# Writing Custom Functions#

This guide will describe how to create custom function classes in SymPy. Custom user defined functions use the same mechanisms as the functions that are included with SymPy such as the common elementary functions like exp() or sin(), special functions like gamma() or Si(), and combinatorial functions and number theory functions like factorial() or primepi(). Consequently, this guide serves both as a guide to end users who want to define their own custom functions and to SymPy developers wishing to extend the functions included with SymPy.

This guide describes how to define complex valued functions, that is functions that map a subset of $$\mathbb{C}^n$$ to $$\mathbb{C}$$. Functions that accept or return other kinds of objects than complex numbers should subclass another class, such as Boolean, MatrixExpr, Expr, or Basic. Some of what is written here will apply to general Basic or Expr subclasses, but much of it only applies to Function subclasses.

## Easy Cases: Fully Symbolic or Fully Evaluated#

Before digging into the more advanced functionality for custom functions, we should mention two common cases, the case where the function is fully symbolic, and the case where the function is fully evaluated. Both of these cases have much simpler alternatives than the full mechanisms described in ths guide.

### The Fully Symbolic Case#

If your function f has no mathematical properties you want to define on it, and should never evaluate on any arguments, you can create an undefined function using Function('f')

>>> from sympy import symbols, Function
>>> x = symbols('x')
>>> f = Function('f')

>>> f(x)
f(x)
>>> f(0)
f(0)


This is useful, for instance, when solving ODEs.

This is also useful if you only wish to create a symbol that depends on another symbol for the purposes of differentiation. By default, SymPy assumes all symbols are independent of one another:

>>> from sympy.abc import x, y
>>> y.diff(x)
0


To make a symbol that depends on another symbol, you can use a function that explicitly depends on that symbol.

>>> y = Function('y')
>>> y(x).diff(x)
Derivative(y(x), x)


If you want your function to have additional behavior, for example, to have a custom derivative, or to evaluate on certain arguments, you should create a custom Function subclass as described below. However, undefined functions do support one additional feature, which is that assumptions can be defined on them, using the same syntax as used by symbols. This defines the assumptions of the output of the function, not the input (that is, it defines the function’s range, not its domain).

>>> g = Function('g', real=True)

>>> g(x)
g(x)
>>> g(x).is_real
True


To make a function’s assumptions depend on its input in some way, you should create a custom Function subclass and define assumptions handlers as described below.

### The Fully Evaluated Case#

At the other end of the spectrum are functions that always evaluate to something no matter what their inputs are. These functions are never left in an unevaluated, symbolic form like f(x).

In this case, you should use a normal Python function using the def keyword:

>>> def f(x):
...     if x == 0:
...         return 0
...     else:
...         return x + 1

>>> f(0)
0
>>> f(1)
2
>>> f(x)
x + 1


If you find yourself defining an eval() method on a Function subclass where you always return a value and never return None, you should consider just using a normal Python function instead, as there is no benefit to using a symbolic Function subclass in that case (see the Best Practices for eval() section below)

Note that in many cases, functions like these can be represented directly using SymPy classes. For example, the above function can be represented symbolically using Piecewise. The Piecewise expression can be evaluated for specific values of x using subs().

>>> from sympy import Piecewise, Eq, pprint
>>> f = Piecewise((0, Eq(x, 0)), (x + 1, True))

>>> pprint(f, use_unicode=True)
⎧  0    for x = 0
⎨
⎩x + 1  otherwise
>>> f.subs(x, 0)
0
>>> f.subs(x, 1)
2


Fully symbolic representations like Piecewise have the advantage that they accurately represent symbolic values. For example, in the above Python def definition of f, f(x) implicitly assumes that x is nonzero. The Piecewise version handles this case correctly and won’t evaluate to the $$x \neq 0$$ case unless x is known to not be zero.

Another option, if you want a function that not only evaluates, but always evaluates to a numerical value, is to use lambdify(). This will convert a SymPy expression into a function that can be evaluated using NumPy.

>>> from sympy import lambdify
>>> func = lambdify(x, Piecewise((0, Eq(x, 0)), (x + 1, True)))
>>> import numpy as np
>>> func(np.arange(5))
array([0., 2., 3., 4., 5.])


Ultimately, the correct tool for the job depends on what you are doing and what exact behavior you want.

## Creating a Custom Function#

The first step to creating a custom function is to subclass Function. The name of the subclass will be the name of the function. Different methods should then be defined on this subclass, depending on what functionality you want to provide.

As a motivating example for this document, let’s create a custom function class representing the versine function. Versine is a trigonometric function which was used historically alongside some of the more familiar trigonometric functions like sine and cosine. It is rarely used today. Versine can be defined by the identity

$\operatorname{versin}(x) = 1 - \cos(x).$

SymPy does not already include versine because it is used so rarely in modern mathematics and because it is so easily defined in terms of the more familiar cosine.

Let us start by subclassing Function.

>>> class versin(Function):
...     pass


At this point, versin has no behaviors defined on it. It is very similar to the undefined functions we discussed above. Note that versin is a class, and versin(x) is an instance of this class.

>>> versin(x)
versin(x)
>>> isinstance(versin(x), versin)
True


Note

All the methods described below are optional. They can be included if you want to define the given behavior, but if they are omitted, SymPy will default to leaving things unevaluated. For example, if you do not define differentiation, diff() will just return an unevaluated Derivative.

### Defining Automatic Evaluation with eval()#

The first and most common thing we might want to define on our custom function is automatic evaluation, that is, the cases where it will return an actual value instead of just remaining unevaluated as-is.

This is done by defining the class method eval(). eval() should take the arguments of the function and return either a value or None. If it returns None, the function will remain unevaluated in that case. This also serves to define the signature of the function (by default, without an eval() method, a Function subclass will accept any number of arguments).

For our function versin, we might recall that $$\cos(n\pi) = (-1)^n$$ for integer $$n$$, so $$\operatorname{versin}(n\pi) = 1 - (-1)^n.$$ We can make versin automatically evaluate to this value when passed an integer multiple of pi:

>>> from sympy import pi, Integer
>>> class versin(Function):
...    @classmethod
...    def eval(cls, x):
...        # If x is an integer multiple of pi, x/pi will cancel and be an Integer
...        n = x/pi
...        if isinstance(n, Integer):
...            return 1 - (-1)**n

>>> versin(pi)
2
>>> versin(2*pi)
0


Here we make use of the fact that if a Python function does not explicitly return a value, it automatically returns None. So in the cases where the if isinstance(n, Integer) statement is not triggered, eval() returns None and versin remains unevaluated.

>>> versin(x*pi)
versin(pi*x)


Note

Function subclasses should not redefine __new__ or __init__. If you want to implement behavior that isn’t possible with eval(), it might make more sense to subclass Expr rather than Function.

eval() can take any number of arguments, including an arbitrary number with *args and optional keyword arguments. The .args of the function will always be the arguments that were passed in by the user. For example

>>> class f(Function):
...     @classmethod
...     def eval(cls, x, y=1, *args):
...         return None

>>> f(1).args
(1,)
>>> f(1, 2).args
(1, 2)
>>> f(1, 2, 3).args
(1, 2, 3)


Finally, note that automatic evaluation on floating-point inputs happens automatically once evalf() is defined, so you do not need to handle it explicitly in eval().

#### Best Practices for eval()#

Certain antipatterns are common when defining eval() methods and should be avoided.

• Don’t just return an expression.

In the above example, we might have been tempted to write

>>> from sympy import cos
>>> class versin(Function):
...     @classmethod
...     def eval(cls, x):
...         # !! Not actually a good eval() method !!
...         return 1 - cos(x)


However, this would make it so that versin(x) would always return 1 - cos(x), regardless of what x is. If all you want is a quick shorthand to 1 - cos(x), that is fine, but would be much simpler and more explicit to just use a Python function as described above. If we defined versin like this, it would never actually be represented as versin(x), and none of the other behavior we define below would matter, because the other behaviors we are going to define on the versin class only apply when the returned object is actually a versin instance. So for example, versin(x).diff(x) would actually just be (1 - cos(x)).diff(x), instead of calling the fdiff() method we define below.

Key Point

The purpose of eval() is not to define what the function is, mathematically, but rather to specify on what inputs it should automatically evaluate. The mathematical definition of a function is determined through the specification of various mathematical properties with the methods outlined below, like numerical evaluation, differentiation, and so on.

If you find yourself doing this, you should think about what you actually want to achieve. If you just want a shorthand function for an expression, it will be simpler to just define a Python function. If you really do want a symbolic function, think about when you want it to evaluate to something else and when you want it to stay unevaluated. One option is to make your function unevaluated in eval() and define a doit() method to evaluate it.

• Avoid too much automatic evaluation.

It is recommended to minimize what is evaluated automatically by eval(). It is typically better to put more advanced simplifications in other methods, like doit(). Remember that whatever you define for automatic evaluation will always evaluate.[1] As in the previous point, if you evaluate every value, there is little point to even having a symbolic function in the first place. For example, we might be tempted to evaluate some trig identities on versin in eval(), but then these identities would always evaluate, and it wouldn’t be possible to represent one half of the identity.

One should also avoid doing anything in eval() that is slow to compute. SymPy generally assumes that it is cheap to create expressions, and if this is not true, it can lead to performance issues.

Finally, it is recommended to avoid performing automatic evaluation in eval() based on assumptions. Instead, eval() should typically only evaluate explicit numerical special values and return None for everything else. You might have noticed in the example above that we used isinstance(n, Integer) instead of checking n.is_integer using the assumptions system. We could have done that instead, which would make versin(n*pi) evaluate even if n = Symbol('n', integer=True). But this is a case where we might not always want evaluation to happen, and if n is a more complicated expression, n.is_integer might be more expensive to compute.

Let’s consider an example. Using the identity $$\cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y)$$, we can derive the identity

$\operatorname{versin}(x + y) = \operatorname{versin}(x)\operatorname{versin}(y) - \operatorname{versin}(x) - \operatorname{versin}(y) - \sin(x)\sin(y) + 1.$

Suppose we decided to automatically expand this in eval():

>>> from sympy import Add, sin
>>> class versin(Function):
...     @classmethod
...     def eval(cls, x):
...         # !! Not actually a good eval() method !!
...             a, b = x.as_two_terms()
...             return (versin(a)*versin(b) - versin(a) - versin(b)
...                     - sin(a)*sin(b) + 1)


This method recursively splits Add terms into two parts and applies the above identity.

>>> x, y, z = symbols('x y z')
>>> versin(x + y)
-sin(x)*sin(y) + versin(x)*versin(y) - versin(x) - versin(y) + 1


But now it’s impossible to represent versin(x + y) without it expanding. This will affect other methods too. For example, suppose we define differentiation (see below):

>>> class versin(Function):
...     @classmethod
...     def eval(cls, x):
...         # !! Not actually a good eval() method !!
...             a, b = x.as_two_terms()
...             return (versin(a)*versin(b) - versin(a) - versin(b)
...                     - sin(a)*sin(b) + 1)
...
...     def fdiff(self, argindex=1):
...         return sin(self.args[0])


We would expect versin(x + y).diff(x) to return sin(x + y), and indeed, if we hadn’t expanded this identity in eval(), it would. But with this version, versin(x + y) gets automatically expanded before diff() gets called, instead we get a more complicated expression:

>>> versin(x + y).diff(x)
sin(x)*versin(y) - sin(x) - sin(y)*cos(x)


And things are even worse than that. Let’s try an Add with three terms:

>>> versin(x + y + z)
(-sin(y)*sin(z) + versin(y)*versin(z) - versin(y) - versin(z) +
1)*versin(x) - sin(x)*sin(y + z) + sin(y)*sin(z) - versin(x) -
versin(y)*versin(z) + versin(y) + versin(z)


We can see that things are getting out of control quite quickly. In fact, versin(Add(*symbols('x:100'))) (versin() on an Add with 100 terms) takes over a second to evaluate, and that’s just to create the expression, without even doing anything with it yet.

Identities like this are better left out of eval and implemented in other methods instead (in the case of this identity, expand_trig()).

• When restricting the input domain: allow None input assumptions.

Our example function $$\operatorname{versin}(x)$$ is a function from $$\mathbb{C}$$ to $$\mathbb{C}$$, so it can accept any input. But suppose we had a function that only made sense with certain inputs. As a second example, let’s define a function divides as

$\begin{split}\operatorname{divides}(m, n) = \begin{cases} 1 & \text{for}\: m \mid n \\ 0 & \text{for}\: m\not\mid n \end{cases}.\end{split}$

That is, divides(m, n) will be 1 if m divides n and 0 otherwise. divides clearly only makes sense if m and n are integers.

We might be tempted to define the eval() method for divides like this:

>>> class divides(Function):
...     @classmethod
...     def eval(cls, m, n):
...         # !! Not actually a good eval() method !!
...
...         # Evaluate for explicit integer m and n. This part is fine.
...         if isinstance(m, Integer) and isinstance(n, Integer):
...             return int(n % m == 0)
...
...         # For symbolic arguments, require m and n to be integer.
...         # If we write the logic this way, we will run into trouble.
...         if not m.is_integer or not n.is_integer:
...             raise TypeError("m and n should be integers")


The problem here is that by using if not m.is_integer, we are requiring m.is_integer to be True. If it is None, it will fail (see the guide on booleans and three-valued logic for details on what it means for an assumption to be None). This is problematic for two reasons. Firstly, it forces the user to define assumptions on any input variable. If the user omits them, it will fail:

>>> n, m = symbols('n m')
>>> print(n.is_integer)
None
>>> divides(m, n)
Traceback (most recent call last):
...
TypeError: m and n should be integers


>>> n, m = symbols('n m', integer=True)
>>> divides(m, n)
divides(m, n)


This may seem like an acceptable restriction, but there is a bigger problem. Sometimes, SymPy’s assumptions system cannot deduce an assumption, even though it is mathematically true. In this case, it will give None (None means both “undefined” and “cannot compute” in SymPy’s assumptions). For example

>>> # n and m are still defined as integer=True as above
>>> divides(2, (m**2 + m)/2)
Traceback (most recent call last):
...
TypeError: m and n should be integers


Here the expression (m**2 + m)/2 is always an integer, but SymPy’s assumptions system is not able to deduce this:

>>> print(((m**2 + m)/2).is_integer)
None


SymPy’s assumptions system is always improving, but there will always be cases like this that it cannot deduce, due to the fundamental computational complexity of the problem, and the fact that the general problem is often undecidable.

Consequently, one should always test negated assumptions for input variables, that is, fail if the assumption is False but allow the assumption to be None.

>>> class divides(Function):
...     @classmethod
...     def eval(cls, m, n):
...         # Evaluate for explicit integer m and n. This part is fine.
...         if isinstance(m, Integer) and isinstance(n, Integer):
...             return int(n % m == 0)
...
...         # For symbolic arguments, require m and n to be integer.
...         # This is the better way to write this logic.
...         if m.is_integer is False or n.is_integer is False:
...             raise TypeError("m and n should be integers")


This still disallows non-integer inputs as desired:

>>> divides(1.5, 1)
Traceback (most recent call last):
...
TypeError: m and n should be integers


But it does not fail in cases where the assumption is None:

>>> divides(2, (m**2 + m)/2)
divides(2, m**2/2 + m/2)
>>> _.subs(m, 2)
0
>>> n, m = symbols('n m') # Redefine n and m without the integer assumption
>>> divides(m, n)
divides(m, n)


Note

This rule of allowing None assumptions only applies to instances where an exception would be raised, such as type checking an input domain. In cases where simplifications or other operations are done, one should treat a None assumption as meaning “can be either True or False” and not perform an operation that might not be mathematically valid.

### Assumptions#

The next thing you might want to define are the assumptions on our function. The assumptions system allows defining what mathematical properties your function has given its inputs, for example, “$$f(x)$$ is positive when $$x$$ is real.”

The guide on the assumptions system goes into the assumptions system in great detail. It is recommended to read through that guide first to understand what the different assumptions mean and how the assumptions system works.

The simplest case is a function that always has a given assumption regardless of its input. In this case, you can define is_assumption directly on the class.

For example, our example divides function is always an integer, because its value is always either 0 or 1:

>>> class divides(Function):
...     is_integer = True
...     is_negative = False

>>> divides(m, n).is_integer
True
>>> divides(m, n).is_nonnegative
True


In general, however, the assumptions of a function depend on the assumptions of its inputs. In this case, you should define an _eval_assumption method.

For our $$\operatorname{versin}(x)$$ example, the function is always in $$[0, 2]$$ when $$x$$ is real, and it is 0 exactly when $$x$$ is an even multiple of $$\pi$$. So versin(x) should be nonnegative whenever x is real and positive whenever x is real and not an even multiple of π. Remember that by default, a function’s domain is all of $$\mathbb{C}$$, and indeed versin(x) makes perfect sense with non-real x.

To see if x is an even multiple of pi, we can use as_independent() to match x structurally as coeff*pi. Pulling apart subexpressions structurally like this in assumptions handlers is preferable to using something like (x/pi).is_even, because that will create a new expression x/pi. The creation of a new expression is much slower. Furthermore, whenever an expression is created, the constructors that are called when creating the expression will often themselves cause assumptions to be queried. If you are not careful, this can lead to infinite recursion. So a good general rule for assumptions handlers is, never create a new expression in an assumptions handler. Always pull apart the args of the function using structural methods like as_independent.

Note that $$\operatorname{versin}(x)$$ can be nonnegative for nonreal $$x$$, for example:

>>> from sympy import I
>>> 1 - cos(pi + I*pi)
1 + cosh(pi)
>>> (1 - cos(pi + I*pi)).evalf()
12.5919532755215


So for the _eval_is_nonnegative handler, we want to return True if x.is_real is True but None if x.is_real is either False or None. It is left as an exercise to the reader to handle the cases for nonreal x that make versin(x) nonnegative, using similar logic from the _eval_is_positive handler.

In the assumptions handler methods, as in all methods, we can access the arguments of the function using self.args.

>>> from sympy.core.logic import fuzzy_and, fuzzy_not
>>> class versin(Function):
...     def _eval_is_nonnegative(self):
...         # versin(x) is nonnegative if x is real
...         x = self.args[0]
...         if x.is_real is True:
...             return True
...
...     def _eval_is_positive(self):
...         # versin(x) is positive iff x is real and not an even multiple of pi
...         x = self.args[0]
...
...         # x.as_independent(pi, as_Add=False) will split x as a Mul of the
...         # form coeff*pi
...         coeff, pi_ = x.as_independent(pi, as_Add=False)
...         # If pi_ = pi, x = coeff*pi. Otherwise x is not (structurally) of
...         # the form coeff*pi.
...         if pi_ == pi:
...             return fuzzy_and([x.is_real, fuzzy_not(coeff.is_even)])
...         elif x.is_real is False:
...             return False
...         # else: return None. We do not know for sure whether x is an even
...         # multiple of pi

>>> versin(1).is_nonnegative
True
>>> versin(2*pi).is_positive
False
>>> versin(3*pi).is_positive
True


Note the use of fuzzy_ functions in the more complicated _eval_is_positive() handler, and the careful handling of the if/elif. It is important when working with assumptions to always be careful about handling three-valued logic correctly. This ensures that the method returns the correct answer when x.is_real or coeff.is_even are None.

Warning

Never define is_assumption as a @property method. Doing so will break the automatic deduction of other assumptions. is_assumption should only ever be defined as a class variable equal to True or False. If the assumption depends on the .args of the function somehow, define the _eval_assumption method.

In this example, it is not necessary to define _eval_is_real() because it is deduced automatically from the other assumptions, since nonnegative -> real. In general, you should avoid defining assumptions that the assumptions system can deduce automatically given its known facts.

>>> versin(1).is_real
True


The assumptions system is often able to deduce more than you might think. For example, from the above, it can deduce that versin(2*n*pi) is zero when n is an integer.

>>> n = symbols('n', integer=True)
>>> versin(2*n*pi).is_zero
True


It’s always worth checking if the assumptions system can deduce something automatically before manually coding it.

Finally, a word of warning: be very careful about correctness when coding assumptions. Make sure to use the exact definitions of the various assumptions, and always check that you’re handling None cases correctly with the fuzzy three-valued logic functions. Incorrect or inconsistent assumptions can lead to subtle bugs. It’s recommended to use unit tests to check all the various cases whenever your function has a nontrivial assumption handler. All functions defined in SymPy itself are required to be extensively tested.

### Numerical Evaluation with evalf()#

Here we show how to define how a function should numerically evaluate to a floating point Float value, for instance, via evalf(). Implementing numerical evaluation enables several behaviors in SymPy. For example, once evalf() is defined, you can plot your function, and things like inequalities can evaluate to explicit values.

If your function has the same name as a function in mpmath, which is the case for most functions included with SymPy, numerical evaluation will happen automatically and you do not need to do anything.

If this is not the case, numerical evaluation can be specified by defining the method _eval_evalf(self, prec), where prec is the binary precision of the input. The method should return the expression evaluated to the given precision, or None if this is not possible.

Note

The prec argument to _eval_evalf() is the binary precision, that is, the number of bits in the floating-point representation. This differs from the first argument to the evalf() method, which is the decimal precision, or dps. For example, the default binary precision of Float is 53, corresponding to a decimal precision of 15. Therefore, if your _eval_evalf() method recursively calls evalf on another expression, it should call expr._eval_evalf(prec) rather than expr.evalf(prec), as the latter will incorrectly use prec as the decimal precision.

We can define numerical evaluation for our example $$\operatorname{versin}(x)$$ function by recursively evaluating $$2\sin^2\left(\frac{x}{2}\right)$$, which is a more numerically stable way of writing $$1 - \cos(x)$$.

>>> from sympy import sin
>>> class versin(Function):
...     def _eval_evalf(self, prec):
...         return (2*sin(self.args[0]/2)**2)._eval_evalf(prec)

>>> versin(1).evalf()
0.459697694131860


Once _eval_evalf() is defined, this enables the automatic evaluation of floating-point inputs. It is not required to implement this manually in eval().

>>> versin(1.)
0.459697694131860


Note that evalf() may be passed any expression, not just one that can be evaluated numerically. In this case, it is expected that the numerical parts of an expression will be evaluated. A general pattern to follow is to recursively call _eval_evalf(prec) on the arguments of the function.

Whenever possible, it’s best to reuse the evalf functionality defined in existing SymPy functions. However, in some cases it will be necessary to use mpmath directly.

### Rewriting and Simplification#

Various simplification functions and methods allow specifying their behavior on custom subclasses. Not every function in SymPy has such hooks. See the documentation of each individual function for details.

#### rewrite()#

The rewrite() method allows rewriting an expression in terms of a specific function or rule. For example,

>>> sin(x).rewrite(cos)
cos(x - pi/2)


To implement rewriting, define a method _eval_rewrite(self, rule, args, **hints), where

• rule is the rule passed to the rewrite() method. Typically rule will be the class of the object to be rewritten to, although for more complex rewrites, it can be anything. Each object that defines _eval_rewrite() defines what rule(s) it supports. Many SymPy functions rewrite to common classes, like expr.rewrite(Add), to perform simplifications or other computations.

• args are the arguments of the function to be used for rewriting. This should be used instead of self.args because any recursive expressions in the args will be rewritten in args (assuming the caller used rewrite(deep=True), which is the default).

• **hints are additional keyword arguments which may be used to specify the behavior of the rewrite. Unknown hints should be ignored as they may be passed to other _eval_rewrite() methods.

The method should return a rewritten expression, using args as the arguments to the function, or None if the expression should be unchanged.

For our versin example, an obvious rewrite we can implement is rewriting versin(x) as 1 - cos(x):

>>> class versin(Function):
...     def _eval_rewrite(self, rule, args, **hints):
...         if rule == cos:
...             return 1 - cos(*args)
>>> versin(x).rewrite(cos)
1 - cos(x)


Once we’ve defined this, simplify() is now able to simplify some expressions containing versin:

>>> from sympy import simplify
>>> simplify(versin(x) + cos(x))
1


#### doit()#

The doit() method is used to evaluate “unevaluated” functions. To define doit() implement doit(self, deep=True, **hints). If deep=True, doit() should recursively call doit() on the arguments. **hints will be any other keyword arguments passed to the user, which should be passed to any recursive calls to doit(). You can use hints to allow the user to specify specific behavior for doit().

The typical usage of doit() in custom Function subclasses is to perform more advanced evaluation which is not performed in eval().

For example, for our divides example, there are several instances that could be simplified using some identities. For example, we defined eval() to evaluate on explicit integers, but we might also want to evaluate examples like divides(k, k*n) where the divisibility is symbolically true. One of the best practices for eval() is to avoid too much automatic evaluation. Automatically evaluating in this case might be considered too much, as it would make use of the assumptions system, which could be expensive. Furthermore, we might want to be able to represent divides(k, k*n) without it always evaluating.

The solution is to implement these more advanced evaluations in doit(). That way, we can explicitly perform them by calling expr.doit(), but they won’t happen by default. An example doit() for divides that performs this simplification (along with the above definition of eval()) might look like this:

Note

If doit() returns a Python int literal, convert it to an Integer so that the returned object is a SymPy type.

>>> from sympy import Integer
>>> class divides(Function):
...     # Define evaluation on basic inputs, as well as type checking that the
...     # inputs are not nonintegral.
...     @classmethod
...     def eval(cls, m, n):
...         # Evaluate for explicit integer m and n.
...         if isinstance(m, Integer) and isinstance(n, Integer):
...             return int(n % m == 0)
...
...         # For symbolic arguments, require m and n to be integer.
...         if m.is_integer is False or n.is_integer is False:
...             raise TypeError("m and n should be integers")
...
...     # Define doit() as further evaluation on symbolic arguments using
...     # assumptions.
...     def doit(self, deep=False, **hints):
...         m, n = self.args
...         # Recursively call doit() on the args whenever deep=True.
...         # Be sure to pass deep=True and **hints through here.
...         if deep:
...            m, n = m.doit(deep=deep, **hints), n.doit(deep=deep, **hints)
...
...         # divides(m, n) is 1 iff n/m is an integer. Note that m and n are
...         # already assumed to be integers because of the logic in eval().
...         isint = (n/m).is_integer
...         if isint is True:
...             return Integer(1)
...         elif isint is False:
...             return Integer(0)
...         else:
...             return divides(m, n)


(Note that this uses the convention that $$k \mid 0$$ for all $$k$$ so that we do not need to check if m or n are nonzero. If we used a different convention we would need to check if m.is_zero and n.is_zero before performing the simplification.)

>>> n, m, k = symbols('n m k', integer=True)
>>> divides(k, k*n)
divides(k, k*n)
>>> divides(k, k*n).doit()
1


Another common way to implement doit() is for it to always return another expression. This effectively treats the function as an “unevaluated” form of another expression.

For example, let’s define a function for fused multiply-add: $$\operatorname{FMA}(x, y, z) = xy + z$$. It may be useful to express this function as a distinct function, e.g., for the purposes of code generation, but it would also be useful in some contexts to “evaluate” FMA(x, y, z) to x*y + z so that it can properly simplify with other expressions.

>>> from sympy import Number
>>> class FMA(Function):
...     """
...     FMA(x, y, z) = x*y + z
...     """
...     @classmethod
...     def eval(cls, x, y, z):
...         # Number is the base class of Integer, Rational, and Float
...         if all(isinstance(i, Number) for i in [x, y, z]):
...            return x*y + z
...
...     def doit(self, deep=True, **hints):
...         x, y, z = self.args
...         # Recursively call doit() on the args whenever deep=True.
...         # Be sure to pass deep=True and **hints through here.
...         if deep:
...             x = x.doit(deep=deep, **hints)
...             y = y.doit(deep=deep, **hints)
...             z = z.doit(deep=deep, **hints)
...         return x*y + z

>>> x, y, z = symbols('x y z')
>>> FMA(x, y, z)
FMA(x, y, z)
>>> FMA(x, y, z).doit()
x*y + z


Most custom functions will not want to define doit() in this way. However, this can provide a happy medium between having a function that always evaluates and a function that never evaluates, producing a function that doesn’t evaluate by default but can be evaluated on demand (see the discussion above).

#### expand()#

The expand() function “expands” an expression in various ways. It is actually a wrapper around several sub-expansion hints. Each function corresponds to a hint to the expand() function/method. A specific expand hint can be defined in a custom function by defining _eval_expand_hint(self, **hints). See the documentation of expand() for details on which hints are defined and the documentation for each specific expand_hint() function (e.g., expand_trig()) for details on what each hint is designed to do.

The **hints keyword arguments are additional hints that may be passed to the expand function to specify additional behavior (these are separate from the predefined hints described in the previous paragraph). Unknown hints should be ignored as they may apply to other functions’ custom expand() methods. A common hint to define is force, where force=True would force an expansion that might not be mathematically valid for all the given input assumptions. For example, expand_log(log(x*y), force=True) produces log(x) + log(y) even though this identity is not true for all complex x and y (typically force=False is the default).

Note that expand() automatically takes care of recursively expanding expressions using its own deep flag, so _eval_expand_* methods should not recursively call expand on the arguments of the function.

For our versin example, we can define rudimentary trig expansion by defining an _eval_expand_trig method, which recursively calls expand_trig() on 1 - cos(x):

>>> from sympy import expand_trig
>>> y = symbols('y')
>>> class versin(Function):
...    def _eval_expand_trig(self, **hints):
...        x = self.args[0]
...        return expand_trig(1 - cos(x))
>>> versin(x + y).expand(trig=True)
sin(x)*sin(y) - cos(x)*cos(y) + 1


A more sophisticated implementation might attempt to rewrite the result of expand_trig(1 - cos(x)) back into versin functions. This is left as an exercise for the reader.

### Differentiation#

To define differentiation via diff(), define a method fdiff(self, argindex). fdiff() should return the derivative of the function, without considering the chain rule, with respect to the argindex-th variable. argindex is indexed starting at 1.

That is, f(x1, ..., xi, ..., xn).fdiff(i) should return $$\frac{d}{d x_i} f(x_1, \ldots, x_i, \ldots, x_n)$$, where $$x_k$$ are independent of one another. diff() will automatically apply the chain rule using the result of fdiff(). User code should use diff() and not call fdiff() directly.

Note

Function subclasses should define differentiation using fdiff(). Subclasses of Expr that aren’t Function subclasses will need to define _eval_derivative() instead. It is not recommended to redefine _eval_derivative() on a Function subclass.

For our $$\operatorname{versin}$$ example function, the derivative is $$\sin(x)$$.

>>> class versin(Function):
...     def fdiff(self, argindex=1):
...         # argindex indexes the args, starting at 1
...         return sin(self.args[0])

>>> versin(x).diff(x)
sin(x)
>>> versin(x**2).diff(x)
2*x*sin(x**2)
>>> versin(x + y).diff(x)
sin(x + y)


As an example of a function that has multiple arguments, consider the fused multiply-add (FMA) example defined above ($$\operatorname{FMA}(x, y, z) = xy + z$$).

We have

$\frac{d}{dx} \operatorname{FMA}(x, y, z) = y,$
$\frac{d}{dy} \operatorname{FMA}(x, y, z) = x,$
$\frac{d}{dz} \operatorname{FMA}(x, y, z) = 1.$

So the fdiff() method for FMA would look like this:

>>> from sympy import Number, symbols
>>> x, y, z = symbols('x y z')
>>> class FMA(Function):
...     """
...     FMA(x, y, z) = x*y + z
...     """
...     def fdiff(self, argindex):
...         # argindex indexes the args, starting at 1
...         x, y, z = self.args
...         if argindex == 1:
...             return y
...         elif argindex == 2:
...             return x
...         elif argindex == 3:
...             return 1

>>> FMA(x, y, z).diff(x)
y
>>> FMA(x, y, z).diff(y)
x
>>> FMA(x, y, z).diff(z)
1
>>> FMA(x**2, x + 1, y).diff(x)
x**2 + 2*x*(x + 1)


To leave a derivative unevaluated, raise sympy.core.function.ArgumentIndexError(self, argindex). This is the default behavior if fdiff() is not defined. Here is an example function $$f(x, y)$$ that is linear in the first argument and has an unevaluated derivative on the second argument.

>>> from sympy.core.function import ArgumentIndexError
>>> class f(Function):
...    @classmethod
...    def eval(cls, x, y):
...        pass
...
...    def fdiff(self, argindex):
...        if argindex == 1:
...           return 1
...        raise ArgumentIndexError(self, argindex)

>>> f(x, y).diff(x)
1
>>> f(x, y).diff(y)
Derivative(f(x, y), y)


### Printing#

You can define how a function prints itself with the varions printers such as the string printer, pretty printers, and LaTeX printer, as well as code printers for various languages like C and Fortran.

In most cases, you will not need to define any printing methods. The default behavior is to print functions using their name. However, in some cases we may want to define special printing for a function.

For example, for our divides example above, we may want the LaTeX printer to print a more mathematical expression. Let’s make the LaTeX printer represent divides(m, n) as \left [ m \middle | n \right ], which looks like $$\left [ m \middle | n \right ]$$ (here $$[P]$$ is the Iverson bracket, which is $$1$$ if $$P$$ is true and $$0$$ if $$P$$ is false).

There are two primary ways to define printing for SymPy objects. One is to define a printer on the printer class. Most classes that are part of the SymPy library should use this method, by defining the printers on the respective classes in sympy.printing. For user code, this may be preferable if you are defining a custom printer, or if you have many custom functions that you want to define printing for. See Example of Custom Printer for an example of how to define a printer in this way.

The other way is to define the printing as a method on the function class. To do this, first look up the printmethod attribute on the printer you want to define the printing for. This is the name of the method you should define for that printer. For the LaTeX printer, LatexPrinter.printmethod is '_latex'. The print method always takes one argument, printer. printer._print should be used to recursively print any other expressions, including the arguments of the function.

So to define our divides LaTeX printer, we will define the function _latex(self, printer) on the class, like this:

>>> from sympy import latex
>>> class divides(Function):
...     def _latex(self, printer):
...         m, n = self.args
...         _m, _n = printer._print(m), printer._print(n)
...         return r'\left [ %s \middle | %s \right ]' % (_m, _n)

>>> print(latex(divides(m, n)))
\left [ m \middle | n \right ]


See Example of Custom Printing Method for more details on how to define printer methods and some pitfalls to avoid. Most importantly, you should always use printer._print() to recursively print the arguments of the function inside of a custom printer.

### Other Methods#

Several other methods can be defined on custom functions to specify various behaviors.

#### inverse()#

The inverse(self, argindex=1) method can be defined to specify the inverse of the function. This is used by solve() and solveset(). The argindex argument is the argument of the function, starting at 1 (similar to the same argument name for the fdiff() method).

inverse() should return a function (not an expression) for the inverse. If the inverse is a larger expression than a single function, it can return a lambda function.

inverse() should only be defined for functions that are one-to-one. In other words, f(x).inverse() is the left inverse of f(x). Defining inverse() on a function that is not one-to-one may result in solve() not giving all possible solutions to an expression containing the function.

Our example versine function is not one-to-one (because cosine is not), but its inverse $$\operatorname{arcversin}$$ is. We may define it as follows (using the same naming convention as other inverse trig functions in SymPy):

>>> class aversin(Function):
...     def inverse(self, argindex=1):
...         return versin


This makes solve() work on aversin(x):

>>> from sympy import solve
>>> solve(aversin(x) - y, x)
[versin(y)]


#### as_real_imag()#

The method as_real_imag() method defines how to split a function into its real and imaginary parts. It is used by various SymPy functions that operate on the real and imaginary parts of an expression separately.

as_real_imag(self, deep=True, **hints) should return a 2-tuple containing the real part and imaginary part of the function. That is expr.as_real_imag() returns (re(expr), im(expr)), where expr == re(expr)

• im(expr)*I, and re(expr)andim(expr) are real.

If deep=True, it should recursively call as_real_imag(deep=True, **hints) on its arguments. As with doit() and the _eval_expand_*() methods, **hints may be any hints to allow the user to specify the behavior of the method. Unknown hints should be ignored and passed through on any recursive calls in case they are meant for other as_real_imag() methods.

For our versin example, we can recursively use the as_real_imag() that is already defined for 1 - cos(x).

>>> class versin(Function):
...     def as_real_imag(self, deep=True, **hints):
...         return (1 - cos(self.args[0])).as_real_imag(deep=deep, **hints)
>>> versin(x).as_real_imag()
(-cos(re(x))*cosh(im(x)) + 1, sin(re(x))*sinh(im(x)))


Defining as_real_imag() also automatically makes expand_complex() work.

>>> versin(x).expand(complex=True)
I*sin(re(x))*sinh(im(x)) - cos(re(x))*cosh(im(x)) + 1


#### Miscellaneous _eval_* methods#

There are many other functions in SymPy whose behavior can be defined on custom functions via a custom _eval_* method, analogous to the ones described above. See the documentation of the specific function for details on how to define each method.

## Complete Examples#

Here are complete examples for the example functions defined in this guide. See the above sections for details on each method.

### Versine#

The versine (versed sine) function is defined as

$\operatorname{versin}(x) = 1 - \cos(x).$

Versine is an example of a simple function defined for all complex numbers. The mathematical definition is simple, which makes it straightforward to define all the above methods on it (in most cases we can just reuse the existing SymPy logic defined on 1 - cos(x)).

#### Definition#

>>> from sympy import Function, cos, expand_trig, Integer, pi, sin
>>> from sympy.core.logic import fuzzy_and, fuzzy_not
>>> class versin(Function):
...     r"""
...     The versine function.
...
...     $\operatorname{versin}(x) = 1 - \cos(x) = 2\sin(x/2)^2.$
...
...     Geometrically, given a standard right triangle with angle x in the
...     unit circle, the versine of x is the positive horizontal distance from
...     the right angle of the triangle to the rightmost point on the unit
...     circle. It was historically used as a more numerically accurate way to
...     compute 1 - cos(x), but it is rarely used today.
...
...     References
...     ==========
...
...     .. [1] https://en.wikipedia.org/wiki/Versine
...     .. [2] https://blogs.scientificamerican.com/roots-of-unity/10-secret-trig-functions-your-math-teachers-never-taught-you/
...     """
...     # Define evaluation on basic inputs.
...     @classmethod
...     def eval(cls, x):
...         # If x is an explicit integer multiple of pi, x/pi will cancel and
...         # be an Integer.
...         n = x/pi
...         if isinstance(n, Integer):
...             return 1 - (-1)**n
...
...     # Define numerical evaluation with evalf().
...     def _eval_evalf(self, prec):
...         return (2*sin(self.args[0]/2)**2)._eval_evalf(prec)
...
...     # Define basic assumptions.
...     def _eval_is_nonnegative(self):
...         # versin(x) is nonnegative if x is real
...         x = self.args[0]
...         if x.is_real is True:
...             return True
...
...     def _eval_is_positive(self):
...         # versin(x) is positive iff x is real and not an even multiple of pi
...         x = self.args[0]
...
...         # x.as_independent(pi, as_Add=False) will split x as a Mul of the
...         # form n*pi
...         coeff, pi_ = x.as_independent(pi, as_Add=False)
...         # If pi_ = pi, x = coeff*pi. Otherwise pi_ = 1 and x is not
...         # (structurally) of the form n*pi.
...         if pi_ == pi:
...             return fuzzy_and([x.is_real, fuzzy_not(coeff.is_even)])
...         elif x.is_real is False:
...             return False
...         # else: return None. We do not know for sure whether x is an even
...         # multiple of pi
...
...     # Define the behavior for various simplification and rewriting
...     # functions.
...     def _eval_rewrite(self, rule, args, **hints):
...         if rule == cos:
...             return 1 - cos(*args)
...         elif rule == sin:
...             return 2*sin(x/2)**2
...
...     def _eval_expand_trig(self, **hints):
...         x = self.args[0]
...         return expand_trig(1 - cos(x))
...
...     def as_real_imag(self, deep=True, **hints):
...         # reuse _eval_rewrite(cos) defined above
...         return self.rewrite(cos).as_real_imag(deep=deep, **hints)
...
...     # Define differentiation.
...     def fdiff(self, argindex=1):
...         return sin(self.args[0])


#### Examples#

Evaluation:

>>> x, y = symbols('x y')
>>> versin(x)
versin(x)
>>> versin(2*pi)
0
>>> versin(1.0)
0.459697694131860


Assumptions:

>>> n = symbols('n', integer=True)
>>> versin(n).is_real
True
>>> versin((2*n + 1)*pi).is_positive
True
>>> versin(2*n*pi).is_zero
True
>>> print(versin(n*pi).is_positive)
None
>>> r = symbols('r', real=True)
>>> print(versin(r).is_positive)
None
>>> nr = symbols('nr', real=False)
>>> print(versin(nr).is_nonnegative)
None


Simplification:

>>> a, b = symbols('a b', real=True)
>>> from sympy import I
>>> versin(x).rewrite(cos)
1 - cos(x)
>>> versin(x).rewrite(sin)
2*sin(x/2)**2
>>> versin(2*x).expand(trig=True)
2 - 2*cos(x)**2
>>> versin(a + b*I).expand(complex=True)
I*sin(a)*sinh(b) - cos(a)*cosh(b) + 1


Differentiation:

>>> versin(x).diff(x)
sin(x)


Solving:

(a more general version of aversin would have all the above methods defined as well)

>>> class aversin(Function):
...     def inverse(self, argindex=1):
...         return versin
>>> from sympy import solve
>>> solve(aversin(x**2) - y, x)
[-sqrt(versin(y)), sqrt(versin(y))]


### divides#

divides is a function defined by

$\begin{split}\operatorname{divides}(m, n) = \begin{cases} 1 & \text{for}\: m \mid n \\ 0 & \text{for}\: m\not\mid n \end{cases},\end{split}$

that is, divides(m, n) is 1 if m divides n and 0 if m does not divide m. It is only defined for integer m and n. For the sake of simplicity, we use the convention that $$m \mid 0$$ for all integer $$m$$.

divides is an example of a function that is only defined for certain input values (integers). divides also gives an example of defining a custom printer (_latex()).

#### Definition#

>>> from sympy import Function, Integer
>>> from sympy.core.logic import fuzzy_not
>>> class divides(Function):
...     r"""
...     $$\operatorname{divides}(m, n) = \begin{cases} 1 & \text{for}\: m \mid n \\ 0 & \text{for}\: m\not\mid n \end{cases}.$$
...
...     That is, divides(m, n) is 1 if m divides n and 0
...     if m does not divide n. It is undefined if m or n are
...     not integers. For simplicity, the convention is used that
...     divides(m, 0) = 1 for all integers m.
...
...     References
...     ==========
...
...     .. [1] https://en.wikipedia.org/wiki/Divisor#Definition
...     """
...     # Define evaluation on basic inputs, as well as type checking that the
...     # inputs are not nonintegral.
...     @classmethod
...     def eval(cls, m, n):
...         # Evaluate for explicit integer m and n.
...         if isinstance(m, Integer) and isinstance(n, Integer):
...             return int(n % m == 0)
...
...         # For symbolic arguments, require m and n to be integer.
...         if m.is_integer is False or n.is_integer is False:
...             raise TypeError("m and n should be integers")
...
...     # Define basic assumptions.
...
...     # divides is always either 0 or 1.
...     is_integer = True
...     is_negative = False
...
...     # Whether divides(m, n) is 0 or 1 depends on m and n. Note that this
...     # method only makes sense because we don't automatically evaluate on
...     # such cases, but instead simplify these cases in doit() below.
...     def _eval_is_zero(self):
...         m, n = self.args
...         if m.is_integer and n.is_integer:
...              return fuzzy_not((n/m).is_integer)
...
...     # Define doit() as further evaluation on symbolic arguments using
...     # assumptions.
...     def doit(self, deep=False, **hints):
...         m, n = self.args
...         # Recursively call doit() on the args whenever deep=True.
...         # Be sure to pass deep=True and **hints through here.
...         if deep:
...            m, n = m.doit(deep=deep, **hints), n.doit(deep=deep, **hints)
...
...         # divides(m, n) is 1 iff n/m is an integer. Note that m and n are
...         # already assumed to be integers because of the logic in eval().
...         isint = (n/m).is_integer
...         if isint is True:
...             return Integer(1)
...         elif isint is False:
...             return Integer(0)
...         else:
...             return divides(m, n)
...
...     # Define LaTeX printing for use with the latex() function and the
...     # Jupyter notebook.
...     def _latex(self, printer):
...         m, n = self.args
...         _m, _n = printer._print(m), printer._print(n)
...         return r'\left [ %s \middle | %s \right ]' % (_m, _n)
...


#### Examples#

Evaluation

>>> from sympy import symbols
>>> n, m, k = symbols('n m k', integer=True)
>>> divides(3, 10)
0
>>> divides(3, 12)
1
>>> divides(m, n).is_integer
True
>>> divides(k, 2*k)
divides(k, 2*k)
>>> divides(k, 2*k).is_zero
False
>>> divides(k, 2*k).doit()
1


Printing:

>>> str(divides(m, n)) # This is using the default str printer
'divides(m, n)'
>>> print(latex(divides(m, n)))
\left [ m \middle | n \right ]


$\operatorname{FMA}(x, y, z) = xy + z.$

It is often implemented in hardware as a single floating-point operation that has better rounding and performance than the equivalent combination of multiplication and addition operations.

FMA is an example of a custom function that is defined as an unevaluated “shorthand” to another function. This is because the doit() method is defined to return x*y + z, meaning the FMA function can easily be evaluated to the expression is represents, but the eval() method does not return anything (except when x, y, and z are all explicit numeric values), meaning that it stays unevaluated by default.

Contrast this with the versine example, which treats versin as a first-class function in its own regard. Even though versin(x) can be expressed in terms of other functions (1 - cos(x)) it does not evaluate on general symbolic inputs in versin.eval(), and versin.doit() is not defined at all.

FMA also represents an example of a continuous function defined on multiple vriables, which demonstrates how argindex works in the fdiff example.

Finally, FMA shows an example of defining some code printers for C and C++ (using the method names from C99CodePrinter.printmethod and CXX11CodePrinter.printmethod), since that is a typical use-case for this function.

The mathematical definition of FMA is very simple and it would be easy to define every method on it, but only a handful are shown here. The versine and divides examples show how to define the other important methods discussed in this guide.

Note that if you want to actually use fused-multiply add for code generation, there is already a version in SymPy sympy.codegen.cfunctions.fma() which is supported by the existing code printers. The version here is only designed to serve as an example.

#### Definition#

>>> from sympy import Number, symbols, Add, Mul
>>> x, y, z = symbols('x y z')
>>> class FMA(Function):
...     """
...     FMA(x, y, z) = x*y + z
...
...     FMA is often defined as a single operation in hardware for better
...     rounding and performance.
...
...     FMA can be evaluated by using the doit() method.
...
...     References
...     ==========
...
...     """
...     # Define automatic evaluation on explicit numbers
...     @classmethod
...     def eval(cls, x, y, z):
...         # Number is the base class of Integer, Rational, and Float
...         if all(isinstance(i, Number) for i in [x, y, z]):
...            return x*y + z
...
...     # Define numerical evaluation with evalf().
...     def _eval_evalf(self, prec):
...         return self.doit(deep=False)._eval_evalf(prec)
...
...     # Define full evaluation to Add and Mul in doit(). This effectively
...     # treats FMA(x, y, z) as just a shorthand for x*y + z that is useful
...     # to have as a separate expression in some contexts and which can be
...     # evaluated to its expanded form in other contexts.
...     def doit(self, deep=True, **hints):
...         x, y, z = self.args
...         # Recursively call doit() on the args whenever deep=True.
...         # Be sure to pass deep=True and **hints through here.
...         if deep:
...             x = x.doit(deep=deep, **hints)
...             y = y.doit(deep=deep, **hints)
...             z = z.doit(deep=deep, **hints)
...         return x*y + z
...
...     # Define FMA.rewrite(Add) and FMA.rewrite(Mul).
...     def _eval_rewrite(self, rule, args, **hints):
...         x, y, z = self.args
...         if rule in [Add, Mul]:
...             return self.doit()
...
...     # Define differentiation.
...     def fdiff(self, argindex):
...         # argindex indexes the args, starting at 1
...         x, y, z = self.args
...         if argindex == 1:
...             return y
...         elif argindex == 2:
...             return x
...         elif argindex == 3:
...             return 1
...
...     # Define code printers for ccode() and cxxcode()
...     def _ccode(self, printer):
...         x, y, z = self.args
...         _x, _y, _z = printer._print(x), printer._print(y), printer._print(z)
...         return "fma(%s, %s, %s)" % (_x, _y, _z)
...
...     def _cxxcode(self, printer):
...         x, y, z = self.args
...         _x, _y, _z = printer._print(x), printer._print(y), printer._print(z)
...         return "std::fma(%s, %s, %s)" % (_x, _y, _z)


#### Examples#

Evaluation:

>>> x, y, z = symbols('x y z')
>>> FMA(2, 3, 4)
10
>>> FMA(x, y, z)
FMA(x, y, z)
>>> FMA(x, y, z).doit()
x*y + z
x*y + z
>>> FMA(2, pi, 1).evalf()
7.28318530717959


Differentiation

>>> FMA(x, x, y).diff(x)
2*x
>>> FMA(x, y, x).diff(x)
y + 1


Code Printers

>>> from sympy import ccode, cxxcode
>>> ccode(FMA(x, y, z))
'fma(x, y, z)'
>>> cxxcode(FMA(x, y, z))
'std::fma(x, y, z)'


• If you have many custom functions that share common logic, you can use a common base class to contain this shared logic. For an example of this, see the source code for the trigonometric functions in SymPy, which use TrigonometricFunction, InverseTrigonometricFunction, and ReciprocalTrigonometricFunction` base classes with some shared logic.