Simplification¶

The generic way to do nontrivial simplifications of expressions is calling simplify() function.

>>> simplify(sin(x)**2 + cos(x)**2)
1
>>> simplify((x**3 + x**2 - x - 1)/(x**2 + 2*x + 1))
x - 1
>>> simplify(gamma(x)/gamma(x - 2))
(x - 2)⋅(x - 1)


There are also more directed simplification functions. These apply very specific rules to the input expression and are typically able to make guarantees about the output. For instance, the factor() function, given a polynomial with rational coefficients in several variables, is guaranteed to produce a factorization into irreducible factors.

The simplify() function applies almost all available in Diofant such specific simplification rules in some heuristics sequence to produce the simplest result.

Tip

The optional measure keyword argument for simplify() lets the user specify the Python function used to determine how “simple” an expression is. The default is count_ops(), which returns the total number of operations in the expression.

That is why it is usually slow. But more important pitfail is that sometimes simplify() doesn’t “simplify” how you might expect, if, for example, it miss some transformation or apply it too early or too late. Lets look on an example

>>> simplify(x**2 + 2*x + 1)
2
x  + 2⋅x + 1
>>> factor(_)
2
(x + 1)


Obviously, the factored form is more “simple”, as it has less arithmetic operations.

The function simplify() is best when used interactively, when you just want to whittle down an expression to a simpler form. You may then choose to apply specific functions once you see what simplify() returns, to get a more precise result. It is also useful when you have no idea what form an expression will take, and you need a catchall function to simplify it.

Rational Functions¶

expand() is one of the most common simplification functions in Diofant. Although it has a lot of scopes, for now, we will consider its function in expanding polynomial expressions.

>>> expand((x + 1)**2)
2
x  + 2⋅x + 1
>>> expand((x + 2)*(x - 3))
2
x  - x - 6


Given a polynomial, expand() will put it into a canonical form of a sum of monomials with help of more directed expansion methods, namely expand_multinomial() and expand_mul().

expand() may not sound like a simplification function. After all, by its very name, it makes expressions bigger, not smaller. Usually this is the case, but often an expression will become smaller upon calling expand() on it due to cancellation.

>>> expand((x + 1)*(x - 2) - (x - 1)*x)
-2


Function factor() takes a multivariate polynomial with rational coefficients and factors it into irreducible factors.

>>> factor(x**3 - x**2 + x - 1)
⎛ 2    ⎞
(x - 1)⋅⎝x  + 1⎠
>>> factor(x**2*z + 4*x*y*z + 4*y**2*z)
2
z⋅(x + 2⋅y)


For polynomials, factor() is the opposite of expand().

Note

The input to factor() and expand() need not be polynomials in the strict sense. They will intelligently factor or expand any kind of expression (though, for example, the factors may not be irreducible if the input is no longer a polynomial over the rationals).

>>> expand((cos(x) + sin(x))**2)
2                           2
sin (x) + 2⋅sin(x)⋅cos(x) + cos (x)
>>> factor(_)
2
(sin(x) + cos(x))


collect() collects common powers of a term in an expression.

>>> x*y + x - 3 + 2*x**2 - z*x**2 + x**3
3    2        2
x  - x ⋅z + 2⋅x  + x⋅y + x - 3
>>> collect(_, x)
3    2
x  + x ⋅(-z + 2) + x⋅(y + 1) - 3


collect() is particularly useful in conjunction with the coeff() method.

>>> _.coeff(x, 2)
-z + 2


cancel() will take any rational function and put it into the standard canonical form, $$p/q$$, where $$p$$ and $$q$$ are expanded polynomials with no common factors.

>>> 1/x + (3*x/2 - 2)/(x - 4)
3⋅x
─── - 2
2        1
─────── + ─
x - 4    x
>>> cancel(_)
2
3⋅x  - 2⋅x - 8
──────────────
2
2⋅x  - 8⋅x

>>> expr = (x*y**2 - 2*x*y*z + x*z**2 + y**2 - 2*y*z + z**2)/(x**2 - 1)
>>> expr
2                2    2            2
x⋅y  - 2⋅x⋅y⋅z + x⋅z  + y  - 2⋅y⋅z + z
───────────────────────────────────────
2
x  - 1
>>> cancel(_)
2            2
y  - 2⋅y⋅z + z
───────────────
x - 1


Note

Since factor() will completely factorize both the numerator and the denominator of an expression, it can also be used to do the same thing:

>>> factor(expr)
2
(y - z)
────────
x - 1


However, it’s less efficient if you are only interested in making sure that the expression is in canceled form.

apart() performs a partial fraction decomposition on a rational function.

>>> (4*x**3 + 21*x**2 + 10*x + 12)/(x**4 + 5*x**3 + 5*x**2 + 4*x)
3       2
4⋅x  + 21⋅x  + 10⋅x + 12
────────────────────────
4      3      2
x  + 5⋅x  + 5⋅x  + 4⋅x
>>> apart(_)
2⋅x - 1       1     3
────────── - ───── + ─
2           x + 4   x
x  + x + 1


Trigonometric Functions¶

To simplify expressions using trigonometric identities, use trigsimp() function.

>>> trigsimp(sin(x)**2 + cos(x)**2)
1
>>> trigsimp(sin(x)**4 - 2*cos(x)**2*sin(x)**2 + cos(x)**4)
cos(4⋅x)   1
──────── + ─
2       2
>>> trigsimp(sin(x)*tan(x)/sec(x))
2
sin (x)


It also works with hyperbolic functions.

>>> trigsimp(cosh(x)**2 + sinh(x)**2)
cosh(2⋅x)
>>> trigsimp(sinh(x)/tanh(x))
cosh(x)


Much like simplify() function, trigsimp() applies various trigonometric identities to the input expression, and then uses a heuristic to return the “best” one.

To expand trigonometric functions, that is, apply the sum or double angle identities, use expand_trig() function.

>>> expand_trig(sin(x + y))
sin(x)⋅cos(y) + sin(y)⋅cos(x)
>>> expand_trig(tan(2*x))
2⋅tan(x)
─────────────
2
- tan (x) + 1


Assumptions¶

By default, Diofant performs all calculations assuming that symbols are commutative with respect to the mutliplication. This assumption makes it easier to treat mathematical problems in full generality.

>>> sqrt(x**2)
____
╱  2
╲╱  x


Yet obviously we can simplify above expression if some additional mathematical properties on x are assumed. This is where assumptions system come into play.

Assumptions are set on Symbol objects when they are created. For instance, we can create a symbol that is assumed to be positive.

>>> p = symbols('p', positive=True)


And then, certain simplifications will be possible:

>>> sqrt(p**2)
p


The assumptions system additionally has deductive capabilities. You might check assumptions on any expression with is_assumption attributes, like is_positive.

>>> p.is_positive
True
>>> (1 + p).is_positive
True
>>> (-p).is_positive
False


Note

False is returned also if certain assumption doesn’t make sense for given object.

In a three-valued logic, used by system, None represents the “unknown” case.

>>> (p - 1).is_positive is None
True


Powers and Logarithms¶

powdenest() function applies identity $$(x^a)^b = x^{a b}$$, from left to right, if assumptions allow.

>>> a, b = symbols('a b', extended_real=True)
>>> powdenest((p**a)**b)
a⋅b
p


powsimp() function reduces expression by combining powers with similar bases and exponent.

>>> powsimp(z**x*z**y)
x + y
z


Again, as for powdenest() above, for the identity $$x^a y^a = (x y)^a$$, that combine bases, we should be careful about assumptions.

>>> q = symbols('q', positive=True)
>>> powsimp(p**a*q**a)
a
(p⋅q)


In general, this identity doesn’t hold. For example, if $$x = y = -1$$ and $$a = 1/2$$.

expand_power_exp() and expand_power_base() functions do reverse of powsimp().

>>> expand_power_exp(x**(y + z))
y  z
x ⋅x
>>> expand_power_base((p*q)**a)
a  a
p ⋅q


Logarithms have similar issues as powers. There are two main identities

1. $$\log{(xy)} = \log{(x)} + \log{(y)}$$
2. $$\log{(x^n)} = n\log{(x)}$$

Neither identity is true for arbitrary complex $$x$$ and $$y$$, due to the branch cut in the complex plane for the complex logarithm.

To apply above identities from left to right, use expand_log(). As for powers, the identities will not be applied unless they are valid with given set of assumptions for symbols.

>>> expand_log(log(p*q))
log(p) + log(q)
>>> expand_log(log(p/q))
log(p) - log(q)
>>> expand_log(log(p**2))
2⋅log(p)
>>> expand_log(log(p**a))
a⋅log(p)
>>> expand_log(log(x*y))
log(x⋅y)


To apply identities from right to left, i.e. do reverse of expand_log(), use logcombine() function.

>>> logcombine(log(p) + log(q))
log(p⋅q)
>>> logcombine(a*log(p))
⎛ a⎞
log⎝p ⎠
>>> logcombine(a*log(z))
a⋅log(z)


Special Functions¶

Diofant implements dozens of special functions, ranging from functions in combinatorics to mathematical physics.

To expand special functions in terms of some identities, use expand_func(). For example the gamma function gamma can be expanded as

>>> expand_func(gamma(x + 3))
x⋅(x + 1)⋅(x + 2)⋅Γ(x)


This method also can help if you would like to rewrite the generalized hypergeometric function hyper or the Meijer G-function meijerg in terms of more standard functions.

>>> expand_func(hyper([1, 1], [2], z))
-log(-z + 1)
─────────────
z
>>> meijerg([[1],[1]], [[1],[]], -z)
╭─╮1, 1 ⎛1  1 │   ⎞
│╶┐     ⎜     │ -z⎟
╰─╯2, 1 ⎝1    │   ⎠
>>> expand_func(_)
z ___
╲╱ ℯ


To simplify combinatorial expressions, involving factorial, binomial or gamma — use combsimp() function.

>>> combsimp(factorial(x)/factorial(x - 3))
x⋅(x - 2)⋅(x - 1)
>>> combsimp(binomial(x + 1, y + 1)/binomial(x, y))
x + 1
─────
y + 1
>>> combsimp(gamma(x)*gamma(1 - x))
π
────────
sin(π⋅x)