Term Rewriting

Term rewriting is a very general class of functionalities which are used to convert expressions of one type in terms of expressions of different kind. For example expanding, combining and converting expressions apply to term rewriting, and also simplification routines can be included here. Currently SymPy has several functions and basic built-in methods for performing various types of rewriting.

Expanding

The simplest rewrite rule is expanding expressions into a _sparse_ form. Expanding has several flavors and include expanding complex valued expressions, arithmetic expand of products and powers but also expanding functions in terms of more general functions is possible. Below are listed all currently available expand rules.

Expanding of arithmetic expressions involving products and powers:

>>> from sympy import *
>>> x, y, z = symbols('x,y,z')
>>> ((x + y)*(x - y)).expand(basic=True)
x**2 - y**2
>>> ((x + y + z)**2).expand(basic=True)
x**2 + 2*x*y + 2*x*z + y**2 + 2*y*z + z**2

Arithmetic expand is done by default in expand() so the keyword basic can be omitted. However you can set basic=False to avoid this type of expand if you use rules described below. This give complete control on what is done with the expression.

Another type of expand rule is expanding complex valued expressions and putting them into a normal form. For this complex keyword is used. Note that it will always perform arithmetic expand to obtain the desired normal form:

>>> (x + I*y).expand(complex=True)
re(x) + I*re(y) + I*im(x) - im(y)

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

Note also that the same behavior can be obtained by using as_real_imag() method. However it will return a tuple containing the real part in the first place and the imaginary part in the other. This can be also done in a two step process by using collect function:

>>> (x + I*y).as_real_imag()
(re(x) - im(y), re(y) + im(x))

>>> collect((x + I*y).expand(complex=True), I, evaluate=False)
{1: re(x) - im(y), I: re(y) + im(x)}

There is also possibility for expanding expressions in terms of expressions of different kind. This is very general type of expanding and usually you would use rewrite() to do specific type of rewrite:

>>> GoldenRatio.expand(func=True)
1/2 + sqrt(5)/2

Common Subexpression Detection and Collection

Before evaluating a large expression, it is often useful to identify common subexpressions, collect them and evaluate them at once. This is implemented in the cse function. Examples:

>>> from sympy import cse, sqrt, sin, pprint
>>> from sympy.abc import x

>>> pprint(cse(sqrt(sin(x))), use_unicode=True)
⎛    ⎡  ________⎤⎞
⎝[], ⎣╲╱ sin(x) ⎦⎠

>>> pprint(cse(sqrt(sin(x)+5)*sqrt(sin(x)+4)), use_unicode=True)
⎛                ⎡  ________   ________⎤⎞
⎝[(x₀, sin(x))], ⎣╲╱ x₀ + 4 ⋅╲╱ x₀ + 5 ⎦⎠

>>> pprint(cse(sqrt(sin(x+1) + 5 + cos(y))*sqrt(sin(x+1) + 4 + cos(y))),
...     use_unicode=True)
⎛                             ⎡  ________   ________⎤⎞
⎝[(x₀, sin(x + 1) + cos(y))], ⎣╲╱ x₀ + 4 ⋅╲╱ x₀ + 5 ⎦⎠

>>> pprint(cse((x-y)*(z-y) + sqrt((x-y)*(z-y))), use_unicode=True)
⎛                          ⎡  ____     ⎤⎞
⎝[(x₀, (x - y)⋅(-y + z))], ⎣╲╱ x₀  + x₀⎦⎠

Optimizations to be performed before and after common subexpressions elimination can be passed in the``optimizations`` optional argument. A set of predefined basic optimizations can be applied by passing optimizations='basic':

>>> pprint(cse((x-y)*(z-y) + sqrt((x-y)*(z-y)), optimizations='basic'),
...     use_unicode=True)
⎛                          ⎡  ____     ⎤⎞
⎝[(x₀, -(x - y)⋅(y - z))], ⎣╲╱ x₀  + x₀⎦⎠

However, these optimizations can be very slow for large expressions. Moreover, if speed is a concern, one can pass the option order='none'. Order of terms will then be dependent on hashing algorithm implementation, but speed will be greatly improved.

More information:

sympy.simplify.cse_main.cse(exprs, symbols=None, optimizations=None, postprocess=None, order='canonical', ignore=(), list=True)

Perform common subexpression elimination on an expression.

Parameters

exprs : list of SymPy expressions, or a single SymPy expression

The expressions to reduce.

symbols : infinite iterator yielding unique Symbols

The symbols used to label the common subexpressions which are pulled out. The numbered_symbols generator is useful. The default is a stream of symbols of the form “x0”, “x1”, etc. This must be an infinite iterator.

optimizations : list of (callable, callable) pairs

The (preprocessor, postprocessor) pairs of external optimization functions. Optionally ‘basic’ can be passed for a set of predefined basic optimizations. Such ‘basic’ optimizations were used by default in old implementation, however they can be really slow on larger expressions. Now, no pre or post optimizations are made by default.

postprocess : a function which accepts the two return values of cse and

returns the desired form of output from cse, e.g. if you want the replacements reversed the function might be the following lambda: lambda r, e: return reversed(r), e

order : string, ‘none’ or ‘canonical’

The order by which Mul and Add arguments are processed. If set to ‘canonical’, arguments will be canonically ordered. If set to ‘none’, ordering will be faster but dependent on expressions hashes, thus machine dependent and variable. For large expressions where speed is a concern, use the setting order=’none’.

ignore : iterable of Symbols

Substitutions containing any Symbol from ignore will be ignored.

list : bool, (default True)

Returns expression in list or else with same type as input (when False).

Returns

replacements : list of (Symbol, expression) pairs

All of the common subexpressions that were replaced. Subexpressions earlier in this list might show up in subexpressions later in this list.

reduced_exprs : list of SymPy expressions

The reduced expressions with all of the replacements above.

Examples

>>> from sympy import cse, SparseMatrix
>>> from sympy.abc import x, y, z, w
>>> cse(((w + x + y + z)*(w + y + z))/(w + x)**3)
([(x0, y + z), (x1, w + x)], [(w + x0)*(x0 + x1)/x1**3])

List of expressions with recursive substitutions:

>>> m = SparseMatrix([x + y, x + y + z])
>>> cse([(x+y)**2, x + y + z, y + z, x + z + y, m])
([(x0, x + y), (x1, x0 + z)], [x0**2, x1, y + z, x1, Matrix([
[x0],
[x1]])])

Note: the type and mutability of input matrices is retained.

>>> isinstance(_[1][-1], SparseMatrix)
True

The user may disallow substitutions containing certain symbols:

>>> cse([y**2*(x + 1), 3*y**2*(x + 1)], ignore=(y,))
([(x0, x + 1)], [x0*y**2, 3*x0*y**2])

The default return value for the reduced expression(s) is a list, even if there is only one expression. The \(list\) flag preserves the type of the input in the output:

>>> cse(x)
([], [x])
>>> cse(x, list=False)
([], x)