all at once. For example, say we had \(x^4 - 4x^3 + 4x^2 - 2x + 3\) and we wanted to replace all instances of \(x\) that have an even power with \(y\), to get \(y^4 - 4x^3 + 4y^2 - 2x + 3\).

>>> expr = x**4 - 4*x**3 + 4*x**2 - 2*x + 3
>>> replacements = [(x**i, y**i) for i in range(5) if i % 2 == 0]
>>> expr.subs(replacements)
-4*x**3 - 2*x + y**4 + 4*y**2 + 3

Converting Strings to SymPy Expressions

The sympify function (that’s sympify, not to be confused with simplify) can be used to convert strings into SymPy expressions.

For example

>>> str_expr = "x**2 + 3*x - 1/2"
>>> expr = sympify(str_expr)
>>> expr
x**2 + 3*x - 1/2
>>> expr.subs(x, 2)
19/2

Warning

sympify uses eval. Don’t use it on unsanitized input.

evalf

To evaluate a numerical expression into a floating point number, use evalf.

>>> expr = sqrt(8)
>>> expr.evalf()
2.82842712474619

SymPy can evaluate floating point expressions to arbitrary precision. By default, 15 digits of precision are used, but you can pass any number as the argument to evalf. Let’s compute the first 100 digits of \(\pi\).

>>> pi.evalf(100)
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068

To numerically evaluate an expression with a Symbol at a point, we might use subs followed by evalf, but it is more efficient and numerically stable to pass the substitution to evalf using the subs flag, which takes a dictionary of Symbol: point pairs.

>>> expr = cos(2*x)
>>> expr.evalf(subs={x: 2.4})
    0.0874989834394464

subs and evalf are good if you want to do simple evaluation, but if you intend to evaluate an expression at many points, there are more efficient ways. See the advanced numerics section for more details.