Lambdify

This module provides convenient functions to transform sympy expressions to lambda functions which can be used to calculate numerical values very fast.

sympy.utilities.lambdify.implemented_function(symfunc, implementation)[source]

Add numerical implementation to function symfunc.

symfunc can be an UndefinedFunction instance, or a name string. In the latter case we create an UndefinedFunction instance with that name.

Be aware that this is a quick workaround, not a general method to create special symbolic functions. If you want to create a symbolic function to be used by all the machinery of SymPy you should subclass the Function class.

Parameters

symfunc : str or UndefinedFunction instance

If str, then create new UndefinedFunction with this as name. If symfunc is an Undefined function, create a new function with the same name and the implemented function attached.

implementation : callable

numerical implementation to be called by evalf() or lambdify

Returns

afunc : sympy.FunctionClass instance

function with attached implementation

Examples

>>> from sympy.abc import x
>>> from sympy.utilities.lambdify import lambdify, implemented_function
>>> from sympy import Function
>>> f = implemented_function('f', lambda x: x+1)
>>> lam_f = lambdify(x, f(x))
>>> lam_f(4)
5
sympy.utilities.lambdify.lambdastr(args, expr, printer=None, dummify=None)[source]

Returns a string that can be evaluated to a lambda function.

Examples

>>> from sympy.abc import x, y, z
>>> from sympy.utilities.lambdify import lambdastr
>>> lambdastr(x, x**2)
'lambda x: (x**2)'
>>> lambdastr((x,y,z), [z,y,x])
'lambda x,y,z: ([z, y, x])'

Although tuples may not appear as arguments to lambda in Python 3, lambdastr will create a lambda function that will unpack the original arguments so that nested arguments can be handled:

>>> lambdastr((x, (y, z)), x + y)
'lambda _0,_1: (lambda x,y,z: (x + y))(_0,_1[0],_1[1])'
sympy.utilities.lambdify.lambdify(args, expr, modules=None, printer=None, use_imps=True, dummify=False)[source]

Convert a SymPy expression into a function that allows for fast numeric evaluation.

Warning

This function uses exec, and thus shouldn’t be used on unsanitized input.

Parameters

args : List[Symbol]

A variable or a list of variables whose nesting represents the nesting of the arguments that will be passed to the function.

Variables can be symbols, undefined functions, or matrix symbols.

>>> from sympy import Eq
>>> from sympy.abc import x, y, z

The list of variables should match the structure of how the arguments will be passed to the function. Simply enclose the parameters as they will be passed in a list.

To call a function like f(x) then [x] should be the first argument to lambdify; for this case a single x can also be used:

>>> f = lambdify(x, x + 1)
>>> f(1)
2
>>> f = lambdify([x], x + 1)
>>> f(1)
2

To call a function like f(x, y) then [x, y] will be the first argument of the lambdify:

>>> f = lambdify([x, y], x + y)
>>> f(1, 1)
2

To call a function with a single 3-element tuple like f((x, y, z)) then [(x, y, z)] will be the first argument of the lambdify:

>>> f = lambdify([(x, y, z)], Eq(z**2, x**2 + y**2))
>>> f((3, 4, 5))
True

If two args will be passed and the first is a scalar but the second is a tuple with two arguments then the items in the list should match that structure:

>>> f = lambdify([x, (y, z)], x + y + z)
>>> f(1, (2, 3))
6

expr : Expr

An expression, list of expressions, or matrix to be evaluated.

Lists may be nested. If the expression is a list, the output will also be a list.

>>> f = lambdify(x, [x, [x + 1, x + 2]])
>>> f(1)
[1, [2, 3]]

If it is a matrix, an array will be returned (for the NumPy module).

>>> from sympy import Matrix
>>> f = lambdify(x, Matrix([x, x + 1]))
>>> f(1)
[[1]
[2]]

Note that the argument order here (variables then expression) is used to emulate the Python lambda keyword. lambdify(x, expr) works (roughly) like lambda x: expr (see How It Works below).

modules : str, optional

Specifies the numeric library to use.

If not specified, modules defaults to:

  • ["scipy", "numpy"] if SciPy is installed

  • ["numpy"] if only NumPy is installed

  • ["math", "mpmath", "sympy"] if neither is installed.

That is, SymPy functions are replaced as far as possible by either scipy or numpy functions if available, and Python’s standard library math, or mpmath functions otherwise.

modules can be one of the following types:

  • The strings "math", "mpmath", "numpy", "numexpr", "scipy", "sympy", or "tensorflow". This uses the corresponding printer and namespace mapping for that module.

  • A module (e.g., math). This uses the global namespace of the module. If the module is one of the above known modules, it will also use the corresponding printer and namespace mapping (i.e., modules=numpy is equivalent to modules="numpy").

  • A dictionary that maps names of SymPy functions to arbitrary functions (e.g., {'sin': custom_sin}).

  • A list that contains a mix of the arguments above, with higher priority given to entries appearing first (e.g., to use the NumPy module but override the sin function with a custom version, you can use [{'sin': custom_sin}, 'numpy']).

dummify : bool, optional

Whether or not the variables in the provided expression that are not valid Python identifiers are substituted with dummy symbols.

This allows for undefined functions like Function('f')(t) to be supplied as arguments. By default, the variables are only dummified if they are not valid Python identifiers.

Set dummify=True to replace all arguments with dummy symbols (if args is not a string) - for example, to ensure that the arguments do not redefine any built-in names.

Explanation

For example, to convert the SymPy expression sin(x) + cos(x) to an equivalent NumPy function that numerically evaluates it:

>>> from sympy import sin, cos, symbols, lambdify
>>> import numpy as np
>>> x = symbols('x')
>>> expr = sin(x) + cos(x)
>>> expr
sin(x) + cos(x)
>>> f = lambdify(x, expr, 'numpy')
>>> a = np.array([1, 2])
>>> f(a)
[1.38177329 0.49315059]

The primary purpose of this function is to provide a bridge from SymPy expressions to numerical libraries such as NumPy, SciPy, NumExpr, mpmath, and tensorflow. In general, SymPy functions do not work with objects from other libraries, such as NumPy arrays, and functions from numeric libraries like NumPy or mpmath do not work on SymPy expressions. lambdify bridges the two by converting a SymPy expression to an equivalent numeric function.

The basic workflow with lambdify is to first create a SymPy expression representing whatever mathematical function you wish to evaluate. This should be done using only SymPy functions and expressions. Then, use lambdify to convert this to an equivalent function for numerical evaluation. For instance, above we created expr using the SymPy symbol x and SymPy functions sin and cos, then converted it to an equivalent NumPy function f, and called it on a NumPy array a.

Examples

>>> from sympy.utilities.lambdify import implemented_function
>>> from sympy import sqrt, sin, Matrix
>>> from sympy import Function
>>> from sympy.abc import w, x, y, z
>>> f = lambdify(x, x**2)
>>> f(2)
4
>>> f = lambdify((x, y, z), [z, y, x])
>>> f(1,2,3)
[3, 2, 1]
>>> f = lambdify(x, sqrt(x))
>>> f(4)
2.0
>>> f = lambdify((x, y), sin(x*y)**2)
>>> f(0, 5)
0.0
>>> row = lambdify((x, y), Matrix((x, x + y)).T, modules='sympy')
>>> row(1, 2)
Matrix([[1, 3]])

lambdify can be used to translate SymPy expressions into mpmath functions. This may be preferable to using evalf (which uses mpmath on the backend) in some cases.

>>> import mpmath
>>> f = lambdify(x, sin(x), 'mpmath')
>>> f(1)
0.8414709848078965

Tuple arguments are handled and the lambdified function should be called with the same type of arguments as were used to create the function:

>>> f = lambdify((x, (y, z)), x + y)
>>> f(1, (2, 4))
3

The flatten function can be used to always work with flattened arguments:

>>> from sympy.utilities.iterables import flatten
>>> args = w, (x, (y, z))
>>> vals = 1, (2, (3, 4))
>>> f = lambdify(flatten(args), w + x + y + z)
>>> f(*flatten(vals))
10

Functions present in expr can also carry their own numerical implementations, in a callable attached to the _imp_ attribute. This can be used with undefined functions using the implemented_function factory:

>>> f = implemented_function(Function('f'), lambda x: x+1)
>>> func = lambdify(x, f(x))
>>> func(4)
5

lambdify always prefers _imp_ implementations to implementations in other namespaces, unless the use_imps input parameter is False.

Usage with Tensorflow:

>>> import tensorflow as tf
>>> from sympy import Max, sin, lambdify
>>> from sympy.abc import x
>>> f = Max(x, sin(x))
>>> func = lambdify(x, f, 'tensorflow')

After tensorflow v2, eager execution is enabled by default. If you want to get the compatible result across tensorflow v1 and v2 as same as this tutorial, run this line.

>>> tf.compat.v1.enable_eager_execution()

If you have eager execution enabled, you can get the result out immediately as you can use numpy.

If you pass tensorflow objects, you may get an EagerTensor object instead of value.

>>> result = func(tf.constant(1.0))
>>> print(result)
tf.Tensor(1.0, shape=(), dtype=float32)
>>> print(result.__class__)
<class 'tensorflow.python.framework.ops.EagerTensor'>

You can use .numpy() to get the numpy value of the tensor.

>>> result.numpy()
1.0
>>> var = tf.Variable(2.0)
>>> result = func(var) # also works for tf.Variable and tf.Placeholder
>>> result.numpy()
2.0

And it works with any shape array.

>>> tensor = tf.constant([[1.0, 2.0], [3.0, 4.0]])
>>> result = func(tensor)
>>> result.numpy()
[[1. 2.]
 [3. 4.]]

Notes

  • For functions involving large array calculations, numexpr can provide a significant speedup over numpy. Please note that the available functions for numexpr are more limited than numpy but can be expanded with implemented_function and user defined subclasses of Function. If specified, numexpr may be the only option in modules. The official list of numexpr functions can be found at: https://numexpr.readthedocs.io/en/latest/user_guide.html#supported-functions

  • In previous versions of SymPy, lambdify replaced Matrix with numpy.matrix by default. As of SymPy 1.0 numpy.array is the default. To get the old default behavior you must pass in [{'ImmutableDenseMatrix':  numpy.matrix}, 'numpy'] to the modules kwarg.

    >>> from sympy import lambdify, Matrix
    >>> from sympy.abc import x, y
    >>> import numpy
    >>> array2mat = [{'ImmutableDenseMatrix': numpy.matrix}, 'numpy']
    >>> f = lambdify((x, y), Matrix([x, y]), modules=array2mat)
    >>> f(1, 2)
    [[1]
     [2]]
    
  • In the above examples, the generated functions can accept scalar values or numpy arrays as arguments. However, in some cases the generated function relies on the input being a numpy array:

    >>> from sympy import Piecewise
    >>> from sympy.testing.pytest import ignore_warnings
    >>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "numpy")
    
    >>> with ignore_warnings(RuntimeWarning):
    ...     f(numpy.array([-1, 0, 1, 2]))
    [-1.   0.   1.   0.5]
    
    >>> f(0)
    Traceback (most recent call last):
        ...
    ZeroDivisionError: division by zero
    

    In such cases, the input should be wrapped in a numpy array:

    >>> with ignore_warnings(RuntimeWarning):
    ...     float(f(numpy.array([0])))
    0.0
    

    Or if numpy functionality is not required another module can be used:

    >>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "math")
    >>> f(0)
    0
    

How It Works

When using this function, it helps a great deal to have an idea of what it is doing. At its core, lambdify is nothing more than a namespace translation, on top of a special printer that makes some corner cases work properly.

To understand lambdify, first we must properly understand how Python namespaces work. Say we had two files. One called sin_cos_sympy.py, with

# sin_cos_sympy.py

from sympy import sin, cos

def sin_cos(x):
    return sin(x) + cos(x)

and one called sin_cos_numpy.py with

# sin_cos_numpy.py

from numpy import sin, cos

def sin_cos(x):
    return sin(x) + cos(x)

The two files define an identical function sin_cos. However, in the first file, sin and cos are defined as the SymPy sin and cos. In the second, they are defined as the NumPy versions.

If we were to import the first file and use the sin_cos function, we would get something like

>>> from sin_cos_sympy import sin_cos 
>>> sin_cos(1) 
cos(1) + sin(1)

On the other hand, if we imported sin_cos from the second file, we would get

>>> from sin_cos_numpy import sin_cos 
>>> sin_cos(1) 
1.38177329068

In the first case we got a symbolic output, because it used the symbolic sin and cos functions from SymPy. In the second, we got a numeric result, because sin_cos used the numeric sin and cos functions from NumPy. But notice that the versions of sin and cos that were used was not inherent to the sin_cos function definition. Both sin_cos definitions are exactly the same. Rather, it was based on the names defined at the module where the sin_cos function was defined.

The key point here is that when function in Python references a name that is not defined in the function, that name is looked up in the “global” namespace of the module where that function is defined.

Now, in Python, we can emulate this behavior without actually writing a file to disk using the exec function. exec takes a string containing a block of Python code, and a dictionary that should contain the global variables of the module. It then executes the code “in” that dictionary, as if it were the module globals. The following is equivalent to the sin_cos defined in sin_cos_sympy.py:

>>> import sympy
>>> module_dictionary = {'sin': sympy.sin, 'cos': sympy.cos}
>>> exec('''
... def sin_cos(x):
...     return sin(x) + cos(x)
... ''', module_dictionary)
>>> sin_cos = module_dictionary['sin_cos']
>>> sin_cos(1)
cos(1) + sin(1)

and similarly with sin_cos_numpy:

>>> import numpy
>>> module_dictionary = {'sin': numpy.sin, 'cos': numpy.cos}
>>> exec('''
... def sin_cos(x):
...     return sin(x) + cos(x)
... ''', module_dictionary)
>>> sin_cos = module_dictionary['sin_cos']
>>> sin_cos(1)
1.38177329068

So now we can get an idea of how lambdify works. The name “lambdify” comes from the fact that we can think of something like lambdify(x, sin(x) + cos(x), 'numpy') as lambda x: sin(x) + cos(x), where sin and cos come from the numpy namespace. This is also why the symbols argument is first in lambdify, as opposed to most SymPy functions where it comes after the expression: to better mimic the lambda keyword.

lambdify takes the input expression (like sin(x) + cos(x)) and

  1. Converts it to a string

  2. Creates a module globals dictionary based on the modules that are passed in (by default, it uses the NumPy module)

  3. Creates the string "def func({vars}): return {expr}", where {vars} is the list of variables separated by commas, and {expr} is the string created in step 1., then exec``s that string with the module globals namespace and returns ``func.

In fact, functions returned by lambdify support inspection. So you can see exactly how they are defined by using inspect.getsource, or ?? if you are using IPython or the Jupyter notebook.

>>> f = lambdify(x, sin(x) + cos(x))
>>> import inspect
>>> print(inspect.getsource(f))
def _lambdifygenerated(x):
    return (sin(x) + cos(x))

This shows us the source code of the function, but not the namespace it was defined in. We can inspect that by looking at the __globals__ attribute of f:

>>> f.__globals__['sin']
<ufunc 'sin'>
>>> f.__globals__['cos']
<ufunc 'cos'>
>>> f.__globals__['sin'] is numpy.sin
True

This shows us that sin and cos in the namespace of f will be numpy.sin and numpy.cos.

Note that there are some convenience layers in each of these steps, but at the core, this is how lambdify works. Step 1 is done using the LambdaPrinter printers defined in the printing module (see sympy.printing.lambdarepr). This allows different SymPy expressions to define how they should be converted to a string for different modules. You can change which printer lambdify uses by passing a custom printer in to the printer argument.

Step 2 is augmented by certain translations. There are default translations for each module, but you can provide your own by passing a list to the modules argument. For instance,

>>> def mysin(x):
...     print('taking the sin of', x)
...     return numpy.sin(x)
...
>>> f = lambdify(x, sin(x), [{'sin': mysin}, 'numpy'])
>>> f(1)
taking the sin of 1
0.8414709848078965

The globals dictionary is generated from the list by merging the dictionary {'sin': mysin} and the module dictionary for NumPy. The merging is done so that earlier items take precedence, which is why mysin is used above instead of numpy.sin.

If you want to modify the way lambdify works for a given function, it is usually easiest to do so by modifying the globals dictionary as such. In more complicated cases, it may be necessary to create and pass in a custom printer.

Finally, step 3 is augmented with certain convenience operations, such as the addition of a docstring.

Understanding how lambdify works can make it easier to avoid certain gotchas when using it. For instance, a common mistake is to create a lambdified function for one module (say, NumPy), and pass it objects from another (say, a SymPy expression).

For instance, say we create

>>> from sympy.abc import x
>>> f = lambdify(x, x + 1, 'numpy')

Now if we pass in a NumPy array, we get that array plus 1

>>> import numpy
>>> a = numpy.array([1, 2])
>>> f(a)
[2 3]

But what happens if you make the mistake of passing in a SymPy expression instead of a NumPy array:

>>> f(x + 1)
x + 2

This worked, but it was only by accident. Now take a different lambdified function:

>>> from sympy import sin
>>> g = lambdify(x, x + sin(x), 'numpy')

This works as expected on NumPy arrays:

>>> g(a)
[1.84147098 2.90929743]

But if we try to pass in a SymPy expression, it fails

>>> try:
...     g(x + 1)
... # NumPy release after 1.17 raises TypeError instead of
... # AttributeError
... except (AttributeError, TypeError):
...     raise AttributeError() 
Traceback (most recent call last):
...
AttributeError:

Now, let’s look at what happened. The reason this fails is that g calls numpy.sin on the input expression, and numpy.sin does not know how to operate on a SymPy object. As a general rule, NumPy functions do not know how to operate on SymPy expressions, and SymPy functions do not know how to operate on NumPy arrays. This is why lambdify exists: to provide a bridge between SymPy and NumPy.

However, why is it that f did work? That’s because f doesn’t call any functions, it only adds 1. So the resulting function that is created, def _lambdifygenerated(x): return x + 1 does not depend on the globals namespace it is defined in. Thus it works, but only by accident. A future version of lambdify may remove this behavior.

Be aware that certain implementation details described here may change in future versions of SymPy. The API of passing in custom modules and printers will not change, but the details of how a lambda function is created may change. However, the basic idea will remain the same, and understanding it will be helpful to understanding the behavior of lambdify.

In general: you should create lambdified functions for one module (say, NumPy), and only pass it input types that are compatible with that module (say, NumPy arrays). Remember that by default, if the module argument is not provided, lambdify creates functions using the NumPy and SciPy namespaces.