Source code for diofant.simplify.radsimp

from collections import defaultdict

from .. import DIOFANT_DEBUG
from ..core import (Add, Derivative, I, Integer, Mul, Pow, Rational,
                    expand_mul, expand_power_base, gcd_terms, symbols, sympify)
from ..core.compatibility import default_sort_key, iterable, ordered
from ..core.exprtools import Factors
from ..core.function import _mexpand
from ..core.mul import _keep_coeff, _unevaluated_Mul
from ..functions import log, sqrt
from ..polys import gcd
from .sqrtdenest import sqrtdenest

[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 (for example, by using :func:`~diofant.core.function.expand` prior to calling this function). 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:`~diofant.core.function.expand_power_base` function. Parameters ========== expr : Expr an expression syms : iterable of Symbol's collected symbols evaluate : boolean First, if ``evaluate`` flag is set (by default), this function will return an expression with collected terms else it will return a dictionary with expressions up to rational powers as keys and collected coefficients as values. Examples ======== This function can collect symbolic coefficients in polynomials or rational expressions. It will manage to find all integer or rational powers of collection variable: >>> collect(a*x**2 + b*x**2 + a*x - b*x + c, x) c + x**2*(a + b) + x*(a - b) The same result can be achieved in dictionary form: >>> d = collect(a*x**2 + b*x**2 + a*x - b*x + c, x, evaluate=False) >>> d[x**2] a + b >>> d[x] a - b >>> d[1] c You can also work with multivariate polynomials. However, remember that this function is greedy so it will care only about a single symbol at time, in specification order: >>> collect(x**2 + y*x**2 + x*y + y + a*y, [x, y]) x**2*(y + 1) + x*y + y*(a + 1) Also more complicated expressions can be used as patterns: >>> 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: >>> collect(a*exp(2*x) + b*exp(2*x), exp(x)) E**(2*x)*(a + b) If you are interested only in collecting specific powers of some symbols then set ``exact`` flag in arguments: >>> collect(a*x**7 + b*x**7, x, exact=True) a*x**7 + b*x**7 >>> collect(a*x**7 + b*x**7, x**7, exact=True) x**7*(a + b) You can also apply this function to differential equations, where derivatives of arbitrary order can be collected. Note that if you collect with respect to a function or a derivative of a function, all derivatives of that function will also be collected. Use ``exact=True`` to prevent this from happening: >>> f = f(x) >>> collect(a*Derivative(f, x) + b*Derivative(f, x), Derivative(f, x)) (a + b)*Derivative(f(x), x) >>> collect(a*Derivative(f, x, 2) + b*Derivative(f, x, 2), f) (a + b)*Derivative(f(x), x, x) >>> collect(a*Derivative(f, x, 2) + b*Derivative(f, x, 2), Derivative(f, x), exact=True) a*Derivative(f(x), x, x) + b*Derivative(f(x), x, x) >>> collect(a*Derivative(f, x) + b*Derivative(f, x) + a*f + b*f, f) f(x)*(a + b) + (a + b)*Derivative(f(x), x) Or you can even match both derivative order and exponent at the same time: >>> collect(a*Derivative(f, x, 2)**2 + b*Derivative(f, x, 2)**2, Derivative(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 See Also ======== collect_const, collect_sqrt, rcollect """ 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 == 1: 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 = Integer(1), 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 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 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 is 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 return [_f for _f in terms if _f], elems, common_expo, has_deriv if evaluate: if expr.is_Mul: return expr.func(*[ 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), Integer(0) for product in summa: terms = [parse_term(i) for i in Mul.make_args(product)] for symbol in syms: if DIOFANT_DEBUG: print("DEBUG: parsing of expression %s with symbol %s " % ( str(terms), str(symbol)) ) result = parse_expression(terms, symbol) if DIOFANT_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 = {k: Add(*v) for k, v in collected.items()} if disliked != 0: collected[Integer(1)] = disliked if order_term is not None: for key, val in collected.items(): collected[key] = val + order_term if func is not None: collected = {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 ======== >>> expr = (x**2*y + x*y + x + y)/(x + y) >>> rcollect(expr, y) (x + y*(x**2 + x + 1))/(x + y) See Also ======== collect, collect_const, collect_sqrt """ 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 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, if non-zero, the terms of the Add will be returned, else the expression itself will be returned as a single term. If ``evaluate`` is True, the expression with any collected terms will be returned. Note: since I = sqrt(-1), it is collected, too. Examples ======== >>> r2, r3, r5 = [sqrt(i) for i in [2, 3, 5]] >>> collect_sqrt(a*r2 + b*r2) sqrt(2)*(a + b) >>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r3) sqrt(2)*(a + b) + sqrt(3)*(a + b) >>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5) sqrt(3)*a + sqrt(5)*b + sqrt(2)*(a + b) If evaluate is False then the arguments will be sorted and returned as a list and a count of the number of sqrt-containing terms will be returned: >>> collect_sqrt(a*r2 + b*r2 + a*r3 + b*r5, evaluate=False) ((sqrt(3)*a, sqrt(5)*b, sqrt(2)*(a + b)), 3) >>> collect_sqrt(a*sqrt(2) + b, evaluate=False) ((b, sqrt(2)*a), 1) >>> collect_sqrt(a + b, evaluate=False) ((a + b,), 0) See Also ======== collect, collect_const, rcollect """ # this step will help to standardize any complex arguments # of sqrts 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.denominator == 2 or m is I): vars.add(m) # we only want radicals, so exclude Number handling; in this case # d will be evaluated d = collect_const(expr, *vars, Numbers=False) hit = expr != d if not evaluate: nrad = 0 # make the evaluated args canonical args = list(ordered(Add.make_args(d))) for i, m in enumerate(args): c, nc = m.args_cnc() for ci in c: # XXX should this be restricted to ci.is_number as above? if ci.is_Pow and ci.exp.is_Rational and ci.exp.denominator == 2 or \ ci is I: nrad += 1 break args[i] *= coeff if not (hit or nrad): args = [Add(*args)] return tuple(args), nrad return coeff*d
[docs]def collect_const(expr, *vars, **kwargs): """A non-greedy collection of terms with similar number coefficients in an Add expr. If ``vars`` is given then only those constants will be targeted. Although any Number can also be targeted, if this is not desired set ``Numbers=False`` and no Float or Rational will be collected. Examples ======== >>> from import s >>> collect_const(sqrt(3) + sqrt(3)*(1 + sqrt(2))) sqrt(3)*(sqrt(2) + 2) >>> collect_const(sqrt(3)*s + sqrt(7)*s + sqrt(3) + sqrt(7)) (sqrt(3) + sqrt(7))*(s + 1) >>> s = sqrt(2) + 2 >>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7)) (sqrt(2) + 3)*(sqrt(3) + sqrt(7)) >>> collect_const(sqrt(3)*s + sqrt(3) + sqrt(7)*s + sqrt(7), sqrt(3)) sqrt(7) + sqrt(3)*(sqrt(2) + 3) + sqrt(7)*(sqrt(2) + 2) The collection is sign-sensitive, giving higher precedence to the unsigned values: >>> collect_const(x - y - z) x - (y + z) >>> collect_const(-y - z) -(y + z) >>> collect_const(2*x - 2*y - 2*z, 2) 2*(x - y - z) >>> collect_const(2*x - 2*y - 2*z, -2) 2*x - 2*(y + z) See Also ======== collect, collect_sqrt, rcollect """ if not expr.is_Add: return expr recurse = False Numbers = kwargs.get('Numbers', True) if not vars: recurse = True vars = set() for a in expr.args: for m in Mul.make_args(a): if m.is_number: vars.add(m) else: vars = sympify(vars) if not Numbers: vars = [v for v in vars if not v.is_Number] vars = list(ordered(vars)) for v in vars: terms = defaultdict(list) Fv = Factors(v) for m in Add.make_args(expr): f = Factors(m) q, r = f.div(Fv) if r.is_one: # only accept this as a true factor if # it didn't change an exponent from an Integer # to a non-Integer, e.g. 2/sqrt(2) -> sqrt(2) # -- we aren't looking for this sort of change fwas = f.factors.copy() fnow = q.factors if not any(k in fwas and fwas[k].is_Integer and not fnow[k].is_Integer for k in fnow): terms[v].append(q.as_expr()) continue terms[Integer(1)].append(m) args = [] hit = False for k in ordered(terms): v = terms[k] if k == 1: args.extend(v) continue if len(v) > 1: v = Add(*v) hit = True if recurse and v != expr: vars.append(v) else: v = v[0] if Numbers and k.is_Number and v.is_Add: args.append(_keep_coeff(k, v, sign=True)) else: args.append(k*v) if hit: expr = Add(*args) if not expr.is_Add: break return expr
[docs]def radsimp(expr, symbolic=True, max_terms=4): r""" Rationalize the denominator by removing square roots. Note: the expression returned from radsimp must be used with caution since if the denominator contains symbols, it will be possible to make substitutions that violate the assumptions of the simplification process: that for a denominator matching a + b*sqrt(c), a != +/-b*sqrt(c). (If there are no symbols, this assumptions is made valid by collecting terms of sqrt(c) so the match variable ``a`` does not contain ``sqrt(c)``.) If you do not want the simplification to occur for symbolic denominators, set ``symbolic`` to False. If there are more than ``max_terms`` radical terms then the expression is returned unchanged. Examples ======== >>> radsimp(1/(I + 1)) (1 - I)/2 >>> radsimp(1/(2 + sqrt(2))) (-sqrt(2) + 2)/2 >>> e = ((2 + 2*sqrt(2))*x + (2 + sqrt(8))*y)/(2 + sqrt(2)) >>> radsimp(e) sqrt(2)*(x + y) No simplification beyond removal of the gcd is done. One might want to polish the result a little, however, by collecting square root terms: >>> r2 = sqrt(2) >>> r5 = sqrt(5) >>> ans = radsimp(1/(y*r2 + x*r2 + a*r5 + b*r5)) >>> pprint(ans, use_unicode=False) ___ ___ ___ ___ \/ 5 *a + \/ 5 *b - \/ 2 *x - \/ 2 *y ------------------------------------------ 2 2 2 2 5*a + 10*a*b + 5*b - 2*x - 4*x*y - 2*y >>> n, d = fraction(ans) >>> pprint(factor_terms(signsimp(collect_sqrt(n))/d, radical=True), use_unicode=False) ___ ___ \/ 5 *(a + b) - \/ 2 *(x + y) ------------------------------------------ 2 2 2 2 5*a + 10*a*b + 5*b - 2*x - 4*x*y - 2*y If radicals in the denominator cannot be removed or there is no denominator, the original expression will be returned. >>> radsimp(sqrt(2)*x + sqrt(2)) sqrt(2)*x + sqrt(2) Results with symbols will not always be valid for all substitutions: >>> eq = 1/(a + b*sqrt(c)) >>> eq.subs({a: b*sqrt(c)}) 1/(2*b*sqrt(c)) >>> 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)) """ from .simplify import signsimp syms = symbols("a:d A:D") def _num(rterms): # return the multiplier that will simplify the expression described # by rterms [(sqrt arg, coeff), ... ] a, b, c, d, A, B, C, D = syms if len(rterms) == 2: reps = dict(zip([A, a, B, b], [j for i in rterms for j in i])) return ( sqrt(A)*a - sqrt(B)*b).xreplace(reps) if len(rterms) == 3: reps = dict(zip([A, a, B, b, C, c], [j for i in rterms for j in i])) return ( (sqrt(A)*a + sqrt(B)*b - sqrt(C)*c)*(2*sqrt(A)*sqrt(B)*a*b - A*a**2 - B*b**2 + C*c**2)).xreplace(reps) elif len(rterms) == 4: reps = dict(zip([A, a, B, b, C, c, D, d], [j for i in rterms for j in i])) return ((sqrt(A)*a + sqrt(B)*b - sqrt(C)*c - sqrt(D)*d)*(2*sqrt(A)*sqrt(B)*a*b - A*a**2 - B*b**2 - 2*sqrt(C)*sqrt(D)*c*d + C*c**2 + D*d**2)*(-8*sqrt(A)*sqrt(B)*sqrt(C)*sqrt(D)*a*b*c*d + A**2*a**4 - 2*A*B*a**2*b**2 - 2*A*C*a**2*c**2 - 2*A*D*a**2*d**2 + B**2*b**4 - 2*B*C*b**2*c**2 - 2*B*D*b**2*d**2 + C**2*c**4 - 2*C*D*c**2*d**2 + D**2*d**4)).xreplace(reps) elif len(rterms) == 1: return sqrt(rterms[0][0]) else: raise NotImplementedError def ispow2(d, log2=False): if not d.is_Pow: return False e = d.exp if e.is_Rational and e.denominator == 2 or symbolic and fraction(e)[1] == 2: return True if log2: q = 1 if e.is_Rational: q = e.denominator elif symbolic: d = fraction(e)[1] if d.is_Integer: q = d if q != 1 and log(q, 2).is_Integer: return True return False def handle(expr): # Handle first reduces to the case # expr = 1/d, where d is an add, or d is base**p/2. # We do this by recursively calling handle on each piece. from .simplify import nsimplify n, d = fraction(expr) if expr.is_Atom or (d.is_Atom and n.is_Atom): return expr elif not n.is_Atom: n = n.func(*[handle(a) for a in n.args]) return _unevaluated_Mul(n, handle(1/d)) elif n != 1: return _unevaluated_Mul(n, handle(1/d)) elif d.is_Mul: return _unevaluated_Mul(*[handle(1/d) for d in d.args]) # By this step, expr is 1/d, and d is not a mul. if not symbolic and d.free_symbols: return expr if ispow2(d): d2 = sqrtdenest(sqrt(d.base))**fraction(d.exp)[0] if d2 != d: return handle(1/d2) elif d.is_Pow and (d.exp.is_integer or d.base.is_positive): # (1/d**i) = (1/d)**i return handle(1/d.base)**d.exp if not (d.is_Add or ispow2(d)): return 1/d.func(*[handle(a) for a in d.args]) # handle 1/d treating d as an Add (though it may not be) keep = True # keep changes that are made # flatten it and collect radicals after checking for special # conditions d = _mexpand(d) # did it change? if d.is_Atom: return 1/d # is it a number that might be handled easily? if d.is_number: _d = nsimplify(d) if _d.is_Number and _d.equals(d): return 1/_d while True: # collect similar terms collected = defaultdict(list) for m in Add.make_args(d): # d might have become non-Add p2 = [] other = [] for i in Mul.make_args(m): if ispow2(i, log2=True): p2.append(i.base if i.exp == Rational(1, 2) else i.base**(2*i.exp)) elif i is I: p2.append(Integer(-1)) else: other.append(i) collected[tuple(ordered(p2))].append(Mul(*other)) rterms = list(ordered(list(collected.items()))) rterms = [(Mul(*i), Add(*j)) for i, j in rterms] nrad = len(rterms) - (1 if rterms[0][0] == 1 else 0) if nrad < 1: break elif nrad > max_terms: # there may have been invalid operations leading to this point # so don't keep changes, e.g. this expression is troublesome # in collecting terms so as not to raise the issue of 2834: # r = sqrt(sqrt(5) + 5) # eq = 1/(sqrt(5)*r + 2*sqrt(5)*sqrt(-sqrt(5) + 5) + 5*r) keep = False break if len(rterms) > 4: # in general, only 4 terms can be removed with repeated squaring # but other considerations can guide selection of radical terms # so that radicals are removed if all(x.is_Integer and (y**2).is_Rational for x, y in rterms): nd, d = rad_rationalize(Integer(1), Add._from_args( [sqrt(x)*y for x, y in rterms])) n *= nd else: # is there anything else that might be attempted? keep = False break from .powsimp import powsimp, powdenest num = powsimp(_num(rterms)) n *= num d *= num d = powdenest(_mexpand(d), force=symbolic) if d.is_Atom: break if not keep: return expr return _unevaluated_Mul(n, 1/d) coeff, expr = expr.as_coeff_Add() expr = expr.normal() old = fraction(expr) n, d = fraction(handle(expr)) if old != (n, d): if not d.is_Atom: was = (n, d) n = signsimp(n, evaluate=False) d = signsimp(d, evaluate=False) u = Factors(_unevaluated_Mul(n, 1/d)) u = _unevaluated_Mul(*[k**v for k, v in u.factors.items()]) n, d = fraction(u) if old == (n, d): n, d = was n = expand_mul(n) if d.is_Number or d.is_Add: n2, d2 = fraction(gcd_terms(_unevaluated_Mul(n, 1/d))) if d2.is_Number or (d2.count_ops() <= d.count_ops()): n, d = [signsimp(i) for i in (n2, d2)] if n.is_Mul and n.args[0].is_Number: n = n.func(*n.args) return coeff + _unevaluated_Mul(n, 1/d)
[docs]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 ======== >>> 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 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. >>> 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: >>> fraction(2*x**(-y)) (2, x**y) >>> fraction(exp(-x)) (1, E**x) >>> fraction(exp(-x), exact=True) (E**(-x), 1) """ expr = sympify(expr) numer, denom = [], [] for term in Mul.make_args(expr): if term.is_Pow and term.is_commutative: b, ex = term.as_base_exp() if ex.is_negative: if ex == -1: 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 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 ======== >>> 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 == Rational(1, 2): 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 _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 ======== >>> _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