Source code for sympy.simplify.simplify

from collections import defaultdict
from operator import itemgetter

from sympy import SYMPY_DEBUG

from sympy.core import (Basic, S, C, Add, Mul, Pow, Rational, Integer,
    Derivative, Wild, Symbol, sympify, expand, expand_mul, expand_func,
    Function, Equality, Dummy, Atom, count_ops, Expr, factor_terms,
    expand_multinomial, expand_power_base)

from sympy.core.compatibility import iterable, reduce
from sympy.core.numbers import igcd, Float
from sympy.core.function import expand_log, count_ops
from sympy.core.mul import _keep_coeff, prod
from sympy.core.rules import Transform

from sympy.functions import gamma, exp, sqrt, log, root, exp_polar
from sympy.utilities.misc import default_sort_key
from sympy.utilities.iterables import flatten, has_variety

from sympy.simplify.cse_main import cse
from sympy.simplify.cse_opts import sub_pre, sub_post
from sympy.simplify.sqrtdenest import sqrtdenest

from sympy.polys import (Poly, together, reduced, cancel, factor,
    ComputationFailed, terms_gcd, lcm, gcd)
from sympy.polys.polytools import _keep_coeff

import sympy.mpmath as mpmath
from functools import reduce

def _mexpand(expr):
    return expand_mul(expand_multinomial(expr))

[docs]def fraction(expr, exact=False): """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 sympy import fraction, Rational, Symbol >>> from sympy.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 sympy import exp >>> fraction(2*x**(-y)) (2, x**y) >>> fraction(exp(-x)) (1, exp(x)) >>> fraction(exp(-x), exact=True) (exp(-x), 1) """ expr = sympify(expr) numer, denom = [], [] for term in Mul.make_args(expr): if term.is_commutative and (term.is_Pow or term.func is exp): b, ex = term.as_base_exp() if ex.is_negative: if ex is S.NegativeOne: denom.append(b) else: denom.append(Pow(b, -ex)) elif ex.is_positive: numer.append(term) elif not exact and ex.is_Mul: n, d = term.as_numer_denom() numer.append(n) denom.append(d) else: numer.append(term) elif term.is_Rational: n, d = term.as_numer_denom() numer.append(n) denom.append(d) else: numer.append(term) return Mul(*numer), Mul(*denom)
def numer(expr): return fraction(expr)[0] def denom(expr): return fraction(expr)[1] def fraction_expand(expr, **hints): return expr.expand(frac=True, **hints) def numer_expand(expr, **hints): a, b = fraction(expr) return a.expand(numer=True, **hints) / b def denom_expand(expr, **hints): a, b = fraction(expr) return a / b.expand(denom=True, **hints) expand_numer = numer_expand expand_denom = denom_expand expand_fraction = fraction_expand
[docs]def separate(expr, deep=False, force=False): """ Deprecated wrapper around ``expand_power_base()``. Use that function instead. """ from sympy.utilities.exceptions import SymPyDeprecationWarning SymPyDeprecationWarning( feature="separate()", useinstead="expand_power_base()", issue=3383, deprecated_since_version="0.7.2", value="Note: in separate() deep " "defaults to False, whereas in expand_power_base(), deep defaults to True.", ).warn() return expand_power_base(sympify(expr), deep=deep, force=force)
[docs]def collect(expr, syms, func=None, evaluate=True, exact=False, distribute_order_term=True): """ 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 :func:`collect`, so user is expected to provide an expression is an appropriate form. This makes :func:`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 :func:`expand_power_base` function. There are two possible types of output. First, if ``evaluate`` flag is set, this function will return an expression with collected terms or else it will return a dictionary with expressions up to rational powers as keys and collected coefficients as values. Examples ======== >>> from sympy import S, collect, expand, factor, Wild >>> from sympy.abc import a, b, c, x, y, z 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[S.One] 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 sympy 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:: >>> 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 :func:`collect` function apply to the exponential function, so you can get:: >>> from sympy import exp >>> collect(a*exp(2*x) + b*exp(2*x), exp(x)) (a + b)*exp(2*x) 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 sympy import Derivative as D, collect, 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) (a + b)*f(x) + (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 .. note:: Arguments are expected to be in expanded form, so you might have to call :func:`expand` prior to calling this function. """ def make_expression(terms): product = [] for term, rat, sym, deriv in terms: if deriv is not None: var, order = deriv while order > 0: term, order = Derivative(term, var), order-1 if sym is None: if rat is S.One: product.append(term) else: product.append(Pow(term, rat)) else: product.append(Pow(term, rat*sym)) return Mul(*product) def parse_derivative(deriv): # scan derivatives tower in the input expression and return # underlying function and maximal differentiation order expr, sym, order = deriv.expr, deriv.variables[0], 1 for s in deriv.variables[1:]: if s == sym: order += 1 else: raise NotImplementedError('Improve MV Derivative support in collect') while isinstance(expr, Derivative): s0 = expr.variables[0] for s in expr.variables: if s != s0: raise NotImplementedError('Improve MV Derivative support in collect') if s0 == sym: expr, order = expr.expr, order+len(expr.variables) else: break return expr, (sym, Rational(order)) def parse_term(expr): """Parses expression expr and outputs tuple (sexpr, rat_expo, sym_expo, deriv) where: - sexpr is the base expression - rat_expo is the rational exponent that sexpr is raised to - sym_expo is the symbolic exponent that sexpr is raised to - deriv contains the derivatives the the expression for example, the output of x would be (x, 1, None, None) the output of 2**x would be (2, 1, x, None) """ rat_expo, sym_expo = S.One, None sexpr, deriv = expr, None if expr.is_Pow: if isinstance(expr.base, Derivative): sexpr, deriv = parse_derivative(expr.base) else: sexpr = expr.base if expr.exp.is_Number: rat_expo = expr.exp else: coeff, tail = expr.exp.as_coeff_Mul() if coeff.is_Number: rat_expo, sym_expo = coeff, tail else: sym_expo = expr.exp elif expr.func is C.exp: arg = expr.args[0] if arg.is_Rational: sexpr, rat_expo = S.Exp1, arg elif arg.is_Mul: coeff, tail = arg.as_coeff_Mul(rational=True) sexpr, rat_expo = C.exp(tail), coeff elif isinstance(expr, Derivative): sexpr, deriv = parse_derivative(expr) return sexpr, rat_expo, sym_expo, deriv def parse_expression(terms, pattern): """Parse terms searching for a pattern. terms is a list of tuples as returned by parse_terms; pattern is an expression treated as a product of factors """ pattern = Mul.make_args(pattern) if len(terms) < len(pattern): # pattern is longer than matched product # so no chance for positive parsing result return None else: pattern = [parse_term(elem) for elem in pattern] terms = terms[:] # need a copy elems, common_expo, has_deriv = [], None, False for elem, e_rat, e_sym, e_ord in pattern: if elem.is_Number and e_rat == 1 and e_sym is None: # a constant is a match for everything continue for j in range(len(terms)): if terms[j] is None: continue term, t_rat, t_sym, t_ord = terms[j] # keeping track of whether one of the terms had # a derivative or not as this will require rebuilding # the expression later if t_ord is not None: has_deriv= True if (term.match(elem) is not None and \ (t_sym == e_sym or t_sym is not None and \ e_sym is not None and \ t_sym.match(e_sym) is not None)): if exact == False: # we don't have to be exact so find common exponent # for both expression's term and pattern's element expo = t_rat / e_rat if common_expo is None: # first time common_expo = expo else: # common exponent was negotiated before so # there is no chance for a pattern match unless # common and current exponents are equal if common_expo != expo: common_expo = 1 else: # we ought to be exact so all fields of # interest must match in every details if e_rat != t_rat or e_ord != t_ord: continue # found common term so remove it from the expression # and try to match next element in the pattern elems.append(terms[j]) terms[j] = None break else: # pattern element not found return None return [_f for _f in terms if _f], elems, common_expo, has_deriv if evaluate: if expr.is_Mul: return Mul(*[ collect(term, syms, func, True, exact, distribute_order_term) for term in expr.args ]) elif expr.is_Pow: b = collect(expr.base, syms, func, True, exact, distribute_order_term) return Pow(b, expr.exp) if iterable(syms): syms = [expand_power_base(i, deep=False) for i in syms] else: syms = [ expand_power_base(syms, deep=False) ] expr = sympify(expr) order_term = None if distribute_order_term: order_term = expr.getO() if order_term is not None: if order_term.has(*syms): order_term = None else: expr = expr.removeO() summa = [expand_power_base(i, deep=False) for i in Add.make_args(expr)] collected, disliked = defaultdict(list), S.Zero for product in summa: terms = [parse_term(i) for i in Mul.make_args(product)] for symbol in syms: if SYMPY_DEBUG: print("DEBUG: parsing of expression %s with symbol %s " % (str(terms), str(symbol))) result = parse_expression(terms, symbol) if SYMPY_DEBUG: print("DEBUG: returned %s" % str(result)) if result is not None: terms, elems, common_expo, has_deriv = result # when there was derivative in current pattern we # will need to rebuild its expression from scratch if not has_deriv: index = 1 for elem in elems: e = elem[1] if elem[2] is not None: e *= elem[2] index *= Pow(elem[0], e) else: index = make_expression(elems) terms = expand_power_base(make_expression(terms), deep=False) index = expand_power_base(index, deep=False) collected[index].append(terms) break else: # none of the patterns matched disliked += product # add terms now for each key collected = dict([(k, Add(*v)) for k, v in collected.items()]) if disliked is not S.Zero: collected[S.One] = disliked if order_term is not None: for key, val in collected.items(): collected[key] = val + order_term if func is not None: collected = dict([ (key, func(val)) for key, val in collected.items() ]) if evaluate: return Add(*[key*val for key, val in collected.items()]) else: return collected
[docs]def rcollect(expr, *vars): """ Recursively collect sums in an expression. Examples ======== >>> from sympy.simplify import rcollect >>> from sympy.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) """ if expr.is_Atom or not expr.has(*vars): return expr else: expr = expr.__class__(*[ rcollect(arg, *vars) for arg in expr.args ]) if expr.is_Add: return collect(expr, vars) else: return expr
[docs]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. 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) == None True """ expr = sympify(expr) if dict: return _separatevars_dict(_separatevars(expr, force), symbols) else: return _separatevars(expr, force)
def _separatevars(expr, force): if len(expr.free_symbols) == 1: 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 = Mul(*args) return expr # get a Pow ready for expansion if expr.is_Pow: expr = Pow(separatevars(expr.base, force=force), expr.exp) # First try other expansion methods expr = expr.expand(mul=False, multinomial=False, force=force) _expr = expr if expr.is_commutative: # factor fails for nc _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: assert all((t.is_Atom for t in symbols)), "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 = dict(((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 list(ret.items()): ret[k] = Mul(*v) return ret
[docs]def ratsimp(expr): """ Put an expression over a common denominator, cancel and reduce. Examples ======== >>> from sympy import ratsimp >>> from sympy.abc import x, y >>> ratsimp(1/x + 1/y) (x + y)/(x*y) """ f, g = cancel(expr).as_numer_denom() try: Q, r = reduced(f, [g], field=True, expand=False) except ComputationFailed: return f/g return Add(*Q) + cancel(r/g)
def ratsimpmodprime(expr, G, *gens, **args): """ Simplifies a rational expression ``expr`` modulo the prime ideal generated by ``G``. ``G`` should be a Groebner basis of the ideal. >>> from sympy.simplify.simplify import ratsimpmodprime >>> from sympy.abc import x, y >>> ratsimpmodprime((x + y**5 + y)/(x - y), [x*y**5 - x - y], x, y, order='lex') (x**2 + x*y + x + y)/(x**2 - x*y) The algorithm computes a rational simplification which minimizes the sum of the total degrees of the numerator and the denominator. References ========== M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial Ideal, http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.6984 (specifically, the second algorithm) """ from sympy.polys import polyoptions as options, parallel_poly_from_expr, degree_list from sympy.polys.polyerrors import PolificationFailed from sympy import monomials, symbols, solve, Monomial from sympy.polys.monomialtools import monomial_div from sympy.core.compatibility import product # usual preparation of polynomials: num, denom = cancel(expr).as_numer_denom() try: polys, opt = parallel_poly_from_expr([num, denom] + G, *gens, **args) except PolificationFailed as exc: return expr domain = opt.domain if domain.has_assoc_Field: opt.domain = domain.get_field() else: raise DomainError("can't compute rational simplification over %s" % domain) # compute only once leading_monomials = [g.LM(opt.order) for g in polys[2:]] def staircase(n): """ Compute all monomials with degree less than ``n`` that are not divisible by any element of ``leading_monomials``. """ S = [] for m in product(*([range(n + 1)] * len(opt.gens))): if sum(m) <= n: if all([monomial_div(m, lmg) is None for lmg in leading_monomials]): S.append(m) return [Monomial(s).as_expr(*opt.gens) for s in S] def _ratsimpmodprime(a, b, N=0, D=0): """ Computes a rational simplification of ``a/b`` which minimizes the sum of the total degrees of the numerator and the denominator. The algorithm proceeds by looking at ``a * d - b * c`` modulo the ideal generated by ``G`` for some ``c`` and ``d`` with degree less than ``a`` and ``b`` respectively. The coefficients of ``c`` and ``d`` are indeterminates and thus the coefficients of the normalform of ``a * d - b * c`` are linear polynomials in these indeterminates. If these linear polynomials, considered as system of equations, have a nontrivial solution, then `\frac{a}{b} \equiv \frac{c}{d}` modulo the ideal generated by ``G``. So, by construction, the degree of ``c`` and ``d`` is less than the degree of ``a`` and ``b``, so a simpler representation has been found. After a simpler representation has been found, the algorithm tries to reduce the degree of the numerator and denominator and returns the result afterwards. """ c, d = a, b steps = 0 while N + D < a.total_degree() + b.total_degree(): M1 = staircase(N) M2 = staircase(D) Cs = symbols("c:%d" % len(M1)) Ds = symbols("d:%d" % len(M2)) c_hat = Poly(sum([Cs[i] * M1[i] for i in range(len(M1))]), opt.gens) d_hat = Poly(sum([Ds[i] * M2[i] for i in range(len(M2))]), opt.gens) r = reduced(a * d_hat - b * c_hat, G, opt.gens, order=opt.order, polys=True)[1] S = r.coeffs() sol = solve(S, Cs + Ds) # If nontrivial solutions exist, solve will give them # parametrized, i.e. the values of some keys will be # exprs. Set these to any value different from 0 to obtain # one nontrivial solution: for key in list(sol.keys()): sol[key] = sol[key].subs(dict(list(zip(Cs + Ds, [1] * (len(Cs) + len(Ds)))))) if sol and not all([s == 0 for s in sol.values()]): c = c_hat.subs(sol) d = d_hat.subs(sol) # The "free" variables occuring before as parameters # might still be in the substituted c, d, so set them # to the value chosen before: c = c.subs(dict(list(zip(Cs + Ds, [1] * (len(Cs) + len(Ds)))))) d = d.subs(dict(list(zip(Cs + Ds, [1] * (len(Cs) + len(Ds)))))) c = Poly(c, opt.gens) d = Poly(d, opt.gens) break N += 1 D += 1 steps += 1 if steps > 0: c, d = _ratsimpmodprime(c, d, N, D - steps) c, d = _ratsimpmodprime(c, d, N - steps, D) return c, d # preprocessing. this improves performance a bit when deg(num) # and deg(denom) are large: num = reduced(num, G, opt.gens, order=opt.order)[1] denom = reduced(denom, G, opt.gens, order=opt.order)[1] c, d = _ratsimpmodprime(Poly(num, opt.gens), Poly(denom, opt.gens)) if not domain.has_Field: c = c.clear_denoms(convert=True)[1] d = d.clear_denoms(convert=True)[1] return c/d
[docs]def trigsimp(expr, deep=False, recursive=False): """ reduces expression by using known trig identities Notes ===== deep: - Apply trigsimp inside all objects with arguments recursive: - Use common subexpression elimination (cse()) and apply trigsimp recursively (this is quite expensive if the expression is large) Examples ======== >>> from sympy import trigsimp, sin, cos, log, cosh, sinh >>> from sympy.abc import x, y >>> e = 2*sin(x)**2 + 2*cos(x)**2 >>> trigsimp(e) 2 >>> trigsimp(log(e)) log(2*sin(x)**2 + 2*cos(x)**2) >>> trigsimp(log(e), deep=True) log(2) """ if not expr.has(C.TrigonometricFunction, C.HyperbolicFunction): return expr if recursive: w, g = cse(expr) g = _trigsimp(g[0], deep) for sub in reversed(w): g = g.subs(sub[0], sub[1]) g = _trigsimp(g, deep) result = g else: result = _trigsimp(expr, deep) return result
def _trigsimp(expr, deep=False): """recursive helper for trigsimp""" a, b, c = list(map(Wild, 'abc')) sin, cos, tan, cot = C.sin, C.cos, C.tan, C.cot sinh, cosh, tanh, coth = C.sinh, C.cosh, C.tanh, C.coth # for the simplifications like sinh/cosh -> tanh: matchers_division = ( (a*sin(b)**c/cos(b)**c, a*tan(b)**c), (a*tan(b)**c*cos(b)**c, a*sin(b)**c), (a*cot(b)**c*sin(b)**c, a*cos(b)**c), (a*tan(b)**c/sin(b)**c, a/cos(b)**c), (a*cot(b)**c/cos(b)**c, a/sin(b)**c), (a*cot(b)**c*tan(b)**c, a), (a*sinh(b)**c/cosh(b)**c, a*tanh(b)**c), (a*tanh(b)**c*cosh(b)**c, a*sinh(b)**c), (a*coth(b)**c*sinh(b)**c, a*cosh(b)**c), (a*tanh(b)**c/sinh(b)**c, a/cosh(b)**c), (a*coth(b)**c/cosh(b)**c, a/sinh(b)**c), (a*coth(b)**c*tanh(b)**c, a) ) # for cos(x)**2 + sin(x)**2 -> 1 matchers_identity = ( (a*sin(b)**2, a - a*cos(b)**2), (a*tan(b)**2, a*(1/cos(b))**2 - a), (a*cot(b)**2, a*(1/sin(b))**2 - a), (a*sin(b + c), a*(sin(b)*cos(c) + sin(c)*cos(b))), (a*cos(b + c), a*(cos(b)*cos(c) - sin(b)*sin(c))), (a*tan(b + c), a*((tan(b) + tan(c))/(1 - tan(b)*tan(c)))), (a*sinh(b)**2, a*cosh(b)**2 - a), (a*tanh(b)**2, a - a*(1/cosh(b))**2), (a*coth(b)**2, a + a*(1/sinh(b))**2), (a*sinh(b + c), a*(sinh(b)*cosh(c) + sinh(c)*cosh(b))), (a*cosh(b + c), a*(cosh(b)*cosh(c) + sinh(b)*sinh(c))), (a*tanh(b + c), a*((tanh(b) + tanh(c))/(1 + tanh(b)*tanh(c)))) ) # Reduce any lingering artefacts, such as sin(x)**2 changing # to 1-cos(x)**2 when sin(x)**2 was "simpler" artifacts = ( (a - a*cos(b)**2 + c, a*sin(b)**2 + c, cos), (a - a*(1/cos(b))**2 + c, -a*tan(b)**2 + c, cos), (a - a*(1/sin(b))**2 + c, -a*cot(b)**2 + c, sin), (a - a*cosh(b)**2 + c, -a*sinh(b)**2 + c, cosh), (a - a*(1/cosh(b))**2 + c, a*tanh(b)**2 + c, cosh), (a + a*(1/sinh(b))**2 + c, a*coth(b)**2 + c, sinh) ) if expr.is_Mul: # do some simplifications like sin/cos -> tan: for pattern, simp in matchers_division: res = expr.match(pattern) if res and res.get(c, 0): # if "a" contains any of sin("b"), cos("b"), tan("b"), cot("b"), # sinh("b"), cosh("b"), tanh("b") or coth("b), # skip the simplification: if res[a].has(C.TrigonometricFunction, C.HyperbolicFunction): continue # simplify and finish: expr = simp.subs(res) break # process below if expr.is_Add: # The types of hyper functions we are looking for # Scan for the terms we need args = [] for term in expr.args: term = _trigsimp(term, deep) for pattern, result in matchers_identity: res = term.match(pattern) if res is not None: term = result.subs(res) break args.append(term) if args != expr.args: expr = Add(*args) # Reduce any lingering artifacts, such as sin(x)**2 changing # to 1 - cos(x)**2 when sin(x)**2 was "simpler" for pattern, result, ex in artifacts: if expr.is_number: break # Substitute a new wild that excludes some function(s) # to help influence a better match. This is because # sometimes, for example, 'a' would match sec(x)**2 a_t = Wild('a', exclude=[ex]) pattern = pattern.subs(a, a_t) result = result.subs(a, a_t) m = expr.match(pattern) while m is not None: if m[a_t] == 0 or -m[a_t] in m[c].args or m[a_t] + m[c] == 0: break expr = result.subs(m) m = expr.match(pattern) return expr elif expr.is_Mul or expr.is_Pow or deep and expr.args: return expr.func(*[_trigsimp(a, deep) for a in expr.args]) return expr
[docs]def collect_sqrt(expr, evaluate=True): """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 the returned expression will be an unevaluated Add with args ordered by default_sort_key. Note: since I = sqrt(-1), it is collected, too. Examples ======== >>> from sympy import sqrt >>> from sympy.simplify.simplify import collect_sqrt >>> from sympy.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(2)*(a + b), sqrt(3)*a, sqrt(5)*b), 3) >>> collect_sqrt(a*sqrt(2) + b, evaluate=False) ((b, sqrt(2)*a), 1) >>> collect_sqrt(a + b, evaluate=False) ((a + b,), 0) """ coeff, expr = expr.as_content_primitive() vars = set() for a in Add.make_args(expr): for m in a.args_cnc()[0]: if m.is_number and (m.is_Pow and m.exp.is_Rational and m.exp.q == 2 or \ m is S.ImaginaryUnit): vars.add(m) vars = list(vars) if not evaluate: vars.sort(key=default_sort_key) vars.reverse() # since it will be reversed below vars.sort(key=count_ops) vars.reverse() d = collect_const(expr, *vars, **dict(first=False)) hit = expr != d d *= coeff if not evaluate: nrad = 0 args = list(Add.make_args(d)) for m in args: c, nc = m.args_cnc() for ci in c: if ci.is_Pow and ci.exp.is_Rational and ci.exp.q == 2 or \ ci is S.ImaginaryUnit: nrad += 1 break if hit or nrad: args.sort(key=default_sort_key) else: args = [Add(*args)] return tuple(args), nrad return d
[docs]def collect_const(expr, *vars, **first): """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. Examples ======== >>> from sympy import sqrt >>> from sympy.abc import a, s >>> from sympy.simplify.simplify import collect_const >>> 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) If no constants are provided then a leading Rational might be returned: >>> collect_const(2*sqrt(3) + 4*a*sqrt(5)) 2*(2*sqrt(5)*a + sqrt(3)) >>> collect_const(2*sqrt(3) + 4*a*sqrt(5), sqrt(3)) 4*sqrt(5)*a + 2*sqrt(3) """ if first.get('first', True): c, p = sympify(expr).as_content_primitive() else: c, p = S.One, expr if c is not S.One: if not vars: return _keep_coeff(c, collect_const(p, *vars, **dict(first=False))) # else don't leave the Rational on the outside return c*collect_const(p, *vars, **dict(first=False)) if not (expr.is_Add or expr.is_Mul): return expr recurse = False if not vars: recurse = True vars = set() for a in Add.make_args(expr): for m in Mul.make_args(a): if m.is_number: vars.add(m) vars = sorted(vars, key=count_ops) # Rationals get autodistributed on Add so don't bother with them vars = [v for v in vars if not v.is_Rational] if not vars: return expr for v in vars: terms = defaultdict(list) for m in Add.make_args(expr): i = [] d = [] for a in Mul.make_args(m): if a == v: d.append(a) else: i.append(a) ai, ad = [Mul(*w) for w in [i, d]] terms[ad].append(ai) args = [] hit = False for k, v in terms.items(): if len(v) > 1: v = Add(*v) hit = True if recurse and v != expr: vars.append(v) else: v = v[0] args.append(k*v) if hit: expr = Add(*args) if not expr.is_Add: break return expr
def _split_gcd(*a): """ split the list of integers `a` into a list of integers a1 having g = gcd(a1) and a list a2 whose elements are not divisible by g Returns g, a1, a2 Examples ======== >>> from sympy.simplify.simplify import _split_gcd >>> _split_gcd(55,35,22,14,77,10) (5, [55, 35, 10], [22, 14, 77]) """ g = a[0] b1 = [g] b2 = [] for x in a[1:]: g1 = gcd(g, x) if g1 == 1: b2.append(x) else: g = g1 b1.append(x) return g, b1, b2 def split_surds(expr): """ split an expression with terms whose squares are rationals into a sum of terms whose surds squared have gcd equal to g and a sum of terms with surds squared prime with g Examples ======== >>> from sympy import sqrt >>> from sympy.simplify.simplify import split_surds >>> split_surds(3*sqrt(3) + sqrt(5)/7 + sqrt(6) + sqrt(10) + sqrt(15)) (3, sqrt(2) + sqrt(5) + 3, sqrt(5)/7 + sqrt(10)) """ args = sorted(expr.args, key=default_sort_key) coeff_muls = [x.as_coeff_Mul() for x in args] surds = [x[1]**2 for x in coeff_muls if x[1].is_Pow] surds.sort(key=default_sort_key) g, b1, b2 = _split_gcd(*surds) g2 = g if not b2 and len(b1) >= 2: b1n = [x/g for x in b1] b1n = [x for x in b1n if x != 1] # only a common factor has been factored; split again g1, b1n, b2 = _split_gcd(*b1n) g2 = g*g1 a1v, a2v = [], [] for c, s in coeff_muls: if s.is_Pow and s.exp == S.Half: s1 = s.base if s1 in b1: a1v.append(c*sqrt(s1/g2)) else: a2v.append(c*s) else: a2v.append(c*s) a = Add(*a1v) b = Add(*a2v) return g2, a, b def rad_rationalize(num, den): """ Rationalize num/den by removing square roots in the denominator; num and den are sum of terms whose squares are rationals Examples ======== >>> from sympy import sqrt >>> from sympy.simplify.simplify import rad_rationalize >>> rad_rationalize(sqrt(3), 1 + sqrt(2)/3) (-sqrt(3) + sqrt(6)/3, -7/9) """ if not den.is_Add: return num, den g, a, b = split_surds(den) a = a*sqrt(g) num = _mexpand((a - b)*num) den = _mexpand(a**2 - b**2) return rad_rationalize(num, den)
[docs]def radsimp(expr, symbolic=True, max_terms=4): """ 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 do not simplify. Examples ======== >>> from sympy import radsimp, sqrt, Symbol, denom, pprint, I >>> from sympy.abc import a, b, c >>> radsimp(1/(I + 1)) (1 - I)/2 >>> radsimp(1/(2 + sqrt(2))) (-sqrt(2) + 2)/2 >>> x,y = list(map(Symbol, 'xy')) >>> e = ((2 + 2*sqrt(2))*x + (2 + sqrt(8))*y)/(2 + sqrt(2)) >>> radsimp(e) sqrt(2)*(x + y) Terms are collected automatically: >>> r2 = sqrt(2) >>> r5 = sqrt(5) >>> pprint(radsimp(1/(y*r2 + x*r2 + a*r5 + b*r5))) ___ ___ \/ 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, the original expression will be returned. If the denominator was 1 then any square roots will also be collected: >>> radsimp(sqrt(2)*x + sqrt(2)) sqrt(2)*(x + 1) 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)) >>> radsimp(eq).subs(a, 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)) """ def handle(expr): if expr.is_Atom or not symbolic and expr.free_symbols: return expr n, d = fraction(expr) if d is S.One: nexpr = expr.func(*[handle(ai) for ai in expr.args]) return nexpr elif d.is_Mul: nargs = [] dargs = [] for di in d.args: ni, di = fraction(handle(1/di)) nargs.append(ni) dargs.append(di) return n*Mul(*nargs)/Mul(*dargs) elif d.is_Add: d = radsimp(d) elif d.is_Pow and d.exp.is_Rational and d.exp.q == 2: d = sqrtdenest(sqrt(d.base))**d.exp.p changed = False while 1: # collect similar terms d, nterms = collect_sqrt(_mexpand(d), evaluate=False) d = Add._from_args(d) if nterms > max_terms: break # check to see if we are done: # - no radical terms # - if there are more than 3 radical terms, or # there 3 radical terms and a constant, use rad_rationalize if not nterms: break if nterms > 3 or nterms == 3 and len(d.args) > 4: if all([(x**2).is_Integer for x in d.args]): nd, d = rad_rationalize(S.One, d) n = _mexpand(n*nd) else: n, d = fraction(expr) break changed = True # now match for a radical if d.is_Add and len(d.args) == 4: r = d.match(a + b*sqrt(c) + D*sqrt(E)) nmul = (a - b*sqrt(c) - D*sqrt(E)).xreplace(r) d = (a**2 - c*b**2 - E*D**2 - 2*b*D*sqrt(c*E)).xreplace(r) n1 = n/d if denom(n1) is not S.One: n = -(-n/d) else: n = n1 n, d = fraction(n*nmul) else: r = d.match(a + b*sqrt(c)) if not r or r[b] == 0: r = d.match(b*sqrt(c)) if r is None: break r[a] = S.Zero va, vb, vc = r[a],r[b],r[c] nmul = va - vb*sqrt(vc) d = va**2 - vc*vb**2 n1 = n/d if denom(n1) is not S.One: n = -(-n/d) else: n = n1 n, d = fraction(n*nmul) nexpr = collect_sqrt(expand_mul(n))/d if changed or nexpr != expr: expr = nexpr return expr a, b, c, D, E, F, G = list(map(Wild, 'abcDEFG')) # do this at the start in case no other change is made since # it is done if a change is made coeff, expr = expr.as_content_primitive() newe = handle(expr) if newe != expr: co, expr = newe.as_content_primitive() coeff *= co else: nexpr, hit = collect_sqrt(expand_mul(expr), evaluate=False) nexpr = Add._from_args(nexpr) if hit and expr.count_ops() >= nexpr.count_ops(): expr = Add(*Add.make_args(nexpr)) return _keep_coeff(coeff, expr)
[docs]def posify(eq): """Return eq (with generic symbols made positive) and a restore dictionary. 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 >>> from sympy.abc import x >>> posify(x + Symbol('p', positive=True) + Symbol('n', negative=True)) (_x + n + p, {_x: x}) >> log(1/x).expand() # should be log(1/x) but it comes back as -log(x) log(1/x) >>> log(posify(1/x)[0]).expand() # take [0] and ignore replacements -log(_x) >>> eq, rep = posify(1/x) >>> log(eq).expand().subs(rep) -log(x) >>> posify([x, 1 + x]) ([_x, _x + 1], {_x: x}) """ eq = sympify(eq) if iterable(eq): f = type(eq) eq = list(eq) syms = set() for e in eq: syms = syms.union(e.atoms(C.Symbol)) reps = {} for s in syms: reps.update(dict((v, k) for k, v in list(posify(s)[1].items()))) for i, e in enumerate(eq): eq[i] = e.subs(reps) return f(eq), dict([(r, s) for s, r in reps.items()]) reps = dict([(s, Dummy(s.name, positive=True)) for s in eq.atoms(Symbol) if s.is_positive is None]) eq = eq.subs(reps) return eq, dict([(r, s) for s, r in reps.items()])
def _polarify(eq, lift, pause=False): from sympy import polar_lift, Integral if eq.is_polar: return eq if eq.is_number and not pause: return polar_lift(eq) if isinstance(eq, Symbol) and not pause and lift: return polar_lift(eq) elif eq.is_Atom: return eq elif eq.is_Add: r = eq.func(*[_polarify(arg, lift, pause=True) for arg in eq.args]) if lift: return polar_lift(r) return r elif eq.is_Function: return eq.func(*[_polarify(arg, lift, pause=False) for arg in eq.args]) elif isinstance(eq, Integral): # Don't lift the integration variable func = _polarify(eq.function, lift, pause=pause) limits = [] for limit in eq.args[1:]: var = _polarify(limit[0], lift=False, pause=pause) rest = _polarify(limit[1:], lift=lift, pause=pause) limits.append((var,) + rest) return Integral(*((func,) + tuple(limits))) else: return eq.func(*[_polarify(arg, lift, pause=pause) for arg in eq.args]) def polarify(eq, subs=True, lift=False): """ Turn all numbers in eq into their polar equivalents (under the standard choice of argument). Note that no attempt is made to guess a formal convention of adding polar numbers, expressions like 1 + x will generally not be altered. Note also that this function does not promote exp(x) to exp_polar(x). If ``subs`` is True, all symbols which are not already polar will be substituted for polar dummies; in this case the function behaves much like posify. If ``lift`` is True, both addition statements and non-polar symbols are changed to their polar_lift()ed versions. Note that lift=True implies subs=False. >>> from sympy import polarify, sin, I >>> from sympy.abc import x, y >>> expr = (-x)**y >>> expr.expand() (-x)**y >>> polarify(expr) ((_x*exp_polar(I*pi))**_y, {_x: x, _y: y}) >>> polarify(expr)[0].expand() _x**_y*exp_polar(_y*I*pi) >>> polarify(x, lift=True) polar_lift(x) >>> polarify(x*(1+y), lift=True) polar_lift(x)*polar_lift(y + 1) Adds are treated carefully: >>> polarify(1 + sin((1 + I)*x)) (sin(_x*polar_lift(1 + I)) + 1, {_x: x}) """ if lift: subs = False eq = _polarify(sympify(eq), lift) if not subs: return eq reps = dict([(s, Dummy(s.name, polar=True)) for s in eq.atoms(Symbol)]) eq = eq.subs(reps) return eq, dict([(r,s) for s, r in reps.items()]) def _unpolarify(eq, exponents_only, pause=False): from sympy import polar_lift, exp, principal_branch, pi if isinstance(eq, bool) or eq.is_Atom: return eq if not pause: if eq.func is exp_polar: return exp(_unpolarify(eq.exp, exponents_only)) if eq.func is principal_branch and eq.args[1] == 2*pi: return _unpolarify(eq.args[0], exponents_only) if ( eq.is_Add or eq.is_Mul or eq.is_Boolean or eq.is_Relational and ( eq.rel_op in ('==', '!=') and 0 in eq.args or eq.rel_op not in ('==', '!=')) ): return eq.func(*[_unpolarify(x, exponents_only) for x in eq.args]) if eq.func is polar_lift: return _unpolarify(eq.args[0], exponents_only) if eq.is_Pow: expo = _unpolarify(eq.exp, exponents_only) base = _unpolarify(eq.base, exponents_only, not (expo.is_integer and not pause)) return base**expo if eq.is_Function and getattr(eq.func, 'unbranched', False): return eq.func(*[_unpolarify(x, exponents_only, exponents_only) for x in eq.args]) return eq.func(*[_unpolarify(x, exponents_only, True) for x in eq.args]) def unpolarify(eq, subs={}, exponents_only=False): """ If p denotes the projection from the Riemann surface of the logarithm to the complex line, return a simplified version eq' of `eq` such that p(eq') == p(eq). Also apply the substitution subs in the end. (This is a convenience, since ``unpolarify``, in a certain sense, undoes polarify.) >>> from sympy import unpolarify, polar_lift, sin, I >>> unpolarify(polar_lift(I + 2)) 2 + I >>> unpolarify(sin(polar_lift(I + 7))) sin(7 + I) """ from sympy import exp_polar, polar_lift if isinstance(eq, bool): return eq eq = sympify(eq) if subs != {}: return unpolarify(eq.subs(subs)) changed = True pause = False if exponents_only: pause = True while changed: changed = False res = _unpolarify(eq, exponents_only, pause) if res != eq: changed = True eq = res if isinstance(res, bool): return res # Finally, replacing Exp(0) by 1 is always correct. # So is polar_lift(0) -> 0. return res.subs({exp_polar(0): 1, polar_lift(0): 0}) def _denest_pow(eq): """ Denest powers. This is a helper function for powdenest that performs the actual transformation. """ b, e = eq.as_base_exp() # denest exp with log terms in exponent if b is S.Exp1 and e.is_Mul: logs = [] other = [] for ei in e.args: if any(ai.func is C.log for ai in Add.make_args(ei)): logs.append(ei) else: other.append(ei) logs = logcombine(Mul(*logs)) return Pow(exp(logs), Mul(*other)) _, be = b.as_base_exp() if be is S.One and not (b.is_Mul or b.is_Rational and b.q != 1 or b.is_positive): return eq # denest eq which is either pos**e or Pow**e or Mul**e or Mul(b1**e1, b2**e2) # handle polar numbers specially polars, nonpolars = [], [] for bb in Mul.make_args(b): if bb.is_polar: polars.append(bb.as_base_exp()) else: nonpolars.append(bb) if len(polars) == 1 and not polars[0][0].is_Mul: return Pow(polars[0][0], polars[0][1]*e)*powdenest(Mul(*nonpolars)**e) elif polars: return Mul(*[powdenest(bb**(ee*e)) for (bb, ee) in polars]) \ *powdenest(Mul(*nonpolars)**e) # see if there is a positive, non-Mul base at the very bottom exponents = [] kernel = eq while kernel.is_Pow: kernel, ex = kernel.as_base_exp() exponents.append(ex) if kernel.is_positive: e = Mul(*exponents) if kernel.is_Mul: b = kernel else: if kernel.is_Integer: # use log to see if there is a power here logkernel = log(kernel) if logkernel.is_Mul: c, logk = logkernel.args e *= c kernel = logk.args[0] return Pow(kernel, e) # if any factor is an atom then there is nothing to be done # but the kernel check may have created a new exponent if any(s.is_Atom for s in Mul.make_args(b)): if exponents: return b**e return eq # let log handle the case of the base of the argument being a mul, e.g. # sqrt(x**(2*i)*y**(6*i)) -> x**i*y**(3**i) if x and y are positive; we # will take the log, expand it, and then factor out the common powers that # now appear as coefficient. We do this manually since terms_gcd pulls out # fractions, terms_gcd(x+x*y/2) -> x*(y + 2)/2 and we don't want the 1/2; # gcd won't pull out numerators from a fraction: gcd(3*x, 9*x/2) -> x but # we want 3*x. Neither work with noncommutatives. def nc_gcd(aa, bb): a, b = [i.as_coeff_Mul() for i in [aa, bb]] c = gcd(a[0], b[0]).as_numer_denom()[0] g = Mul(*(a[1].args_cnc(cset=True)[0] & b[1].args_cnc(cset=True)[0])) return _keep_coeff(c, g) glogb = expand_log(log(b)) if glogb.is_Add: args = glogb.args g = reduce(nc_gcd, args) if g != 1: cg, rg = g.as_coeff_Mul() glogb = _keep_coeff(cg, rg*Add(*[a/g for a in args])) # now put the log back together again if glogb.func is C.log or not glogb.is_Mul: if glogb.args[0].is_Pow or glogb.args[0].func is exp: glogb = _denest_pow(glogb.args[0]) if (abs(glogb.exp) < 1) is True: return Pow(glogb.base, glogb.exp*e) return eq # the log(b) was a Mul so join any adds with logcombine add= [] other = [] for a in glogb.args: if a.is_Add: add.append(a) else: other.append(a) return Pow(exp(logcombine(Mul(*add))), e*Mul(*other))
[docs]def powdenest(eq, force=False, polar=False): r""" 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 sympy.abc import a, b, x, y, z >>> from sympy 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) >>> p = Symbol('p', positive=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(((p**(2*a))**(3*y))**x) p**(6*a*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 """ if force: eq, rep = posify(eq) return powdenest(eq, force=False).xreplace(rep) if polar: eq, rep = polarify(eq) return unpolarify(powdenest(unpolarify(eq, exponents_only=True)), rep) new = powsimp(sympify(eq)) return new.xreplace(Transform(_denest_pow, filter=lambda m: m.is_Pow or m.func is exp))
_y = Dummy('y')
[docs]def powsimp(expr, deep=False, combine='all', force=False, measure=count_ops): """ 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 sympy import powsimp, exp, log, symbols >>> from sympy.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(exp(x)*exp(y)) >>> powsimp(log(exp(x)*exp(y)), deep=True) x + y Radicals with Mul bases will be combined if combine='exp' >>> from sympy 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)) """ def recurse(arg, **kwargs): _deep = kwargs.get('deep', deep) _combine = kwargs.get('combine', combine) _force = kwargs.get('force', force) _measure = kwargs.get('measure', measure) return powsimp(arg, _deep, _combine, _force, _measure) expr = sympify(expr) if not isinstance(expr, Basic) or expr.is_Atom or expr in (exp_polar(0), exp_polar(1)): return expr if deep or expr.is_Add or expr.is_Mul and _y not in expr.args: expr = expr.func(*[recurse(w) for w in expr.args]) if expr.is_Pow: return recurse(expr*_y, deep=False)/_y if not expr.is_Mul: return expr # handle the Mul if combine in ('exp', 'all'): # Collect base/exp data, while maintaining order in the # non-commutative parts of the product c_powers = defaultdict(list) nc_part = [] newexpr = [] for term in expr.args: if term.is_commutative: b, e = term.as_base_exp() if deep: b, e = [recurse(i) for i in [b, e]] c_powers[b].append(e) else: # This is the logic that combines exponents for equal, # but non-commutative bases: A**x*A**y == A**(x+y). if nc_part: b1, e1 = nc_part[-1].as_base_exp() b2, e2 = term.as_base_exp() if (b1 == b2 and e1.is_commutative and e2.is_commutative): nc_part[-1] = Pow(b1, Add(e1, e2)) continue nc_part.append(term) # add up exponents of common bases for b, e in c_powers.items(): c_powers[b] = Add(*e) # check for base and inverted base pairs be = list(c_powers.items()) skip = set() # skip if we already saw them for b, e in be: if b in skip: continue bpos = b.is_positive or b.is_polar if bpos: binv = 1/b if b != binv and binv in c_powers: if b.as_numer_denom()[0] is S.One: c_powers.pop(b) c_powers[binv] -= e else: skip.add(binv) e = c_powers.pop(binv) c_powers[b] -= e # filter c_powers and convert to a list c_powers = [(b, e) for b, e in c_powers.items() if e] # ============================================================== # check for Mul bases of Rational powers that can be combined with # separated bases, e.g. x*sqrt(x*y)*sqrt(x*sqrt(x*y)) -> (x*sqrt(x*y))**(3/2) # ---------------- helper functions def ratq(x): '''Return Rational part of x's exponent as it appears in the bkey. ''' return bkey(x)[0][1] def bkey(b, e=None): '''Return (b**s, c.q), c.p where e -> c*s. If e is not given then it will be taken by using as_base_exp() on the input b. e.g. x**3/2 -> (x, 2), 3 x**y -> (x**y, 1), 1 x**(2*y/3) -> (x**y, 3), 2 exp(x/2) -> (exp(a), 2), 1 ''' if e is not None: # coming from c_powers or from below if e.is_Integer: return (b, S.One), e elif e.is_Rational: return (b, Integer(e.q)), Integer(e.p) else: c, m = e.as_coeff_Mul(rational=True) if c is not S.One: return (b**m, Integer(c.q)), Integer(c.p) else: return (b**e, S.One), S.One else: return bkey(*b.as_base_exp()) def update(b): '''Decide what to do with base, b. If its exponent is now an integer multiple of the Rational denominator, then remove it and put the factors of its base in the common_b dictionary or update the existing bases if necessary. If it has been zeroed out, simply remove the base. ''' newe, r = divmod(common_b[b], b[1]) if not r: common_b.pop(b) if newe: for m in Mul.make_args(b[0]**newe): b, e = bkey(m) if b not in common_b: common_b[b] = 0 common_b[b] += e if b[1] != 1: bases.append(b) # ---------------- end of helper functions # assemble a dictionary of the factors having a Rational power common_b = {} done = [] bases = [] for b, e in c_powers: b, e = bkey(b, e) common_b[b] = e if b[1] != 1 and b[0].is_Mul: bases.append(b) bases.sort(key=default_sort_key) # this makes tie-breaking canonical bases.sort(key=measure, reverse= True) # handle longest first for base in bases: if base not in common_b: # it may have been removed already continue b, exponent = base last = False # True when no factor of base is a radical qlcm = 1 # the lcm of the radical denominators while True: bstart = b qstart = qlcm bb = [] # list of factors ee = [] # (factor's exponent, current value of that exponent in common_b) for bi in Mul.make_args(b): bib, bie = bkey(bi) if bib not in common_b or common_b[bib] < bie: ee = bb = [] # failed break ee.append([bie, common_b[bib]]) bb.append(bib) if ee: # find the number of extractions possible # e.g. [(1, 2), (2, 2)] -> min(2/1, 2/2) -> 1 min1 = ee[0][1]/ee[0][0] for i in range(len(ee)): rat = ee[i][1]/ee[i][0] if rat < 1: break min1 = min(min1, rat) else: # update base factor counts # e.g. if ee = [(2, 5), (3, 6)] then min1 = 2 # and the new base counts will be 5-2*2 and 6-2*3 for i in range(len(bb)): common_b[bb[i]] -= min1*ee[i][0] update(bb[i]) # update the count of the base # e.g. x**2*y*sqrt(x*sqrt(y)) the count of x*sqrt(y) # will increase by 4 to give bkey (x*sqrt(y), 2, 5) common_b[base] += min1*qstart*exponent if (last # no more radicals in base or len(common_b) == 1 # nothing left to join with or all(k[1] == 1 for k in common_b) # no radicals left in common_b ): break # see what we can exponentiate base by to remove any radicals # so we know what to search for # e.g. if base were x**(1/2)*y**(1/3) then we should exponentiate # by 6 and look for powers of x and y in the ratio of 2 to 3 qlcm = lcm([ratq(bi) for bi in Mul.make_args(bstart)]) if qlcm == 1: break # we are done b = bstart**qlcm qlcm *= qstart if all(ratq(bi) == 1 for bi in Mul.make_args(b)): last = True # we are going to be done after this next pass # this base no longer can find anything to join with and # since it was longer than any other we are done with it b, q = base done.append((b, common_b.pop(base)*Rational(1, q))) # update c_powers and get ready to continue with powsimp c_powers = done # there may be terms still in common_b that were bases that were # identified as needing processing, so remove those, too for (b, q), e in list(common_b.items()): if (b.is_Pow or b.func is exp) and \ q is not S.One and not b.exp.is_Rational: b, be = b.as_base_exp() b = b**(be/q) else: b = root(b, q) c_powers.append((b, e)) check = len(c_powers) c_powers = dict(c_powers) assert len(c_powers) == check # there should have been no duplicates # ============================================================== # rebuild the expression newexpr = Mul(*(newexpr + [Pow(b, e) for b, e in c_powers.items()])) if combine == 'exp': return Mul(newexpr, Mul(*nc_part)) else: return recurse(Mul(*nc_part), combine='base')*\ recurse(newexpr, combine='base') elif combine == 'base': # Build c_powers and nc_part. These must both be lists not # dicts because exp's are not combined. c_powers = [] nc_part = [] for term in expr.args: if term.is_commutative: c_powers.append(list(term.as_base_exp())) else: # This is the logic that combines bases that are # different and non-commutative, but with equal and # commutative exponents: A**x*B**x == (A*B)**x. if nc_part: b1, e1 = nc_part[-1].as_base_exp() b2, e2 = term.as_base_exp() if (e1 == e2 and e2.is_commutative): nc_part[-1] = Pow(Mul(b1, b2), e1) continue nc_part.append(term) # Pull out numerical coefficients from exponent if assumptions allow # e.g., 2**(2*x) => 4**x for i in range(len(c_powers)): b, e = c_powers[i] if not (b.is_nonnegative or e.is_integer or force or b.is_polar): continue exp_c, exp_t = e.as_coeff_Mul(rational=True) if exp_c is not S.One and exp_t is not S.One: c_powers[i] = [Pow(b, exp_c), exp_t] # Combine bases whenever they have the same exponent and # assumptions allow # first gather the potential bases under the common exponent c_exp = defaultdict(list) for b, e in c_powers: if deep: e = recurse(e) c_exp[e].append(b) del c_powers # Merge back in the results of the above to form a new product c_powers = defaultdict(list) for e in c_exp: bases = c_exp[e] # calculate the new base for e if len(bases) == 1: new_base = bases[0] elif e.is_integer or force: new_base = Mul(*bases) else: # see which ones can be joined unk=[] nonneg=[] neg=[] for bi in bases: if bi.is_negative: neg.append(bi) elif bi.is_nonnegative: nonneg.append(bi) elif bi.is_polar: nonneg.append(bi) # polar can be treated like non-negative else: unk.append(bi) if len(unk) == 1 and not neg or len(neg) == 1 and not unk: # a single neg or a single unk can join the rest nonneg.extend(unk + neg) unk = neg = [] elif neg: # their negative signs cancel in pairs neg = [-w for w in neg] if len(neg) % 2: unk.append(S.NegativeOne) # these shouldn't be joined for b in unk: c_powers[b].append(e) # here is a new joined base new_base = Mul(*(nonneg + neg)) # if there are positive parts they will just get separated again # unless some change is made def _terms(e): # return the number of terms of this expression # when multiplied out -- assuming no joining of terms if e.is_Add: return sum([_terms(ai) for ai in e.args]) if e.is_Mul: return prod([_terms(mi) for mi in e.args]) return 1 xnew_base = expand_mul(new_base, deep=False) if len(Add.make_args(xnew_base)) < _terms(new_base): new_base = factor_terms(xnew_base) c_powers[new_base].append(e) # break out the powers from c_powers now c_part = [Pow(b, ei) for b, e in c_powers.items() for ei in e] # we're done return Mul(*(c_part + nc_part)) else: raise ValueError("combine must be one of ('all', 'exp', 'base').")
[docs]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. 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) 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
[docs]def hypersimilar(f, g, k): """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. 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)
from sympy.utilities.timeutils import timethis @timethis('combsimp')
[docs]def combsimp(expr): r""" 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 result is be given in terms of binomials and factorials. 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 more familiar functions. If the initial expression contained any combinatorial functions, the result is expressed using binomial coefficients and gamma functions. If the initial expression consisted of gamma functions alone, 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: 1. http://functions.wolfram.com/GammaBetaErf/Pochhammer/17/01/02/ 2. http://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/0005/ Examples ======== >>> from sympy.simplify import combsimp >>> from sympy import factorial, binomial >>> from sympy.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) """ factorial = C.factorial binomial = C.binomial gamma = C.gamma # as a rule of thumb, if the expression contained gammas initially, it # probably makes sense to retain them as_gamma = not expr.has(factorial, binomial) class rf(Function): @classmethod def eval(cls, a, b): if b.is_Integer: if not b: return S.One n, result = int(b), S.One if n > 0: for i in range(n): result *= a + i return result elif n < 0: for i in range(1, -n + 1): result *= a - i return 1/result else: if b.is_Add: c, _b = b.as_coeff_Add() if c.is_Integer: if c > 0: return rf(a, _b)*rf(a + _b, c) elif c < 0: return rf(a, _b)/rf(a + _b + c, -c) if a.is_Add: c, _a = a.as_coeff_Add() if c.is_Integer: if c > 0: return rf(_a, b)*rf(_a + b, c)/rf(_a, c) elif c < 0: return rf(_a, b)*rf(_a + c, -c)/rf(_a + b + c, -c) expr = expr.replace(binomial, lambda n, k: rf((n - k + 1).expand(), k.expand())/rf(1, k.expand())) expr = expr.replace(factorial, lambda n: rf(1, n.expand())) expr = expr.replace(gamma, lambda n: rf(1, (n - 1).expand())) if as_gamma: expr = expr.replace(rf, lambda a, b: gamma(a + b)/gamma(a)) else: expr = expr.replace(rf, lambda a, b: binomial(a + b - 1, b)*factorial(b)) def rule(n, k): coeff, rewrite = S.One, False cn, _n = n.as_coeff_Add() if _n and cn.is_Integer and cn: coeff *= rf(_n + 1, cn)/rf(_n - k + 1, cn) rewrite = True n = _n # this sort of binomial has already been removed by # rising factorials but is left here in case the order # of rule application is changed if k.is_Add: ck, _k = k.as_coeff_Add() if _k and ck.is_Integer and ck: coeff *= rf(n - ck - _k + 1, ck)/rf(_k + 1, ck) rewrite = True k = _k if rewrite: return coeff*binomial(n, k) expr = expr.replace(binomial, rule) def rule_gamma(expr): """ Simplify products of gamma functions further. """ from itertools import count from sympy.core.compatibility import permutations if expr.is_Atom: return expr expr = expr.func(*[rule_gamma(x) for x in expr.args]) if not expr.is_Mul: return expr numer_gammas = [] denom_gammas = [] denom_others = [] newargs, numer_others = expr.args_cnc() # order newargs canonically cexpr = expr.func(*newargs) newargs = list(cexpr._sorted_args) if not cexpr.is_Atom else [cexpr] del cexpr while newargs: arg = newargs.pop() b, e = arg.as_base_exp() if e.is_Integer: n = abs(e) if isinstance(b, gamma): barg = b.args[0] if e > 0: numer_gammas.extend([barg]*n) elif e < 0: denom_gammas.extend([barg]*n) else: if e > 0: numer_others.extend([b]*n) elif e < 0: denom_others.extend([b]*n) else: numer_others.append(arg) # Try to reduce the number of gamma factors by applying the # reflection formula gamma(x)*gamma(1-x) = pi/sin(pi*x) for gammas, numer, denom in [(numer_gammas, numer_others, denom_others), (denom_gammas, denom_others, numer_others)]: new = [] while gammas: g1 = gammas.pop() if g1.is_integer: new.append(g1) continue for i, g2 in enumerate(gammas): n = g1 + g2 - 1 if not n.is_Integer: continue append = False numer.append(S.Pi) denom.append(C.sin(S.Pi*g1)) gammas.pop(i) if n > 0: for k in range(n): numer.append(1 - g1 + k) elif n < 0: for k in range(-n): denom.append(-g1 - k) break else: new.append(g1) # /!\ updating IN PLACE gammas[:] = new # Try to reduce the number of gamma factors by applying the # multiplication theorem. def _run(coeffs): # find runs in coeffs such that the difference in terms (mod 1) # of t1, t2, ..., tn is 1/n from sympy.utilities.iterables import uniq u = uniq(coeffs) for i in range(len(u)): dj = ([((u[j] - u[i]) % 1, j) for j in range(i + 1, len(u))]) for one, j in dj: if one.p == 1 and one.q != 1: n = one.q got = [i] get = list(range(1, n)) for d, j in dj: m = n*d if m.is_Integer and m in get: get.remove(m) got.append(j) if not get: break else: continue for i, j in enumerate(got): c = u[j] coeffs.remove(c) got[i] = c return one.q, got[0], got[1:] def _mult_thm(gammas, numer, denom): # pull off and analyze the leading coefficient from each gamma arg # looking for runs in those Rationals # expr -> coeff + resid -> rats[resid] = coeff rats = {} for g in gammas: c, resid = g.as_coeff_Add() rats.setdefault(resid, []).append(c) # look for runs in Rationals for each resid keys = sorted(rats, key=default_sort_key) for resid in keys: coeffs = list(sorted(rats[resid])) new = [] while True: run = _run(coeffs) if run is None: break # process the sequence that was found: # 1) convert all the gamma functions to have the right # argument (could be off by an integer) # 2) append the factors corresponding to the theorem # 3) append the new gamma function n, ui, other = run # (1) for u in other: con = resid + u - 1 for k in range(int(u - ui)): numer.append(con - k) con = n*(resid + ui) # for (2) and (3) # (2) numer.append((2*S.Pi)**(S(n - 1)/2)* n**(S(1)/2 - con)) # (3) new.append(con) # restore resid to coeffs rats[resid] = [resid + c for c in coeffs] + new # rebuild the gamma arguments g = [] for resid in keys: g += rats[resid] # /!\ updating IN PLACE gammas[:] = g for l, numer, denom in [(numer_gammas, numer_others, denom_others), (denom_gammas, denom_others, numer_others)]: _mult_thm(l, numer, denom) # Try to reduce the number of gammas by using the duplication # theorem to cancel an upper and lower. # e.g. gamma(2*s)/gamma(s) = gamma(s)*gamma(s+1/2)*C/gamma(s) # (in principle this can also be done with with factors other than two, # but two is special in that we need only matching numer and denom, not # several in numer). for ng, dg, no, do in [(numer_gammas, denom_gammas, numer_others, denom_others), (denom_gammas, numer_gammas, denom_others, numer_others)]: while True: for x in ng: for y in dg: n = x - 2*y if n.is_Integer: break else: continue break else: break ng.remove(x) dg.remove(y) if n > 0: for k in range(n): no.append(2*y + k) elif n < 0: for k in range(-n): do.append(2*y - 1 - k) ng.append(y + S(1)/2) no.append(2**(2*y - 1)) do.append(sqrt(S.Pi)) # Try to absorb factors into the gammas. # This code (in particular repeated calls to find_fuzzy) can be very # slow. def find_fuzzy(l, x): S1, T1 = compute_ST(x) for y in l: S2, T2 = inv[y] if T1 != T2 or (not S1.intersection(S2) and \ (S1 != set() or S2 != set())): continue # XXX we want some simplification (e.g. cancel or # simplify) but no matter what it's slow. a = len(cancel(x/y).free_symbols) b = len(x.free_symbols) c = len(y.free_symbols) # TODO is there a better heuristic? if a == 0 and (b > 0 or c > 0): return y # We thus try to avoid expensive calls by building the following # "invariants": For every factor or gamma function argument # - the set of free symbols S # - the set of functional components T # We will only try to absorb if T1==T2 and (S1 intersect S2 != emptyset # or S1 == S2 == emptyset) inv = {} def compute_ST(expr): from sympy import Function, Pow if expr in inv: return inv[expr] return (expr.free_symbols, expr.atoms(Function).union( set(e.exp for e in expr.atoms(Pow)))) def update_ST(expr): inv[expr] = compute_ST(expr) for expr in numer_gammas + denom_gammas + numer_others + denom_others: update_ST(expr) for gammas, numer, denom in [(numer_gammas, numer_others, denom_others), (denom_gammas, denom_others, numer_others)]: new = [] while gammas: g = gammas.pop() cont = True while cont: cont = False y = find_fuzzy(numer, g) if y is not None: numer.remove(y) if y != g: numer.append(y/g) update_ST(y/g) g += 1 cont = True y = find_fuzzy(numer, 1/(g - 1)) if y is not None: numer.remove(y) if y != 1/(g - 1): numer.append((g - 1)*y) update_ST((g - 1)*y) g -= 1 cont = True y = find_fuzzy(denom, 1/g) if y is not None: denom.remove(y) if y != 1/g: denom.append(y*g) update_ST(y*g) g += 1 cont = True y = find_fuzzy(denom, g - 1) if y is not None: denom.remove(y) if y != g - 1: numer.append((g - 1)/y) update_ST((g - 1)/y) g -= 1 cont = True new.append(g) # /!\ updating IN PLACE gammas[:] = new return C.Mul(*[gamma(g) for g in numer_gammas]) \ / C.Mul(*[gamma(g) for g in denom_gammas]) \ * C.Mul(*numer_others) / C.Mul(*denom_others) # (for some reason we cannot use Basic.replace in this case) expr = rule_gamma(expr) return factor(expr)
def signsimp(expr, evaluate=True): """Make all Add sub-expressions canonical wrt sign. 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 >>> from sympy.abc import x, y >>> 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 >>> n**3 (-1 + 1/x)**3 >>> signsimp(_) -(1 - 1/x)**3 By default, signsimp doesn't 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)) """ expr = sympify(expr) if not isinstance(expr, Expr) or expr.is_Atom: return expr e = sub_post(sub_pre(expr)) if not isinstance(e, Expr) or e.is_Atom: return e if e.is_Add: return Add(*[signsimp(a) for a in e.args]) if evaluate: e = e.xreplace(dict([(m, -(-m)) for m in e.atoms(Mul) if -(-m) != m])) return e
[docs]def simplify(expr, ratio=1.7, measure=count_ops): """ 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 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 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 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 >>> b = trigsimp(a) >>> b (x**2 + x)/x >>> c = cancel(b) >>> c 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 can't 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 don't 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**(log(1/a) + 1)) >>> count_ops(g) 8 >>> count_ops(h) 6 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)) ADD + DIV + 2*LOG + MUL + POW >>> 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) 15 >>> 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. """ from sympy.simplify.hyperexpand import hyperexpand from sympy.functions.special.bessel import BesselBase original_expr = expr = sympify(expr) expr = signsimp(expr) if not isinstance(expr, Basic): # XXX: temporary hack return expr if isinstance(expr, Atom): return expr if isinstance(expr, C.Relational): return expr.__class__(simplify(expr.lhs, ratio=ratio), simplify(expr.rhs, ratio=ratio)) # 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. 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) if expr.is_commutative is False: expr1 = factor_terms(together(powsimp(expr))) if ratio is S.Infinity: return expr1 return shorter(expr1, expr) expr0 = powsimp(expr) expr1 = cancel(expr0) expr2 = together(expr1.expand(), deep=True) # sometimes factors in the denominators need to be allowed to join # factors in numerators (see issue 3270) n, d = expr.as_numer_denom() if (n, d) != fraction(expr): expr0b = powsimp(n)/powsimp(d) if expr0b != expr0: expr1b = cancel(expr0b) expr2b = together(expr1b.expand(), deep=True) if shorter(expr2b, expr) == expr2b: expr1, expr2 = expr1b, expr2b if ratio is S.Infinity: expr = expr2 else: expr = shorter(expr2, expr1, expr) if not isinstance(expr, Basic): # XXX: temporary hack return expr # hyperexpand automatically only works on hypergeometric terms expr = hyperexpand(expr) if expr.has(BesselBase): expr = besselsimp(expr) if expr.has(C.TrigonometricFunction) or expr.has(C.HyperbolicFunction): expr = trigsimp(expr, deep=True) if expr.has(C.log): expr = shorter(expand_log(expr, deep=True), logcombine(expr)) if expr.has(C.CombinatorialFunction, gamma): expr = combsimp(expr) expr = powsimp(expr, combine='exp', deep=True) short = shorter(expr, powsimp(factor_terms(expr))) if short != expr: # get rid of hollow 2-arg Mul factorization from sympy.core.rules import Transform 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 = shorter(short.xreplace(hollow_mul), expr) 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 = expr.as_numer_denom() if d != 0: expr = -n/(-d) if measure(expr) > ratio*measure(original_expr): return original_expr if original_expr.is_Matrix: expr = matrixify(expr) return expr
def _real_to_rational(expr): """ Replace all reals in expr with rationals. >>> from sympy import nsimplify >>> from sympy.abc import x >>> nsimplify(.76 + .1*x**.5, rational=True) sqrt(x)/10 + 19/25 """ p = expr reps = {} for r in p.atoms(C.Float): newr = nsimplify(r, rational=False) if not newr.is_Rational or \ r.is_finite and not newr.is_finite: newr = r if newr < 0: s = -1 newr *= s else: s = 1 d = Pow(10, int((mpmath.log(newr)/mpmath.log(10)))) newr = s*Rational(str(newr/d))*d reps[r] = newr return p.subs(reps, simultaneous=True)
[docs]def nsimplify(expr, constants=[], tolerance=None, full=False, rational=None): """ 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 sympy 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]) exp(-pi/2) >>> nsimplify(pi, tolerance=0.01) 22/7 See Also ======== sympy.core.function.nfloat """ expr = sympify(expr) if rational or expr.free_symbols: return _real_to_rational(expr) # sympy's default tolarance for Rationals is 15; other numbers may have # lower tolerances set, so use them to pick the largest tolerance if none # was given tolerance = tolerance or 10**-min([15] + [mpmath.libmp.libmpf.prec_to_dps(n._prec) for n in expr.atoms(Float)]) 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() # Must be numerical if not ((re.is_Float or re.is_Integer) and (im.is_Float or im.is_Integer)): 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.findpoly(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 expr.is_finite is False and not xv 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) 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)
[docs]def logcombine(expr, force=False): """ Takes logarithms and combines them using the following rules: - log(x)+log(y) == log(x*y) - a*log(x) == log(x**a) These identities are only valid if x and y are positive and if a is real, so the function will not combine the terms unless the arguments have the proper assumptions on them. Use logcombine(func, force=True) to automatically assume that the arguments of logs are positive and that coefficients are real. Note that this will not change any assumptions already in place, so if the coefficient is imaginary or the argument negative, combine will still not combine the equations. Change the assumptions on the variables to make them combine. Examples ======== >>> from sympy import Symbol, symbols, log, logcombine >>> 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) """ # Try to make (a+bi)*log(x) == a*log(x)+bi*log(x). This needs to be a # separate function call to avoid infinite recursion. expr = expand_mul(expr) return _logcombine(expr, force)
def _logcombine(expr, force=False): """ Does the main work for logcombine, it's a separate function to avoid an infinite recursion. See the docstrings of logcombine() for help. """ def _getlogargs(expr): """ Returns the arguments of the logarithm in an expression. Examples ======== _getlogargs(a*log(x*y)) x*y """ if expr.func is log: return [expr.args[0]] else: args = [] for i in expr.args: if i.func is log: args.append(_getlogargs(i)) return flatten(args) return None if expr.is_Number or expr.is_NumberSymbol or type(expr) == C.Integral: return expr if isinstance(expr, Equality): retval = Equality(_logcombine(expr.lhs-expr.rhs, force),\ Integer(0)) # If logcombine couldn't do much with the equality, try to make it like # it was. Hopefully extract_additively won't become smart enough to # take logs apart :) right = retval.lhs.extract_additively(expr.lhs) if right: return Equality(expr.lhs, _logcombine(-right, force)) else: return retval if expr.is_Add: argslist = 1 notlogs = 0 coeflogs = 0 for i in expr.args: if i.func is log: if (i.args[0].is_positive or (force and not \ i.args[0].is_nonpositive)): argslist *= _logcombine(i.args[0], force) else: notlogs += i elif i.is_Mul and any([getattr(t,'func', False)==log for t in i.args]): largs = _getlogargs(i) assert len(largs) != 0 loglargs = 1 for j in largs: loglargs *= log(j) if all(getattr(t,'is_positive') for t in largs)\ and getattr(i.extract_multiplicatively(loglargs),'is_real', False)\ or (force\ and not all(getattr(t,'is_nonpositive') for t in largs)\ and not getattr(i.extract_multiplicatively(loglargs),\ 'is_real')==False): coeflogs += _logcombine(i, force) else: notlogs += i elif i.has(log): notlogs += _logcombine(i, force) else: notlogs += i if notlogs + log(argslist) + coeflogs == expr: return expr else: alllogs = _logcombine(log(argslist) + coeflogs, force) return notlogs + alllogs if expr.is_Mul: a = Wild('a') x = Wild('x') coef = expr.match(a*log(x)) if coef\ and (coef[a].is_real\ or expr.is_Number\ or expr.is_NumberSymbol\ or type(coef[a]) in (int, float)\ or (force\ and not coef[a].is_imaginary))\ and (coef[a].func != log\ or force\ or (not getattr(coef[a],'is_real')==False\ and getattr(x, 'is_positive'))): return log(coef[x]**coef[a]) else: return _logcombine(expr.args[0], force)*reduce(lambda x, y:\ _logcombine(x, force)*_logcombine(y, force),\ expr.args[1:], S.One) if expr.is_Function: return expr.func(*[_logcombine(t, force) for t in expr.args]) if expr.is_Pow: return _logcombine(expr.args[0], force)**\ _logcombine(expr.args[1], force) return expr
[docs]def besselsimp(expr): """ 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. After that, functions of half-integer order are rewritten using trigonometric functions. >>> 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)) """ from sympy import besselj, besseli, jn, I, pi, Dummy # TODO # - extension to more types of functions # (at least rewriting functions of half integer order should be straight # forward also for Y and K) # - 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 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(1)/2: return fro(nu, z) return unpolarify(fro(nu, z0).rewrite(besselj).rewrite(jn).expand(func=True)).subs(z0, z) return repl expr = expr.replace(besselj, expander(besselj)) expr = expr.replace(besseli, expander(besseli)) return expr