# Basics of expressions in SymPy¶

SymPy is all about construction and manipulation of expressions. By the term expression we mean mathematical expressions represented in the Python language using SymPy’s classes and objects. Expressions may consist of symbols, numbers, functions and function applications (and many other) and operators binding them together (addiction, subtraction, multiplication, division, exponentiation).

Suppose we want to construct an expression for $$x + 1$$:

>>> x = Symbol('x')

>>> x + 1
x + 1

>>> type(_)


Entering x + 1 gave us an instance of Add class. This expression consists of a symbol (x), a number (1) and addition operator, which is represented by the topmost class (Add). This was the simplest way of entering an expression for $$x + 1$$. We could also enter:

>>> y = Symbol('y')

>>> x - y + 17 + y - 16 + sin(pi)
x + 1


In this case SymPy automatically rewrote the input expression and gave its canonical form, which is x + 1 once again. This is a very important behavior: all expressions are subject to automatic evaluation, during which SymPy tries to find a canonical form for expressions, but it doesn’t apply “heroic” measures to achieve this goal. For example the following expression:

>>> (x**2 - 1)/(x - 1)
2
x  - 1
──────
x - 1


is left unsimplified. This is because automatic canonicalization would lose important information about this expression ($$x \not= 1$$). We can use cancel() remove common factors from the numerator and the denominator:

>>> cancel(_)
x + 1


SymPy never applies any transformations automatically that could cause information loss or that would result in results that are valid only almost everywhere. Consider the following expression:

>>> log(x*y)
log(x⋅y)


We know that $$\log(x y)$$ is equivalent to $$\log x + \log y$$ and there is expand() that is supposed be able to do this:

>>> expand(_)
log(x⋅y)


Unfortunately nothing interesting happened. This is because the formula stated above is not universally valid, e.g.:

>>> log((-2)*(-3))
log(6)
>>> log(-2) + log(-3)
log(2) + log(3) + 2⋅ⅈ⋅π


It is possible to ignore such cases and expand forcibly:

>>> expand(log(x*y), force=True)
log(x) + log(y)


Many other expression manipulation function also support force option. Usually a better way is to assign additional knowledge with an expression:

>>> var('a,b', positive=True)
(a, b)

>>> log(a*b)
log(a⋅b)

>>> expand(_)
log(a) + log(b)


In this case force=True wasn’t necessary, because we gave sufficient information to expand() so that it was able to decide that the expansion rule is valid universally for this expression.

## Arithmetic operators¶

Arithmetic operators +, -, *, /, ** are mapped to combinations of three core SymPy’s classes: Add, Mul and Pow, and work the following way:

>>> x + y
x + y
>>> type(_)

x + y
>>> type(_)

x + y
>>> type(_)

• x - y uses Add and Mul classes, and __sub__ method:

>>> x - y
x - y
>>> type(_)
>>> __.args
(-y, x)
>>> type(_)
<class 'sympy.core.mul.Mul'>

>>> x.__sub__(y)
x - y
>>> type(_)
>>> __.args
(-y, x)
>>> type(_)
<class 'sympy.core.mul.Mul'>

x - y
>>> type(_)
>>> __.args
(-y, x)
>>> type(_)
<class 'sympy.core.mul.Mul'>

• x*y uses Mul class and __mul__ method:

>>> x*y
x*y
>>> type(_)
<class 'sympy.core.mul.Mul'>

>>> x.__mul__(y)
x*y
>>> type(_)
<class 'sympy.core.mul.Mul'>

>>> Mul(x, y)
x*y
>>> type(_)
<class 'sympy.core.mul.Mul'>

• x/y uses Pow and Mul classes and __div__ method:

>>> x/y
x
─
y
>>> type(_)
<class 'sympy.core.mul.Mul'>
>>> __.args
⎛   1⎞
⎜x, ─⎟
⎝   y⎠
>>> type(_)
<class 'sympy.core.pow.Pow'>

>>> x.__div__(y)
x
─
y
>>> type(_)
<class 'sympy.core.mul.Mul'>
>>> __.args
⎛   1⎞
⎜x, ─⎟
⎝   y⎠
>>> type(_)
<class 'sympy.core.pow.Pow'>

>>> Mul(x, 1/y)
x
─
y
>>> type(_)
<class 'sympy.core.mul.Mul'>
>>> __.args
⎛   1⎞
⎜x, ─⎟
⎝   y⎠
>>> type(_)
<class 'sympy.core.pow.Pow'>

• x**y uses Pow class and __pow__ method:

>>> x**y
y
x
>>> type(_)
<class 'sympy.core.pow.Pow'>

>>> x.__pow__(y)
y
x
>>> type(_)
<class 'sympy.core.pow.Pow'>

>>> Pow(x, y)
y
x
>>> type(_)
<class 'sympy.core.pow.Pow'>


When the first argument is not an instance SymPy’s class, e.g. as in 1 - x, then Python falls back to __r*__ methods, which are also implemented in all SymPy’s classes:

>>> (1).__sub__(x)
NotImplemented

>>> x.__rsub__(1)
-x + 1
>>> 1 - x
-x + 1


1. Construct an expression for $$1 + x + x^2 + \ldots + x^{10}$$. Can you construct this expression in a different way? Write a function that could generate an expression for $$1 + x + x^2 + \ldots + x^n$$ for any integer $$n >= 0$$. Extend this function to allow $$n < 0$$.

(solution)

2. Write a function that can compute nested powers, e.g. $$x^x$$, $$x^{x^x}$$ and so on. The function should take two parameters: an expression and a positive integer $$n$$ that specifies the depth.

(solution)

## Building blocks of expressions¶

Expressions can consist of instances of subclasses of Expr class. This includes:

• numbers:

>>> Integer(2)
2
>>> Rational(1, 2)
1/2
>>> Float("1e-1000")
1.00000000000000e-1000

• symbols:

>>> Symbol('x')
x
>>> Dummy('y')
y

• numer symbols:

>>> pi
π
>>> E
ℯ
>>> Catalan
Catalan

• functions:

>>> Function('f')
f
>>> sin
sin
>>> cos
cos

• function applications:

>>> Function('f')(x)
f(x)
>>> sin(x)
sin(x)
>>> cos(x)
cos(x)

• operators:

>>> Add(x, y, z)
x + y + z
>>> Mul(x, y, z)
x⋅y⋅z
>>> Pow(x, y)
y
x
>>> Or(x, y, z)
x ∨ y ∨ z
>>> And(x, y, z)
x ∧ y ∧ z

• unevaluated operators:

>>> Derivative(1/x, x)
d ⎛1⎞
──⎜─⎟
dx⎝x⎠
>>> Integral(1/x, x)
⌠
⎮ 1
⎮ ─ dx
⎮ x
⌡
>>> Sum(1/k, (k, 1, n))
n
___
\
\   1
)  ─
/   k
/__,
k = 1

• other:

>>> Poly(x**2 + y, x)
Poly(x**2 + y, x, domain='ZZ[y]')
>>> RootOf(z**5 + z + 3, 2)
⎛ 5           ⎞
RootOf⎝z  + z + 3, 2⎠


This list isn’t at all complete and we included only few classes that SymPy implements that can be used as expression building blocks. Besides those, SymPy has also very many classes that represent entities that can’t be used for constructing expressions, but can be useful as containers of expressions or as utilities for expression building blocks.

1. Expressions implement a doit() method. For most types expressions it doesn’t do anything useful, but in the case of unevaluated operators, it executes an action assigned to to an unevaluated operator (it differentiates, integrates, etc.). Take advantage of doit() and write a function that generates integral tables for a few polynomials, rational functions and elementary functions.

(solution)

## Foreign types in SymPy¶

SymPy internally expects that all objects it works with are instances of subclasses of Basic class. So why x + 1 works without raising an exception? The number 1 is not a SymPy’s type, but:

>>> type(1)
<type 'int'>


it’s a built-in type. SymPy implements sympify() function for the task of converting foreign types to SymPy’s types (yes, Python’s built-in types are also considered as foreign). All SymPy’s classes, methods and functions use sympify() and this is the reason why you can safely write x + 1 instead of more verbose and less convenient x + Integer(1). Note that not all functions return instances of SymPy’s types. Usually, if a function is supposed to return a property of an expression, it will use built-in Python’s types, e.g.:

>>> Poly(x**2 + y).degree(y)
1
>>> type(_)
<type 'int'>


>>> sympify(1)
1
>>> type(_)
<class 'sympy.core.numbers.One'>

>>> sympify(117)
117
>>> type(_)
<class 'sympy.core.numbers.Integer'>

>>> sympify(0.5)
0.500000000000000
>>> type(_)
<class 'sympy.core.numbers.Float'>

>>> from fractions import Fraction

>>> sympify(Fraction(1, 2))
1/2
>>> type(_)
<class 'sympy.core.numbers.Rational'>


SymPy implements explicit sympification rules, heuristics based on __int__, __float__ and other attributes, and in the worst case scenario it falls back to parsing string representation of an object. This usually works fine, but sometimes sympify() can be wrong:

>>> from gmpy import mpz, mpq

>>> sympify(mpz(117))
117.000000000000
>>>> type(_)
<class 'sympy.core.numbers.Float'>

>>> sympify(mpq(1, 2))
0.500000000000000
>>>> type(_)
<class 'sympy.core.numbers.Float'>


This happens because sympify() doesn’t know about either mpz or mpq, and it first looks for __float__ attribute, which is implemented by both those types. Getting float for exact value isn’t very useful so let’s extend sympify() and add support for mpz. The way to achieve this is to add a new entry to converter dictionary. converter takes types as keys and sympification functions as values. Before we extend this dict, we have to resolve a little problem with mpz:

>>> mpz
<built-in function mpz>


which isn’t a type but a function. We can use a little trick here and take the type of some mpz object:

>>> type(mpz(1))
<type 'mpz'>


Let’s now add an entry to converter for mpz:

>>> from sympy.core.sympify import converter

>>> def mpz_to_Integer(obj):
...     return Integer(int(obj))
...
...

>>> converter[type(mpz(1))] = mpz_to_Integer


We could use lambda as well. Now we can sympify mpz:

>>> sympify(mpz(117))
117
>>> type(_)
<class 'sympy.core.numbers.Integer'>


Similar things should be done for mpq. Let’s try one more type:

>>> import numpy

>>> ar = numpy.array([1, 2, 3])
>>> sympify(ar)

>>> sympify(ar)
Traceback (most recent call last):
...
SympifyError: SympifyError: "could not parse u'[1 2 3]'"


sympify() isn’t aware of numpy.ndarray and heuristics didn’t work, so it computed string representation of ar using str() and tried to parse is, which failed because:

>>> str(ar)
[1 2 3]


We might be tempted to add support for numpy.ndarray to sympify() by treating NumPy’s arrays (at least a subset of) as SymPy’s matrices, but matrices aren’t sympifiable:

>>> Matrix(3, 3, lambda i, j: i + j)
⎡0  1  2⎤
⎢       ⎥
⎢1  2  3⎥
⎢       ⎥
⎣2  3  4⎦
>>> sympify(_)
Traceback (most recent call last):
...
SympifyError: SympifyError: 'Matrix cannot be sympified'


We will explain this odd behavior later.

1. Add support for mpq to sympify().

(solution)

2. SymPy implements Tuple class, which provides functionality of Python’s built-in tuple, but is a subclass of Basic. Take advantage of this and make sympify() work for row NumPy arrays, for which it should return instances of Tuple. Raise SympifyError for other classes of arrays.

(solution)

## The role of symbols¶

Let’s now talk about the most important part of expressions: symbols. Symbols are placeholders, abstract entities that can be filled in with whatever content we want (unless there are explicit restrictions given). For example in expression x + 1 we have one symbol x. Let’s start fresh Python’s interpreter and issue:

>>> from sympy import *
>>> init_printing()


We want to start work with our very advanced x + 1 expression, so we may be tempted to simply write:

>>> x + 1
Traceback (most recent call last):
...
NameError: name 'x' is not defined


For users that come from other symbolic mathematics systems, this behavior may seem odd, because in those systems, symbols are constructed implicitly when necessary. In general purpose programming language like Python, we have to define all objects we want to use before we actually use them. So, the first thing we have to always do is to construct symbols and assign them to Python’s variables:

>>> x = Symbol('x')

>>> x + 1
x + 1


Now it worked. Symbols are independent of variables, so nothing prevents you from issuing:

>>> t = Symbol('a')


Well, besides taste. It’s also perfectly valid to create symbols containing special characters:

>>> Symbol('+')
+


_ and ^ characters in symbols have special meaning and are used to denote subscripts and superscripts, respectively:

>>> Symbol('x_1')
x₁
>>> Symbol('x^1')
x¹


If you need more symbols in your expression, you have to define and assign them all before using them. Later you can reuse existing symbols for other purposes. To make life easier, SymPy provides several methods for constructing symbols. The most low-level method is to use Symbol class, as we have been doing it before. However, if you need more symbols, then your can use symbols():

>>> symbols('x,y,z')
(x, y, z)


It takes a textual specification of symbols and returns a tuple with constructed symbols. symbols() supports several syntaxes and can make your life much simpler, when it comes to constructing symbols. First of all, commas can be followed by or completely replaced by whitespace:

>>> symbols('x, y, z')
(x, y, z)
>>> symbols('x y z')
(x, y, z)


If you need indexed symbols, then use range syntax:

>>> symbols("x:5")
(x₀, x₁, x₂, x₃, x₄)
>>> symbols('x5:10')
(x₅, x₆, x₇, x₈, x₉)


You can also create consecutive symbols with lexicographic syntax:

>>> symbols('a:d')
(a, b, c, d)


Note that range syntax simulates range()‘s behavior, so it is exclusive, lexicographic syntax is inclusive, because it makes more sense in this case.

When we issue:

>>> symbols('u,v')
(u, v)


we may be tempted to use u and v:

>>> u
Traceback (most recent call last):
...
NameError: name 'u' is not defined

>>> v
Traceback (most recent call last):
...
NameError: name 'v' is not defined


We got NameError, because we constructed those symbols, but we didn’t assign them to any variables. This solves the problem:

>>> u, v = symbols('u,v')
>>> u, v
u, v


but is a little redundant, because we have to repeat the same information twice. To save time and typing effort, SymPy has another function var() for constructing symbols, which has exactly the same syntax and semantics as symbols(), but it also injects constructed symbols into the global namespace, making this function very useful in interactive sessions:

>>> del u, v
>>> var('u,v)
(u, v)

>>> u + v
u + v


We don’t allow to use var() in SymPy’s library code. There is one more way of constructing symbols, which is related to indexed symbols. Sometimes we don’t know in advance how many symbols will be required to solve a certain problem. For this case, SymPy has numbered_symbols() generator:

>>> X = numbered_symbols('x')

>>> X.next()
x₀

>>> [ X.next() for i in xrange(5) ]
[x₁, x₂, x₃, x₄, x₅]


1. Implement a function that would generate an expression for $$x_1^1 + x_2^2 + \ldots + x_n^n$$. This function would take two arguments: base name for indexed symbols and integer exponent $$n >= 1$$. What’s the best approach among the four presented above?

(solution)

## Obtaining parts of expressions¶

We already know how to construct expressions, but how to get parts of complex expressions? The most basic and low-level way of decomposing expressions is to use args property:

>>> x + y + 1
x + y + 1
>>> _.args
(1, y, x)
>>> map(type)
[<class 'sympy.core.numbers.One'>, <class 'sympy.core.symbol.Symbol'>, <class 'sympy.core.symbol.Symbol'>]


args always gives a tuple of instances of SymPy’s classes. One should notice the weird order of elements, which doesn’t match printing order. This happens for classes that in which order of arguments is insignificant. The most notable examples of such class are Add and Mul (for commutative part). In this particular case we can use as_ordered_terms() method to get args in printing order:

>>> (x + y + 1).as_ordered_terms()
[x, y, 1]


When dealing which classes that have fixed order of arguments, printing order and args order match:

>>> Derivative(sin(x), x, x)
2
d
─────(sin(x))
dx dx

>>> _.args
(sin(x), x, x)


Lets suppose that Cls represents any SymPy’s class and expr is an instance of this class (expr = Cls()). Then the following holds:

Cls(*expr.args) == expr


This is very useful invariant, because we can easily decompose, modify and rebuild expressions of various kinds in SymPy exactly the same way. This invariant is being used in all functions that manipulation expressions.

Let’s now use args to something a little more interesting than simple decomposition of expressions. Working with expressions, one may be interested in the depth of such expressions. By viewing expressions as n-ary trees, by depth we understand the longest path in a tree.

Trees consist of branches and leafs. In SymPy, leafs of expressions are instances of subclasses of Atom class (numbers, symbols, special constants):

>>> Integer(10)
10
>>> isinstance(_, Atom)
True

>>> pi
π
>>> isinstance(_, Atom)
True


Atoms can be also recognized by the fact that their args are empty. Note, however, that this is an implementation detail, and one should use either isinstance() built-in function or is_Atom property to recognize atoms properly. Everything else than an Atom is a branch.

Let’s implement depth() function:

from sympy.core import Atom

def depth(expr):
if isinstance(expr, Atom):
return 1
else:
return 1 + max([ depth(arg) for arg in expr.args ])


The implementation is straightforward. First we check if the input expression is an atom. In this case we return 1 and terminate recursion. Otherwise depth() recurses for every argument of expr and returns 1 plus maximum of depths of all branches.

Let’s see depth() in action:

>>> depth(x)
1
>>> depth(x + 1)
2
>>> depth(x + sin(x))
3
>>> depth(x + sin(x) + sin(cos(x)))
4


All those examples work as expected. However, not everything is perfect with this function. Let’s look at the following phenomenon:

>>> depth(Integer(117))
1
>>> depth(117)
Traceback (most recent call last):
...
AttributeError: 'int' object has no attribute 'args'


117 is an instance of Python’s built-in type int, but this type is not a subclass of Atom, so Python choses the other branch in depth() and this must fail. Before the last example we pass only instances of SymPy’s expression to depth(). If we want depth() to work also for non-SymPy types, we have to sympify expr with sympify() before using it.

1. Change depth() so that it sympifies its input argument. Rewrite depth() so that is calls sympify() only once.

(solution)

2. Add support for iterable containers to depth(). Containers should be treated as branches and have depth defined the same way.

(solution)

## Immutability of expressions¶

Expressions in SymPy are immutable and cannot be modified by an in-place operation. This means that a function will always return an object, and the original expression will not be modified. Consider the following code:

>>> var('x,y,a,b')
(x, y, a, b)

>>> original = 3*x + 4*y
>>> modified = original.subs({x: a, y: b})

>>> original
3*x + 4*y
>>> modified
3*a + 4*b


The output shows that the subs() method gave a new expression with symbol x replaced with symbol a and symbol y replaced with symbol b. The original expression wasn’t modified. This behavior applies to all classes that are subclasses of Basic. An exception to immutability rule is Matrix, which allows in-place modifications, but it is not a subclass of Basic:

>>> Matrix.mro()
[<class 'sympy.matrices.matrices.Matrix'>, <type 'object'>]


Be also aware of the fact that SymPy’s symbols aren’t Python’s variables (they just can be assigned to Python’s variables), so if you issue:

>>> u = Symbol('u')
>>> v = u
>>> v += 1
>>> v
u + 1


then in-place operator += constructed an new instance of Add and left the original expression stored in variable u unchanged:

>>> u
u


For efficiency reason, any in-place operator used on elements of a matrix, modifies the matrix in-place and doesn’t waste memory for unnecessary copies.

1. This is the first time we used subs(). This is a very important method and we will talk more about it later. However, we can also use subs() to generate some cool looking expressions. Start with x**x expression and substitute in it x**x for x. What do you get? (make sure you use pretty printer) Can you achieve the same effect without subs()?

(solution)

## Comparing expressions with ==¶

Consider the following two expressions:

>>> f = (x + 1)**2
>>> f
2
(x + y)

>>> g = x**2 + 2*x + 1
>>> g
2
x  + 2⋅x + 1


We should remember from calculus 101 that those two expressions are equivalent, because we can use binomial theorem to expand f and we will get g. However in SymPy:

>>> f == g
False


This is correct result, because SymPy implements structural understanding of == operator, not semantic. So, for SymPy f and g are very different expressions.

What to do if we have two variables and we want to know if their contents are equivalent, but not necessarily structurally equal? There is no simple answer to this question in general. In the particular case of f and g, it is sufficient to issue:

>>> expand(f) == expand(g)
True


or, based on $$f = g \equiv f - g = 0$$ equivalence:

>>> expand(f - g) == 0
True


In case of more complicated expression, e.g. those involving elementary or special functions, this approach may be insufficient. For example:

>>> u = sin(x)**2 - 1
>>> v = cos(x)**2

>>> u == v
False
>>> expand(u - v) == 0
False


In this case we have to use more advanced term rewriting function:

>>> simplify(u - v) == 0
True


## The meaning of expressions¶

Expressions don’t have any meaning assigned to them by default. Thus $$x + 1$$ is simply an expression, not a function or a univariate polynomial. Meaning is assigned when we use expressions in a context, e.g.:

>>> div(x**2 - y, x - y)
⎛        2    ⎞
⎝x + y, y  - y⎠


In this case, x**2 - y and x - y where treated as multivariate polynomials in variables x and y (in this order). We could change this understanding and ask explicitly for polynomials in variables y and x. This makes div() return a different result:

>>> div(x**2 - y, x - y, y, x)
⎛    2    ⎞
⎝1, x  - x⎠


Quite often SymPy is capable of deriving the most useful understanding of expressions in a given context. However, there are situations when expressions simply don’t carry enough information to make SymPy perform computations without telling it explicitly what to do:

>>> roots(x**2 - y)
Traceback (most recent call last):
...
PolynomialError: multivariate polynomials are not supported


Here we have to tell roots() in which variable roots should be computed:

>>> roots(x**2 - y, x)
⎧   ⎽⎽⎽       ⎽⎽⎽   ⎫
⎨-╲╱ y : 1, ╲╱ y : 1⎬
⎩                   ⎭


Of course the choice of y is also a valid one, assuming that this is what you really want. This of course doesn’t apply only to polynomials.

## Turning strings into expressions¶

Suppose we saved the following expression:

>>> var('x,y')

>>> expr = x**2 + sin(y) + S(1)/2
>>> expr
2            1
x  + sin(y) + ─
2


by printing it with sstr() printer and storing to a file:

>>> sstr(expr)
x**2 + sin(y) + 1/2

>>> with open("expression.txt", "w") as f:
...     f.write(_)
...
...


We used this kind of printer because we wanted the file to be fairly readable. Now we want to restore the original expression. First we have to read the text form from the file:

>>> with open("expression.txt") as f:
...
...

>>> text_form
x**2 + sin(y) + 1/2
>>> type(_)
<type 'str'>


We could try to try to use eval() on text_form but this doesn’t give expected results:

>>> eval(text_form)
2
x  + sin(y) + 0.5


This happens because 1/2 isn’t understood by Python as rational number and is equivalent to a problem we had when entering expressions of this kind in interactive sessions.

To overcome this problem we have to use sympify(), which implements tokenize–based parser that allows us to handle this issue:

>>> sympify(text_form)
2            1
x  + sin(y) + ─
2
>>> _ == expr
True


Let’s now consider a more interesting problem. Suppose we define our own function:

>>> class my_func(Function):
...     """Returns zero for integer values. """
...
...     @classmethod
...     def eval(cls, arg):
...         if arg.is_Number:
...             return 2*arg
...
...


This function gives twice the input argument if the argument is a number and doesn’t do anything for all other classes of arguments:

>>> my_func(117)
234
>>> my_func(S(1)/2)
1

>>> my_func(x)
my_func(x)
>>> _.subs(x, 2.1)
4.20000000000000

>>> my_func(1) + 1
3


Let’s create an expression that contains my_func():

>>> expr = my_func(x) + 1
>>> expr
my_func(x) + 1

>>> _.subs(x, 1)
3


Now we will print it using sstr() printer and sympify the result:

>>> sympified = sympify(sstr(expr))
>>> sympified
my_func(x) + 1


We can use subs() method to quickly verify the expression is correct:

>>> sympified.subs(x, 1)
my_func(1) + 1


This is not exactly what we expected. This happens because:

>>> expr == sympified
False

>>> expr.args
(1, my_func(x))
>>> type(_) is my_func
True

>>> sympified.args
(1, my_func(x))
>>> type(_) is my_func
False


sympify() evaluates the given string in the context of from sympy import * and is not aware of user defined names. We can explicitly pass a mapping between names and values to it:

>>> sympify(sstr(expr), {'my_func': my_func})
my_func(x) + 1
>>> _.subs(x, 1)
3


This time we got the desired result. This shows that we have to be careful when working with expressions encoded as strings. This happens to be even more tricky when we put assumptions on symbols. Do you remember the example in which we tried to expand $$\log(a b)$$? Lets do it once again:

>>> var('a,b', positive=True)
(a, b)
>>> log(a*b).expand()
log(a) + log(b)


This worked as previously. However, let’s now print $$\log(a b)$$, sympify the resulting string and expand the restored expression:

>>> sympify(sstr(log(a*b))).expand()
log(a⋅b)


This didn’t work, because sympify() doesn’t know what a and b are, so it assumed that those are symbols and it created them implicitly. This issue is similar to what we already experienced with my_func().

The most reliable approach to storing expression is to use pickle module. In the case of $$\log(a b)$$ it works like this:

>>> import pickle
>>> pickled = pickle.dumps(log(a*b))
>>> expr.expand()
log(a) + log(b)


Unfortunately, due to pickle‘s limitations, this doesn’t work for user defined functions like my_func():

>>> pickle.dumps(my_func(x))
Traceback (most recent call last):
...
`

1. Construct a polynomial of degree, let’s say, 1000. Use both techniques to save and restore this expression. Compare speed of those approaches. Verify that the result is correct.

(solution)