# Simplify¶

## simplify¶

diofant.simplify.simplify.simplify(expr, ratio=1.7, measure=<function count_ops>, fu=False)[source]

Simplifies the given expression.

Simplification is not a well defined term and the exact strategies this function tries can change in the future versions of Diofant. If your algorithm relies on “simplification” (whatever it is), try to determine what you need exactly - is it powsimp()?, radsimp()?, together()?, logcombine()?, or something else? And use this particular function directly, because those are well defined and thus your algorithm will be robust.

Nonetheless, especially for interactive use, or when you don’t know anything about the structure of the expression, simplify() tries to apply intelligent heuristics to make the input expression “simpler”. For example:

>>> from diofant import simplify, cos, sin
>>> from diofant.abc import x, y
>>> a = (x + x**2)/(x*sin(y)**2 + x*cos(y)**2)
>>> a
(x**2 + x)/(x*sin(y)**2 + x*cos(y)**2)
>>> simplify(a)
x + 1


Note that we could have obtained the same result by using specific simplification functions:

>>> from diofant import trigsimp, cancel
>>> trigsimp(a)
(x**2 + x)/x
>>> cancel(_)
x + 1


In some cases, applying simplify() may actually result in some more complicated expression. The default ratio=1.7 prevents more extreme cases: if (result length)/(input length) > ratio, then input is returned unmodified. The measure parameter lets you specify the function used to determine how complex an expression is. The function should take a single argument as an expression and return a number such that if expression a is more complex than expression b, then measure(a) > measure(b). The default measure function is count_ops(), which returns the total number of operations in the expression.

For example, if ratio=1, simplify output can’t be longer than input.

>>> from diofant import sqrt, simplify, count_ops, oo
>>> root = 1/(sqrt(2)+3)


Since simplify(root) would result in a slightly longer expression, root is returned unchanged instead:

>>> simplify(root, ratio=1) == root
True


If ratio=oo, simplify will be applied anyway:

>>> count_ops(simplify(root, ratio=oo)) > count_ops(root)
True


Note that the shortest expression is not necessary the simplest, so setting ratio to 1 may not be a good idea. Heuristically, the default value ratio=1.7 seems like a reasonable choice.

You can easily define your own measure function based on what you feel should represent the “size” or “complexity” of the input expression. Note that some choices, such as lambda expr: len(str(expr)) may appear to be good metrics, but have other problems (in this case, the measure function may slow down simplify too much for very large expressions). If you don’t know what a good metric would be, the default, count_ops, is a good one.

For example:

>>> from diofant import symbols, log
>>> a, b = symbols('a b', positive=True)
>>> g = log(a) + log(b) + log(a)*log(1/b)
>>> h = simplify(g)
>>> h
log(a*b**(-log(a) + 1))
>>> count_ops(g)
8
>>> count_ops(h)
5


So you can see that h is simpler than g using the count_ops metric. However, we may not like how simplify (in this case, using logcombine) has created the b**(log(1/a) + 1) term. A simple way to reduce this would be to give more weight to powers as operations in count_ops. We can do this by using the visual=True option:

>>> print(count_ops(g, visual=True))
2*ADD + DIV + 4*LOG + MUL
>>> print(count_ops(h, visual=True))
2*LOG + MUL + POW + SUB

>>> from diofant import Symbol, S
>>> def my_measure(expr):
...     POW = Symbol('POW')
...     # Discourage powers by giving POW a weight of 10
...     count = count_ops(expr, visual=True).subs(POW, 10)
...     # Every other operation gets a weight of 1 (the default)
...     count = count.replace(Symbol, type(S.One))
...     return count
>>> my_measure(g)
8
>>> my_measure(h)
14
>>> 15./8 > 1.7  # 1.7 is the default ratio
True
>>> simplify(g, measure=my_measure)
-log(a)*log(b) + log(a) + log(b)


Note that because simplify() internally tries many different simplification strategies and then compares them using the measure function, we get a completely different result that is still different from the input expression by doing this.

## separatevars¶

diofant.simplify.simplify.separatevars(expr, symbols=[], dict=False, force=False)[source]

Separates variables in an expression, if possible. By default, it separates with respect to all symbols in an expression and collects constant coefficients that are independent of symbols.

If dict=True then the separated terms will be returned in a dictionary keyed to their corresponding symbols. By default, all symbols in the expression will appear as keys; if symbols are provided, then all those symbols will be used as keys, and any terms in the expression containing other symbols or non-symbols will be returned keyed to the string ‘coeff’. (Passing None for symbols will return the expression in a dictionary keyed to ‘coeff’.)

If force=True, then bases of powers will be separated regardless of assumptions on the symbols involved.

Notes

The order of the factors is determined by Mul, so that the separated expressions may not necessarily be grouped together.

Although factoring is necessary to separate variables in some expressions, it is not necessary in all cases, so one should not count on the returned factors being factored.

Examples

>>> from diofant.abc import x, y, z, alpha
>>> from diofant import separatevars, sin
>>> separatevars((x*y)**y)
(x*y)**y
>>> separatevars((x*y)**y, force=True)
x**y*y**y

>>> e = 2*x**2*z*sin(y)+2*z*x**2
>>> separatevars(e)
2*x**2*z*(sin(y) + 1)
>>> separatevars(e, symbols=(x, y), dict=True)
{'coeff': 2*z, x: x**2, y: sin(y) + 1}
>>> separatevars(e, [x, y, alpha], dict=True)
{'coeff': 2*z, alpha: 1, x: x**2, y: sin(y) + 1}


If the expression is not really separable, or is only partially separable, separatevars will do the best it can to separate it by using factoring.

>>> separatevars(x + x*y - 3*x**2)
-x*(3*x - y - 1)


If the expression is not separable then expr is returned unchanged or (if dict=True) then None is returned.

>>> eq = 2*x + y*sin(x)
>>> separatevars(eq) == eq
True
>>> separatevars(2*x + y*sin(x), symbols=(x, y), dict=True) == None
True


## nthroot¶

diofant.simplify.simplify.nthroot(expr, n, max_len=4, prec=15)[source]

compute a real nth-root of a sum of surds

Parameters: expr : sum of surds n : integer max_len : maximum number of surds passed as constants to nsimplify

Notes

First nsimplify is used to get a candidate root; if it is not a root the minimal polynomial is computed; the answer is one of its roots.

Examples

>>> from diofant.simplify.simplify import nthroot
>>> from diofant import sqrt
>>> nthroot(90 + 34*sqrt(7), 3)
sqrt(7) + 3


## besselsimp¶

diofant.simplify.simplify.besselsimp(expr)[source]

Simplify bessel-type functions.

This routine tries to simplify bessel-type functions. Currently it only works on the Bessel J and I functions, however. It works by looking at all such functions in turn, and eliminating factors of “I” and “-1” (actually their polar equivalents) in front of the argument. Then, functions of half-integer order are rewritten using trigonometric functions and functions of integer order (> 1) are rewritten using functions of low order. Finally, if the expression was changed, compute factorization of the result with factor().

>>> from diofant import besselj, besseli, besselsimp, polar_lift, I, Rational
>>> from diofant.abc import z, nu
>>> besselsimp(besselj(nu, z*polar_lift(-1)))
E**(I*pi*nu)*besselj(nu, z)
>>> besselsimp(besseli(nu, z*polar_lift(-I)))
E**(-I*pi*nu/2)*besselj(nu, z)
>>> besselsimp(besseli(Rational(-1, 2), z))
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
>>> besselsimp(z*besseli(0, z) + z*(besseli(2, z))/2 + besseli(1, z))
3*z*besseli(0, z)/2


## hypersimp¶

diofant.simplify.simplify.hypersimp(f, k)[source]

Given combinatorial term f(k) simplify its consecutive term ratio i.e. f(k+1)/f(k). The input term can be composed of functions and integer sequences which have equivalent representation in terms of gamma special function.

Notes

The algorithm [R10351035] performs three basic steps:

1. Rewrite all functions in terms of gamma, if possible.

2. Rewrite all occurrences of gamma in terms of products
of gamma and rising factorial with integer,  absolute
constant exponent.

3. Perform simplification of nested fractions, powers
and if the resulting expression is a quotient of
polynomials, reduce their total degree.


If f(k) is hypergeometric then as result we arrive with a quotient of polynomials of minimal degree. Otherwise None is returned.

References

 [R10351035] (1, 2) W. Koepf, Algorithms for m-fold Hypergeometric Summation, Journal of Symbolic Computation (1995) 20, 399-417

## hypersimilar¶

diofant.simplify.simplify.hypersimilar(f, g, k)[source]

Returns True if ‘f’ and ‘g’ are hyper-similar.

Similarity in hypergeometric sense means that a quotient of f(k) and g(k) is a rational function in k. This procedure is useful in solving recurrence relations.

## nsimplify¶

diofant.simplify.simplify.nsimplify(expr, constants=[], tolerance=None, full=False, rational=None)[source]

Find a simple representation for a number or, if there are free symbols or if rational=True, then replace Floats with their Rational equivalents. If no change is made and rational is not False then Floats will at least be converted to Rationals.

For numerical expressions, a simple formula that numerically matches the given numerical expression is sought (and the input should be possible to evalf to a precision of at least 30 digits).

Optionally, a list of (rationally independent) constants to include in the formula may be given.

A lower tolerance may be set to find less exact matches. If no tolerance is given then the least precise value will set the tolerance (e.g. Floats default to 15 digits of precision, so would be tolerance=10**-15).

With full=True, a more extensive search is performed (this is useful to find simpler numbers when the tolerance is set low).

Examples

>>> from diofant import nsimplify, sqrt, GoldenRatio, exp, I, exp, pi
>>> nsimplify(4/(1+sqrt(5)), [GoldenRatio])
-2 + 2*GoldenRatio
>>> nsimplify((1/(exp(3*pi*I/5)+1)))
1/2 - I*sqrt(sqrt(5)/10 + 1/4)
>>> nsimplify(I**I, [pi])
E**(-pi/2)
>>> nsimplify(pi, tolerance=0.01)
22/7


## posify¶

diofant.simplify.simplify.posify(eq)[source]

Return eq (with generic symbols made positive) and a dictionary containing the mapping between the old and new symbols.

Any symbol that has positive=None will be replaced with a positive dummy symbol having the same name. This replacement will allow more symbolic processing of expressions, especially those involving powers and logarithms.

A dictionary that can be sent to subs to restore eq to its original symbols is also returned.

>>> from diofant import posify, Symbol, log, solve
>>> from diofant.abc import x
>>> posify(x + Symbol('p', positive=True) + Symbol('n', negative=True))
(n + p + _x, {_x: x})

>>> eq = 1/x
>>> log(eq).expand()
log(1/x)
>>> log(posify(eq)[0]).expand()
-log(_x)
>>> p, rep = posify(eq)
>>> log(p).expand().subs(rep)
-log(x)


It is possible to apply the same transformations to an iterable of expressions:

>>> eq = x**2 - 4
>>> solve(eq, x)
[{x: -2}, {x: 2}]
>>> eq_x, reps = posify([eq, x]); eq_x
[_x**2 - 4, _x]
>>> solve(*eq_x)
[{_x: 2}]


## logcombine¶

diofant.simplify.simplify.logcombine(expr, force=False)[source]

Takes logarithms and combines them using the following rules:

• log(x) + log(y) == log(x*y) if both are not negative
• a*log(x) == log(x**a) if x is positive and a is real

If force is True then the assumptions above will be assumed to hold if there is no assumption already in place on a quantity. For example, if a is imaginary or the argument negative, force will not perform a combination but if a is a symbol with no assumptions the change will take place.

posify
replace all symbols with symbols having positive assumptions

Examples

>>> from diofant import Symbol, symbols, log, logcombine, I
>>> from diofant.abc import a, x, y, z
>>> logcombine(a*log(x) + log(y) - log(z))
a*log(x) + log(y) - log(z)
>>> logcombine(a*log(x) + log(y) - log(z), force=True)
log(x**a*y/z)
>>> x, y, z = symbols('x y z', positive=True)
>>> a = Symbol('a', extended_real=True)
>>> logcombine(a*log(x) + log(y) - log(z))
log(x**a*y/z)


The transformation is limited to factors and/or terms that contain logs, so the result depends on the initial state of expansion:

>>> eq = (2 + 3*I)*log(x)
>>> logcombine(eq, force=True) == eq
True
>>> logcombine(eq.expand(), force=True)
log(x**2) + I*log(x**3)


diofant.simplify.radsimp.radsimp(expr, symbolic=True, max_terms=4)[source]

Rationalize the denominator by removing square roots.

Note: the expression returned from radsimp must be used with caution since if the denominator contains symbols, it will be possible to make substitutions that violate the assumptions of the simplification process: that for a denominator matching a + b*sqrt(c), a != +/-b*sqrt(c). (If there are no symbols, this assumptions is made valid by collecting terms of sqrt(c) so the match variable a does not contain sqrt(c).) If you do not want the simplification to occur for symbolic denominators, set symbolic to False.

If there are more than max_terms radical terms then the expression is returned unchanged.

Examples

>>> from diofant import radsimp, sqrt, Symbol, denom, pprint, I
>>> from diofant import factor_terms, fraction, signsimp
>>> from diofant.abc import a, b, c

>>> radsimp(1/(I + 1))
(1 - I)/2
(-sqrt(2) + 2)/2
>>> x, y = map(Symbol, 'xy')
>>> e = ((2 + 2*sqrt(2))*x + (2 + sqrt(8))*y)/(2 + sqrt(2))
sqrt(2)*(x + y)


No simplification beyond removal of the gcd is done. One might want to polish the result a little, however, by collecting square root terms:

>>> r2 = sqrt(2)
>>> r5 = sqrt(5)
>>> ans = radsimp(1/(y*r2 + x*r2 + a*r5 + b*r5))
>>> pprint(ans, use_unicode=False)
___       ___       ___       ___
\/ 5 *a + \/ 5 *b - \/ 2 *x - \/ 2 *y
------------------------------------------
2               2      2              2
5*a  + 10*a*b + 5*b  - 2*x  - 4*x*y - 2*y

>>> n, d = fraction(ans)
___             ___
\/ 5 *(a + b) - \/ 2 *(x + y)
------------------------------------------
2               2      2              2
5*a  + 10*a*b + 5*b  - 2*x  - 4*x*y - 2*y


If radicals in the denominator cannot be removed or there is no denominator, the original expression will be returned.

>>> radsimp(sqrt(2)*x + sqrt(2))
sqrt(2)*x + sqrt(2)


Results with symbols will not always be valid for all substitutions:

>>> eq = 1/(a + b*sqrt(c))
>>> eq.subs(a, b*sqrt(c))
1/(2*b*sqrt(c))
nan


If symbolic=False, symbolic denominators will not be transformed (but numeric denominators will still be processed):

>>> radsimp(eq, symbolic=False)
1/(a + b*sqrt(c))


diofant.simplify.radsimp.rad_rationalize(num, den)[source]

Rationalize num/den by removing square roots in the denominator; num and den are sum of terms whose squares are rationals

Examples

>>> from diofant import sqrt
(-sqrt(3) + sqrt(6)/3, -7/9)


### collect¶

diofant.simplify.radsimp.collect(expr, syms, func=None, evaluate=True, exact=False, distribute_order_term=True)[source]

Collect additive terms of an expression.

This function collects additive terms of an expression with respect to a list of expression up to powers with rational exponents. By the term symbol here are meant arbitrary expressions, which can contain powers, products, sums etc. In other words symbol is a pattern which will be searched for in the expression’s terms.

The input expression is not expanded by collect(), so user is expected to provide an expression is an appropriate form (for example, by using expand() prior to calling this function). This makes collect() more predictable as there is no magic happening behind the scenes. However, it is important to note, that powers of products are converted to products of powers using the expand_power_base() function.

Parameters: expr : Expr an expression syms : iterable of Symbol’s collected symbols evaluate : boolean First, if evaluate flag is set (by default), this function will return an expression with collected terms else it will return a dictionary with expressions up to rational powers as keys and collected coefficients as values.

Examples

>>> from diofant import collect, expand, factor
>>> from diofant.abc import a, b, c, x, y


This function can collect symbolic coefficients in polynomials or rational expressions. It will manage to find all integer or rational powers of collection variable:

>>> collect(a*x**2 + b*x**2 + a*x - b*x + c, x)
c + x**2*(a + b) + x*(a - b)


The same result can be achieved in dictionary form:

>>> d = collect(a*x**2 + b*x**2 + a*x - b*x + c, x, evaluate=False)
>>> d[x**2]
a + b
>>> d[x]
a - b
>>> d[1]
c


You can also work with multivariate polynomials. However, remember that this function is greedy so it will care only about a single symbol at time, in specification order:

>>> collect(x**2 + y*x**2 + x*y + y + a*y, [x, y])
x**2*(y + 1) + x*y + y*(a + 1)


Also more complicated expressions can be used as patterns:

>>> from diofant import sin, log
>>> collect(a*sin(2*x) + b*sin(2*x), sin(2*x))
(a + b)*sin(2*x)

>>> collect(a*x*log(x) + b*(x*log(x)), x*log(x))
x*(a + b)*log(x)


You can use wildcards in the pattern:

>>> from diofant import Wild
>>> w = Wild('w1')
>>> collect(a*x**y - b*x**y, w**y)
x**y*(a - b)


It is also possible to work with symbolic powers, although it has more complicated behavior, because in this case power’s base and symbolic part of the exponent are treated as a single symbol:

>>> collect(a*x**c + b*x**c, x)
a*x**c + b*x**c
>>> collect(a*x**c + b*x**c, x**c)
x**c*(a + b)


However if you incorporate rationals to the exponents, then you will get well known behavior:

>>> collect(a*x**(2*c) + b*x**(2*c), x**c)
x**(2*c)*(a + b)


Note also that all previously stated facts about collect() function apply to the exponential function, so you can get:

>>> from diofant import exp
>>> collect(a*exp(2*x) + b*exp(2*x), exp(x))
E**(2*x)*(a + b)


If you are interested only in collecting specific powers of some symbols then set exact flag in arguments:

>>> collect(a*x**7 + b*x**7, x, exact=True)
a*x**7 + b*x**7
>>> collect(a*x**7 + b*x**7, x**7, exact=True)
x**7*(a + b)


You can also apply this function to differential equations, where derivatives of arbitrary order can be collected. Note that if you collect with respect to a function or a derivative of a function, all derivatives of that function will also be collected. Use exact=True to prevent this from happening:

>>> from diofant import Derivative as D, Function
>>> f = Function('f')(x)

>>> collect(a*D(f, x) + b*D(f, x), D(f, x))
(a + b)*Derivative(f(x), x)

>>> collect(a*D(D(f, x), x) + b*D(D(f, x), x), f)
(a + b)*Derivative(f(x), x, x)

>>> collect(a*D(D(f, x), x) + b*D(D(f, x), x), D(f, x), exact=True)
a*Derivative(f(x), x, x) + b*Derivative(f(x), x, x)

>>> collect(a*D(f, x) + b*D(f, x) + a*f + b*f, f)
f(x)*(a + b) + (a + b)*Derivative(f(x), x)


Or you can even match both derivative order and exponent at the same time:

>>> collect(a*D(D(f, x), x)**2 + b*D(D(f, x), x)**2, D(f, x))
(a + b)*Derivative(f(x), x, x)**2


Finally, you can apply a function to each of the collected coefficients. For example you can factorize symbolic coefficients of polynomial:

>>> f = expand((x + a + 1)**3)

>>> collect(f, x, factor)
x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + (a + 1)**3

diofant.simplify.radsimp.rcollect(expr, *vars)[source]

Recursively collect sums in an expression.

Examples

>>> from diofant.simplify import rcollect
>>> from diofant.abc import x, y

>>> expr = (x**2*y + x*y + x + y)/(x + y)

>>> rcollect(expr, y)
(x + y*(x**2 + x + 1))/(x + y)


### collect_sqrt¶

diofant.simplify.radsimp.collect_sqrt(expr, evaluate=True)[source]

Return expr with terms having common square roots collected together. If evaluate is False a count indicating the number of sqrt-containing terms will be returned and, if non-zero, the terms of the Add will be returned, else the expression itself will be returned as a single term. If evaluate is True, the expression with any collected terms will be returned.

Note: since I = sqrt(-1), it is collected, too.

Examples

>>> from diofant import sqrt
>>> from diofant.abc import a, b

>>> r2, r3, r5 = [sqrt(i) for i in [2, 3, 5]]
>>> collect_sqrt(a*r2 + b*r2)
sqrt(2)*(a + b)
>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r3)
sqrt(2)*(a + b) + sqrt(3)*(a + b)
>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5)
sqrt(3)*a + sqrt(5)*b + sqrt(2)*(a + b)


If evaluate is False then the arguments will be sorted and returned as a list and a count of the number of sqrt-containing terms will be returned:

>>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5, evaluate=False)
((sqrt(3)*a, sqrt(5)*b, sqrt(2)*(a + b)), 3)
>>> collect_sqrt(a*sqrt(2) + b, evaluate=False)
((b, sqrt(2)*a), 1)
>>> collect_sqrt(a + b, evaluate=False)
((a + b,), 0)


### collect_const¶

diofant.simplify.radsimp.collect_const(expr, *vars, **kwargs)[source]

A non-greedy collection of terms with similar number coefficients in an Add expr. If vars is given then only those constants will be targeted. Although any Number can also be targeted, if this is not desired set Numbers=False and no Float or Rational will be collected.

Examples

>>> from diofant import sqrt
>>> from diofant.abc import a, s, x, y, z
>>> collect_const(sqrt(3) + sqrt(3)*(1 + sqrt(2)))
sqrt(3)*(sqrt(2) + 2)
>>> collect_const(sqrt(3)*s + sqrt(7)*s + sqrt(3) + sqrt(7))
(sqrt(3) + sqrt(7))*(s + 1)
>>> s = sqrt(2) + 2
>>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7))
(sqrt(2) + 3)*(sqrt(3) + sqrt(7))
>>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7), sqrt(3))
sqrt(7) + sqrt(3)*(sqrt(2) + 3) + sqrt(7)*(sqrt(2) + 2)


The collection is sign-sensitive, giving higher precedence to the unsigned values:

>>> collect_const(x - y - z)
x - (y + z)
>>> collect_const(-y - z)
-(y + z)
>>> collect_const(2*x - 2*y - 2*z, 2)
2*(x - y - z)
>>> collect_const(2*x - 2*y - 2*z, -2)
2*x - 2*(y + z)


### fraction¶

diofant.simplify.radsimp.fraction(expr, exact=False)[source]

Returns a pair with expression’s numerator and denominator.

If the given expression is not a fraction then this function will return the tuple (expr, 1).

This function will not make any attempt to simplify nested fractions or to do any term rewriting at all.

If only one of the numerator/denominator pair is needed then use numer(expr) or denom(expr) functions respectively.

>>> from diofant import fraction, Rational, Symbol
>>> from diofant.abc import x, y

>>> fraction(x/y)
(x, y)
>>> fraction(x)
(x, 1)

>>> fraction(1/y**2)
(1, y**2)

>>> fraction(x*y/2)
(x*y, 2)
>>> fraction(Rational(1, 2))
(1, 2)


This function will also work fine with assumptions:

>>> k = Symbol('k', negative=True)
>>> fraction(x * y**k)
(x, y**(-k))


If we know nothing about sign of some exponent and ‘exact’ flag is unset, then structure this exponent’s structure will be analyzed and pretty fraction will be returned:

>>> from diofant import exp
>>> fraction(2*x**(-y))
(2, x**y)

>>> fraction(exp(-x))
(1, E**x)

>>> fraction(exp(-x), exact=True)
(E**(-x), 1)


## Ratsimp¶

### ratsimp¶

diofant.simplify.ratsimp.ratsimp(expr)[source]

Put an expression over a common denominator, cancel and reduce.

Examples

>>> from diofant import ratsimp
>>> from diofant.abc import x, y
>>> ratsimp(1/x + 1/y)
(x + y)/(x*y)


## Trigonometric simplification¶

### trigsimp¶

diofant.simplify.trigsimp.trigsimp(expr, **opts)[source]

reduces expression by using known trig identities

Notes

method: - Determine the method to use. Valid choices are ‘matching’ (default), ‘groebner’, ‘combined’, and ‘fu’. If ‘matching’, simplify the expression recursively by targeting common patterns. If ‘groebner’, apply an experimental groebner basis algorithm. In this case further options are forwarded to trigsimp_groebner, please refer to its docstring. If ‘combined’, first run the groebner basis algorithm with small default parameters, then run the ‘matching’ algorithm. ‘fu’ runs the collection of trigonometric transformations described by Fu, et al.

Examples

>>> from diofant import trigsimp, sin, cos, log
>>> from diofant.abc import x, y
>>> e = 2*sin(x)**2 + 2*cos(x)**2
>>> trigsimp(e)
2


Simplification occurs wherever trigonometric functions are located.

>>> trigsimp(log(e))
log(2)


Using $$method="groebner"$$ (or $$"combined"$$) might lead to greater simplification.

The old trigsimp routine can be accessed as with method ‘old’.

>>> from diofant import coth, tanh
>>> t = 3*tanh(x)**7 - 2/coth(x)**7
>>> trigsimp(t, method='old') == t
True
>>> trigsimp(t)
tanh(x)**7


### futrig¶

diofant.simplify.trigsimp.futrig(e, **kwargs)[source]

Return simplified e using Fu-like transformations. This is not the “Fu” algorithm. This is called by default from trigsimp. By default, hyperbolics subexpressions will be simplified, but this can be disabled by setting hyper=False.

Examples

>>> from diofant import trigsimp, tan, sinh, tanh
>>> from diofant.simplify.trigsimp import futrig
>>> from diofant.abc import x
>>> trigsimp(1/tan(x)**2)
tan(x)**(-2)

>>> futrig(sinh(x)/tanh(x))
cosh(x)


### fu¶

diofant.simplify.fu.fu(rv, measure=<function <lambda>>)[source]

Attempt to simplify expression by using transformation rules given in the algorithm by Fu et al.

fu() will try to minimize the objective function measure. By default this first minimizes the number of trig terms and then minimizes the number of total operations.

References

Examples

>>> from diofant.simplify.fu import fu
>>> from diofant import cos, sin, tan, pi, S, sqrt
>>> from diofant.abc import x, y, a, b

>>> fu(sin(50)**2 + cos(50)**2 + sin(pi/6))
3/2
>>> fu(sqrt(6)*cos(x) + sqrt(2)*sin(x))
2*sqrt(2)*sin(x + pi/3)


CTR1 example

>>> eq = sin(x)**4 - cos(y)**2 + sin(y)**2 + 2*cos(x)**2
>>> fu(eq)
cos(x)**4 - 2*cos(y)**2 + 2


CTR2 example

>>> fu(S.Half - cos(2*x)/2)
sin(x)**2


CTR3 example

>>> fu(sin(a)*(cos(b) - sin(b)) + cos(a)*(sin(b) + cos(b)))
sqrt(2)*sin(a + b + pi/4)


CTR4 example

>>> fu(sqrt(3)*cos(x)/2 + sin(x)/2)
sin(x + pi/3)


Example 1

>>> fu(1-sin(2*x)**2/4-sin(y)**2-cos(x)**4)
-cos(x)**2 + cos(y)**2


Example 2

>>> fu(cos(4*pi/9))
sin(pi/18)
>>> fu(cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9))
1/16


Example 3

>>> fu(tan(7*pi/18)+tan(5*pi/18)-sqrt(3)*tan(5*pi/18)*tan(7*pi/18))
-sqrt(3)


Objective function example

>>> fu(sin(x)/cos(x))  # default objective function
tan(x)
>>> fu(sin(x)/cos(x), measure=lambda x: -x.count_ops()) # maximize op count
sin(x)/cos(x)


## Power simplification¶

### powsimp¶

diofant.simplify.powsimp.powsimp(expr, deep=False, combine='all', force=False, measure=<function count_ops>)[source]

reduces expression by combining powers with similar bases and exponents.

Notes

If deep is True then powsimp() will also simplify arguments of functions. By default deep is set to False.

If force is True then bases will be combined without checking for assumptions, e.g. sqrt(x)*sqrt(y) -> sqrt(x*y) which is not true if x and y are both negative.

You can make powsimp() only combine bases or only combine exponents by changing combine=’base’ or combine=’exp’. By default, combine=’all’, which does both. combine=’base’ will only combine:

 a   a          a                          2x      x
x * y  =>  (x*y)   as well as things like 2   =>  4


and combine=’exp’ will only combine

 a   b      (a + b)
x * x  =>  x


combine=’exp’ will strictly only combine exponents in the way that used to be automatic. Also use deep=True if you need the old behavior.

When combine=’all’, ‘exp’ is evaluated first. Consider the first example below for when there could be an ambiguity relating to this. This is done so things like the second example can be completely combined. If you want ‘base’ combined first, do something like powsimp(powsimp(expr, combine=’base’), combine=’exp’).

Examples

>>> from diofant import powsimp, exp, log, symbols
>>> from diofant.abc import x, y, z, n
>>> powsimp(x**y*x**z*y**z, combine='all')
x**(y + z)*y**z
>>> powsimp(x**y*x**z*y**z, combine='exp')
x**(y + z)*y**z
>>> powsimp(x**y*x**z*y**z, combine='base', force=True)
x**y*(x*y)**z

>>> powsimp(x**z*x**y*n**z*n**y, combine='all', force=True)
(n*x)**(y + z)
>>> powsimp(x**z*x**y*n**z*n**y, combine='exp')
n**(y + z)*x**(y + z)
>>> powsimp(x**z*x**y*n**z*n**y, combine='base', force=True)
(n*x)**y*(n*x)**z

>>> x, y = symbols('x y', positive=True)
>>> powsimp(log(exp(x)*exp(y)))
log(E**x*E**y)
>>> powsimp(log(exp(x)*exp(y)), deep=True)
x + y


Radicals with Mul bases will be combined if combine=’exp’

>>> from diofant import sqrt, Mul
>>> x, y = symbols('x y')


Two radicals are automatically joined through Mul:

>>> a = sqrt(x*sqrt(y))
>>> a*a**3 == a**4
True


But if an integer power of that radical has been autoexpanded then Mul does not join the resulting factors:

>>> a**4  # auto expands to a Mul, no longer a Pow
x**2*y
>>> _*a  # so Mul doesn't combine them
x**2*y*sqrt(x*sqrt(y))
>>> powsimp(_)  # but powsimp will
(x*sqrt(y))**(5/2)
>>> powsimp(x*y*a)  # but won't when doing so would violate assumptions
x*y*sqrt(x*sqrt(y))


### powdenest¶

diofant.simplify.powsimp.powdenest(eq, force=False, polar=False)[source]

Collect exponents on powers as assumptions allow.

Given (bb**be)**e, this can be simplified as follows:
• if bb is positive, or
• e is an integer, or
• |be| < 1 then this simplifies to bb**(be*e)

Given a product of powers raised to a power, (bb1**be1 * bb2**be2...)**e, simplification can be done as follows:

• if e is positive, the gcd of all bei can be joined with e;
• all non-negative bb can be separated from those that are negative and their gcd can be joined with e; autosimplification already handles this separation.
• integer factors from powers that have integers in the denominator of the exponent can be removed from any term and the gcd of such integers can be joined with e

Setting force to True will make symbols that are not explicitly negative behave as though they are positive, resulting in more denesting.

Setting polar to True will do simplifications on the Riemann surface of the logarithm, also resulting in more denestings.

When there are sums of logs in exp() then a product of powers may be obtained e.g. exp(3*(log(a) + 2*log(b))) - > a**3*b**6.

Examples

>>> from diofant.abc import a, b, x, y, z
>>> from diofant import Symbol, exp, log, sqrt, symbols, powdenest

>>> powdenest((x**(2*a/3))**(3*x))
(x**(2*a/3))**(3*x)
>>> powdenest(exp(3*x*log(2)))
2**(3*x)


Assumptions may prevent expansion:

>>> powdenest(sqrt(x**2))
sqrt(x**2)

>>> p = symbols('p', positive=True)
>>> powdenest(sqrt(p**2))
p


No other expansion is done.

>>> i, j = symbols('i j', integer=True)
>>> powdenest((x**x)**(i + j))  # -X-> (x**x)**i*(x**x)**j
x**(x*(i + j))


But exp() will be denested by moving all non-log terms outside of the function; this may result in the collapsing of the exp to a power with a different base:

>>> powdenest(exp(3*y*log(x)))
x**(3*y)
>>> powdenest(exp(y*(log(a) + log(b))))
(a*b)**y
>>> powdenest(exp(3*(log(a) + log(b))))
a**3*b**3


If assumptions allow, symbols can also be moved to the outermost exponent:

>>> i = Symbol('i', integer=True)
>>> powdenest(((x**(2*i))**(3*y))**x)
((x**(2*i))**(3*y))**x
>>> powdenest(((x**(2*i))**(3*y))**x, force=True)
x**(6*i*x*y)

>>> powdenest(((x**(2*a/3))**(3*y/i))**x)
((x**(2*a/3))**(3*y/i))**x
>>> powdenest((x**(2*i)*y**(4*i))**z, force=True)
(x*y**2)**(2*i*z)

>>> n = Symbol('n', negative=True)

>>> powdenest((x**i)**y, force=True)
x**(i*y)
>>> powdenest((n**i)**x, force=True)
(n**i)**x


## Combinatorial simplification¶

### combsimp¶

diofant.simplify.combsimp.combsimp(expr)[source]

Simplify combinatorial expressions.

This function takes as input an expression containing factorials, binomials, Pochhammer symbol and other “combinatorial” functions, and tries to minimize the number of those functions and reduce the size of their arguments.

The algorithm works by rewriting all combinatorial functions as expressions involving rising factorials (Pochhammer symbols) and applies recurrence relations and other transformations applicable to rising factorials, to reduce their arguments, possibly letting the resulting rising factorial to cancel. Rising factorials with the second argument being an integer are expanded into polynomial forms and finally all other rising factorial are rewritten in terms of more familiar functions. If the initial expression consisted of gamma functions alone, the result is expressed in terms of gamma functions. If the initial expression consists of gamma function with some other combinatorial, the result is expressed in terms of gamma functions.

If the result is expressed using gamma functions, the following three additional steps are performed:

1. Reduce the number of gammas by applying the reflection theorem gamma(x)*gamma(1-x) == pi/sin(pi*x).
2. Reduce the number of gammas by applying the multiplication theorem gamma(x)*gamma(x+1/n)*…*gamma(x+(n-1)/n) == C*gamma(n*x).
3. Reduce the number of prefactors by absorbing them into gammas, where possible.

All transformation rules can be found (or was derived from) here:

Examples

>>> from diofant.simplify import combsimp
>>> from diofant import factorial, binomial
>>> from diofant.abc import n, k

>>> combsimp(factorial(n)/factorial(n - 3))
n*(n - 2)*(n - 1)
>>> combsimp(binomial(n+1, k+1)/binomial(n, k))
(n + 1)/(k + 1)


## Square Root Denest¶

### sqrtdenest¶

diofant.simplify.sqrtdenest.sqrtdenest(expr, max_iter=3)[source]

Denests sqrts in an expression that contain other square roots if possible, otherwise returns the expr unchanged. This is based on the algorithms of [1].

References

 [R10381038] D. J. Jeffrey and A. D. Rich, ‘Symplifying Square Roots of Square Roots by Denesting’ (available at http://www.cybertester.com/data/denest.pdf)

Examples

>>> from diofant.simplify.sqrtdenest import sqrtdenest
>>> from diofant import sqrt
>>> sqrtdenest(sqrt(5 + 2 * sqrt(6)))
sqrt(2) + sqrt(3)


diofant.simplify.sqrtdenest.unrad(eq, *syms, **flags)[source]

Remove radicals with symbolic arguments and return (eq, cov), None or raise an error:

None is returned if there are no radicals to remove.

NotImplementedError is raised if there are radicals and they cannot be removed or if the relationship between the original symbols and the change of variable needed to rewrite the system as a polynomial cannot be solved.

Otherwise the tuple, (eq, cov), is returned where:

eq, cov
eq is an equation without radicals (in the symbol(s) of
interest) whose solutions are a superset of the solutions to the
original expression. eq might be re-written in terms of a new
variable; the relationship to the original variables is given by
cov which is a list containing v and v**p - b where
p is the power needed to clear the radical and b is the
radical now expressed as a polynomial in the symbols of interest.
For example, for sqrt(2 - x) the tuple would be
(c, c**2 - 2 + x). The solutions of eq will contain
solutions to the original equation (if there are any).

syms
an iterable of symbols which, if provided, will limit the focus of radical removal: only radicals with one or more of the symbols of interest will be cleared. All free symbols are used if syms is not set.

flags are used internally for communication during recursive calls. Two options are also recognized:

take, when defined, is interpreted as a single-argument function
that returns True if a given Pow should be handled.


Radicals can be removed from an expression if:

*   all bases of the radicals are the same; a change of variables is
done in this case.
*   if all radicals appear in one term of the expression
*   there are only 4 terms with sqrt() factors or there are less than
four terms having sqrt() factors
*   there are only two terms with radicals


Examples

>>> from diofant.solvers.solvers import unrad
>>> from diofant.abc import x
>>> from diofant import cbrt, sqrt, Rational, root, real_roots, solve

>>> unrad(sqrt(x)*cbrt(x) + 2)
(x**5 - 64, [])
>>> unrad(sqrt(x) + root(x + 1, 3))
(x**3 - x**2 - 2*x - 1, [])
>>> eq = sqrt(x) + root(x, 3) - 2
(_p**3 + _p**2 - 2, [_p, -x + _p**6])


## Common Subexpresion Elimination¶

### cse¶

diofant.simplify.cse_main.cse(exprs, symbols=None, optimizations=None, postprocess=None, order='canonical')[source]

Perform common subexpression elimination on an expression.

Parameters: exprs : list of diofant expressions, or a single diofant 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’. 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 diofant expressions The reduced expressions with all of the replacements above.

Examples

>>> from diofant import cse, SparseMatrix
>>> from diofant.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])


Note that currently, y + z will not get substituted if -y - z is used.

>>> cse(((w + x + y + z)*(w - y - z))/(w + x)**3)
([(x0, w + x)], [(w - y - z)*(x0 + y + z)/x0**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


### opt_cse¶

diofant.simplify.cse_main.opt_cse(exprs, order='canonical')[source]

Find optimization opportunities in Adds, Muls, Pows and negative coefficient Muls

Parameters: exprs : list of diofant expressions The expressions to optimize. order : string, ‘none’ or ‘canonical’ The order by which Mul and Add arguments are processed. For large expressions where speed is a concern, use the setting order=’none’. opt_subs : dictionary of expression substitutions The expression substitutions which can be useful to optimize CSE.

Examples

>>> from diofant.simplify.cse_main import opt_cse
>>> from diofant.abc import x
>>> opt_subs = opt_cse([x**-2])
>>> opt_subs
{x**(-2): 1/(x**2)}


### tree_cse¶

diofant.simplify.cse_main.tree_cse(exprs, symbols, opt_subs={}, order='canonical')[source]

Perform raw CSE on expression tree, taking opt_subs into account.

Parameters: exprs : list of diofant expressions The expressions to reduce. symbols : infinite iterator yielding unique Symbols The symbols used to label the common subexpressions which are pulled out. opt_subs : dictionary of expression substitutions The expressions to be substituted before any CSE action is performed. order : string, ‘none’ or ‘canonical’ The order by which Mul and Add arguments are processed. For large expressions where speed is a concern, use the setting order=’none’.

## Hypergeometric Function Expansion¶

### hyperexpand¶

diofant.simplify.hyperexpand.hyperexpand(f, allow_hyper=False, rewrite='default')[source]

Expand hypergeometric functions. If allow_hyper is True, allow partial simplification (that is a result different from input, but still containing hypergeometric functions).

Examples

>>> from diofant.simplify.hyperexpand import hyperexpand
>>> from diofant.functions import hyper
>>> from diofant.abc import z
>>> hyperexpand(hyper([], [], z))
E**z


Non-hyperegeometric parts of the expression and hypergeometric expressions that are not recognised are left unchanged:

>>> hyperexpand(1 + hyper([1, 1, 1], [], z))
hyper((1, 1, 1), (), z) + 1


## Traversal Tools¶

### use¶

diofant.simplify.traversaltools.use(expr, func, level=0, args=(), kwargs={})[source]

Use func to transform expr at the given level.

Examples

>>> from diofant import use, expand
>>> from diofant.abc import x, y

>>> f = (x + y)**2*x + 1

>>> use(f, expand, level=2)
x*(x**2 + 2*x*y + y**2) + 1
>>> expand(f)
x**3 + 2*x**2*y + x*y**2 + 1


## EPath Tools¶

### EPath class¶

class diofant.simplify.epathtools.EPath[source]

Manipulate expressions using paths.

EPath grammar in EBNF notation:

literal   ::= /[A-Za-z_][A-Za-z_0-9]*/
number    ::= /-?\d+/
type      ::= literal
attribute ::= literal "?"
all       ::= "*"
slice     ::= "[" number? (":" number? (":" number?)?)? "]"
range     ::= all | slice
query     ::= (type | attribute) ("|" (type | attribute))*
selector  ::= range | query range?
path      ::= "/" selector ("/" selector)*

apply(expr, func, args=None, kwargs=None)[source]

Modify parts of an expression selected by a path.

Examples

>>> from diofant.simplify.epathtools import EPath
>>> from diofant import sin, cos, E
>>> from diofant.abc import x, y, z, t

>>> path = EPath("/*/[0]/Symbol")
>>> expr = [((x, 1), 2), ((3, y), z)]

>>> path.apply(expr, lambda expr: expr**2)
[((x**2, 1), 2), ((3, y**2), z)]

>>> path = EPath("/*/*/Symbol")
>>> expr = t + sin(x + 1) + cos(x + y + E)

>>> path.apply(expr, lambda expr: 2*expr)
t + sin(2*x + 1) + cos(2*x + 2*y + E)

select(expr)[source]

Retrieve parts of an expression selected by a path.

Examples

>>> from diofant.simplify.epathtools import EPath
>>> from diofant import sin, cos, E
>>> from diofant.abc import x, y, z, t

>>> path = EPath("/*/[0]/Symbol")
>>> expr = [((x, 1), 2), ((3, y), z)]

>>> path.select(expr)
[x, y]

>>> path = EPath("/*/*/Symbol")
>>> expr = t + sin(x + 1) + cos(x + y + E)

>>> path.select(expr)
[x, x, y]


### epath¶

diofant.simplify.epathtools.epath(path, expr=None, func=None, args=None, kwargs=None)[source]

Manipulate parts of an expression selected by a path.

This function allows to manipulate large nested expressions in single line of code, utilizing techniques to those applied in XML processing standards (e.g. XPath).

If func is None, epath() retrieves elements selected by the path. Otherwise it applies func to each matching element.

Note that it is more efficient to create an EPath object and use the select and apply methods of that object, since this will compile the path string only once. This function should only be used as a convenient shortcut for interactive use.

This is the supported syntax:

• select all: /*
Equivalent of for arg in args:.
• select slice: /[0] or /[1:5] or /[1:5:2]
Supports standard Python’s slice syntax.
• select by type: /list or /list|tuple
Emulates isinstance.
• select by attribute: /__iter__?
Emulates hasattr.
Parameters: path : str | EPath A path as a string or a compiled EPath. expr : Basic | iterable An expression or a container of expressions. func : callable (optional) A callable that will be applied to matching parts. args : tuple (optional) Additional positional arguments to func. kwargs : dict (optional) Additional keyword arguments to func.

Examples

>>> from diofant.simplify.epathtools import epath
>>> from diofant import sin, cos, E
>>> from diofant.abc import x, y, z, t

>>> path = "/*/[0]/Symbol"
>>> expr = [((x, 1), 2), ((3, y), z)]

>>> epath(path, expr)
[x, y]
>>> epath(path, expr, lambda expr: expr**2)
[((x**2, 1), 2), ((3, y**2), z)]

>>> path = "/*/*/Symbol"
>>> expr = t + sin(x + 1) + cos(x + y + E)

>>> epath(path, expr)
[x, x, y]
>>> epath(path, expr, lambda expr: 2*expr)
t + sin(2*x + 1) + cos(2*x + 2*y + E)