from sympy.core import S, Function, diff, Tuple, Dummy, Mul
from sympy.core.basic import Basic, as_Basic
from sympy.core.numbers import Rational, NumberSymbol, _illegal
from sympy.core.parameters import global_parameters
from sympy.core.relational import (Lt, Gt, Eq, Ne, Relational,
_canonical, _canonical_coeff)
from sympy.core.sorting import ordered
from sympy.functions.elementary.miscellaneous import Max, Min
from sympy.logic.boolalg import (And, Boolean, distribute_and_over_or, Not,
true, false, Or, ITE, simplify_logic, to_cnf, distribute_or_over_and)
from sympy.utilities.iterables import uniq, sift, common_prefix
from sympy.utilities.misc import filldedent, func_name
from itertools import product
Undefined = S.NaN # Piecewise()
class ExprCondPair(Tuple):
"""Represents an expression, condition pair."""
def __new__(cls, expr, cond):
expr = as_Basic(expr)
if cond == True:
return Tuple.__new__(cls, expr, true)
elif cond == False:
return Tuple.__new__(cls, expr, false)
elif isinstance(cond, Basic) and cond.has(Piecewise):
cond = piecewise_fold(cond)
if isinstance(cond, Piecewise):
cond = cond.rewrite(ITE)
if not isinstance(cond, Boolean):
raise TypeError(filldedent('''
Second argument must be a Boolean,
not `%s`''' % func_name(cond)))
return Tuple.__new__(cls, expr, cond)
@property
def expr(self):
"""
Returns the expression of this pair.
"""
return self.args[0]
@property
def cond(self):
"""
Returns the condition of this pair.
"""
return self.args[1]
@property
def is_commutative(self):
return self.expr.is_commutative
def __iter__(self):
yield self.expr
yield self.cond
def _eval_simplify(self, **kwargs):
return self.func(*[a.simplify(**kwargs) for a in self.args])
class Piecewise(Function):
"""
Represents a piecewise function.
Usage:
Piecewise( (expr,cond), (expr,cond), ... )
- Each argument is a 2-tuple defining an expression and condition
- The conds are evaluated in turn returning the first that is True.
If any of the evaluated conds are not explicitly False,
e.g. ``x < 1``, the function is returned in symbolic form.
- If the function is evaluated at a place where all conditions are False,
nan will be returned.
- Pairs where the cond is explicitly False, will be removed and no pair
appearing after a True condition will ever be retained. If a single
pair with a True condition remains, it will be returned, even when
evaluation is False.
Examples
========
>>> from sympy import Piecewise, log, piecewise_fold
>>> from sympy.abc import x, y
>>> f = x**2
>>> g = log(x)
>>> p = Piecewise((0, x < -1), (f, x <= 1), (g, True))
>>> p.subs(x,1)
1
>>> p.subs(x,5)
log(5)
Booleans can contain Piecewise elements:
>>> cond = (x < y).subs(x, Piecewise((2, x < 0), (3, True))); cond
Piecewise((2, x < 0), (3, True)) < y
The folded version of this results in a Piecewise whose
expressions are Booleans:
>>> folded_cond = piecewise_fold(cond); folded_cond
Piecewise((2 < y, x < 0), (3 < y, True))
When a Boolean containing Piecewise (like cond) or a Piecewise
with Boolean expressions (like folded_cond) is used as a condition,
it is converted to an equivalent :class:`~.ITE` object:
>>> Piecewise((1, folded_cond))
Piecewise((1, ITE(x < 0, y > 2, y > 3)))
When a condition is an ``ITE``, it will be converted to a simplified
Boolean expression:
>>> piecewise_fold(_)
Piecewise((1, ((x >= 0) | (y > 2)) & ((y > 3) | (x < 0))))
See Also
========
piecewise_fold
piecewise_exclusive
ITE
"""
nargs = None
is_Piecewise = True
def __new__(cls, *args, **options):
if len(args) == 0:
raise TypeError("At least one (expr, cond) pair expected.")
# (Try to) sympify args first
newargs = []
for ec in args:
# ec could be a ExprCondPair or a tuple
pair = ExprCondPair(*getattr(ec, 'args', ec))
cond = pair.cond
if cond is false:
continue
newargs.append(pair)
if cond is true:
break
eval = options.pop('evaluate', global_parameters.evaluate)
if eval:
r = cls.eval(*newargs)
if r is not None:
return r
elif len(newargs) == 1 and newargs[0].cond == True:
return newargs[0].expr
return Basic.__new__(cls, *newargs, **options)
@classmethod
def eval(cls, *_args):
"""Either return a modified version of the args or, if no
modifications were made, return None.
Modifications that are made here:
1. relationals are made canonical
2. any False conditions are dropped
3. any repeat of a previous condition is ignored
4. any args past one with a true condition are dropped
If there are no args left, nan will be returned.
If there is a single arg with a True condition, its
corresponding expression will be returned.
EXAMPLES
========
>>> from sympy import Piecewise
>>> from sympy.abc import x
>>> cond = -x < -1
>>> args = [(1, cond), (4, cond), (3, False), (2, True), (5, x < 1)]
>>> Piecewise(*args, evaluate=False)
Piecewise((1, -x < -1), (4, -x < -1), (2, True))
>>> Piecewise(*args)
Piecewise((1, x > 1), (2, True))
"""
if not _args:
return Undefined
if len(_args) == 1 and _args[0][-1] == True:
return _args[0][0]
newargs = _piecewise_collapse_arguments(_args)
# some conditions may have been redundant
missing = len(newargs) != len(_args)
# some conditions may have changed
same = all(a == b for a, b in zip(newargs, _args))
# if either change happened we return the expr with the
# updated args
if not newargs:
raise ValueError(filldedent('''
There are no conditions (or none that
are not trivially false) to define an
expression.'''))
if missing or not same:
return cls(*newargs)
def doit(self, **hints):
"""
Evaluate this piecewise function.
"""
newargs = []
for e, c in self.args:
if hints.get('deep', True):
if isinstance(e, Basic):
newe = e.doit(**hints)
if newe != self:
e = newe
if isinstance(c, Basic):
c = c.doit(**hints)
newargs.append((e, c))
return self.func(*newargs)
def _eval_simplify(self, **kwargs):
return piecewise_simplify(self, **kwargs)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
for e, c in self.args:
if c == True or c.subs(x, 0) == True:
return e.as_leading_term(x)
def _eval_adjoint(self):
return self.func(*[(e.adjoint(), c) for e, c in self.args])
def _eval_conjugate(self):
return self.func(*[(e.conjugate(), c) for e, c in self.args])
def _eval_derivative(self, x):
return self.func(*[(diff(e, x), c) for e, c in self.args])
def _eval_evalf(self, prec):
return self.func(*[(e._evalf(prec), c) for e, c in self.args])
def _eval_is_meromorphic(self, x, a):
# Conditions often implicitly assume that the argument is real.
# Hence, there needs to be some check for as_set.
if not a.is_real:
return None
# Then, scan ExprCondPairs in the given order to find a piece that would contain a,
# possibly as a boundary point.
for e, c in self.args:
cond = c.subs(x, a)
if cond.is_Relational:
return None
if a in c.as_set().boundary:
return None
# Apply expression if a is an interior point of the domain of e.
if cond:
return e._eval_is_meromorphic(x, a)
def piecewise_integrate(self, x, **kwargs):
"""Return the Piecewise with each expression being
replaced with its antiderivative. To obtain a continuous
antiderivative, use the :func:`~.integrate` function or method.
Examples
========
>>> from sympy import Piecewise
>>> from sympy.abc import x
>>> p = Piecewise((0, x < 0), (1, x < 1), (2, True))
>>> p.piecewise_integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x, True))
Note that this does not give a continuous function, e.g.
at x = 1 the 3rd condition applies and the antiderivative
there is 2*x so the value of the antiderivative is 2:
>>> anti = _
>>> anti.subs(x, 1)
2
The continuous derivative accounts for the integral *up to*
the point of interest, however:
>>> p.integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x - 1, True))
>>> _.subs(x, 1)
1
See Also
========
Piecewise._eval_integral
"""
from sympy.integrals import integrate
return self.func(*[(integrate(e, x, **kwargs), c) for e, c in self.args])
def _handle_irel(self, x, handler):
"""Return either None (if the conditions of self depend only on x) else
a Piecewise expression whose expressions (handled by the handler that
was passed) are paired with the governing x-independent relationals,
e.g. Piecewise((A, a(x) & b(y)), (B, c(x) | c(y)) ->
Piecewise(
(handler(Piecewise((A, a(x) & True), (B, c(x) | True)), b(y) & c(y)),
(handler(Piecewise((A, a(x) & True), (B, c(x) | False)), b(y)),
(handler(Piecewise((A, a(x) & False), (B, c(x) | True)), c(y)),
(handler(Piecewise((A, a(x) & False), (B, c(x) | False)), True))
"""
# identify governing relationals
rel = self.atoms(Relational)
irel = list(ordered([r for r in rel if x not in r.free_symbols
and r not in (S.true, S.false)]))
if irel:
args = {}
exprinorder = []
for truth in product((1, 0), repeat=len(irel)):
reps = dict(zip(irel, truth))
# only store the true conditions since the false are implied
# when they appear lower in the Piecewise args
if 1 not in truth:
cond = None # flag this one so it doesn't get combined
else:
andargs = Tuple(*[i for i in reps if reps[i]])
free = list(andargs.free_symbols)
if len(free) == 1:
from sympy.solvers.inequalities import (
reduce_inequalities, _solve_inequality)
try:
t = reduce_inequalities(andargs, free[0])
# ValueError when there are potentially
# nonvanishing imaginary parts
except (ValueError, NotImplementedError):
# at least isolate free symbol on left
t = And(*[_solve_inequality(
a, free[0], linear=True)
for a in andargs])
else:
t = And(*andargs)
if t is S.false:
continue # an impossible combination
cond = t
expr = handler(self.xreplace(reps))
if isinstance(expr, self.func) and len(expr.args) == 1:
expr, econd = expr.args[0]
cond = And(econd, True if cond is None else cond)
# the ec pairs are being collected since all possibilities
# are being enumerated, but don't put the last one in since
# its expr might match a previous expression and it
# must appear last in the args
if cond is not None:
args.setdefault(expr, []).append(cond)
# but since we only store the true conditions we must maintain
# the order so that the expression with the most true values
# comes first
exprinorder.append(expr)
# convert collected conditions as args of Or
for k in args:
args[k] = Or(*args[k])
# take them in the order obtained
args = [(e, args[e]) for e in uniq(exprinorder)]
# add in the last arg
args.append((expr, True))
return Piecewise(*args)
def _eval_integral(self, x, _first=True, **kwargs):
"""Return the indefinite integral of the
Piecewise such that subsequent substitution of x with a
value will give the value of the integral (not including
the constant of integration) up to that point. To only
integrate the individual parts of Piecewise, use the
``piecewise_integrate`` method.
Examples
========
>>> from sympy import Piecewise
>>> from sympy.abc import x
>>> p = Piecewise((0, x < 0), (1, x < 1), (2, True))
>>> p.integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x - 1, True))
>>> p.piecewise_integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x, True))
See Also
========
Piecewise.piecewise_integrate
"""
from sympy.integrals.integrals import integrate
if _first:
def handler(ipw):
if isinstance(ipw, self.func):
return ipw._eval_integral(x, _first=False, **kwargs)
else:
return ipw.integrate(x, **kwargs)
irv = self._handle_irel(x, handler)
if irv is not None:
return irv
# handle a Piecewise from -oo to oo with and no x-independent relationals
# -----------------------------------------------------------------------
ok, abei = self._intervals(x)
if not ok:
from sympy.integrals.integrals import Integral
return Integral(self, x) # unevaluated
pieces = [(a, b) for a, b, _, _ in abei]
oo = S.Infinity
done = [(-oo, oo, -1)]
for k, p in enumerate(pieces):
if p == (-oo, oo):
# all undone intervals will get this key
for j, (a, b, i) in enumerate(done):
if i == -1:
done[j] = a, b, k
break # nothing else to consider
N = len(done) - 1
for j, (a, b, i) in enumerate(reversed(done)):
if i == -1:
j = N - j
done[j: j + 1] = _clip(p, (a, b), k)
done = [(a, b, i) for a, b, i in done if a != b]
# append an arg if there is a hole so a reference to
# argument -1 will give Undefined
if any(i == -1 for (a, b, i) in done):
abei.append((-oo, oo, Undefined, -1))
# return the sum of the intervals
args = []
sum = None
for a, b, i in done:
anti = integrate(abei[i][-2], x, **kwargs)
if sum is None:
sum = anti
else:
sum = sum.subs(x, a)
e = anti._eval_interval(x, a, x)
if sum.has(*_illegal) or e.has(*_illegal):
sum = anti
else:
sum += e
# see if we know whether b is contained in original
# condition
if b is S.Infinity:
cond = True
elif self.args[abei[i][-1]].cond.subs(x, b) == False:
cond = (x < b)
else:
cond = (x <= b)
args.append((sum, cond))
return Piecewise(*args)
def _eval_interval(self, sym, a, b, _first=True):
"""Evaluates the function along the sym in a given interval [a, b]"""
# FIXME: Currently complex intervals are not supported. A possible
# replacement algorithm, discussed in issue 5227, can be found in the
# following papers;
# http://portal.acm.org/citation.cfm?id=281649
# http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.70.4127&rep=rep1&type=pdf
if a is None or b is None:
# In this case, it is just simple substitution
return super()._eval_interval(sym, a, b)
else:
x, lo, hi = map(as_Basic, (sym, a, b))
if _first: # get only x-dependent relationals
def handler(ipw):
if isinstance(ipw, self.func):
return ipw._eval_interval(x, lo, hi, _first=None)
else:
return ipw._eval_interval(x, lo, hi)
irv = self._handle_irel(x, handler)
if irv is not None:
return irv
if (lo < hi) is S.false or (
lo is S.Infinity or hi is S.NegativeInfinity):
rv = self._eval_interval(x, hi, lo, _first=False)
if isinstance(rv, Piecewise):
rv = Piecewise(*[(-e, c) for e, c in rv.args])
else:
rv = -rv
return rv
if (lo < hi) is S.true or (
hi is S.Infinity or lo is S.NegativeInfinity):
pass
else:
_a = Dummy('lo')
_b = Dummy('hi')
a = lo if lo.is_comparable else _a
b = hi if hi.is_comparable else _b
pos = self._eval_interval(x, a, b, _first=False)
if a == _a and b == _b:
# it's purely symbolic so just swap lo and hi and
# change the sign to get the value for when lo > hi
neg, pos = (-pos.xreplace({_a: hi, _b: lo}),
pos.xreplace({_a: lo, _b: hi}))
else:
# at least one of the bounds was comparable, so allow
# _eval_interval to use that information when computing
# the interval with lo and hi reversed
neg, pos = (-self._eval_interval(x, hi, lo, _first=False),
pos.xreplace({_a: lo, _b: hi}))
# allow simplification based on ordering of lo and hi
p = Dummy('', positive=True)
if lo.is_Symbol:
pos = pos.xreplace({lo: hi - p}).xreplace({p: hi - lo})
neg = neg.xreplace({lo: hi + p}).xreplace({p: lo - hi})
elif hi.is_Symbol:
pos = pos.xreplace({hi: lo + p}).xreplace({p: hi - lo})
neg = neg.xreplace({hi: lo - p}).xreplace({p: lo - hi})
# evaluate limits that may have unevaluate Min/Max
touch = lambda _: _.replace(
lambda x: isinstance(x, (Min, Max)),
lambda x: x.func(*x.args))
neg = touch(neg)
pos = touch(pos)
# assemble return expression; make the first condition be Lt
# b/c then the first expression will look the same whether
# the lo or hi limit is symbolic
if a == _a: # the lower limit was symbolic
rv = Piecewise(
(pos,
lo < hi),
(neg,
True))
else:
rv = Piecewise(
(neg,
hi < lo),
(pos,
True))
if rv == Undefined:
raise ValueError("Can't integrate across undefined region.")
if any(isinstance(i, Piecewise) for i in (pos, neg)):
rv = piecewise_fold(rv)
return rv
# handle a Piecewise with lo <= hi and no x-independent relationals
# -----------------------------------------------------------------
ok, abei = self._intervals(x)
if not ok:
from sympy.integrals.integrals import Integral
# not being able to do the interval of f(x) can
# be stated as not being able to do the integral
# of f'(x) over the same range
return Integral(self.diff(x), (x, lo, hi)) # unevaluated
pieces = [(a, b) for a, b, _, _ in abei]
done = [(lo, hi, -1)]
oo = S.Infinity
for k, p in enumerate(pieces):
if p[:2] == (-oo, oo):
# all undone intervals will get this key
for j, (a, b, i) in enumerate(done):
if i == -1:
done[j] = a, b, k
break # nothing else to consider
N = len(done) - 1
for j, (a, b, i) in enumerate(reversed(done)):
if i == -1:
j = N - j
done[j: j + 1] = _clip(p, (a, b), k)
done = [(a, b, i) for a, b, i in done if a != b]
# return the sum of the intervals
sum = S.Zero
upto = None
for a, b, i in done:
if i == -1:
if upto is None:
return Undefined
# TODO simplify hi <= upto
return Piecewise((sum, hi <= upto), (Undefined, True))
sum += abei[i][-2]._eval_interval(x, a, b)
upto = b
return sum
def _intervals(self, sym, err_on_Eq=False):
r"""Return a bool and a message (when bool is False), else a
list of unique tuples, (a, b, e, i), where a and b
are the lower and upper bounds in which the expression e of
argument i in self is defined and $a < b$ (when involving
numbers) or $a \le b$ when involving symbols.
If there are any relationals not involving sym, or any
relational cannot be solved for sym, the bool will be False
a message be given as the second return value. The calling
routine should have removed such relationals before calling
this routine.
The evaluated conditions will be returned as ranges.
Discontinuous ranges will be returned separately with
identical expressions. The first condition that evaluates to
True will be returned as the last tuple with a, b = -oo, oo.
"""
from sympy.solvers.inequalities import _solve_inequality
assert isinstance(self, Piecewise)
def nonsymfail(cond):
return False, filldedent('''
A condition not involving
%s appeared: %s''' % (sym, cond))
def _solve_relational(r):
if sym not in r.free_symbols:
return nonsymfail(r)
try:
rv = _solve_inequality(r, sym)
except NotImplementedError:
return False, 'Unable to solve relational %s for %s.' % (r, sym)
if isinstance(rv, Relational):
free = rv.args[1].free_symbols
if rv.args[0] != sym or sym in free:
return False, 'Unable to solve relational %s for %s.' % (r, sym)
if rv.rel_op == '==':
# this equality has been affirmed to have the form
# Eq(sym, rhs) where rhs is sym-free; it represents
# a zero-width interval which will be ignored
# whether it is an isolated condition or contained
# within an And or an Or
rv = S.false
elif rv.rel_op == '!=':
try:
rv = Or(sym < rv.rhs, sym > rv.rhs)
except TypeError:
# e.g. x != I ==> all real x satisfy
rv = S.true
elif rv == (S.NegativeInfinity < sym) & (sym < S.Infinity):
rv = S.true
return True, rv
args = list(self.args)
# make self canonical wrt Relationals
keys = self.atoms(Relational)
reps = {}
for r in keys:
ok, s = _solve_relational(r)
if ok != True:
return False, ok
reps[r] = s
# process args individually so if any evaluate, their position
# in the original Piecewise will be known
args = [i.xreplace(reps) for i in self.args]
# precondition args
expr_cond = []
default = idefault = None
for i, (expr, cond) in enumerate(args):
if cond is S.false:
continue
if cond is S.true:
default = expr
idefault = i
break
if isinstance(cond, Eq):
# unanticipated condition, but it is here in case a
# replacement caused an Eq to appear
if err_on_Eq:
return False, 'encountered Eq condition: %s' % cond
continue # zero width interval
cond = to_cnf(cond)
if isinstance(cond, And):
cond = distribute_or_over_and(cond)
if isinstance(cond, Or):
expr_cond.extend(
[(i, expr, o) for o in cond.args
if not isinstance(o, Eq)])
elif cond is not S.false:
expr_cond.append((i, expr, cond))
elif cond is S.true:
default = expr
idefault = i
break
# determine intervals represented by conditions
int_expr = []
for iarg, expr, cond in expr_cond:
if isinstance(cond, And):
lower = S.NegativeInfinity
upper = S.Infinity
exclude = []
for cond2 in cond.args:
if not isinstance(cond2, Relational):
return False, 'expecting only Relationals'
if isinstance(cond2, Eq):
lower = upper # ignore
if err_on_Eq:
return False, 'encountered secondary Eq condition'
break
elif isinstance(cond2, Ne):
l, r = cond2.args
if l == sym:
exclude.append(r)
elif r == sym:
exclude.append(l)
else:
return nonsymfail(cond2)
continue
elif cond2.lts == sym:
upper = Min(cond2.gts, upper)
elif cond2.gts == sym:
lower = Max(cond2.lts, lower)
else:
return nonsymfail(cond2) # should never get here
if exclude:
exclude = list(ordered(exclude))
newcond = []
for i, e in enumerate(exclude):
if e < lower == True or e > upper == True:
continue
if not newcond:
newcond.append((None, lower)) # add a primer
newcond.append((newcond[-1][1], e))
newcond.append((newcond[-1][1], upper))
newcond.pop(0) # remove the primer
expr_cond.extend([(iarg, expr, And(i[0] < sym, sym < i[1])) for i in newcond])
continue
elif isinstance(cond, Relational) and cond.rel_op != '!=':
lower, upper = cond.lts, cond.gts # part 1: initialize with givens
if cond.lts == sym: # part 1a: expand the side ...
lower = S.NegativeInfinity # e.g. x <= 0 ---> -oo <= 0
elif cond.gts == sym: # part 1a: ... that can be expanded
upper = S.Infinity # e.g. x >= 0 ---> oo >= 0
else:
return nonsymfail(cond)
else:
return False, 'unrecognized condition: %s' % cond
lower, upper = lower, Max(lower, upper)
if err_on_Eq and lower == upper:
return False, 'encountered Eq condition'
if (lower >= upper) is not S.true:
int_expr.append((lower, upper, expr, iarg))
if default is not None:
int_expr.append(
(S.NegativeInfinity, S.Infinity, default, idefault))
return True, list(uniq(int_expr))
def _eval_nseries(self, x, n, logx, cdir=0):
args = [(ec.expr._eval_nseries(x, n, logx), ec.cond) for ec in self.args]
return self.func(*args)
def _eval_power(self, s):
return self.func(*[(e**s, c) for e, c in self.args])
def _eval_subs(self, old, new):
# this is strictly not necessary, but we can keep track
# of whether True or False conditions arise and be
# somewhat more efficient by avoiding other substitutions
# and avoiding invalid conditions that appear after a
# True condition
args = list(self.args)
args_exist = False
for i, (e, c) in enumerate(args):
c = c._subs(old, new)
if c != False:
args_exist = True
e = e._subs(old, new)
args[i] = (e, c)
if c == True:
break
if not args_exist:
args = ((Undefined, True),)
return self.func(*args)
def _eval_transpose(self):
return self.func(*[(e.transpose(), c) for e, c in self.args])
def _eval_template_is_attr(self, is_attr):
b = None
for expr, _ in self.args:
a = getattr(expr, is_attr)
if a is None:
return
if b is None:
b = a
elif b is not a:
return
return b
_eval_is_finite = lambda self: self._eval_template_is_attr(
'is_finite')
_eval_is_complex = lambda self: self._eval_template_is_attr('is_complex')
_eval_is_even = lambda self: self._eval_template_is_attr('is_even')
_eval_is_imaginary = lambda self: self._eval_template_is_attr(
'is_imaginary')
_eval_is_integer = lambda self: self._eval_template_is_attr('is_integer')
_eval_is_irrational = lambda self: self._eval_template_is_attr(
'is_irrational')
_eval_is_negative = lambda self: self._eval_template_is_attr('is_negative')
_eval_is_nonnegative = lambda self: self._eval_template_is_attr(
'is_nonnegative')
_eval_is_nonpositive = lambda self: self._eval_template_is_attr(
'is_nonpositive')
_eval_is_nonzero = lambda self: self._eval_template_is_attr(
'is_nonzero')
_eval_is_odd = lambda self: self._eval_template_is_attr('is_odd')
_eval_is_polar = lambda self: self._eval_template_is_attr('is_polar')
_eval_is_positive = lambda self: self._eval_template_is_attr('is_positive')
_eval_is_extended_real = lambda self: self._eval_template_is_attr(
'is_extended_real')
_eval_is_extended_positive = lambda self: self._eval_template_is_attr(
'is_extended_positive')
_eval_is_extended_negative = lambda self: self._eval_template_is_attr(
'is_extended_negative')
_eval_is_extended_nonzero = lambda self: self._eval_template_is_attr(
'is_extended_nonzero')
_eval_is_extended_nonpositive = lambda self: self._eval_template_is_attr(
'is_extended_nonpositive')
_eval_is_extended_nonnegative = lambda self: self._eval_template_is_attr(
'is_extended_nonnegative')
_eval_is_real = lambda self: self._eval_template_is_attr('is_real')
_eval_is_zero = lambda self: self._eval_template_is_attr(
'is_zero')
@classmethod
def __eval_cond(cls, cond):
"""Return the truth value of the condition."""
if cond == True:
return True
if isinstance(cond, Eq):
try:
diff = cond.lhs - cond.rhs
if diff.is_commutative:
return diff.is_zero
except TypeError:
pass
def as_expr_set_pairs(self, domain=None):
"""Return tuples for each argument of self that give
the expression and the interval in which it is valid
which is contained within the given domain.
If a condition cannot be converted to a set, an error
will be raised. The variable of the conditions is
assumed to be real; sets of real values are returned.
Examples
========
>>> from sympy import Piecewise, Interval
>>> from sympy.abc import x
>>> p = Piecewise(
... (1, x < 2),
... (2,(x > 0) & (x < 4)),
... (3, True))
>>> p.as_expr_set_pairs()
[(1, Interval.open(-oo, 2)),
(2, Interval.Ropen(2, 4)),
(3, Interval(4, oo))]
>>> p.as_expr_set_pairs(Interval(0, 3))
[(1, Interval.Ropen(0, 2)),
(2, Interval(2, 3))]
"""
if domain is None:
domain = S.Reals
exp_sets = []
U = domain
complex = not domain.is_subset(S.Reals)
cond_free = set()
for expr, cond in self.args:
cond_free |= cond.free_symbols
if len(cond_free) > 1:
raise NotImplementedError(filldedent('''
multivariate conditions are not handled.'''))
if complex:
for i in cond.atoms(Relational):
if not isinstance(i, (Eq, Ne)):
raise ValueError(filldedent('''
Inequalities in the complex domain are
not supported. Try the real domain by
setting domain=S.Reals'''))
cond_int = U.intersect(cond.as_set())
U = U - cond_int
if cond_int != S.EmptySet:
exp_sets.append((expr, cond_int))
return exp_sets
def _eval_rewrite_as_ITE(self, *args, **kwargs):
byfree = {}
args = list(args)
default = any(c == True for b, c in args)
for i, (b, c) in enumerate(args):
if not isinstance(b, Boolean) and b != True:
raise TypeError(filldedent('''
Expecting Boolean or bool but got `%s`
''' % func_name(b)))
if c == True:
break
# loop over independent conditions for this b
for c in c.args if isinstance(c, Or) else [c]:
free = c.free_symbols
x = free.pop()
try:
byfree[x] = byfree.setdefault(
x, S.EmptySet).union(c.as_set())
except NotImplementedError:
if not default:
raise NotImplementedError(filldedent('''
A method to determine whether a multivariate
conditional is consistent with a complete coverage
of all variables has not been implemented so the
rewrite is being stopped after encountering `%s`.
This error would not occur if a default expression
like `(foo, True)` were given.
''' % c))
if byfree[x] in (S.UniversalSet, S.Reals):
# collapse the ith condition to True and break
args[i] = list(args[i])
c = args[i][1] = True
break
if c == True:
break
if c != True:
raise ValueError(filldedent('''
Conditions must cover all reals or a final default
condition `(foo, True)` must be given.
'''))
last, _ = args[i] # ignore all past ith arg
for a, c in reversed(args[:i]):
last = ITE(c, a, last)
return _canonical(last)
def _eval_rewrite_as_KroneckerDelta(self, *args):
from sympy.functions.special.tensor_functions import KroneckerDelta
rules = {
And: [False, False],
Or: [True, True],
Not: [True, False],
Eq: [None, None],
Ne: [None, None]
}
class UnrecognizedCondition(Exception):
pass
def rewrite(cond):
if isinstance(cond, Eq):
return KroneckerDelta(*cond.args)
if isinstance(cond, Ne):
return 1 - KroneckerDelta(*cond.args)
cls, args = type(cond), cond.args
if cls not in rules:
raise UnrecognizedCondition(cls)
b1, b2 = rules[cls]
k = Mul(*[1 - rewrite(c) for c in args]) if b1 else Mul(*[rewrite(c) for c in args])
if b2:
return 1 - k
return k
conditions = []
true_value = None
for value, cond in args:
if type(cond) in rules:
conditions.append((value, cond))
elif cond is S.true:
if true_value is None:
true_value = value
else:
return
if true_value is not None:
result = true_value
for value, cond in conditions[::-1]:
try:
k = rewrite(cond)
result = k * value + (1 - k) * result
except UnrecognizedCondition:
return
return result
def piecewise_fold(expr, evaluate=True):
"""
Takes an expression containing a piecewise function and returns the
expression in piecewise form. In addition, any ITE conditions are
rewritten in negation normal form and simplified.
The final Piecewise is evaluated (default) but if the raw form
is desired, send ``evaluate=False``; if trivial evaluation is
desired, send ``evaluate=None`` and duplicate conditions and
processing of True and False will be handled.
Examples
========
>>> from sympy import Piecewise, piecewise_fold, S
>>> from sympy.abc import x
>>> p = Piecewise((x, x < 1), (1, S(1) <= x))
>>> piecewise_fold(x*p)
Piecewise((x**2, x < 1), (x, True))
See Also
========
Piecewise
piecewise_exclusive
"""
if not isinstance(expr, Basic) or not expr.has(Piecewise):
return expr
new_args = []
if isinstance(expr, (ExprCondPair, Piecewise)):
for e, c in expr.args:
if not isinstance(e, Piecewise):
e = piecewise_fold(e)
# we don't keep Piecewise in condition because
# it has to be checked to see that it's complete
# and we convert it to ITE at that time
assert not c.has(Piecewise) # pragma: no cover
if isinstance(c, ITE):
c = c.to_nnf()
c = simplify_logic(c, form='cnf')
if isinstance(e, Piecewise):
new_args.extend([(piecewise_fold(ei), And(ci, c))
for ei, ci in e.args])
else:
new_args.append((e, c))
else:
# Given
# P1 = Piecewise((e11, c1), (e12, c2), A)
# P2 = Piecewise((e21, c1), (e22, c2), B)
# ...
# the folding of f(P1, P2) is trivially
# Piecewise(
# (f(e11, e21), c1),
# (f(e12, e22), c2),
# (f(Piecewise(A), Piecewise(B)), True))
# Certain objects end up rewriting themselves as thus, so
# we do that grouping before the more generic folding.
# The following applies this idea when f = Add or f = Mul
# (and the expression is commutative).
if expr.is_Add or expr.is_Mul and expr.is_commutative:
p, args = sift(expr.args, lambda x: x.is_Piecewise, binary=True)
pc = sift(p, lambda x: tuple([c for e,c in x.args]))
for c in list(ordered(pc)):
if len(pc[c]) > 1:
pargs = [list(i.args) for i in pc[c]]
# the first one is the same; there may be more
com = common_prefix(*[
[i.cond for i in j] for j in pargs])
n = len(com)
collected = []
for i in range(n):
collected.append((
expr.func(*[ai[i].expr for ai in pargs]),
com[i]))
remains = []
for a in pargs:
if n == len(a): # no more args
continue
if a[n].cond == True: # no longer Piecewise
remains.append(a[n].expr)
else: # restore the remaining Piecewise
remains.append(
Piecewise(*a[n:], evaluate=False))
if remains:
collected.append((expr.func(*remains), True))
args.append(Piecewise(*collected, evaluate=False))
continue
args.extend(pc[c])
else:
args = expr.args
# fold
folded = list(map(piecewise_fold, args))
for ec in product(*[
(i.args if isinstance(i, Piecewise) else
[(i, true)]) for i in folded]):
e, c = zip(*ec)
new_args.append((expr.func(*e), And(*c)))
if evaluate is None:
# don't return duplicate conditions, otherwise don't evaluate
new_args = list(reversed([(e, c) for c, e in {
c: e for e, c in reversed(new_args)}.items()]))
rv = Piecewise(*new_args, evaluate=evaluate)
if evaluate is None and len(rv.args) == 1 and rv.args[0].cond == True:
return rv.args[0].expr
if any(s.expr.has(Piecewise) for p in rv.atoms(Piecewise) for s in p.args):
return piecewise_fold(rv)
return rv
def _clip(A, B, k):
"""Return interval B as intervals that are covered by A (keyed
to k) and all other intervals of B not covered by A keyed to -1.
The reference point of each interval is the rhs; if the lhs is
greater than the rhs then an interval of zero width interval will
result, e.g. (4, 1) is treated like (1, 1).
Examples
========
>>> from sympy.functions.elementary.piecewise import _clip
>>> from sympy import Tuple
>>> A = Tuple(1, 3)
>>> B = Tuple(2, 4)
>>> _clip(A, B, 0)
[(2, 3, 0), (3, 4, -1)]
Interpretation: interval portion (2, 3) of interval (2, 4) is
covered by interval (1, 3) and is keyed to 0 as requested;
interval (3, 4) was not covered by (1, 3) and is keyed to -1.
"""
a, b = B
c, d = A
c, d = Min(Max(c, a), b), Min(Max(d, a), b)
a, b = Min(a, b), b
p = []
if a != c:
p.append((a, c, -1))
else:
pass
if c != d:
p.append((c, d, k))
else:
pass
if b != d:
if d == c and p and p[-1][-1] == -1:
p[-1] = p[-1][0], b, -1
else:
p.append((d, b, -1))
else:
pass
return p
def piecewise_simplify_arguments(expr, **kwargs):
from sympy.simplify.simplify import simplify
# simplify conditions
f1 = expr.args[0].cond.free_symbols
args = None
if len(f1) == 1 and not expr.atoms(Eq):
x = f1.pop()
# this won't return intervals involving Eq
# and it won't handle symbols treated as
# booleans
ok, abe_ = expr._intervals(x, err_on_Eq=True)
def include(c, x, a):
"return True if c.subs(x, a) is True, else False"
try:
return c.subs(x, a) == True
except TypeError:
return False
if ok:
args = []
covered = S.EmptySet
from sympy.sets.sets import Interval
for a, b, e, i in abe_:
c = expr.args[i].cond
incl_a = include(c, x, a)
incl_b = include(c, x, b)
iv = Interval(a, b, not incl_a, not incl_b)
cset = iv - covered
if not cset:
continue
if incl_a and incl_b:
if a.is_infinite and b.is_infinite:
c = S.true
elif b.is_infinite:
c = (x >= a)
elif a in covered or a.is_infinite:
c = (x <= b)
else:
c = And(a <= x, x <= b)
elif incl_a:
if a in covered or a.is_infinite:
c = (x < b)
else:
c = And(a <= x, x < b)
elif incl_b:
if b.is_infinite:
c = (x > a)
else:
c = (x <= b)
else:
if a in covered:
c = (x < b)
else:
c = And(a < x, x < b)
covered |= iv
if a is S.NegativeInfinity and incl_a:
covered |= {S.NegativeInfinity}
if b is S.Infinity and incl_b:
covered |= {S.Infinity}
args.append((e, c))
if not S.Reals.is_subset(covered):
args.append((Undefined, True))
if args is None:
args = list(expr.args)
for i in range(len(args)):
e, c = args[i]
if isinstance(c, Basic):
c = simplify(c, **kwargs)
args[i] = (e, c)
# simplify expressions
doit = kwargs.pop('doit', None)
for i in range(len(args)):
e, c = args[i]
if isinstance(e, Basic):
# Skip doit to avoid growth at every call for some integrals
# and sums, see sympy/sympy#17165
newe = simplify(e, doit=False, **kwargs)
if newe != e:
e = newe
args[i] = (e, c)
# restore kwargs flag
if doit is not None:
kwargs['doit'] = doit
return Piecewise(*args)
def _piecewise_collapse_arguments(_args):
newargs = [] # the unevaluated conditions
current_cond = set() # the conditions up to a given e, c pair
for expr, cond in _args:
cond = cond.replace(
lambda _: _.is_Relational, _canonical_coeff)
# Check here if expr is a Piecewise and collapse if one of
# the conds in expr matches cond. This allows the collapsing
# of Piecewise((Piecewise((x,x<0)),x<0)) to Piecewise((x,x<0)).
# This is important when using piecewise_fold to simplify
# multiple Piecewise instances having the same conds.
# Eventually, this code should be able to collapse Piecewise's
# having different intervals, but this will probably require
# using the new assumptions.
if isinstance(expr, Piecewise):
unmatching = []
for i, (e, c) in enumerate(expr.args):
if c in current_cond:
# this would already have triggered
continue
if c == cond:
if c != True:
# nothing past this condition will ever
# trigger and only those args before this
# that didn't match a previous condition
# could possibly trigger
if unmatching:
expr = Piecewise(*(
unmatching + [(e, c)]))
else:
expr = e
break
else:
unmatching.append((e, c))
# check for condition repeats
got = False
# -- if an And contains a condition that was
# already encountered, then the And will be
# False: if the previous condition was False
# then the And will be False and if the previous
# condition is True then then we wouldn't get to
# this point. In either case, we can skip this condition.
for i in ([cond] +
(list(cond.args) if isinstance(cond, And) else
[])):
if i in current_cond:
got = True
break
if got:
continue
# -- if not(c) is already in current_cond then c is
# a redundant condition in an And. This does not
# apply to Or, however: (e1, c), (e2, Or(~c, d))
# is not (e1, c), (e2, d) because if c and d are
# both False this would give no results when the
# true answer should be (e2, True)
if isinstance(cond, And):
nonredundant = []
for c in cond.args:
if isinstance(c, Relational):
if c.negated.canonical in current_cond:
continue
# if a strict inequality appears after
# a non-strict one, then the condition is
# redundant
if isinstance(c, (Lt, Gt)) and (
c.weak in current_cond):
cond = False
break
nonredundant.append(c)
else:
cond = cond.func(*nonredundant)
elif isinstance(cond, Relational):
if cond.negated.canonical in current_cond:
cond = S.true
current_cond.add(cond)
# collect successive e,c pairs when exprs or cond match
if newargs:
if newargs[-1].expr == expr:
orcond = Or(cond, newargs[-1].cond)
if isinstance(orcond, (And, Or)):
orcond = distribute_and_over_or(orcond)
newargs[-1] = ExprCondPair(expr, orcond)
continue
elif newargs[-1].cond == cond:
continue
newargs.append(ExprCondPair(expr, cond))
return newargs
_blessed = lambda e: getattr(e.lhs, '_diff_wrt', False) and (
getattr(e.rhs, '_diff_wrt', None) or
isinstance(e.rhs, (Rational, NumberSymbol)))
def piecewise_simplify(expr, **kwargs):
expr = piecewise_simplify_arguments(expr, **kwargs)
if not isinstance(expr, Piecewise):
return expr
args = list(expr.args)
args = _piecewise_simplify_eq_and(args)
args = _piecewise_simplify_equal_to_next_segment(args)
return Piecewise(*args)
def _piecewise_simplify_equal_to_next_segment(args):
"""
See if expressions valid for an Equal expression happens to evaluate
to the same function as in the next piecewise segment, see:
https://github.com/sympy/sympy/issues/8458
"""
prevexpr = None
for i, (expr, cond) in reversed(list(enumerate(args))):
if prevexpr is not None:
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Eq), binary=True)
elif isinstance(cond, Eq):
eqs, other = [cond], []
else:
eqs = other = []
_prevexpr = prevexpr
_expr = expr
if eqs and not other:
eqs = list(ordered(eqs))
for e in eqs:
# allow 2 args to collapse into 1 for any e
# otherwise limit simplification to only simple-arg
# Eq instances
if len(args) == 2 or _blessed(e):
_prevexpr = _prevexpr.subs(*e.args)
_expr = _expr.subs(*e.args)
# Did it evaluate to the same?
if _prevexpr == _expr:
# Set the expression for the Not equal section to the same
# as the next. These will be merged when creating the new
# Piecewise
args[i] = args[i].func(args[i + 1][0], cond)
else:
# Update the expression that we compare against
prevexpr = expr
else:
prevexpr = expr
return args
def _piecewise_simplify_eq_and(args):
"""
Try to simplify conditions and the expression for
equalities that are part of the condition, e.g.
Piecewise((n, And(Eq(n,0), Eq(n + m, 0))), (1, True))
-> Piecewise((0, And(Eq(n, 0), Eq(m, 0))), (1, True))
"""
for i, (expr, cond) in enumerate(args):
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Eq), binary=True)
elif isinstance(cond, Eq):
eqs, other = [cond], []
else:
eqs = other = []
if eqs:
eqs = list(ordered(eqs))
for j, e in enumerate(eqs):
# these blessed lhs objects behave like Symbols
# and the rhs are simple replacements for the "symbols"
if _blessed(e):
expr = expr.subs(*e.args)
eqs[j + 1:] = [ei.subs(*e.args) for ei in eqs[j + 1:]]
other = [ei.subs(*e.args) for ei in other]
cond = And(*(eqs + other))
args[i] = args[i].func(expr, cond)
return args
def piecewise_exclusive(expr, *, skip_nan=False, deep=True):
"""
Rewrite :class:`Piecewise` with mutually exclusive conditions.
Explanation
===========
SymPy represents the conditions of a :class:`Piecewise` in an
"if-elif"-fashion, allowing more than one condition to be simultaneously
True. The interpretation is that the first condition that is True is the
case that holds. While this is a useful representation computationally it
is not how a piecewise formula is typically shown in a mathematical text.
The :func:`piecewise_exclusive` function can be used to rewrite any
:class:`Piecewise` with more typical mutually exclusive conditions.
Note that further manipulation of the resulting :class:`Piecewise`, e.g.
simplifying it, will most likely make it non-exclusive. Hence, this is
primarily a function to be used in conjunction with printing the Piecewise
or if one would like to reorder the expression-condition pairs.
If it is not possible to determine that all possibilities are covered by
the different cases of the :class:`Piecewise` then a final
:class:`~sympy.core.numbers.NaN` case will be included explicitly. This
can be prevented by passing ``skip_nan=True``.
Examples
========
>>> from sympy import piecewise_exclusive, Symbol, Piecewise, S
>>> x = Symbol('x', real=True)
>>> p = Piecewise((0, x < 0), (S.Half, x <= 0), (1, True))
>>> piecewise_exclusive(p)
Piecewise((0, x < 0), (1/2, Eq(x, 0)), (1, x > 0))
>>> piecewise_exclusive(Piecewise((2, x > 1)))
Piecewise((2, x > 1), (nan, x <= 1))
>>> piecewise_exclusive(Piecewise((2, x > 1)), skip_nan=True)
Piecewise((2, x > 1))
Parameters
==========
expr: a SymPy expression.
Any :class:`Piecewise` in the expression will be rewritten.
skip_nan: ``bool`` (default ``False``)
If ``skip_nan`` is set to ``True`` then a final
:class:`~sympy.core.numbers.NaN` case will not be included.
deep: ``bool`` (default ``True``)
If ``deep`` is ``True`` then :func:`piecewise_exclusive` will rewrite
any :class:`Piecewise` subexpressions in ``expr`` rather than just
rewriting ``expr`` itself.
Returns
=======
An expression equivalent to ``expr`` but where all :class:`Piecewise` have
been rewritten with mutually exclusive conditions.
See Also
========
Piecewise
piecewise_fold
"""
def make_exclusive(*pwargs):
cumcond = false
newargs = []
# Handle the first n-1 cases
for expr_i, cond_i in pwargs[:-1]:
cancond = And(cond_i, Not(cumcond)).simplify()
cumcond = Or(cond_i, cumcond).simplify()
newargs.append((expr_i, cancond))
# For the nth case defer simplification of cumcond
expr_n, cond_n = pwargs[-1]
cancond_n = And(cond_n, Not(cumcond)).simplify()
newargs.append((expr_n, cancond_n))
if not skip_nan:
cumcond = Or(cond_n, cumcond).simplify()
if cumcond is not true:
newargs.append((Undefined, Not(cumcond).simplify()))
return Piecewise(*newargs, evaluate=False)
if deep:
return expr.replace(Piecewise, make_exclusive)
elif isinstance(expr, Piecewise):
return make_exclusive(*expr.args)
else:
return expr