from collections import defaultdict from sympy.concrete.products import Product from sympy.concrete.summations import Sum from sympy.core import (Basic, S, Add, Mul, Pow, Symbol, sympify, expand_func, Function, Dummy, Expr, factor_terms, expand_power_exp, Eq) from sympy.core.exprtools import factor_nc from sympy.core.parameters import global_parameters from sympy.core.function import (expand_log, count_ops, _mexpand, nfloat, expand_mul, expand) from sympy.core.numbers import Float, I, pi, Rational from sympy.core.relational import Relational from sympy.core.rules import Transform from sympy.core.sorting import ordered from sympy.core.sympify import _sympify from sympy.core.traversal import bottom_up as _bottom_up, walk as _walk from sympy.functions import gamma, exp, sqrt, log, exp_polar, re from sympy.functions.combinatorial.factorials import CombinatorialFunction from sympy.functions.elementary.complexes import unpolarify, Abs, sign from sympy.functions.elementary.exponential import ExpBase from sympy.functions.elementary.hyperbolic import HyperbolicFunction from sympy.functions.elementary.integers import ceiling from sympy.functions.elementary.piecewise import (Piecewise, piecewise_fold, piecewise_simplify) from sympy.functions.elementary.trigonometric import TrigonometricFunction from sympy.functions.special.bessel import (BesselBase, besselj, besseli, besselk, bessely, jn) from sympy.functions.special.tensor_functions import KroneckerDelta from sympy.integrals.integrals import Integral from sympy.matrices.expressions import (MatrixExpr, MatAdd, MatMul, MatPow, MatrixSymbol) from sympy.polys import together, cancel, factor from sympy.polys.numberfields.minpoly import _is_sum_surds, _minimal_polynomial_sq from sympy.simplify.combsimp import combsimp from sympy.simplify.cse_opts import sub_pre, sub_post from sympy.simplify.hyperexpand import hyperexpand from sympy.simplify.powsimp import powsimp from sympy.simplify.radsimp import radsimp, fraction, collect_abs from sympy.simplify.sqrtdenest import sqrtdenest from sympy.simplify.trigsimp import trigsimp, exptrigsimp from sympy.utilities.decorator import deprecated from sympy.utilities.iterables import has_variety, sift, subsets, iterable from sympy.utilities.misc import as_int import mpmath def separatevars(expr, symbols=[], dict=False, force=False): """ 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. Explanation =========== 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 sympy.abc import x, y, z, alpha >>> from sympy 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) is None True """ expr = sympify(expr) if dict: return _separatevars_dict(_separatevars(expr, force), symbols) else: return _separatevars(expr, force) def _separatevars(expr, force): if isinstance(expr, Abs): arg = expr.args[0] if arg.is_Mul and not arg.is_number: s = separatevars(arg, dict=True, force=force) if s is not None: return Mul(*map(expr.func, s.values())) else: return expr if len(expr.free_symbols) < 2: return expr # don't destroy a Mul since much of the work may already be done if expr.is_Mul: args = list(expr.args) changed = False for i, a in enumerate(args): args[i] = separatevars(a, force) changed = changed or args[i] != a if changed: expr = expr.func(*args) return expr # get a Pow ready for expansion if expr.is_Pow and expr.base != S.Exp1: expr = Pow(separatevars(expr.base, force=force), expr.exp) # First try other expansion methods expr = expr.expand(mul=False, multinomial=False, force=force) _expr, reps = posify(expr) if force else (expr, {}) expr = factor(_expr).subs(reps) if not expr.is_Add: return expr # Find any common coefficients to pull out args = list(expr.args) commonc = args[0].args_cnc(cset=True, warn=False)[0] for i in args[1:]: commonc &= i.args_cnc(cset=True, warn=False)[0] commonc = Mul(*commonc) commonc = commonc.as_coeff_Mul()[1] # ignore constants commonc_set = commonc.args_cnc(cset=True, warn=False)[0] # remove them for i, a in enumerate(args): c, nc = a.args_cnc(cset=True, warn=False) c = c - commonc_set args[i] = Mul(*c)*Mul(*nc) nonsepar = Add(*args) if len(nonsepar.free_symbols) > 1: _expr = nonsepar _expr, reps = posify(_expr) if force else (_expr, {}) _expr = (factor(_expr)).subs(reps) if not _expr.is_Add: nonsepar = _expr return commonc*nonsepar def _separatevars_dict(expr, symbols): if symbols: if not all(t.is_Atom for t in symbols): raise ValueError("symbols must be Atoms.") symbols = list(symbols) elif symbols is None: return {'coeff': expr} else: symbols = list(expr.free_symbols) if not symbols: return None ret = {i: [] for i in symbols + ['coeff']} for i in Mul.make_args(expr): expsym = i.free_symbols intersection = set(symbols).intersection(expsym) if len(intersection) > 1: return None if len(intersection) == 0: # There are no symbols, so it is part of the coefficient ret['coeff'].append(i) else: ret[intersection.pop()].append(i) # rebuild for k, v in ret.items(): ret[k] = Mul(*v) return ret def posify(eq): """Return ``eq`` (with generic symbols made positive) and a dictionary containing the mapping between the old and new symbols. Explanation =========== 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 sympy import posify, Symbol, log, solve >>> from sympy.abc import x >>> posify(x + Symbol('p', positive=True) + Symbol('n', negative=True)) (_x + n + p, {_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) [-2, 2] >>> eq_x, reps = posify([eq, x]); eq_x [_x**2 - 4, _x] >>> solve(*eq_x) [2] """ eq = sympify(eq) if iterable(eq): f = type(eq) eq = list(eq) syms = set() for e in eq: syms = syms.union(e.atoms(Symbol)) reps = {} for s in syms: reps.update({v: k for k, v in posify(s)[1].items()}) for i, e in enumerate(eq): eq[i] = e.subs(reps) return f(eq), {r: s for s, r in reps.items()} reps = {s: Dummy(s.name, positive=True, **s.assumptions0) for s in eq.free_symbols if s.is_positive is None} eq = eq.subs(reps) return eq, {r: s for s, r in reps.items()} def hypersimp(f, k): """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. Explanation =========== The algorithm 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. For more information on the implemented algorithm refer to: 1. W. Koepf, Algorithms for m-fold Hypergeometric Summation, Journal of Symbolic Computation (1995) 20, 399-417 """ f = sympify(f) g = f.subs(k, k + 1) / f g = g.rewrite(gamma) if g.has(Piecewise): g = piecewise_fold(g) g = g.args[-1][0] g = expand_func(g) g = powsimp(g, deep=True, combine='exp') if g.is_rational_function(k): return simplify(g, ratio=S.Infinity) else: return None def hypersimilar(f, g, k): """ Returns True if ``f`` and ``g`` are hyper-similar. Explanation =========== 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. For more information see hypersimp(). """ f, g = list(map(sympify, (f, g))) h = (f/g).rewrite(gamma) h = h.expand(func=True, basic=False) return h.is_rational_function(k) def signsimp(expr, evaluate=None): """Make all Add sub-expressions canonical wrt sign. Explanation =========== If an Add subexpression, ``a``, can have a sign extracted, as determined by could_extract_minus_sign, it is replaced with Mul(-1, a, evaluate=False). This allows signs to be extracted from powers and products. Examples ======== >>> from sympy import signsimp, exp, symbols >>> from sympy.abc import x, y >>> i = symbols('i', odd=True) >>> n = -1 + 1/x >>> n/x/(-n)**2 - 1/n/x (-1 + 1/x)/(x*(1 - 1/x)**2) - 1/(x*(-1 + 1/x)) >>> signsimp(_) 0 >>> x*n + x*-n x*(-1 + 1/x) + x*(1 - 1/x) >>> signsimp(_) 0 Since powers automatically handle leading signs >>> (-2)**i -2**i signsimp can be used to put the base of a power with an integer exponent into canonical form: >>> n**i (-1 + 1/x)**i By default, signsimp does not leave behind any hollow simplification: if making an Add canonical wrt sign didn't change the expression, the original Add is restored. If this is not desired then the keyword ``evaluate`` can be set to False: >>> e = exp(y - x) >>> signsimp(e) == e True >>> signsimp(e, evaluate=False) exp(-(x - y)) """ if evaluate is None: evaluate = global_parameters.evaluate expr = sympify(expr) if not isinstance(expr, (Expr, Relational)) or expr.is_Atom: return expr # get rid of an pre-existing unevaluation regarding sign e = expr.replace(lambda x: x.is_Mul and -(-x) != x, lambda x: -(-x)) e = sub_post(sub_pre(e)) if not isinstance(e, (Expr, Relational)) or e.is_Atom: return e if e.is_Add: rv = e.func(*[signsimp(a) for a in e.args]) if not evaluate and isinstance(rv, Add ) and rv.could_extract_minus_sign(): return Mul(S.NegativeOne, -rv, evaluate=False) return rv if evaluate: e = e.replace(lambda x: x.is_Mul and -(-x) != x, lambda x: -(-x)) return e def simplify(expr, ratio=1.7, measure=count_ops, rational=False, inverse=False, doit=True, **kwargs): """Simplifies the given expression. Explanation =========== Simplification is not a well defined term and the exact strategies this function tries can change in the future versions of SymPy. 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 do not know anything about the structure of the expression, simplify() tries to apply intelligent heuristics to make the input expression "simpler". For example: >>> from sympy import simplify, cos, sin >>> from sympy.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 sympy import trigsimp, cancel >>> trigsimp(a) (x**2 + x)/x >>> cancel(_) x + 1 In some cases, applying :func:`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 :func:`~.count_ops`, which returns the total number of operations in the expression. For example, if ``ratio=1``, ``simplify`` output cannot be longer than input. :: >>> from sympy 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 do not know what a good metric would be, the default, ``count_ops``, is a good one. For example: >>> from sympy 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**(1 - log(a))) >>> 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 sympy 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. If ``rational=True``, Floats will be recast as Rationals before simplification. If ``rational=None``, Floats will be recast as Rationals but the result will be recast as Floats. If rational=False(default) then nothing will be done to the Floats. If ``inverse=True``, it will be assumed that a composition of inverse functions, such as sin and asin, can be cancelled in any order. For example, ``asin(sin(x))`` will yield ``x`` without checking whether x belongs to the set where this relation is true. The default is False. Note that ``simplify()`` automatically calls ``doit()`` on the final expression. You can avoid this behavior by passing ``doit=False`` as an argument. Also, it should be noted that simplifying a boolean expression is not well defined. If the expression prefers automatic evaluation (such as :obj:`~.Eq()` or :obj:`~.Or()`), simplification will return ``True`` or ``False`` if truth value can be determined. If the expression is not evaluated by default (such as :obj:`~.Predicate()`), simplification will not reduce it and you should use :func:`~.refine()` or :func:`~.ask()` function. This inconsistency will be resolved in future version. See Also ======== sympy.assumptions.refine.refine : Simplification using assumptions. sympy.assumptions.ask.ask : Query for boolean expressions using assumptions. """ def shorter(*choices): """ Return the choice that has the fewest ops. In case of a tie, the expression listed first is selected. """ if not has_variety(choices): return choices[0] return min(choices, key=measure) def done(e): rv = e.doit() if doit else e return shorter(rv, collect_abs(rv)) expr = sympify(expr, rational=rational) kwargs = { "ratio": kwargs.get('ratio', ratio), "measure": kwargs.get('measure', measure), "rational": kwargs.get('rational', rational), "inverse": kwargs.get('inverse', inverse), "doit": kwargs.get('doit', doit)} # no routine for Expr needs to check for is_zero if isinstance(expr, Expr) and expr.is_zero: return S.Zero if not expr.is_Number else expr _eval_simplify = getattr(expr, '_eval_simplify', None) if _eval_simplify is not None: return _eval_simplify(**kwargs) original_expr = expr = collect_abs(signsimp(expr)) if not isinstance(expr, Basic) or not expr.args: # XXX: temporary hack return expr if inverse and expr.has(Function): expr = inversecombine(expr) if not expr.args: # simplified to atomic return expr # do deep simplification handled = Add, Mul, Pow, ExpBase expr = expr.replace( # here, checking for x.args is not enough because Basic has # args but Basic does not always play well with replace, e.g. # when simultaneous is True found expressions will be masked # off with a Dummy but not all Basic objects in an expression # can be replaced with a Dummy lambda x: isinstance(x, Expr) and x.args and not isinstance( x, handled), lambda x: x.func(*[simplify(i, **kwargs) for i in x.args]), simultaneous=False) if not isinstance(expr, handled): return done(expr) if not expr.is_commutative: expr = nc_simplify(expr) # TODO: Apply different strategies, considering expression pattern: # is it a purely rational function? Is there any trigonometric function?... # See also https://github.com/sympy/sympy/pull/185. # rationalize Floats floats = False if rational is not False and expr.has(Float): floats = True expr = nsimplify(expr, rational=True) expr = _bottom_up(expr, lambda w: getattr(w, 'normal', lambda: w)()) expr = Mul(*powsimp(expr).as_content_primitive()) _e = cancel(expr) expr1 = shorter(_e, _mexpand(_e).cancel()) # issue 6829 expr2 = shorter(together(expr, deep=True), together(expr1, deep=True)) if ratio is S.Infinity: expr = expr2 else: expr = shorter(expr2, expr1, expr) if not isinstance(expr, Basic): # XXX: temporary hack return expr expr = factor_terms(expr, sign=False) # must come before `Piecewise` since this introduces more `Piecewise` terms if expr.has(sign): expr = expr.rewrite(Abs) # Deal with Piecewise separately to avoid recursive growth of expressions if expr.has(Piecewise): # Fold into a single Piecewise expr = piecewise_fold(expr) # Apply doit, if doit=True expr = done(expr) # Still a Piecewise? if expr.has(Piecewise): # Fold into a single Piecewise, in case doit lead to some # expressions being Piecewise expr = piecewise_fold(expr) # kroneckersimp also affects Piecewise if expr.has(KroneckerDelta): expr = kroneckersimp(expr) # Still a Piecewise? if expr.has(Piecewise): # Do not apply doit on the segments as it has already # been done above, but simplify expr = piecewise_simplify(expr, deep=True, doit=False) # Still a Piecewise? if expr.has(Piecewise): # Try factor common terms expr = shorter(expr, factor_terms(expr)) # As all expressions have been simplified above with the # complete simplify, nothing more needs to be done here return expr # hyperexpand automatically only works on hypergeometric terms # Do this after the Piecewise part to avoid recursive expansion expr = hyperexpand(expr) if expr.has(KroneckerDelta): expr = kroneckersimp(expr) if expr.has(BesselBase): expr = besselsimp(expr) if expr.has(TrigonometricFunction, HyperbolicFunction): expr = trigsimp(expr, deep=True) if expr.has(log): expr = shorter(expand_log(expr, deep=True), logcombine(expr)) if expr.has(CombinatorialFunction, gamma): # expression with gamma functions or non-integer arguments is # automatically passed to gammasimp expr = combsimp(expr) if expr.has(Sum): expr = sum_simplify(expr, **kwargs) if expr.has(Integral): expr = expr.xreplace({ i: factor_terms(i) for i in expr.atoms(Integral)}) if expr.has(Product): expr = product_simplify(expr, **kwargs) from sympy.physics.units import Quantity if expr.has(Quantity): from sympy.physics.units.util import quantity_simplify expr = quantity_simplify(expr) short = shorter(powsimp(expr, combine='exp', deep=True), powsimp(expr), expr) short = shorter(short, cancel(short)) short = shorter(short, factor_terms(short), expand_power_exp(expand_mul(short))) if short.has(TrigonometricFunction, HyperbolicFunction, ExpBase, exp): short = exptrigsimp(short) # get rid of hollow 2-arg Mul factorization hollow_mul = Transform( lambda x: Mul(*x.args), lambda x: x.is_Mul and len(x.args) == 2 and x.args[0].is_Number and x.args[1].is_Add and x.is_commutative) expr = short.xreplace(hollow_mul) numer, denom = expr.as_numer_denom() if denom.is_Add: n, d = fraction(radsimp(1/denom, symbolic=False, max_terms=1)) if n is not S.One: expr = (numer*n).expand()/d if expr.could_extract_minus_sign(): n, d = fraction(expr) if d != 0: expr = signsimp(-n/(-d)) if measure(expr) > ratio*measure(original_expr): expr = original_expr # restore floats if floats and rational is None: expr = nfloat(expr, exponent=False) return done(expr) def sum_simplify(s, **kwargs): """Main function for Sum simplification""" if not isinstance(s, Add): s = s.xreplace({a: sum_simplify(a, **kwargs) for a in s.atoms(Add) if a.has(Sum)}) s = expand(s) if not isinstance(s, Add): return s terms = s.args s_t = [] # Sum Terms o_t = [] # Other Terms for term in terms: sum_terms, other = sift(Mul.make_args(term), lambda i: isinstance(i, Sum), binary=True) if not sum_terms: o_t.append(term) continue other = [Mul(*other)] s_t.append(Mul(*(other + [s._eval_simplify(**kwargs) for s in sum_terms]))) result = Add(sum_combine(s_t), *o_t) return result def sum_combine(s_t): """Helper function for Sum simplification Attempts to simplify a list of sums, by combining limits / sum function's returns the simplified sum """ used = [False] * len(s_t) for method in range(2): for i, s_term1 in enumerate(s_t): if not used[i]: for j, s_term2 in enumerate(s_t): if not used[j] and i != j: temp = sum_add(s_term1, s_term2, method) if isinstance(temp, (Sum, Mul)): s_t[i] = temp s_term1 = s_t[i] used[j] = True result = S.Zero for i, s_term in enumerate(s_t): if not used[i]: result = Add(result, s_term) return result def factor_sum(self, limits=None, radical=False, clear=False, fraction=False, sign=True): """Return Sum with constant factors extracted. If ``limits`` is specified then ``self`` is the summand; the other keywords are passed to ``factor_terms``. Examples ======== >>> from sympy import Sum >>> from sympy.abc import x, y >>> from sympy.simplify.simplify import factor_sum >>> s = Sum(x*y, (x, 1, 3)) >>> factor_sum(s) y*Sum(x, (x, 1, 3)) >>> factor_sum(s.function, s.limits) y*Sum(x, (x, 1, 3)) """ # XXX deprecate in favor of direct call to factor_terms kwargs = {"radical": radical, "clear": clear, "fraction": fraction, "sign": sign} expr = Sum(self, *limits) if limits else self return factor_terms(expr, **kwargs) def sum_add(self, other, method=0): """Helper function for Sum simplification""" #we know this is something in terms of a constant * a sum #so we temporarily put the constants inside for simplification #then simplify the result def __refactor(val): args = Mul.make_args(val) sumv = next(x for x in args if isinstance(x, Sum)) constant = Mul(*[x for x in args if x != sumv]) return Sum(constant * sumv.function, *sumv.limits) if isinstance(self, Mul): rself = __refactor(self) else: rself = self if isinstance(other, Mul): rother = __refactor(other) else: rother = other if type(rself) is type(rother): if method == 0: if rself.limits == rother.limits: return factor_sum(Sum(rself.function + rother.function, *rself.limits)) elif method == 1: if simplify(rself.function - rother.function) == 0: if len(rself.limits) == len(rother.limits) == 1: i = rself.limits[0][0] x1 = rself.limits[0][1] y1 = rself.limits[0][2] j = rother.limits[0][0] x2 = rother.limits[0][1] y2 = rother.limits[0][2] if i == j: if x2 == y1 + 1: return factor_sum(Sum(rself.function, (i, x1, y2))) elif x1 == y2 + 1: return factor_sum(Sum(rself.function, (i, x2, y1))) return Add(self, other) def product_simplify(s, **kwargs): """Main function for Product simplification""" terms = Mul.make_args(s) p_t = [] # Product Terms o_t = [] # Other Terms deep = kwargs.get('deep', True) for term in terms: if isinstance(term, Product): if deep: p_t.append(Product(term.function.simplify(**kwargs), *term.limits)) else: p_t.append(term) else: o_t.append(term) used = [False] * len(p_t) for method in range(2): for i, p_term1 in enumerate(p_t): if not used[i]: for j, p_term2 in enumerate(p_t): if not used[j] and i != j: tmp_prod = product_mul(p_term1, p_term2, method) if isinstance(tmp_prod, Product): p_t[i] = tmp_prod used[j] = True result = Mul(*o_t) for i, p_term in enumerate(p_t): if not used[i]: result = Mul(result, p_term) return result def product_mul(self, other, method=0): """Helper function for Product simplification""" if type(self) is type(other): if method == 0: if self.limits == other.limits: return Product(self.function * other.function, *self.limits) elif method == 1: if simplify(self.function - other.function) == 0: if len(self.limits) == len(other.limits) == 1: i = self.limits[0][0] x1 = self.limits[0][1] y1 = self.limits[0][2] j = other.limits[0][0] x2 = other.limits[0][1] y2 = other.limits[0][2] if i == j: if x2 == y1 + 1: return Product(self.function, (i, x1, y2)) elif x1 == y2 + 1: return Product(self.function, (i, x2, y1)) return Mul(self, other) def _nthroot_solve(p, n, prec): """ helper function for ``nthroot`` It denests ``p**Rational(1, n)`` using its minimal polynomial """ from sympy.solvers import solve while n % 2 == 0: p = sqrtdenest(sqrt(p)) n = n // 2 if n == 1: return p pn = p**Rational(1, n) x = Symbol('x') f = _minimal_polynomial_sq(p, n, x) if f is None: return None sols = solve(f, x) for sol in sols: if abs(sol - pn).n() < 1./10**prec: sol = sqrtdenest(sol) if _mexpand(sol**n) == p: return sol def logcombine(expr, force=False): """ Takes logarithms and combines them using the following rules: - log(x) + log(y) == log(x*y) if both are positive - 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. Examples ======== >>> from sympy import Symbol, symbols, log, logcombine, I >>> from sympy.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', 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) See Also ======== posify: replace all symbols with symbols having positive assumptions sympy.core.function.expand_log: expand the logarithms of products and powers; the opposite of logcombine """ def f(rv): if not (rv.is_Add or rv.is_Mul): return rv def gooda(a): # bool to tell whether the leading ``a`` in ``a*log(x)`` # could appear as log(x**a) return (a is not S.NegativeOne and # -1 *could* go, but we disallow (a.is_extended_real or force and a.is_extended_real is not False)) def goodlog(l): # bool to tell whether log ``l``'s argument can combine with others a = l.args[0] return a.is_positive or force and a.is_nonpositive is not False other = [] logs = [] log1 = defaultdict(list) for a in Add.make_args(rv): if isinstance(a, log) and goodlog(a): log1[()].append(([], a)) elif not a.is_Mul: other.append(a) else: ot = [] co = [] lo = [] for ai in a.args: if ai.is_Rational and ai < 0: ot.append(S.NegativeOne) co.append(-ai) elif isinstance(ai, log) and goodlog(ai): lo.append(ai) elif gooda(ai): co.append(ai) else: ot.append(ai) if len(lo) > 1: logs.append((ot, co, lo)) elif lo: log1[tuple(ot)].append((co, lo[0])) else: other.append(a) # if there is only one log in other, put it with the # good logs if len(other) == 1 and isinstance(other[0], log): log1[()].append(([], other.pop())) # if there is only one log at each coefficient and none have # an exponent to place inside the log then there is nothing to do if not logs and all(len(log1[k]) == 1 and log1[k][0] == [] for k in log1): return rv # collapse multi-logs as far as possible in a canonical way # TODO: see if x*log(a)+x*log(a)*log(b) -> x*log(a)*(1+log(b))? # -- in this case, it's unambiguous, but if it were were a log(c) in # each term then it's arbitrary whether they are grouped by log(a) or # by log(c). So for now, just leave this alone; it's probably better to # let the user decide for o, e, l in logs: l = list(ordered(l)) e = log(l.pop(0).args[0]**Mul(*e)) while l: li = l.pop(0) e = log(li.args[0]**e) c, l = Mul(*o), e if isinstance(l, log): # it should be, but check to be sure log1[(c,)].append(([], l)) else: other.append(c*l) # logs that have the same coefficient can multiply for k in list(log1.keys()): log1[Mul(*k)] = log(logcombine(Mul(*[ l.args[0]**Mul(*c) for c, l in log1.pop(k)]), force=force), evaluate=False) # logs that have oppositely signed coefficients can divide for k in ordered(list(log1.keys())): if k not in log1: # already popped as -k continue if -k in log1: # figure out which has the minus sign; the one with # more op counts should be the one num, den = k, -k if num.count_ops() > den.count_ops(): num, den = den, num other.append( num*log(log1.pop(num).args[0]/log1.pop(den).args[0], evaluate=False)) else: other.append(k*log1.pop(k)) return Add(*other) return _bottom_up(expr, f) def inversecombine(expr): """Simplify the composition of a function and its inverse. Explanation =========== No attention is paid to whether the inverse is a left inverse or a right inverse; thus, the result will in general not be equivalent to the original expression. Examples ======== >>> from sympy.simplify.simplify import inversecombine >>> from sympy import asin, sin, log, exp >>> from sympy.abc import x >>> inversecombine(asin(sin(x))) x >>> inversecombine(2*log(exp(3*x))) 6*x """ def f(rv): if isinstance(rv, log): if isinstance(rv.args[0], exp) or (rv.args[0].is_Pow and rv.args[0].base == S.Exp1): rv = rv.args[0].exp elif rv.is_Function and hasattr(rv, "inverse"): if (len(rv.args) == 1 and len(rv.args[0].args) == 1 and isinstance(rv.args[0], rv.inverse(argindex=1))): rv = rv.args[0].args[0] if rv.is_Pow and rv.base == S.Exp1: if isinstance(rv.exp, log): rv = rv.exp.args[0] return rv return _bottom_up(expr, f) def kroneckersimp(expr): """ Simplify expressions with KroneckerDelta. The only simplification currently attempted is to identify multiplicative cancellation: Examples ======== >>> from sympy import KroneckerDelta, kroneckersimp >>> from sympy.abc import i >>> kroneckersimp(1 + KroneckerDelta(0, i) * KroneckerDelta(1, i)) 1 """ def args_cancel(args1, args2): for i1 in range(2): for i2 in range(2): a1 = args1[i1] a2 = args2[i2] a3 = args1[(i1 + 1) % 2] a4 = args2[(i2 + 1) % 2] if Eq(a1, a2) is S.true and Eq(a3, a4) is S.false: return True return False def cancel_kronecker_mul(m): args = m.args deltas = [a for a in args if isinstance(a, KroneckerDelta)] for delta1, delta2 in subsets(deltas, 2): args1 = delta1.args args2 = delta2.args if args_cancel(args1, args2): return S.Zero * m # In case of oo etc return m if not expr.has(KroneckerDelta): return expr if expr.has(Piecewise): expr = expr.rewrite(KroneckerDelta) newexpr = expr expr = None while newexpr != expr: expr = newexpr newexpr = expr.replace(lambda e: isinstance(e, Mul), cancel_kronecker_mul) return expr def besselsimp(expr): """ Simplify bessel-type functions. Explanation =========== 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 strigonometric 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 sympy import besselj, besseli, besselsimp, polar_lift, I, S >>> from sympy.abc import z, nu >>> besselsimp(besselj(nu, z*polar_lift(-1))) exp(I*pi*nu)*besselj(nu, z) >>> besselsimp(besseli(nu, z*polar_lift(-I))) exp(-I*pi*nu/2)*besselj(nu, z) >>> besselsimp(besseli(S(-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 """ # TODO # - better algorithm? # - simplify (cos(pi*b)*besselj(b,z) - besselj(-b,z))/sin(pi*b) ... # - use contiguity relations? def replacer(fro, to, factors): factors = set(factors) def repl(nu, z): if factors.intersection(Mul.make_args(z)): return to(nu, z) return fro(nu, z) return repl def torewrite(fro, to): def tofunc(nu, z): return fro(nu, z).rewrite(to) return tofunc def tominus(fro): def tofunc(nu, z): return exp(I*pi*nu)*fro(nu, exp_polar(-I*pi)*z) return tofunc orig_expr = expr ifactors = [I, exp_polar(I*pi/2), exp_polar(-I*pi/2)] expr = expr.replace( besselj, replacer(besselj, torewrite(besselj, besseli), ifactors)) expr = expr.replace( besseli, replacer(besseli, torewrite(besseli, besselj), ifactors)) minusfactors = [-1, exp_polar(I*pi)] expr = expr.replace( besselj, replacer(besselj, tominus(besselj), minusfactors)) expr = expr.replace( besseli, replacer(besseli, tominus(besseli), minusfactors)) z0 = Dummy('z') def expander(fro): def repl(nu, z): if (nu % 1) == S.Half: return simplify(trigsimp(unpolarify( fro(nu, z0).rewrite(besselj).rewrite(jn).expand( func=True)).subs(z0, z))) elif nu.is_Integer and nu > 1: return fro(nu, z).expand(func=True) return fro(nu, z) return repl expr = expr.replace(besselj, expander(besselj)) expr = expr.replace(bessely, expander(bessely)) expr = expr.replace(besseli, expander(besseli)) expr = expr.replace(besselk, expander(besselk)) def _bessel_simp_recursion(expr): def _use_recursion(bessel, expr): while True: bessels = expr.find(lambda x: isinstance(x, bessel)) try: for ba in sorted(bessels, key=lambda x: re(x.args[0])): a, x = ba.args bap1 = bessel(a+1, x) bap2 = bessel(a+2, x) if expr.has(bap1) and expr.has(bap2): expr = expr.subs(ba, 2*(a+1)/x*bap1 - bap2) break else: return expr except (ValueError, TypeError): return expr if expr.has(besselj): expr = _use_recursion(besselj, expr) if expr.has(bessely): expr = _use_recursion(bessely, expr) return expr expr = _bessel_simp_recursion(expr) if expr != orig_expr: expr = expr.factor() return expr def nthroot(expr, n, max_len=4, prec=15): """ 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`` Algorithm ========= 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 sympy.simplify.simplify import nthroot >>> from sympy import sqrt >>> nthroot(90 + 34*sqrt(7), 3) sqrt(7) + 3 """ expr = sympify(expr) n = sympify(n) p = expr**Rational(1, n) if not n.is_integer: return p if not _is_sum_surds(expr): return p surds = [] coeff_muls = [x.as_coeff_Mul() for x in expr.args] for x, y in coeff_muls: if not x.is_rational: return p if y is S.One: continue if not (y.is_Pow and y.exp == S.Half and y.base.is_integer): return p surds.append(y) surds.sort() surds = surds[:max_len] if expr < 0 and n % 2 == 1: p = (-expr)**Rational(1, n) a = nsimplify(p, constants=surds) res = a if _mexpand(a**n) == _mexpand(-expr) else p return -res a = nsimplify(p, constants=surds) if _mexpand(a) is not _mexpand(p) and _mexpand(a**n) == _mexpand(expr): return _mexpand(a) expr = _nthroot_solve(expr, n, prec) if expr is None: return p return expr def nsimplify(expr, constants=(), tolerance=None, full=False, rational=None, rational_conversion='base10'): """ 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. Explanation =========== 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). When converting to rational, if rational_conversion='base10' (the default), then convert floats to rationals using their base-10 (string) representation. When rational_conversion='exact' it uses the exact, base-2 representation. Examples ======== >>> from sympy import nsimplify, sqrt, GoldenRatio, exp, I, 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]) exp(-pi/2) >>> nsimplify(pi, tolerance=0.01) 22/7 >>> nsimplify(0.333333333333333, rational=True, rational_conversion='exact') 6004799503160655/18014398509481984 >>> nsimplify(0.333333333333333, rational=True) 1/3 See Also ======== sympy.core.function.nfloat """ try: return sympify(as_int(expr)) except (TypeError, ValueError): pass expr = sympify(expr).xreplace({ Float('inf'): S.Infinity, Float('-inf'): S.NegativeInfinity, }) if expr is S.Infinity or expr is S.NegativeInfinity: return expr if rational or expr.free_symbols: return _real_to_rational(expr, tolerance, rational_conversion) # SymPy's default tolerance for Rationals is 15; other numbers may have # lower tolerances set, so use them to pick the largest tolerance if None # was given if tolerance is None: tolerance = 10**-min([15] + [mpmath.libmp.libmpf.prec_to_dps(n._prec) for n in expr.atoms(Float)]) # XXX should prec be set independent of tolerance or should it be computed # from tolerance? prec = 30 bprec = int(prec*3.33) constants_dict = {} for constant in constants: constant = sympify(constant) v = constant.evalf(prec) if not v.is_Float: raise ValueError("constants must be real-valued") constants_dict[str(constant)] = v._to_mpmath(bprec) exprval = expr.evalf(prec, chop=True) re, im = exprval.as_real_imag() # safety check to make sure that this evaluated to a number if not (re.is_Number and im.is_Number): return expr def nsimplify_real(x): orig = mpmath.mp.dps xv = x._to_mpmath(bprec) try: # We'll be happy with low precision if a simple fraction if not (tolerance or full): mpmath.mp.dps = 15 rat = mpmath.pslq([xv, 1]) if rat is not None: return Rational(-int(rat[1]), int(rat[0])) mpmath.mp.dps = prec newexpr = mpmath.identify(xv, constants=constants_dict, tol=tolerance, full=full) if not newexpr: raise ValueError if full: newexpr = newexpr[0] expr = sympify(newexpr) if x and not expr: # don't let x become 0 raise ValueError if expr.is_finite is False and xv not in [mpmath.inf, mpmath.ninf]: raise ValueError return expr finally: # even though there are returns above, this is executed # before leaving mpmath.mp.dps = orig try: if re: re = nsimplify_real(re) if im: im = nsimplify_real(im) except ValueError: if rational is None: return _real_to_rational(expr, rational_conversion=rational_conversion) return expr rv = re + im*S.ImaginaryUnit # if there was a change or rational is explicitly not wanted # return the value, else return the Rational representation if rv != expr or rational is False: return rv return _real_to_rational(expr, rational_conversion=rational_conversion) def _real_to_rational(expr, tolerance=None, rational_conversion='base10'): """ Replace all reals in expr with rationals. Examples ======== >>> from sympy.simplify.simplify import _real_to_rational >>> from sympy.abc import x >>> _real_to_rational(.76 + .1*x**.5) sqrt(x)/10 + 19/25 If rational_conversion='base10', this uses the base-10 string. If rational_conversion='exact', the exact, base-2 representation is used. >>> _real_to_rational(0.333333333333333, rational_conversion='exact') 6004799503160655/18014398509481984 >>> _real_to_rational(0.333333333333333) 1/3 """ expr = _sympify(expr) inf = Float('inf') p = expr reps = {} reduce_num = None if tolerance is not None and tolerance < 1: reduce_num = ceiling(1/tolerance) for fl in p.atoms(Float): key = fl if reduce_num is not None: r = Rational(fl).limit_denominator(reduce_num) elif (tolerance is not None and tolerance >= 1 and fl.is_Integer is False): r = Rational(tolerance*round(fl/tolerance) ).limit_denominator(int(tolerance)) else: if rational_conversion == 'exact': r = Rational(fl) reps[key] = r continue elif rational_conversion != 'base10': raise ValueError("rational_conversion must be 'base10' or 'exact'") r = nsimplify(fl, rational=False) # e.g. log(3).n() -> log(3) instead of a Rational if fl and not r: r = Rational(fl) elif not r.is_Rational: if fl in (inf, -inf): r = S.ComplexInfinity elif fl < 0: fl = -fl d = Pow(10, int(mpmath.log(fl)/mpmath.log(10))) r = -Rational(str(fl/d))*d elif fl > 0: d = Pow(10, int(mpmath.log(fl)/mpmath.log(10))) r = Rational(str(fl/d))*d else: r = S.Zero reps[key] = r return p.subs(reps, simultaneous=True) def clear_coefficients(expr, rhs=S.Zero): """Return `p, r` where `p` is the expression obtained when Rational additive and multiplicative coefficients of `expr` have been stripped away in a naive fashion (i.e. without simplification). The operations needed to remove the coefficients will be applied to `rhs` and returned as `r`. Examples ======== >>> from sympy.simplify.simplify import clear_coefficients >>> from sympy.abc import x, y >>> from sympy import Dummy >>> expr = 4*y*(6*x + 3) >>> clear_coefficients(expr - 2) (y*(2*x + 1), 1/6) When solving 2 or more expressions like `expr = a`, `expr = b`, etc..., it is advantageous to provide a Dummy symbol for `rhs` and simply replace it with `a`, `b`, etc... in `r`. >>> rhs = Dummy('rhs') >>> clear_coefficients(expr, rhs) (y*(2*x + 1), _rhs/12) >>> _[1].subs(rhs, 2) 1/6 """ was = None free = expr.free_symbols if expr.is_Rational: return (S.Zero, rhs - expr) while expr and was != expr: was = expr m, expr = ( expr.as_content_primitive() if free else factor_terms(expr).as_coeff_Mul(rational=True)) rhs /= m c, expr = expr.as_coeff_Add(rational=True) rhs -= c expr = signsimp(expr, evaluate = False) if expr.could_extract_minus_sign(): expr = -expr rhs = -rhs return expr, rhs def nc_simplify(expr, deep=True): ''' Simplify a non-commutative expression composed of multiplication and raising to a power by grouping repeated subterms into one power. Priority is given to simplifications that give the fewest number of arguments in the end (for example, in a*b*a*b*c*a*b*c simplifying to (a*b)**2*c*a*b*c gives 5 arguments while a*b*(a*b*c)**2 has 3). If ``expr`` is a sum of such terms, the sum of the simplified terms is returned. Keyword argument ``deep`` controls whether or not subexpressions nested deeper inside the main expression are simplified. See examples below. Setting `deep` to `False` can save time on nested expressions that do not need simplifying on all levels. Examples ======== >>> from sympy import symbols >>> from sympy.simplify.simplify import nc_simplify >>> a, b, c = symbols("a b c", commutative=False) >>> nc_simplify(a*b*a*b*c*a*b*c) a*b*(a*b*c)**2 >>> expr = a**2*b*a**4*b*a**4 >>> nc_simplify(expr) a**2*(b*a**4)**2 >>> nc_simplify(a*b*a*b*c**2*(a*b)**2*c**2) ((a*b)**2*c**2)**2 >>> nc_simplify(a*b*a*b + 2*a*c*a**2*c*a**2*c*a) (a*b)**2 + 2*(a*c*a)**3 >>> nc_simplify(b**-1*a**-1*(a*b)**2) a*b >>> nc_simplify(a**-1*b**-1*c*a) (b*a)**(-1)*c*a >>> expr = (a*b*a*b)**2*a*c*a*c >>> nc_simplify(expr) (a*b)**4*(a*c)**2 >>> nc_simplify(expr, deep=False) (a*b*a*b)**2*(a*c)**2 ''' if isinstance(expr, MatrixExpr): expr = expr.doit(inv_expand=False) _Add, _Mul, _Pow, _Symbol = MatAdd, MatMul, MatPow, MatrixSymbol else: _Add, _Mul, _Pow, _Symbol = Add, Mul, Pow, Symbol # =========== Auxiliary functions ======================== def _overlaps(args): # Calculate a list of lists m such that m[i][j] contains the lengths # of all possible overlaps between args[:i+1] and args[i+1+j:]. # An overlap is a suffix of the prefix that matches a prefix # of the suffix. # For example, let expr=c*a*b*a*b*a*b*a*b. Then m[3][0] contains # the lengths of overlaps of c*a*b*a*b with a*b*a*b. The overlaps # are a*b*a*b, a*b and the empty word so that m[3][0]=[4,2,0]. # All overlaps rather than only the longest one are recorded # because this information helps calculate other overlap lengths. m = [[([1, 0] if a == args[0] else [0]) for a in args[1:]]] for i in range(1, len(args)): overlaps = [] j = 0 for j in range(len(args) - i - 1): overlap = [] for v in m[i-1][j+1]: if j + i + 1 + v < len(args) and args[i] == args[j+i+1+v]: overlap.append(v + 1) overlap += [0] overlaps.append(overlap) m.append(overlaps) return m def _reduce_inverses(_args): # replace consecutive negative powers by an inverse # of a product of positive powers, e.g. a**-1*b**-1*c # will simplify to (a*b)**-1*c; # return that new args list and the number of negative # powers in it (inv_tot) inv_tot = 0 # total number of inverses inverses = [] args = [] for arg in _args: if isinstance(arg, _Pow) and arg.args[1].is_extended_negative: inverses = [arg**-1] + inverses inv_tot += 1 else: if len(inverses) == 1: args.append(inverses[0]**-1) elif len(inverses) > 1: args.append(_Pow(_Mul(*inverses), -1)) inv_tot -= len(inverses) - 1 inverses = [] args.append(arg) if inverses: args.append(_Pow(_Mul(*inverses), -1)) inv_tot -= len(inverses) - 1 return inv_tot, tuple(args) def get_score(s): # compute the number of arguments of s # (including in nested expressions) overall # but ignore exponents if isinstance(s, _Pow): return get_score(s.args[0]) elif isinstance(s, (_Add, _Mul)): return sum([get_score(a) for a in s.args]) return 1 def compare(s, alt_s): # compare two possible simplifications and return a # "better" one if s != alt_s and get_score(alt_s) < get_score(s): return alt_s return s # ======================================================== if not isinstance(expr, (_Add, _Mul, _Pow)) or expr.is_commutative: return expr args = expr.args[:] if isinstance(expr, _Pow): if deep: return _Pow(nc_simplify(args[0]), args[1]).doit() else: return expr elif isinstance(expr, _Add): return _Add(*[nc_simplify(a, deep=deep) for a in args]).doit() else: # get the non-commutative part c_args, args = expr.args_cnc() com_coeff = Mul(*c_args) if com_coeff != 1: return com_coeff*nc_simplify(expr/com_coeff, deep=deep) inv_tot, args = _reduce_inverses(args) # if most arguments are negative, work with the inverse # of the expression, e.g. a**-1*b*a**-1*c**-1 will become # (c*a*b**-1*a)**-1 at the end so can work with c*a*b**-1*a invert = False if inv_tot > len(args)/2: invert = True args = [a**-1 for a in args[::-1]] if deep: args = tuple(nc_simplify(a) for a in args) m = _overlaps(args) # simps will be {subterm: end} where `end` is the ending # index of a sequence of repetitions of subterm; # this is for not wasting time with subterms that are part # of longer, already considered sequences simps = {} post = 1 pre = 1 # the simplification coefficient is the number of # arguments by which contracting a given sequence # would reduce the word; e.g. in a*b*a*b*c*a*b*c, # contracting a*b*a*b to (a*b)**2 removes 3 arguments # while a*b*c*a*b*c to (a*b*c)**2 removes 6. It's # better to contract the latter so simplification # with a maximum simplification coefficient will be chosen max_simp_coeff = 0 simp = None # information about future simplification for i in range(1, len(args)): simp_coeff = 0 l = 0 # length of a subterm p = 0 # the power of a subterm if i < len(args) - 1: rep = m[i][0] start = i # starting index of the repeated sequence end = i+1 # ending index of the repeated sequence if i == len(args)-1 or rep == [0]: # no subterm is repeated at this stage, at least as # far as the arguments are concerned - there may be # a repetition if powers are taken into account if (isinstance(args[i], _Pow) and not isinstance(args[i].args[0], _Symbol)): subterm = args[i].args[0].args l = len(subterm) if args[i-l:i] == subterm: # e.g. a*b in a*b*(a*b)**2 is not repeated # in args (= [a, b, (a*b)**2]) but it # can be matched here p += 1 start -= l if args[i+1:i+1+l] == subterm: # e.g. a*b in (a*b)**2*a*b p += 1 end += l if p: p += args[i].args[1] else: continue else: l = rep[0] # length of the longest repeated subterm at this point start -= l - 1 subterm = args[start:end] p = 2 end += l if subterm in simps and simps[subterm] >= start: # the subterm is part of a sequence that # has already been considered continue # count how many times it's repeated while end < len(args): if l in m[end-1][0]: p += 1 end += l elif isinstance(args[end], _Pow) and args[end].args[0].args == subterm: # for cases like a*b*a*b*(a*b)**2*a*b p += args[end].args[1] end += 1 else: break # see if another match can be made, e.g. # for b*a**2 in b*a**2*b*a**3 or a*b in # a**2*b*a*b pre_exp = 0 pre_arg = 1 if start - l >= 0 and args[start-l+1:start] == subterm[1:]: if isinstance(subterm[0], _Pow): pre_arg = subterm[0].args[0] exp = subterm[0].args[1] else: pre_arg = subterm[0] exp = 1 if isinstance(args[start-l], _Pow) and args[start-l].args[0] == pre_arg: pre_exp = args[start-l].args[1] - exp start -= l p += 1 elif args[start-l] == pre_arg: pre_exp = 1 - exp start -= l p += 1 post_exp = 0 post_arg = 1 if end + l - 1 < len(args) and args[end:end+l-1] == subterm[:-1]: if isinstance(subterm[-1], _Pow): post_arg = subterm[-1].args[0] exp = subterm[-1].args[1] else: post_arg = subterm[-1] exp = 1 if isinstance(args[end+l-1], _Pow) and args[end+l-1].args[0] == post_arg: post_exp = args[end+l-1].args[1] - exp end += l p += 1 elif args[end+l-1] == post_arg: post_exp = 1 - exp end += l p += 1 # Consider a*b*a**2*b*a**2*b*a: # b*a**2 is explicitly repeated, but note # that in this case a*b*a is also repeated # so there are two possible simplifications: # a*(b*a**2)**3*a**-1 or (a*b*a)**3 # The latter is obviously simpler. # But in a*b*a**2*b**2*a**2 the simplifications are # a*(b*a**2)**2 and (a*b*a)**3*a in which case # it's better to stick with the shorter subterm if post_exp and exp % 2 == 0 and start > 0: exp = exp/2 _pre_exp = 1 _post_exp = 1 if isinstance(args[start-1], _Pow) and args[start-1].args[0] == post_arg: _post_exp = post_exp + exp _pre_exp = args[start-1].args[1] - exp elif args[start-1] == post_arg: _post_exp = post_exp + exp _pre_exp = 1 - exp if _pre_exp == 0 or _post_exp == 0: if not pre_exp: start -= 1 post_exp = _post_exp pre_exp = _pre_exp pre_arg = post_arg subterm = (post_arg**exp,) + subterm[:-1] + (post_arg**exp,) simp_coeff += end-start if post_exp: simp_coeff -= 1 if pre_exp: simp_coeff -= 1 simps[subterm] = end if simp_coeff > max_simp_coeff: max_simp_coeff = simp_coeff simp = (start, _Mul(*subterm), p, end, l) pre = pre_arg**pre_exp post = post_arg**post_exp if simp: subterm = _Pow(nc_simplify(simp[1], deep=deep), simp[2]) pre = nc_simplify(_Mul(*args[:simp[0]])*pre, deep=deep) post = post*nc_simplify(_Mul(*args[simp[3]:]), deep=deep) simp = pre*subterm*post if pre != 1 or post != 1: # new simplifications may be possible but no need # to recurse over arguments simp = nc_simplify(simp, deep=False) else: simp = _Mul(*args) if invert: simp = _Pow(simp, -1) # see if factor_nc(expr) is simplified better if not isinstance(expr, MatrixExpr): f_expr = factor_nc(expr) if f_expr != expr: alt_simp = nc_simplify(f_expr, deep=deep) simp = compare(simp, alt_simp) else: simp = simp.doit(inv_expand=False) return simp def dotprodsimp(expr, withsimp=False): """Simplification for a sum of products targeted at the kind of blowup that occurs during summation of products. Intended to reduce expression blowup during matrix multiplication or other similar operations. Only works with algebraic expressions and does not recurse into non. Parameters ========== withsimp : bool, optional Specifies whether a flag should be returned along with the expression to indicate roughly whether simplification was successful. It is used in ``MatrixArithmetic._eval_pow_by_recursion`` to avoid attempting to simplify an expression repetitively which does not simplify. """ def count_ops_alg(expr): """Optimized count algebraic operations with no recursion into non-algebraic args that ``core.function.count_ops`` does. Also returns whether rational functions may be present according to negative exponents of powers or non-number fractions. Returns ======= ops, ratfunc : int, bool ``ops`` is the number of algebraic operations starting at the top level expression (not recursing into non-alg children). ``ratfunc`` specifies whether the expression MAY contain rational functions which ``cancel`` MIGHT optimize. """ ops = 0 args = [expr] ratfunc = False while args: a = args.pop() if not isinstance(a, Basic): continue if a.is_Rational: if a is not S.One: # -1/3 = NEG + DIV ops += bool (a.p < 0) + bool (a.q != 1) elif a.is_Mul: if a.could_extract_minus_sign(): ops += 1 if a.args[0] is S.NegativeOne: a = a.as_two_terms()[1] else: a = -a n, d = fraction(a) if n.is_Integer: ops += 1 + bool (n < 0) args.append(d) # won't be -Mul but could be Add elif d is not S.One: if not d.is_Integer: args.append(d) ratfunc=True ops += 1 args.append(n) # could be -Mul else: ops += len(a.args) - 1 args.extend(a.args) elif a.is_Add: laargs = len(a.args) negs = 0 for ai in a.args: if ai.could_extract_minus_sign(): negs += 1 ai = -ai args.append(ai) ops += laargs - (negs != laargs) # -x - y = NEG + SUB elif a.is_Pow: ops += 1 args.append(a.base) if not ratfunc: ratfunc = a.exp.is_negative is not False return ops, ratfunc def nonalg_subs_dummies(expr, dummies): """Substitute dummy variables for non-algebraic expressions to avoid evaluation of non-algebraic terms that ``polys.polytools.cancel`` does. """ if not expr.args: return expr if expr.is_Add or expr.is_Mul or expr.is_Pow: args = None for i, a in enumerate(expr.args): c = nonalg_subs_dummies(a, dummies) if c is a: continue if args is None: args = list(expr.args) args[i] = c if args is None: return expr return expr.func(*args) return dummies.setdefault(expr, Dummy()) simplified = False # doesn't really mean simplified, rather "can simplify again" if isinstance(expr, Basic) and (expr.is_Add or expr.is_Mul or expr.is_Pow): expr2 = expr.expand(deep=True, modulus=None, power_base=False, power_exp=False, mul=True, log=False, multinomial=True, basic=False) if expr2 != expr: expr = expr2 simplified = True exprops, ratfunc = count_ops_alg(expr) if exprops >= 6: # empirically tested cutoff for expensive simplification if ratfunc: dummies = {} expr2 = nonalg_subs_dummies(expr, dummies) if expr2 is expr or count_ops_alg(expr2)[0] >= 6: # check again after substitution expr3 = cancel(expr2) if expr3 != expr2: expr = expr3.subs([(d, e) for e, d in dummies.items()]) simplified = True # very special case: x/(x-1) - 1/(x-1) -> 1 elif (exprops == 5 and expr.is_Add and expr.args [0].is_Mul and expr.args [1].is_Mul and expr.args [0].args [-1].is_Pow and expr.args [1].args [-1].is_Pow and expr.args [0].args [-1].exp is S.NegativeOne and expr.args [1].args [-1].exp is S.NegativeOne): expr2 = together (expr) expr2ops = count_ops_alg(expr2)[0] if expr2ops < exprops: expr = expr2 simplified = True else: simplified = True return (expr, simplified) if withsimp else expr bottom_up = deprecated( """ Using bottom_up from the sympy.simplify.simplify submodule is deprecated. Instead, use bottom_up from the top-level sympy namespace, like sympy.bottom_up """, deprecated_since_version="1.10", active_deprecations_target="deprecated-traversal-functions-moved", )(_bottom_up) # XXX: This function really should either be private API or exported in the # top-level sympy/__init__.py walk = deprecated( """ Using walk from the sympy.simplify.simplify submodule is deprecated. Instead, use walk from sympy.core.traversal.walk """, deprecated_since_version="1.10", active_deprecations_target="deprecated-traversal-functions-moved", )(_walk)