# Source code for sympy.functions.elementary.piecewise

from __future__ import print_function, division

from sympy.core import Basic, S, Function, diff, Tuple
from sympy.core.relational import Equality, Relational
from sympy.functions.elementary.miscellaneous import Max, Min
from sympy.logic.boolalg import (And, Boolean, distribute_and_over_or, Not, Or,
true, false)
from sympy.core.compatibility import default_sort_key, range

[docs]class ExprCondPair(Tuple):
"""Represents an expression, condition pair."""

def __new__(cls, expr, cond):
if cond == True:
return Tuple.__new__(cls, expr, true)
elif cond == False:
return Tuple.__new__(cls, expr, false)
return Tuple.__new__(cls, expr, cond)

@property
[docs]    def expr(self):
"""
Returns the expression of this pair.
"""
return self.args[0]

@property
[docs]    def cond(self):
"""
Returns the condition of this pair.
"""
return self.args[1]

@property
[docs]    def free_symbols(self):
"""
Return the free symbols of this pair.
"""
# Overload Basic.free_symbols because self.args[1] may contain non-Basic
result = self.expr.free_symbols
if hasattr(self.cond, 'free_symbols'):
result |= self.cond.free_symbols
return result

@property
def is_commutative(self):
return self.expr.is_commutative

def __iter__(self):
yield self.expr
yield self.cond

[docs]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 determined 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,
a ValueError exception will be raised.
- Pairs where the cond is explicitly False, will be removed.

Examples
========

>>> from sympy import Piecewise, log
>>> from sympy.abc import x
>>> 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)

========

piecewise_fold
"""

nargs = None
is_Piecewise = True

def __new__(cls, *args, **options):
# (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 == false:
continue
if not isinstance(cond, (bool, Relational, Boolean)):
raise TypeError(
"Cond %s is of type %s, but must be a Relational,"
" Boolean, or a built-in bool." % (cond, type(cond)))
newargs.append(pair)
if cond == True:
break

if options.pop('evaluate', True):
r = cls.eval(*newargs)
else:
r = None

if r is None:
return Basic.__new__(cls, *newargs, **options)
else:
return r

@classmethod
def eval(cls, *args):
# Check for situations where we can evaluate the Piecewise object.
# 1) Hit an unevaluable cond (e.g. x<1) -> keep object
# 2) Hit a true condition -> return that expr
# 3) Remove false conditions, if no conditions left -> raise ValueError
all_conds_evaled = True    # Do all conds eval to a bool?
piecewise_again = False    # Should we pass args to Piecewise again?
non_false_ecpairs = []
or1 = Or(*[cond for (_, cond) in args if cond != true])
for expr, cond in args:
# 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):
or2 = Or(*[c for (_, c) in expr.args if c != true])
for e, c in expr.args:
# Don't collapse if cond is "True" as this leads to
# incorrect simplifications with nested Piecewises.
if c == cond and (or1 == or2 or cond != true):
expr = e
piecewise_again = True
cond_eval = cls.__eval_cond(cond)
if cond_eval is None:
all_conds_evaled = False
elif cond_eval:
if all_conds_evaled:
return expr
if len(non_false_ecpairs) != 0:
if non_false_ecpairs[-1].cond == cond:
continue
elif non_false_ecpairs[-1].expr == expr:
newcond = Or(cond, non_false_ecpairs[-1].cond)
if isinstance(newcond, (And, Or)):
newcond = distribute_and_over_or(newcond)
non_false_ecpairs[-1] = ExprCondPair(expr, newcond)
continue
non_false_ecpairs.append(ExprCondPair(expr, cond))
if len(non_false_ecpairs) != len(args) or piecewise_again:
return cls(*non_false_ecpairs)

return None

[docs]    def doit(self, **hints):
"""
Evaluate this piecewise function.
"""
newargs = []
for e, c in self.args:
if hints.get('deep', True):
if isinstance(e, Basic):
e = e.doit(**hints)
if isinstance(c, Basic):
c = c.doit(**hints)
newargs.append((e, c))
return self.func(*newargs)

for e, c in self.args:
if c == True or c.subs(x, 0) == True:

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_integral(self, x):
from sympy.integrals import integrate
return self.func(*[(integrate(e, x), c) for e, c in self.args])

def _eval_interval(self, sym, a, b):
"""Evaluates the function along the sym in a given interval ab"""
# 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

if a is None or b is None:
# In this case, it is just simple substitution
return piecewise_fold(
super(Piecewise, self)._eval_interval(sym, a, b))

mul = 1
if (a == b) == True:
return S.Zero
elif (a > b) == True:
a, b, mul = b, a, -1
elif (a <= b) != True:
newargs = []
for e, c in self.args:
intervals = self._sort_expr_cond(
sym, S.NegativeInfinity, S.Infinity, c)
values = []
for lower, upper, expr in intervals:
if (a < lower) == True:
mid = lower
rep = b
val = e._eval_interval(sym, mid, b)
val += self._eval_interval(sym, a, mid)
elif (a > upper) == True:
mid = upper
rep = b
val = e._eval_interval(sym, mid, b)
val += self._eval_interval(sym, a, mid)
elif (a >= lower) == True and (a <= upper) == True:
rep = b
val = e._eval_interval(sym, a, b)
elif (b < lower) == True:
mid = lower
rep = a
val = e._eval_interval(sym, a, mid)
val += self._eval_interval(sym, mid, b)
elif (b > upper) == True:
mid = upper
rep = a
val = e._eval_interval(sym, a, mid)
val += self._eval_interval(sym, mid, b)
elif ((b >= lower) == True) and ((b <= upper) == True):
rep = a
val = e._eval_interval(sym, a, b)
else:
raise NotImplementedError(
"""The evaluation of a Piecewise interval when both the lower
and the upper limit are symbolic is not yet implemented.""")
values.append(val)
if len(set(values)) == 1:
try:
c = c.subs(sym, rep)
except AttributeError:
pass
e = values[0]
newargs.append((e, c))
else:
for i in range(len(values)):
newargs.append((values[i], (c == True and i == len(values) - 1) or
And(rep >= intervals[i][0], rep <= intervals[i][1])))
return self.func(*newargs)

# Determine what intervals the expr,cond pairs affect.
int_expr = self._sort_expr_cond(sym, a, b)

# Finally run through the intervals and sum the evaluation.
ret_fun = 0
for int_a, int_b, expr in int_expr:
if isinstance(expr, Piecewise):
# If we still have a Piecewise by now, _sort_expr_cond would
# already have determined that its conditions are independent
# of the integration variable, thus we just use substitution.
ret_fun += piecewise_fold(
super(Piecewise, expr)._eval_interval(sym, Max(a, int_a), Min(b, int_b)))
else:
ret_fun += expr._eval_interval(sym, Max(a, int_a), Min(b, int_b))
return mul * ret_fun

def _sort_expr_cond(self, sym, a, b, targetcond=None):
"""Determine what intervals the expr, cond pairs affect.

1) If cond is True, then log it as default
1.1) Currently if cond can't be evaluated, throw NotImplementedError.
2) For each inequality, if previous cond defines part of the interval
update the new conds interval.
-  eg x < 1, x < 3 -> [oo,1],[1,3] instead of [oo,1],[oo,3]
3) Sort the intervals to make it easier to find correct exprs

Under normal use, we return the expr,cond pairs in increasing order
along the real axis corresponding to the symbol sym.  If targetcond
is given, we return a list of (lowerbound, upperbound) pairs for
this condition."""
from sympy.solvers.inequalities import _solve_inequality
default = None
int_expr = []
expr_cond = []
or_cond = False
or_intervals = []
independent_expr_cond = []
for expr, cond in self.args:
if isinstance(cond, Or):
for cond2 in sorted(cond.args, key=default_sort_key):
expr_cond.append((expr, cond2))
else:
expr_cond.append((expr, cond))
if cond == True:
break
for expr, cond in expr_cond:
if cond == True:
independent_expr_cond.append((expr, cond))
default = self.func(*independent_expr_cond)
break
orig_cond = cond
if sym not in cond.free_symbols:
independent_expr_cond.append((expr, cond))
continue
elif isinstance(cond, Equality):
continue
elif isinstance(cond, And):
lower = S.NegativeInfinity
upper = S.Infinity
for cond2 in cond.args:
if sym not in [cond2.lts, cond2.gts]:
cond2 = _solve_inequality(cond2, sym)
if cond2.lts == sym:
upper = Min(cond2.gts, upper)
elif cond2.gts == sym:
lower = Max(cond2.lts, lower)
else:
raise NotImplementedError(
"Unable to handle interval evaluation of expression.")
else:
if sym not in [cond.lts, cond.gts]:
cond = _solve_inequality(cond, sym)
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:
raise NotImplementedError(
"Unable to handle interval evaluation of expression.")

# part 1b: Reduce (-)infinity to what was passed in.
lower, upper = Max(a, lower), Min(b, upper)

for n in range(len(int_expr)):
# Part 2: remove any interval overlap.  For any conflicts, the
# its bounds accordingly.
if self.__eval_cond(lower < int_expr[n][1]) and \
self.__eval_cond(lower >= int_expr[n][0]):
lower = int_expr[n][1]
elif len(int_expr[n][1].free_symbols) and \
self.__eval_cond(lower >= int_expr[n][0]):
if self.__eval_cond(lower == int_expr[n][0]):
lower = int_expr[n][1]
else:
int_expr[n][1] = Min(lower, int_expr[n][1])
elif len(int_expr[n][0].free_symbols) and \
self.__eval_cond(upper == int_expr[n][1]):
upper = Min(upper, int_expr[n][0])
elif len(int_expr[n][1].free_symbols) and \
(lower >= int_expr[n][0]) != True and \
(int_expr[n][1] == Min(lower, upper)) != True:
upper = Min(upper, int_expr[n][0])
elif self.__eval_cond(upper > int_expr[n][0]) and \
self.__eval_cond(upper <= int_expr[n][1]):
upper = int_expr[n][0]
elif len(int_expr[n][0].free_symbols) and \
self.__eval_cond(upper < int_expr[n][1]):
int_expr[n][0] = Max(upper, int_expr[n][0])

if self.__eval_cond(lower >= upper) != True:  # Is it still an interval?
int_expr.append([lower, upper, expr])
if orig_cond == targetcond:
return [(lower, upper, None)]
elif isinstance(targetcond, Or) and cond in targetcond.args:
or_cond = Or(or_cond, cond)
or_intervals.append((lower, upper, None))
if or_cond == targetcond:
or_intervals.sort(key=lambda x: x[0])
return or_intervals

int_expr.sort(key=lambda x: x[1].sort_key(
) if x[1].is_number else S.NegativeInfinity.sort_key())
int_expr.sort(key=lambda x: x[0].sort_key(
) if x[0].is_number else S.Infinity.sort_key())

for n in range(len(int_expr)):
if len(int_expr[n][0].free_symbols) or len(int_expr[n][1].free_symbols):
if isinstance(int_expr[n][1], Min) or int_expr[n][1] == b:
newval = Min(*int_expr[n][:-1])
if n > 0 and int_expr[n][0] == int_expr[n - 1][1]:
int_expr[n - 1][1] = newval
int_expr[n][0] = newval
else:
newval = Max(*int_expr[n][:-1])
if n < len(int_expr) - 1 and int_expr[n][1] == int_expr[n + 1][0]:
int_expr[n + 1][0] = newval
int_expr[n][1] = newval

# Add holes to list of intervals if there is a default value,
# otherwise raise a ValueError.
holes = []
curr_low = a
for int_a, int_b, expr in int_expr:
if (curr_low < int_a) == True:
holes.append([curr_low, Min(b, int_a), default])
elif (curr_low >= int_a) != True:
holes.append([curr_low, Min(b, int_a), default])
curr_low = Min(b, int_b)
if (curr_low < b) == True:
holes.append([Min(b, curr_low), b, default])
elif (curr_low >= b) != True:
holes.append([Min(b, curr_low), b, default])

if holes and default is not None:
int_expr.extend(holes)
if targetcond == True:
return [(h[0], h[1], None) for h in holes]
elif holes and default is None:
raise ValueError("Called interval evaluation over piecewise "
"function on undefined intervals %s" %
", ".join([str((h[0], h[1])) for h in holes]))

return int_expr

def _eval_nseries(self, x, n, logx):
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):
"""
Piecewise conditions may contain bool which are not of Basic type.
"""
args = list(self.args)
for i, (e, c) in enumerate(args):
if isinstance(c, bool):
pass
elif isinstance(c, Basic):
c = c._subs(old, new)
if c != False:
e = e._subs(old, new)
args[i] = e, c
if c == True:
return self.func(*args)

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, when_multiple=None):
b = None
for expr, _ in self.args:
a = getattr(expr, is_attr)
if a is None:
return None
if b is None:
b = a
elif b is not a:
return when_multiple
return b

_eval_is_finite = lambda self: self._eval_template_is_attr(
'is_finite', when_multiple=False)
_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', when_multiple=True)
_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_real = lambda self: self._eval_template_is_attr('is_real')
_eval_is_zero = lambda self: self._eval_template_is_attr(
'is_zero', when_multiple=False)

@classmethod
def __eval_cond(cls, cond):
"""Return the truth value of the condition."""
from sympy.solvers.solvers import checksol
if cond == True:
return True
if isinstance(cond, Equality):
if checksol(cond, {}, minimal=True):
# the equality is trivially solved
return True
diff = cond.lhs - cond.rhs
if diff.is_commutative:
return diff.is_zero
return None

def as_expr_set_pairs(self):
exp_sets = []
U = S.Reals
for expr, cond in self.args:
cond_int = U.intersect(cond.as_set())
U = U - cond_int
exp_sets.append((expr, cond_int))
return exp_sets

[docs]def piecewise_fold(expr):
"""
Takes an expression containing a piecewise function and returns the
expression in piecewise form.

Examples
========

>>> from sympy import Piecewise, piecewise_fold, sympify as 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, 1 <= x))

========

Piecewise
"""
if not isinstance(expr, Basic) or not expr.has(Piecewise):
return expr
new_args = list(map(piecewise_fold, expr.args))
if expr.func is ExprCondPair:
return ExprCondPair(*new_args)
piecewise_args = []
for n, arg in enumerate(new_args):
if isinstance(arg, Piecewise):
piecewise_args.append(n)
if len(piecewise_args) > 0:
n = piecewise_args[0]
new_args = [(expr.func(*(new_args[:n] + [e] + new_args[n + 1:])), c)
for e, c in new_args[n].args]
if isinstance(expr, Boolean):
# If expr is Boolean, we must return some kind of PiecewiseBoolean.
# This is constructed by means of Or, And and Not.
# piecewise_fold(0 < Piecewise( (sin(x), x<0), (cos(x), True)))
# can't return Piecewise((0 < sin(x), x < 0), (0 < cos(x), True))
# but instead Or(And(x < 0, 0 < sin(x)), And(0 < cos(x), Not(x<0)))
other = True
rtn = False
for e, c in new_args:
rtn = Or(rtn, And(other, c, e))
other = And(other, Not(c))
if len(piecewise_args) > 1:
return piecewise_fold(rtn)
return rtn
if len(piecewise_args) > 1:
return piecewise_fold(Piecewise(*new_args))
return Piecewise(*new_args)
else:
return expr.func(*new_args)