Writing Tests

The most important thing for a mathematical library like SymPy is correctness. Functions should never return mathematically incorrect results. Correctness is always the top concern, even if it comes at the cost of things like performance or modularity.

Consequently, all functionality in SymPy is tested extensively. This guide goes over how tests in SymPy are written.

Testing Policies

In order to ensure the high standard of correctness, SymPy has the following rules that apply to all pull requests:

  1. All new functionality must be tested. Tests should aim to cover all possible cases to best ensure correctness. This means not only maximizing code coverage, but also covering all possible corner cases.

  2. Every pull request must pass all tests before it can be merged. The tests are automatically run by the GitHub Actions CI on every pull request. If any tests fail, the CI will fail with a red ❌. These failures must be addressed before the pull request can be merged.

  3. Bug fixes should be accompanied by a regression test.

Basics for Writing Tests

Tests are located alongside the code in tests/ directories, in files named test_<thing>.py. In most cases, if you modified sympy/<submodule>/<file>.py then the test for the functionality will go in sympy/<submodule>/tests/test_<file>.py. For example, the tests for the functions in sympy/simplify/sqrtdenest.py are in sympy/simplify/tests/test_sqrtdenest.py. There are some exceptions to this rule, so in general try to find where the existing tests are for a function and add your tests alongside them. If you are adding tests for a new function, follow the general pattern of tests in the module you are adding to.

Tests follow a simple pattern, which should be apparent from reading the existing test files. Tests are in functions that start with test_ and contain lines like

assert function(arguments) == result

For example

# from sympy/functions/elementary/tests/test_trigonometric.py

def test_cos_series():
    assert cos(x).series(x, 0, 9) == \
        1 - x**2/2 + x**4/24 - x**6/720 + x**8/40320 + O(x**9)

New test cases can be added to an existing test function if it is relevant, or you can create a new test function.

Running Tests

The basic way to run the tests is to use


to run the tests, and


to run the doctests. Note that the full test suite can take some time to run, so typically you should just run a subset of the tests, e.g., corresponding to the module you modified. You can do this by passing the name of the submodules or tests files to the test command. For example,

./bin/test solvers

will run only the tests for the solvers.

If you want, you can also use pytest to run the tests instead of the ./bin/test tool, for example

pytest -m 'not slow' sympy/solvers

Another option is to just push your code up to GitHub and let the tests run on the CI. The GitHub Actions CI will run all the tests. However, it can take some time to finish, so it is usually advisable to run at least the basic tests before committing to avoid having to wait.

Debugging Test Failures on GitHub Actions

When you see a test failure on CI, like

_________________ sympy/printing/pretty/tests/test_pretty.py:test_upretty_sub_super _________________
Traceback (most recent call last):
  File "/home/oscar/current/sympy/sympy.git/sympy/printing/pretty/tests/test_pretty.py", line 317, in test_upretty_sub_super
    assert upretty( Symbol('beta_1_2') ) == 'β₁₂'

The bit in between _________________ is the name of the test. You can reproduce the test locally by copying and pasting this:

./bin/test sympy/printing/pretty/tests/test_pretty.py::test_upretty_sub_super


pytest sympy/printing/pretty/tests/test_pretty.py::test_upretty_sub_super

The test also shows the file and line number (in this example, 317 in sympy/printing/pretty/tests/test_pretty.py) of the assertion that fails, so you can look it up to see what the test is testing.

Sometimes when you do this, you will not be able to reproduce the test failure locally. Some common causes of this are:

  • You may need to merge the latest master into your branch to reproduce the failure (GitHub Actions will always merge your branch with the latest master before running the tests).

  • Something about the CI testing environment may be different from yours (this is especially likely for tests that depend on optional dependencies. Check which versions of relevant packages are installed at the top of the CI log.

  • It’s possible that some other test that ran prior to yours may have somehow influenced your test. SymPy is not supposed to have global state, but sometimes some state can sneak in on accident. The only way to check this is to run the exact same test command that was run on CI.

  • A test may fail sporadically. Try rerunning the test multiple times. The beginning of the test log on CI prints the random seed, which can be passed to ./bin/test --seed, and the PYTHONHASHSEED environment variable, which may be helpful for reproducing such failures.

It is also sometimes possible that a failure on CI may be unrelated to your branch. We only merge branches that have passing CI, so that master always ideally has passing tests. But sometimes a failure can slip in. Typically this is either because the failure is sporadic (see the previous bullet), and it wasn’t noticed, or because some optional dependency was updated which broken an optional dependency test. If a test failure seems like it is unrelated to your change, check if the CI builds for master and if CI builds on other recent PRs have the same failure. If they do, this is likely the case. If they don’t, you should check more carefully if your change is causing the failure, even if it seems unrelated.

When there is a CI failure in the master branch, be aware that your pull request cannot be merged until it is fixed. This is not required, but if you know how to fix it, please do this to help everyone (if you do this, do it in a separate pull request so that it can be merged expeditiously).

Regression Tests

Regression tests are tests that would fail before a bug fix but now pass. Often you can use a code example from an issue as a test case, although it is also OK to simplify such examples or to write your own, so long as it tests the issue in question.

For example, consider issue #21177, which identified the following wrong result:

>>> residue(cot(pi*x)/((x - 1)*(x - 2) + 1), x, S(3)/2 - sqrt(3)*I/2) 
>>> residue(cot(pi*x)/(x**2 - 3*x + 3), x, S(3)/2 - sqrt(3)*I/2) 

Here the first expression was correct but the second was not. In the issue, the cause of the issue was identified in the as_leading_term method, and several other related issues were also found.

In the corresponding pull request (#21253), several regression tests were added. For example (from that PR):

# In sympy/functions/elementary/tests/test_trigonometric.py

def test_tan():
    # <This test was already existing. The following was added to the end>

    # https://github.com/sympy/sympy/issues/21177
    f = tan(pi*(x + S(3)/2))/(3*x)
    assert f.as_leading_term(x) == -1/(3*pi*x**2)
# In sympy/core/tests/test_expr.py

def test_as_leading_term():
    # <This test was already existing. The following was added to the end>

    # https://github.com/sympy/sympy/issues/21177
    f = -3*x + (x + Rational(3, 2) - sqrt(3)*S.ImaginaryUnit/2)**2\
        - Rational(3, 2) + 3*sqrt(3)*S.ImaginaryUnit/2
    assert f.as_leading_term(x) == \
        (3*sqrt(3)*x - 3*S.ImaginaryUnit*x)/(sqrt(3) + 3*S.ImaginaryUnit)

    # https://github.com/sympy/sympy/issues/21245
    f = 1 - x - x**2
    fi = (1 + sqrt(5))/2
    assert f.subs(x, y + 1/fi).as_leading_term(y) == \
        (-36*sqrt(5)*y - 80*y)/(16*sqrt(5) + 36)
# In sympy/series/tests/test_residues.py

def test_issue_21177():
    r = -sqrt(3)*tanh(sqrt(3)*pi/2)/3
    a = residue(cot(pi*x)/((x - 1)*(x - 2) + 1), x, S(3)/2 - sqrt(3)*I/2)
    b = residue(cot(pi*x)/(x**2 - 3*x + 3), x, S(3)/2 - sqrt(3)*I/2)
    assert a == r
    assert (b - a).cancel() == 0

This example shows some important aspects of regression tests:

  • Tests should be added for the underlying fix, not just the originally reported issue. The originally reported issue in this example was with the residue() function but the underlying issue was with the as_leading_term() method.

  • At the same time, it can also be beneficial to add a test for the high-level issue as reported. This ensures that residue itself won’t break in the future, even if the implementation details of it change so that it no longer uses the same code path that was fixed.

  • This example does not show it, but in some cases it may be prudent to simplify the originally reported issue for the test case. For example, sometimes users will include unnecessary details in the report that don’t actually matter for the reproduction of the issue (like unnecessary assumptions on symbols), or make the input expression too large or have too many unnecessary constant symbols. This is especially important to do if the code from the originally stated issue is slow to compute. If the same thing can be tested with a test that runs more quickly, this should be preferred.

  • Regression tests should also be added for additional bugs that are identified in the issue. In this example, the second test (the test added to test_as_leading_term()) was identified as a related problem in a comment on the issue.

  • It is useful to cross-reference the issue number in a regression test, either using a comment or in the test name. A comment is preferred if the test is being added to an existing test.

Regression tests aren’t just for bug fixes. They should also be used for new features, to make sure the newly implemented functionality remains implemented and correct.

Special Types of Tests

Most tests will be of the form assert function(input) == output. However, there are other types of things that you might want to test that should be tested in certain ways.

Testing Exceptions

To test that a function raises a given exception, use sympy.testing.pytest.raises. raises() takes an exception class and a lambda. For example

from sympy.testing.pytest.raises
raises(TypeError, lambda: cos(x, y)

Remember to include the lambda. Otherwise, the code will be executed immediately and will raise the exception, causing the test to fail.

raises(TypeError, cos(x, y)) # This test will fail

raises can also be used as a context manager, like

with raises(TypeError):
    cos(x, y)

However, be careful using this form, as it can only check one expression at a time. If the code under context manager raises multiple exceptions, only the first one will actually be tested

with raises(TypeError):
   cos(x, y)

The lambda form is generally better because it avoids this problem, although if you are testing something that cannot be represented in a lambda you will need to use the context manager form.

Testing Warnings

Warnings can be tested with the sympy.testing.pytest.warns() context manager. Note that SymPyDeprecationWarning is special and should be tested with warns_deprecated_sympy() instead (see below).

The context manager should take a warning class (warnings.warn() uses UserWarning by default), and, optionally, a regular expression that the warning message should match as the match keyword argument.

from sympy.testing.pytest import warns
with warns(UserWarning):

with warns(UserWarning, match=r'warning'):

Any test functionality that emits a warning should use warns(). That way, no warnings are actually emitted during the tests themselves. This includes warnings coming from external libraries.

Warnings within SymPy itself should be used very sparingly. Aside from deprecation warnings, warnings are generally not used in SymPy, as they may be too annoying for users, especially those who use SymPy as a library, to be warranted.

When you do use them, you must set the stacklevel parameter in the warning so that it shows the user code that called the function that emitted the warning. If the stacklevel parameter is impossible to set correctly, use warns(test_stacklevel=False) to disable the check in warns that stacklevel is used properly. warns(SymPyDeprecationWarning, test_stacklevel=False) must be used in place of warns_deprecated_sympy() if this applies to a SymPyDeprecationWarning

Test Deprecated Functionality

Deprecated functionality should be tested with the sympy.testing.pytest.warns_deprecated_sympy() context manager.

The only purpose of this context manager is to test that the deprecation warning itself is functioning correctly. This should be the only place in the test suite where deprecated functionality is called. All other tests should use non-deprecated functionality. If it is impossible to avoid deprecated functionality, this may be a sign that the functionality should not actually be deprecated.

The deprecation policy page goes into detail about how to add a deprecation to a function.

For example,

from sympy.testing.pytest import warns_deprecated_sympy
x = symbols('x')

# expr_free_symbols is deprecated
def test_deprecated_expr_free_symbols():
    with warns_deprecated_sympy():
        assert x.expr_free_symbols == {x}

If code is using deprecated functionality from another library, this code should be updated. Until then, the normal warns() context manager should be used in the corresponding tests to prevent the warning from being emitted.

Testing that Something is Unchanged

The normal test style of

assert function(input) == output

works for most types of tests. However, it doesn’t work in the case where a SymPy object should remain unchanged. Consider the following example:

assert sin(pi) == 0
assert sin(pi/2) == 1
assert sin(1) == sin(1)

The first two tests here are fine. The test that sin returns the corresponding special value for the inputs pi and pi/2. However, the last test nominally checks that sin(1) doesn’t return anything. But upon closer inspection, we see that it doesn’t do that at all. sin(1) could in fact return anything. It could return complete nonsense or even a wrong answer like 0. The test would still pass, because all it is doing is checking that the result of sin(1) equals the result of sin(1), which it always will so long as it always returns the same thing.

We really want to check that sin(1) remains unevaluated. The sympy.core.expr.unchanged helper will do this.

Use it like

from sympy.core.expr import unchanged

def test_sin_1_unevaluated():
    assert unchanged(sin, 1)

This test now actually checks the correct thing. If sin(1) were made to return some value, the test would fail.

Testing Expressions with Dummy

Expressions that return Dummy cannot be tested with == directly, due to the nature of Dummy. In such cases, use the dummy_eq() method. For example:

# from

def test_factorial_rewrite():
    n = Symbol('n', integer=True)
    k = Symbol('k', integer=True, nonnegative=True)

    assert factorial(n).rewrite(gamma) == gamma(n + 1)
    _i = Dummy('i')
    assert factorial(k).rewrite(Product).dummy_eq(Product(_i, (_i, 1, k)))
    assert factorial(n).rewrite(Product) == factorial(n)

Consistency Checks

Checking a set of known inputs and outputs can only get you so far. A test like

assert function(input) == expression

will check that function(input) returns expression, but it doesn’t check that expression itself is actually mathematically correct.

However, depending on what function is, sometimes a consistency check can be done to check that expression itself is correct. This typically boils down to “computing expression in two different ways”. If both ways agree, there is a pretty high chance it is correct, as it is unlikely that two completely different methods will produce the same wrong answer.

For example, the inverse of indefinite integration is differentiation. The tests for integrals can be checked for consistency by seeing if the derivative of the result produces the original integrand:

expr = sin(x)*exp(x)
expected == exp(x)*sin(x)/2 - exp(x)*cos(x)/2

# The test for integrate()
assert integrate(expr, x) == expected
# The consistency check that the test itself is correct
assert diff(expected, x) == expr

The implementation for diff is very simple compared to integrate, and it is tested separately, so this confirms the answer is correct.

Of course, one could also just confirm the answer by hand, and this is what most tests in SymPy do. But a consistency check does not hurt, especially when it is easy to do.

The use of consistency checks in the SymPy test suite is not, itself, consistent. Some modules make heavy use of them, e.g., every test in the ODE module checks itself using checkodesol(), for instance. Other modules do not use consistency checks in their tests at all, although some of these could be updated to do so. In some cases, there are no reasonable consistency checks and other sources of truth must be used to verify the test outputs.

When making heavy use of consistency checks, it’s often a good idea to factor out the logic into a helper function in the test file to avoid duplication. Helper functions should start with an underscore so they aren’t mistaken for test functions by the test runner.

Random Tests

Another way that tests can check themselves for consistency is to check the expressions on random numerical inputs. The helper functions in sympy.core.random can be used for this. See the tests in sympy/functions/special/ which make heavy use of this functionality.

If you add a random test, be sure to run the test multiple times to ensure that it always passes. Random tests can be reproduced by using the random seed printed at the top of the tests. For example

========================================================================== test process starts ==========================================================================
executable:         /Users/aaronmeurer/anaconda3/bin/python  (3.9.13-final-0) [CPython]
architecture:       64-bit
cache:              yes
ground types:       gmpy 2.1.2
numpy:              1.22.4
random seed:        7357232
hash randomization: on (PYTHONHASHSEED=3923913114)

Here the random seed is 7357232. It can be reproduced with

./bin/test --seed 7357232

In general you may need to use the same Python version and architecture as shown in the test header to reproduce a random test failure. You may also in some situations, need to run the tests using the exact same input arguments (i.e., running the full test suite or running only a subset) in order to reproduce a test that fails randomly.

Skipping Tests

Tests can be skipped using the sympy.testing.pytest.SKIP decorator or using the sympy.testing.pytest.skip() function. Note that tests that are skipped because they are expected to fail should use the @XFAIL decorator instead (see below). Test that are skipped because they are too slow should use the @slow decorator instead.

Tests that are skipped unconditionally should be avoided. Such a test is almost completely useless, as it will never be actually run. The only reason to skip a test unconditionally is if it would otherwise be @XFAIL or @slow but cannot use one of those decorators for some reason.

Both @SKIP() and skip() should include a message that explains why the test is being skipped, like skip('numpy not installed').

The typical usage of skipping a test is when a test depends on an optional dependency.

Such tests are generally written like

from sympy.external import import_module

# numpy will be None if NumPy is not installed
numpy = import_module('numpy')

def test_func():
    if not numpy:
       skip('numpy is not installed')

    assert func(...) == ...

When the test is written in this way, the test will not fail when NumPy is not installed, which is important since NumPy is not a hard dependency of SymPy. See also Writing Tests with External Dependencies below.

Marking Tests as Expected to Fail

Some tests in SymPy are expected to fail. They are written so that when the functionality the check is finally implemented, a test is already written for it.

Tests that are expected to fail are called XFAIL tests. They show up as f in the test runner when they fail as expected and X when they pass (or “XPASS”). A test that XPASSes should have its @XFAIL decorator removed so that it becomes a normal test.

To XFAIL a test, add the sympy.testing.pytest.XFAIL decorator to it

from sympy.testing.pytest import XFAIL

def test_failing_integral():
    assert integrate(sqrt(x**2 + 1/x**2), x) == x*sqrt(x**2 + x**(-2))*(sqrt(x**4 + 1) - atanh(sqrt(x**4 + 1)))/(2*sqrt(x**4 + 1))

Care should be taken when writing an XFAIL test so that it actually passes when the functionality starts working. If you mistype the output, for example, the test may never pass. For example, the integral in the above test might start working, but return a result in a slightly different form than the one being checked. A more robust test would be

from sympy.testing.pytest import XFAIL

def test_failing_integral():
    # Should be x*sqrt(x**2 + x**(-2))*(sqrt(x**4 + 1) - atanh(sqrt(x**4 + 1)))/(2*sqrt(x**4 + 1))
    assert not integrate(sqrt(x**2 + 1/x**2), x).has(Integral)

This will cause the test to XPASS once the integral starts working, at which time the test can be updated with the actual output of integrate() (which can be compared against the expected output).

Marking Tests as Slow

A test that is slow to run should be marked with the @slow decorator from sympy.testing.pytest.slow. The @slow decorator should be used for tests that take more than a minute to run. Tests that hang should use @SKIP instead of @slow. The slow tests will be run automatically in a separate CI job, but are skipped by default. You can manually run the slow tests with

./bin/test --slow

Writing Tests with External Dependencies

When writing a test for a function that uses one of SymPy’s optional dependencies, the test should be written in a way that makes it so that the test does not fail when the module is not installed.

The way to do this is to use sympy.external.import_module(). This will import the module if it is installed and return None otherwise.

sympy.testing.pytest.skip should be used to skip tests when the module in question is not installed (see Skipping Tests above). This can be done at the module level if the entire test file should be skippped, or in each individual function.

You should also make sure the test is run in the “Optional Dependencies” CI run. To do this, edit bin/test_optional_dependencies.py and make sure the test is included (most SymPy submodules that test optional dependencies are already included automatically).

If the optional dependency is new, add it to the list of packages that are installed in the optional dependencies build in .github/workflows/runtests.yml, and add it to the optional dependencies document at doc/src/contributing/dependencies.md.

Note that it is not necessary to do any of this when using mpmath, as it is already a hard dependency of SymPy and will always be installed.


Every public function should have a docstring, and every docstring should have a examples. Code examples are all tested, which is why they are also sometimes called doctests. The docstring style guide has more details on how to format examples in docstrings.

To run the doctests, use the


command. This command can also take arguments to test a specific file or submodule, similar to bin/test.

Doctests should be written in a self-contained manner, with each doctest acting like a fresh Python session. This means that each doctest must manually import each function used in the doctest and define the symbols used. This may seem verbose, but it is helpful to users who are new to SymPy or even to Python who may not know where different functions come from. It also makes it easy for a user to copy and paste an example into a Python session of their own (the HTML documentation includes a button in the top right of every code example that copies the whole example to the clipboard).

For example

>>> from sympy import Function, dsolve, cos, sin
>>> from sympy.abc import x
>>> f = Function('f')
>>> dsolve(cos(f(x)) - (x*sin(f(x)) - f(x)**2)*f(x).diff(x),
...        f(x), hint='1st_exact')
Eq(x*cos(f(x)) + f(x)**3/3, C1)

The doctest output should look exactly as it would in a python session, with >>> before the inputs and the outputs after. The doctester tests that the output string matches, unlike normal tests which typically check that the Python objects are the same with ==. Consequently, the output needs to look exactly the same as it does in a Python session.

Like tests, all doctests must pass for a change to be accepted. However, when writing doctests, it is important to remember that doctests should not be thought of as tests. Rather, they are examples that happen to be tested.

Therefore, you should always think about what will make a good, readable example when writing doctests. Doctests do not need to extensively cover all possible inputs, and should not include corner or extreme cases unless they are important for users to be aware of.

Everything that is tested in a doctest should also be tested in a normal test. You should always be free to remove or change a doctest example at any time if it improves the documentation (to contrast, a normal test should never be changed or removed, except in certain exceptional situations).

This also means that doctests should be written first and foremost in a way that makes them understandable by someone reading the documentation. It can sometimes be tempting to write a doctest in some indirect way to please the doctester, but this should be avoided if it makes the example harder to understand. For example

>>> from sympy import sin, cos, trigsimp, symbols
>>> x = symbols('x')
>>> result = trigsimp(sin(x)*cos(x))
>>> result == sin(2*x)/2

This passes the doctest, and something along these lines would be fine a normal test. But in a docstring example, it is much clearer to just show the actual output

>>> from sympy import sin, cos, trigsimp, symbols
>>> x = symbols('x')
>>> trigsimp(sin(x)*cos(x))

Of course, in some situations, the full output is unwieldy and showing it would make the example harder to read, so this sort of thing may be appropriate. Use your best judgment, keeping in mind that the understandability of the doctest as a documentation example is the most important thing. In some extreme instances, it may be preferable to just skip testing an example (see below) rather than writing it in a convoluted way that is difficult to read just to please the doctester.

Here are some additional tips for writing doctests:

  • Long input lines can be broken into multiple lines by using ... as a continuation prompt, as in the example above. The doctest runner also allows long outputs to be line wrapped (it ignores newlines in the output).

  • Common symbol names can be imported from sympy.abc. Uncommon symbol names or symbols that use assumptions should be defined using symbols.

    >>> from sympy.abc import x, y
    >>> x + y
    x + y
    >>> from sympy import symbols, sqrt
    >>> a, b = symbols('a b', positive=True)
    >>> sqrt((a + b)**2)
    a + b
  • If a test shows a traceback, everything between Traceback (most recent call last): and the last line with the exception message should be replaced with ..., like

    >>> from sympy import Integer
    >>> Integer('a')
    Traceback (most recent call last):
    ValueError: invalid literal for int() with base 10: 'a'
  • ... is special in that whenever it appears in the output of an example, the doctester will allow it to replace any amount of text. It should also be used in instances where the exact output differs between runs, like

    >>> from sympy import simplify
    >>> simplify
    <function simplify at ...>

    Here the actual output is something like <function simplify at 0x10e997790> but the 0x10e997790 is a memory address which will differ with every Python session.

    ... in outputs should be used sparingly, as it prevents the doctest from actually checking that part of the output. It also may not be clear to the reader of the documentation what it is meant. Note that it’s fine if the output of a doctest is updated to something else in the future. ... should not be used in an attempt to “future-proof” doctest output. Also note that the doctester already automatically handles things like whitespace-only differences in the output and floating-point values.

  • You can line break output lines. The doctester automatically ignores whitespace-only differences in the output, which includes newlines. Long lines should be broken so that they do not extend beyond the page in the HTML documentation (and so that the source code does not have lines longer than 80 characters). For example:

    >>> ((x + 1)**10).expand()
    x**10 + 10*x**9 + 45*x**8 + 120*x**7 + 210*x**6 + 252*x**5 + 210*x**4 +
    120*x**3 + 45*x**2 + 10*x + 1
  • Another option if a doctest cannot pass is to skip it, by adding # doctest:+SKIP to the end of the input line, like

    >>> import random
    >>> random.random()      # doctest: +SKIP

    The # doctest:+SKIP part will be automatically hidden in the HTML documentation. When skipping a doctest, always be sure to test the output manually, as the doctester will not check it for you.

    # doctest:+SKIP should be used sparingly. Ideally a doctest should only be skipped when it is impossible to run it. A doctest that is skipped will never be tested, meaning it may become outdated (i.e., incorrect), which will be confusing to users.

  • Doctests that require a dependency to run should not be skipped with # doctest: +SKIP. Instead, use the @doctest_depends_on decorator on the function to indicate which libraries should be installed for the doctest to run.

  • If the test output includes a blank line, use <BLANKLINE> in place of the blank line. Otherwise the doctester will think that the output ends at the blank line. <BLANKLINE> will be automatically hidden in the HTML documentation. This is not common as most SymPy objects do not print with blank lines.

  • Avoid using pprint() in doctest examples. If you need to show an expression in an easier to read way, you can include it inline as LaTeX math using dollar signs. If you absolutely must use pprint(), always use pprint(use_unicode=False) as the Unicode characters used for pretty printing do not always render correctly in the HTML documentation.

  • If you want to show that something returns None use print, like

    >>> from sympy import Symbol
    >>> x = Symbol('x', positive=True)
    >>> x.is_real
    >>> x = Symbol('x', real=True)
    >>> x.is_positive # Shows nothing, because it is None
    >>> print(x.is_positive)
  • You can add short comments to doctests, either at the end of a line or by themselves after >>>. However, these should typically be only a few words long. Detailed explanations of what is happening in the doctest should go in the surrounding text.

  • Dictionaries and sets are automatically sorted by the doctester, and any expressions are automatically sorted so that the order of terms is always printed in the same way. Usually you can just include the output that the doctester “expects” it and it will always pass subsequently.

    >>> {'b': 1, 'a': 2}
    {'a': 2, 'b': 1}
    >>> {'b', 'a'}
    {'a', 'b'}
    >>> y + x
    x + y

Updating Existing Tests

Sometimes when you change something or fix a bug, some existing tests will fail. If this happens, you should check the test to see why it is failing. In many cases, the test will be checking for something you didn’t consider, or your change has an unexpected side effect that broke something else. When this happens, you may need to revisit your change. If you are unsure what to do, you should discuss it on the issue or pull request.

If the test that fails is a code quality test, that usually means you just need to fix your code so that it satisfies the code quality check (e.g., remove trailing whitespace).

Occasionally, however, it can happen that the test fails but there is nothing wrong. In this case, the test should be updated. The most common instance of this is a test that checks for a specific expression, but the function now returns a different, but mathematically equivalent expression. This is especially common with doctests, since they check not just the output expression but the way it is printed.

If a function output is mathematically equivalent, the existing test can be updated with the new output. However, even when doing this, you should be careful:

  • Carefully check that the new output is indeed the same. Manually check something like if the difference of old and new expressions simplifies to 0. Sometimes, two expressions are equivalent for some assumptions but not for all, so check that the two expressions are really the same for all complex numbers. This can particularly happen with expressions involving square roots or other radicals. You can check random numbers, or use the equals() method to do this.

  • If the new output is considerably more complicated than the old output, then it may not be a good idea to update the test, even if they are mathematically equivalent. Instead, you may need to adjust the change so that the function still returns the simpler result.

  • It’s not common, but it can happen that an existing test is itself incorrect. If a test is plain wrong, it should just be deleted, and updated.

In any case, when updating an existing test, you should always explain the rationale for doing so in a commit message or in a pull request comment. Do not explain the change in a code comment or documentation. Code comments and documentation should only refer to the code as it is. Discussion of changes belongs in the commit messages or issue tracker. Code comments that talk about how the code used to be will only become confusing and won’t actually be relevant anymore once the change is made.

Again, the default should be to not change existing tests. The tests exist for a reason, and changing them defeats the purpose of having them in the first place. The exception to this rule is doctests, which are allowed to change or be removed if they improve the documentation, as the primary purpose of doctests is to serve as examples for users.

Code Quality Checks

SymPy has several code quality checks that must pass. The first job that is run on the CI on a pull request is the code quality checks. If this job fails, none of the other tests are run. Your PR may be ignored by reviewers until they are fixed.

The code quality checks are all straightforward to fix. You can run the checks locally using

./bin/test quality


flake8 sympy

This second command requires you to install flake8. Make sure you have the latest version of flake8 and its dependencies pycodestyle and pyflakes installed. Sometimes newer versions of these packages will add new checks and if you have an older version installed you won’t see the checks for them.

The ./bin/test quality check tests for very basic code quality things. The most common of these that will cause the test to fail is trailing whitespace. Trailing whitespace is when a line of code has spaces at the end of it. These spaces do nothing, and they only cause the code diff to be polluted. The best way to handle trailing whitespace is to configure your text editor to automatically strip trailing whitespace when you save. You can also use the ./bin/strip_whitepace command in the SymPy repo.

The flake8 command will check the code for basic code errors like undefined variables. These are restricted by the configuration in setup.cfg to only check for things that are logical errors. The usual flake8 checks for cosmetic style errors are disabled. In rare situations, a flake8 warning will be a false positive. If this happens, add a # noqa: <CODE> comment to the corresponding line, where <CODE> is the code for the error from https://flake8.pycqa.org/en/latest/user/error-codes.html. For example, code that uses multipledispatch will need to use

def funcname(arg1, arg2): # noqa: F811

def funcname(arg1, arg2): # noqa: F811

to avoid warnings about redefining the same function multiple times.

Tests Style Guide

In most cases, tests should be written in a way that matches the surrounding tests in the same test file.

A few important stylistic points should be followed when writing tests:

  • Test functions should start with test_. If they do not, the test runner will not test them. Any helper functions which are not test functions should not start with test_. Usually it is best to start test helper functions with an underscore. If you find yourself reusing the same helper function for many test files, consider whether it should be moved to somewhere like sympy.testing.

  • Format expressions using the same whitespace that would be produced by str() (e.g., spaces around binary + and -, no spaces around * and **, space after comma, no redundant parentheses, etc.)

  • Avoid the use of Float values in test cases. Unless the test is explicitly testing the result of a function on floating-point inputs, test expressions should use exact values.

    In particular, avoid using integer division like 1/2 that will create a float value (see the gotchas section of the tutorial). For example:

    # BAD
    assert expand((x + 1/2)**2) == x**2 + x + 1/4
    # GOOD
    assert expand((x + S(1)/2)**2) == x**2 + x + S(1)/4

    If you do actually intend to explicitly test an expression with a floating-point value, use a float (like 0.5 instead of 1/2) so that it is clear this is intentional and not accidental.

  • Symbols may be defined at the top of the test file or within each test function. Symbols with assumptions that are defined at the top of the test file should be named in a way that makes it clear they have an assumption (e.g., xp = Symbol('x', positive=True)). It is often best to define symbols that have assumptions inside each test function so that they are not accidentally reused in another test that doesn’t expect them to have the assumption defined (which can often change the behavior of the test).

  • Test files are typically named corresponding to the code file they test (e.g., sympy/core/tests/test_symbol.py has the tests for sympy/core/symbol.py). However, this rule can be broken if there are tests that don’t exactly correspond to a specific code file.

  • Avoid using string forms of expressions in tests (obviously strings should be used in the printing tests; this rule applies to other types of tests). This makes the test depend on the exact printing output, rather than just the expression output. This makes the test harder to read, and if the printer is ever changed in some way, the test would have be updated.

    For example:

    # BAD
    assert str(expand((x + 2)**3)) == 'x**3 + 6*x**2 + 12*x + 8'
    # GOOD
    assert expand((x + 2)**3) == x**3 + 6*x**2 + 12*x + 8

    Similarly, do not parse the string form of an expression for input (unless the test is explicitly testing parsing strings). Just create the expression directly. Even if this requires creating many symbols or extensive use of S() to wrap rationals, this is still cleaner.

    # BAD
    expr = sympify('a*b*c*d*e')
    assert expr.count_ops() == 4
    # GOOD
    a, b, c, d, e = symbols('a b c d e')
    expr = a*b*c*d*e
    assert expr.count_ops() == 4
  • Use is True, is False and is None when testing assumptions. Don’t rely on truthiness, as it’s easy to forget that None is considered false by Python.

    # BAD
    assert not x.is_real
    # GOOD
    assert x.is_real is False

Test Coverage

To generate a test coverage report, first install coverage.py (e.g., with pip install coverage). Then run


This will run the test suite and analyze which lines of the codebase are covered by at least one test. Note that this will take longer than running the tests normally with ./bin/test because the coverage tooling makes Python run a little bit slower. You can also run a subset of the tests, e.g., ./bin/coverage_report.py sympy/solvers.

Once the tests are done, the coverage report will be in covhtml, which you can view by opening covhtml/index.html. Each file will show which lines were covered by a test (in green) and which were not covered by any test (in red).

Lines that are not covered by any test should have a test added for them, if possible. Note that 100% coverage is generally impossible. There may be a line of defensive code that checks if something has gone wrong, but which would only be triggered if there is a bug. Or there may be some functionality that is simply too hard to test (e.g., some code that interfaces with external dependencies), or that is only triggered when a given optional dependency is installed. However, if a line of code can be tested, it should be. And, for instance, the test files themselves should have 100% coverage. If a line in a test file is not covered, that generally indicates a mistake (see https://nedbatchelder.com/blog/202008/you_should_include_your_tests_in_coverage.html).

Also be aware that coverage is not the end of the story. While a line of code that is not tested has no guarantees of being correct, a line of code that is covered is not guaranteed to be correct either. Maybe it is only tested for general inputs, but not for corner cases. Sometimes code may have a conditional, like if a or b, and a is always true in every test, so that the b condition is never tested. And of course, just because a line of code is executed, doesn’t mean that is correct. The test needs to actually check that the output of the function is what it is supposed to be. Test coverage is just one part of ensuring the correctness of a codebase. See https://nedbatchelder.com/blog/200710/flaws_in_coverage_measurement.html.

Hypothesis Testing

Property based tests can now be created using the Hypothesis library. Tests should be added to the test_hypothesis.py file in the respective tests subdirectory. If the file does not exist, create one. Below is an example of hypothesis test for modular arithmetic:

from hypothesis import given
from hypothesis import strategies as st
from sympy import symbols
from sympy import Mod

@given(a = st.integers(), p = st.integers().filter(lambda p: p != 0), i = st.integers(),
j = st.integers().filter(lambda j: j != 0))
def test_modular(a, p, i, j):
    x, y = symbols('x y')
    value = Mod(x, y).subs({x: a, y: p})
    assert value == a % p