piecewise.py

# sage.doctest: needs sage.symbolic
r"""
Piecewise functions

This module implement piecewise functions in a single variable. See
:mod:`sage.sets.real_set` for more information about how to construct
subsets of the real line for the domains.

EXAMPLES::

    sage: f = piecewise([((0,1), x^3), ([-1,0], -x^2)]);  f
    piecewise(x|-->x^3 on (0, 1), x|-->-x^2 on [-1, 0]; x)
    sage: 2*f
    2*piecewise(x|-->x^3 on (0, 1), x|-->-x^2 on [-1, 0]; x)
    sage: f(x=1/2)
    1/8
    sage: plot(f)    # not tested

.. TODO::

    Implement max/min location and values,

AUTHORS:

- David Joyner (2006-04): initial version

- David Joyner (2006-09): added __eq__, extend_by_zero_to, unextend,
  convolution, trapezoid, trapezoid_integral_approximation,
  riemann_sum, riemann_sum_integral_approximation, tangent_line fixed
  bugs in __mul__, __add__

- David Joyner (2007-03): adding Hann filter for FS, added general FS
  filter methods for computing and plotting, added options to plotting
  of FS (eg, specifying rgb values are now allowed). Fixed bug in
  documentation reported by Pablo De Napoli.

- David Joyner (2007-09): bug fixes due to behaviour of
  SymbolicArithmetic

- David Joyner (2008-04): fixed docstring bugs reported by J Morrow; added
  support for Laplace transform of functions with infinite support.

- David Joyner (2008-07): fixed a left multiplication bug reported by
  C. Boncelet (by defining __rmul__ = __mul__).

- Paul Butler (2009-01): added indefinite integration and default_variable

- Volker Braun (2013): Complete rewrite

- Ralf Stephan (2015): Rewrite of convolution() and other calculus
  functions; many doctest adaptations

- Eric Gourgoulhon (2017): Improve documentation and user interface of
  Fourier series

TESTS::

    sage: fast_callable(f, vars=[x])(0.5)
    0.125000000000...
"""

# ****************************************************************************
#       Copyright (C) 2006 William Stein <wstein@gmail.com>
#                     2006 David Joyner <wdjoyner@gmail.com>
#                     2013 Volker Braun <vbraun.name@gmail.com>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  https://www.gnu.org/licenses/
# ****************************************************************************

from sage.misc.lazy_import import lazy_import
from sage.rings.infinity import minus_infinity, infinity
from sage.sets.real_set import RealSet
from sage.symbolic.function import BuiltinFunction

lazy_import('sage.symbolic.ring', 'SR')


class PiecewiseFunction(BuiltinFunction):
    def __init__(self):
        """
        Piecewise function.

        EXAMPLES::

            sage: var('x, y')
            (x, y)
            sage: f = piecewise([((0,1), x^2*y), ([-1,0], -x*y^2)], var=x);  f
            piecewise(x|-->x^2*y on (0, 1), x|-->-x*y^2 on [-1, 0]; x)
            sage: f(1/2)
            1/4*y
            sage: f(-1/2)
            1/2*y^2
        """
        BuiltinFunction.__init__(self, "piecewise",
                                 latex_name='piecewise',
                                 conversions=dict(), nargs=2)

    def __call__(self, function_pieces, **kwds):
        r"""
        Piecewise functions.

        INPUT:

        - ``function_pieces`` -- list of pairs consisting of a
          domain and a symbolic function

        - ``var=x`` -- a symbolic variable or ``None`` (default); the
          real variable in which the function is piecewise in

        OUTPUT:

        A piecewise-defined function. A :exc:`ValueError` will be raised
        if the domains of the pieces are not pairwise disjoint.

        EXAMPLES::

            sage: my_abs = piecewise([((-1, 0), -x), ([0, 1], x)], var=x);  my_abs
            piecewise(x|-->-x on (-1, 0), x|-->x on [0, 1]; x)
            sage: [ my_abs(i/5) for i in range(-4, 5)]
            [4/5, 3/5, 2/5, 1/5, 0, 1/5, 2/5, 3/5, 4/5]

        TESTS::

            sage: piecewise([([-1, 0], -x), ([0, 1], x)], var=x)
            Traceback (most recent call last):
            ...
            ValueError: domains must be pairwise disjoint

            sage: step = piecewise([((-1, 0), -1), ([0, 0], 0), ((0, 1), 1)], var=x);  step
            piecewise(x|-->-1 on (-1, 0), x|-->0 on {0}, x|-->1 on (0, 1); x)
            sage: step(-1/2), step(0), step(1/2)
            (-1, 0, 1)
        """
        from types import FunctionType
        var = kwds.pop('var', None)
        parameters = []
        domain_list = []
        for piece in function_pieces:
            domain, function = piece
            if not isinstance(domain, RealSet):
                domain = RealSet(domain)
            if domain.is_empty():
                continue
            if isinstance(function, FunctionType):
                if var is None:
                    var = SR.var('x')
                if function.__code__.co_argcount == 0:
                    function = function()
                else:
                    function = function(var)
            function = SR(function)
            if var is None and len(function.variables()) > 0:
                var = function.variables()[0]
            parameters.append((domain, function))
            domain_list.append(domain)
        if not RealSet.are_pairwise_disjoint(*domain_list):
            raise ValueError('domains must be pairwise disjoint')
        if var is None:
            var = self.default_variable()
        parameters = SR._force_pyobject(tuple(parameters), recursive=False)
        return BuiltinFunction.__call__(self, parameters, var, **kwds)

    def _print_(self, parameters, variable):
        """
        Return a string representation.

        OUTPUT: string

        EXAMPLES::

            sage: p = piecewise([((-2, 0), -x), ([0, 4], x)], var=x)
            sage: str(p)    # indirect doctest
            'piecewise(x|-->-x on (-2, 0), x|-->x on [0, 4]; x)'
        """
        s = 'piecewise('
        # NOTE : could use ⟼ instead of |-->
        args = (f'{variable}|-->{func} on {domain}'
                for domain, func in parameters)
        s += ', '.join(args) + f'; {variable})'
        return s

    def _subs_(self, subs_map, options, parameters, x):
        """
        Callback from Pynac ``subs()``.

        EXAMPLES:

        If the substitution changes the piecewise variable, it must
        evaluate to a number so that we know which component we are
        on::

            sage: p = piecewise([((-2, 0), -x), ([0, 4], x)], var=x)
            sage: p.subs(x=-1)
            1
            sage: (10+p).subs(x=-1)
            11
            sage: p.subs(x=pi)
            pi

        Auxiliary variables can be substituted arbitrarily::

            sage: var('x,y')
            (x, y)
            sage: p = piecewise([((-2, 0), -x^y), ([0, 2], x-y)], var=x);  p
            piecewise(x|-->-x^y on (-2, 0), x|-->x - y on [0, 2]; x)
            sage: p.subs(y=sin(y))
            piecewise(x|-->-x^sin(y) on (-2, 0), x|-->x - sin(y) on [0, 2]; x)

        One can change the variable as follows::

            sage: p = piecewise([((-2, 0), -x), ([0, 4], x)], var=x)
            sage: y = SR.var('y')
            sage: p(y)
            piecewise(y|-->-y on (-2, 0), y|-->y on [0, 4]; y)
        """
        point = subs_map.apply_to(x, 0)

        if point.is_symbol():  # avoid to compare with x (see #37925)
            new_params = [(domain, subs_map.apply_to(func, 0))
                          for domain, func in parameters]
            return piecewise(new_params, var=point)

        if (point.is_numeric() or point.is_constant()) and point.is_real():
            if hasattr(point, 'pyobject'):
                # unwrap any numeric values
                point = point.pyobject()
            for domain, func in parameters:
                if domain.contains(point):
                    return subs_map.apply_to(func, 0)
            raise ValueError(f'point {point} is not in the domain')

        raise ValueError('substitution not allowed')

    @staticmethod
    def in_operands(ex):
        """
        Return whether a symbolic expression contains a piecewise
        function as operand.

        INPUT:

        - ``ex`` -- a symbolic expression

        OUTPUT: boolean

        EXAMPLES::

            sage: f = piecewise([([0,0], sin(x)), ((0,2), cos(x))]);  f
            piecewise(x|-->sin(x) on {0}, x|-->cos(x) on (0, 2); x)
            sage: piecewise.in_operands(f)
            True
            sage: piecewise.in_operands(1+sin(f))
            True
            sage: piecewise.in_operands(1+sin(0*f))
            False
        """
        def is_piecewise(ex):
            if ex.operator() is piecewise:
                return True
            for op in ex.operands():
                if is_piecewise(op):
                    return True
            return False
        return is_piecewise(ex)

    @staticmethod
    def simplify(ex):
        """
        Combine piecewise operands into single piecewise function.

        OUTPUT:

        A piecewise function whose operands are not piecewise if
        possible, that is, as long as the piecewise variable is the same.

        EXAMPLES::

            sage: f = piecewise([([0,0], sin(x)), ((0,2), cos(x))])
            sage: piecewise.simplify(f)
            Traceback (most recent call last):
            ...
            NotImplementedError
        """
        raise NotImplementedError

    def _tderivative_(self, parameters, variable, *args, **kwds):
        """
        Return the derivative of the piecewise function by applying the
        derivative to each piece.

        EXAMPLES::

            sage: f = piecewise([[(-1,1), x**2], [(1,3), x**3]])
            sage: f.diff()
            piecewise(x|-->2*x on (-1, 1), x|-->3*x^2 on (1, 3); x)
            sage: f.diff(x,x)
            piecewise(x|-->2 on (-1, 1), x|-->6*x on (1, 3); x)

        This still fails miserably::

            sage: y = SR.var('y')
            sage: f = piecewise([[(-6,0), x+y], [(0,8), x*y]],var=x)
            sage: f.derivative(x)  # known bug
            piecewise(x|-->1 on (-6, 0), x|-->y on (0, 8); x)
            sage: f.derivative(y)  # known bug
            piecewise(x|-->1 on (-6, 0), x|-->x on (0, 8); x)

        TESTS::

            sage: f = piecewise([((-oo, -1), 0), ((-1, 1), exp(-1/(1 - x^2))), ((1, oo), 0)])
            sage: f.diff()
            piecewise(x|-->0 on (-oo, -1), x|-->-2*x*e^(1/(x^2 - 1))/(x^2 - 1)^2 on (-1, 1), x|-->0 on (1, +oo); x)
        """
        return piecewise([(domain, func.derivative(*args))
                          for domain, func in parameters],
                         var=variable)

    class EvaluationMethods:

        def __pow__(self, parameters, variable, n):
            """
            Return the `n`-th power of the piecewise function by applying the
            operation to each piece.

            INPUT:

            - ``n`` -- number or symbolic expression

            EXAMPLES::

                sage: f1(x) = -abs(x) + 1; f2(x) = abs(x - 2) - 1
                sage: f = piecewise([[(-1,1), f1], [(1,3), f2]])
                sage: (f^2).integral(definite=True)
                4/3
            """
            return piecewise(zip(self.domains(),
                                 [ex**n for ex in self.expressions()]),
                             var=variable)

        def expression_at(self, parameters, variable, point):
            """
            Return the expression defining the piecewise function at
            ``value``.

            INPUT:

            - ``point`` -- a real number

            OUTPUT:

            The symbolic expression defining the function value at the
            given ``point``.

            EXAMPLES::

                sage: f = piecewise([([0,0], sin(x)), ((0,2), cos(x))]);  f
                piecewise(x|-->sin(x) on {0}, x|-->cos(x) on (0, 2); x)
                sage: f.expression_at(0)
                sin(x)
                sage: f.expression_at(1)
                cos(x)
                sage: f.expression_at(2)
                Traceback (most recent call last):
                ...
                ValueError: point is not in the domain
            """
            for domain, func in parameters:
                if domain.contains(point):
                    return func
            raise ValueError('point is not in the domain')

        which_function = expression_at

        def domains(self, parameters, variable):
            """
            Return the individual domains.

            See also :meth:`~expressions`.

            OUTPUT:

            The collection of domains of the component functions as a
            tuple of :class:`~sage.sets.real_set.RealSet`.

            EXAMPLES::

                sage: f = piecewise([([0,0], sin(x)), ((0,2), cos(x))]);  f
                piecewise(x|-->sin(x) on {0}, x|-->cos(x) on (0, 2); x)
                sage: f.domains()
                ({0}, (0, 2))
            """
            return tuple(dom for dom, fun in parameters)

        def domain(self, parameters, variable):
            """
            Return the domain.

            OUTPUT:

            The union of the domains of the individual pieces as a
            :class:`~sage.sets.real_set.RealSet`.

            EXAMPLES::

                sage: f = piecewise([([0,0], sin(x)), ((0,2), cos(x))]);  f
                piecewise(x|-->sin(x) on {0}, x|-->cos(x) on (0, 2); x)
                sage: f.domain()
                [0, 2)
            """
            intervals = []
            for domain, func in parameters:
                intervals += list(domain)
            return RealSet(*intervals)

        def __len__(self, parameters, variable):
            """
            Return the number of "pieces".

            OUTPUT: integer

            EXAMPLES::

                sage: f = piecewise([([0,0], sin(x)), ((0,2), cos(x))]);  f
                piecewise(x|-->sin(x) on {0}, x|-->cos(x) on (0, 2); x)
                sage: len(f)
                2
            """
            return len(parameters)

        def expressions(self, parameters, variable):
            """
            Return the individual domains.

            See also :meth:`~domains`.

            OUTPUT: the collection of expressions of the component functions

            EXAMPLES::

                sage: f = piecewise([([0,0], sin(x)), ((0,2), cos(x))]);  f
                piecewise(x|-->sin(x) on {0}, x|-->cos(x) on (0, 2); x)
                sage: f.expressions()
                (sin(x), cos(x))
            """
            return tuple(fun for dom, fun in parameters)

        def items(self, parameters, variable):
            """
            Iterate over the pieces of the piecewise function.

            .. NOTE::

                You should probably use :meth:`pieces` instead, which
                offers a nicer interface.

            OUTPUT:

            This method iterates over pieces of the piecewise
            function, each represented by a pair. The first element is
            the support, and the second the function over that
            support.

            EXAMPLES::

                sage: f = piecewise([([0,0], sin(x)), ((0,2), cos(x))])
                sage: for support, function in f.items():
                ....:     print('support is {0}, function is {1}'.format(support, function))
                support is {0}, function is sin(x)
                support is (0, 2), function is cos(x)
            """
            yield from parameters

        def __call__(self, parameters, variable, value=None, **kwds):
            """
            Call the piecewise function.

            EXAMPLES::

                sage: f = piecewise([([0,0], sin(x)), ((0,2), cos(x))]);  f
                piecewise(x|-->sin(x) on {0}, x|-->cos(x) on (0, 2); x)
                sage: f(0)
                0
                sage: f(1)
                cos(1)
                sage: f(2)
                Traceback (most recent call last):
                ...
                ValueError: point 2 is not in the domain
            """
            self = piecewise(parameters, var=variable)
            substitution = dict()
            for k, v in kwds.items():
                substitution[SR.var(k)] = v
            if value is not None:
                substitution[variable] = value
            return self.subs(substitution)

        def _fast_callable_(self, parameters, variable, etb):
            """
            Override the ``fast_callable``.

            OUTPUT:

            A :class:`~sage.ext.fast_callable.ExpressionCall`
            representing the piecewise function in the expression
            tree.

            EXAMPLES::

                sage: p = piecewise([((-1, 0), -x), ([0, 1], x)], var=x)
                sage: from sage.ext.fast_callable import ExpressionTreeBuilder
                sage: etb = ExpressionTreeBuilder(vars=['x'])
                sage: p._fast_callable_(etb)
                {piecewise(x|-->-x on (-1, 0), x|-->x on [0, 1]; x)}(v_0)
            """
            self = piecewise(parameters, var=variable)
            return etb.call(self, variable)

        def restriction(self, parameters, variable, restricted_domain):
            """
            Restrict the domain.

            INPUT:

            - ``restricted_domain`` -- a
              :class:`~sage.sets.real_set.RealSet` or something that
              defines one.

            OUTPUT: a new piecewise function obtained by restricting the domain

            EXAMPLES::

                sage: f = piecewise([((-oo, oo), x)]);  f
                piecewise(x|-->x on (-oo, +oo); x)
                sage: f.restriction([[-1,1], [3,3]])
                piecewise(x|-->x on [-1, 1] ∪ {3}; x)
            """
            restricted_domain = RealSet(*restricted_domain)
            new_param = []
            for domain, func in parameters:
                domain = domain.intersection(restricted_domain)
                new_param.append((domain, func))
            return piecewise(new_param, var=variable)

        def extension(self, parameters, variable, extension, extension_domain=None):
            """
            Extend the function.

            INPUT:

            - ``extension`` -- a symbolic expression

            - ``extension_domain`` -- a
              :class:`~sage.sets.real_set.RealSet` or ``None``
              (default). The domain of the extension. By default, the
              entire complement of the current domain.

            EXAMPLES::

                sage: f = piecewise([((-1,1), x)]);  f
                piecewise(x|-->x on (-1, 1); x)
                sage: f(3)
                Traceback (most recent call last):
                ...
                ValueError: point 3 is not in the domain

                sage: g = f.extension(0);  g
                piecewise(x|-->x on (-1, 1), x|-->0 on (-oo, -1] ∪ [1, +oo); x)
                sage: g(3)
                0

                sage: h = f.extension(1, RealSet.unbounded_above_closed(1));  h
                piecewise(x|-->x on (-1, 1), x|-->1 on [1, +oo); x)
                sage: h(3)
                1
            """
            self = piecewise(parameters, var=variable)
            if extension_domain is None:
                extension_domain = self.domain().complement()
            ext = ((extension_domain, SR(extension)),)
            return piecewise(parameters + ext, var=variable)

        def unextend_zero(self, parameters, variable):
            """
            Remove zero pieces.

            EXAMPLES::

                sage: f = piecewise([((-1,1), x)]);  f
                piecewise(x|-->x on (-1, 1); x)
                sage: g = f.extension(0);  g
                piecewise(x|-->x on (-1, 1), x|-->0 on (-oo, -1] ∪ [1, +oo); x)
                sage: g(3)
                0
                sage: h = g.unextend_zero()
                sage: bool(h == f)
                True
            """
            result = [(domain, func) for domain, func in parameters
                      if func != 0]
            if len(result) == len(self):
                return self
            return piecewise(result, var=variable)

        def pieces(self, parameters, variable):
            """
            Return the "pieces".

            OUTPUT:

            A tuple of piecewise functions, each having only a single
            expression.

            EXAMPLES::

                sage: p = piecewise([((-1, 0), -x), ([0, 1], x)], var=x)
                sage: p.pieces()
                (piecewise(x|-->-x on (-1, 0); x),
                 piecewise(x|-->x on [0, 1]; x))
            """
            return tuple(piecewise([(domain, func)], var=variable)
                         for domain, func in parameters)

        def end_points(self, parameters, variable) -> list:
            """
            Return a list of all interval endpoints for this function.

            EXAMPLES::

                sage: f1(x) = 1
                sage: f2(x) = 1 - x
                sage: f3(x) = x^2 - 5
                sage: f = piecewise([[(0,1), f1], [(1,2), f2], [(2,3), f3]])
                sage: f.end_points()
                [0, 1, 2, 3]
                sage: f = piecewise([([0,0], sin(x)), ((0,2), cos(x))]);  f
                piecewise(x|-->sin(x) on {0}, x|-->cos(x) on (0, 2); x)
                sage: f.end_points()
                [0, 2]
            """
            s = set()
            for domain, func in parameters:
                for interval in domain:
                    s.add(interval.lower())
                    s.add(interval.upper())
            s.discard(minus_infinity)
            s.discard(infinity)
            return sorted(s)

        def piecewise_add(self, parameters, variable, other):
            """
            Return a new piecewise function with domain the union
            of the original domains and functions summed. Undefined
            intervals in the union domain get function value `0`.

            EXAMPLES::

                sage: f = piecewise([([0,1], 1), ((2,3), x)])
                sage: g = piecewise([((1/2, 2), x)])
                sage: f.piecewise_add(g).unextend_zero()
                piecewise(x|-->1 on (0, 1/2],
                          x|-->x + 1 on (1/2, 1],
                          x|-->x on (1, 2) ∪ (2, 3); x)
            """
            points = ([minus_infinity] +
                      sorted(set(self.end_points() + other.end_points())) +
                      [infinity])
            domain = []
            funcs = []
            contains_lower = False
            contains_upper = False
            for i in range(len(points) - 1):
                a, b = points[i], points[i + 1]
                try:
                    contains_lower = (self.domain().contains(a) or
                        other.domain().contains(a)) and not contains_upper
                    contains_upper = (self.domain().contains(b) or
                        other.domain().contains(b))
                    if contains_lower:
                        if contains_upper:
                            rs = RealSet.closed(a, b)
                        else:
                            rs = RealSet.closed_open(a, b)
                    else:
                        if contains_upper:
                            rs = RealSet.open_closed(a, b)
                        else:
                            rs = RealSet.open(a, b)
                    point = (b + a) / 2
                except ValueError:
                    if a == minus_infinity and b == infinity:
                        rs = RealSet.open(minus_infinity, infinity)
                        point = 0
                    elif a == minus_infinity:
                        if contains_lower:
                            rs = RealSet.unbounded_below_closed(b)
                        else:
                            rs = RealSet.unbounded_below_open(b)
                        point = b - 1
                    elif b == infinity:
                        if contains_upper:
                            rs = RealSet.unbounded_above_closed(a)
                        else:
                            rs = RealSet.unbounded_above_open(a)
                        point = a + 1
                    else:
                        raise
                try:
                    ex1 = self.expression_at(point)
                except ValueError:
                    ex1 = 0
                try:
                    ex2 = other.expression_at(point)
                except ValueError:
                    ex2 = 0
                ex = ex1 + ex2
                if i > 0 and funcs[-1] == ex:
                    # extend the previous domain
                    rs += domain[-1]
                    domain[-1] = rs
                else:
                    domain += rs
                    funcs.append(ex)
            return piecewise(zip(domain, funcs))

        def integral(self, parameters, variable, x=None, a=None, b=None, definite=False, **kwds):
            r"""
            By default, return the indefinite integral of the function.

            If ``definite=True`` is given, returns the definite integral.

            AUTHOR:

            - Paul Butler

            EXAMPLES::

                sage: f1(x) = 1 - x
                sage: f = piecewise([((0,1), 1), ((1,2), f1)])
                sage: f.integral(definite=True)
                1/2

            ::

                sage: f1(x) = -1
                sage: f2(x) = 2
                sage: f = piecewise([((0,pi/2), f1), ((pi/2,pi), f2)])
                sage: f.integral(definite=True)
                1/2*pi

                sage: f1(x) = 2
                sage: f2(x) = 3 - x
                sage: f = piecewise([[(-2, 0), f1], [(0, 3), f2]])
                sage: f.integral()
                piecewise(x|-->2*x + 4 on (-2, 0),
                          x|-->-1/2*x^2 + 3*x + 4 on (0, 3); x)

                sage: f1(y) = -1
                sage: f2(y) = y + 3
                sage: f3(y) = -y - 1
                sage: f4(y) = y^2 - 1
                sage: f5(y) = 3
                sage: f = piecewise([[[-4,-3], f1], [(-3,-2), f2], [[-2,0], f3],
                ....:                [(0,2), f4], [[2,3], f5]])
                sage: F = f.integral(y); F
                piecewise(y|-->-y - 4 on [-4, -3],
                          y|-->1/2*y^2 + 3*y + 7/2 on (-3, -2),
                          y|-->-1/2*y^2 - y - 1/2 on [-2, 0],
                          y|-->1/3*y^3 - y - 1/2 on (0, 2),
                          y|-->3*y - 35/6 on [2, 3]; y)

            Ensure results are consistent with FTC::

                sage: F(-3) - F(-4)
                -1
                sage: F(-1) - F(-3)
                1
                sage: F(2) - F(0)
                2/3
                sage: f.integral(y, 0, 2)
                2/3
                sage: F(3) - F(-4)
                19/6
                sage: f.integral(y, -4, 3)
                19/6
                sage: f.integral(definite=True)
                19/6

            ::

                sage: f1(y) = (y+3)^2
                sage: f2(y) = y+3
                sage: f3(y) = 3
                sage: f = piecewise([[(-infinity, -3), f1], [(-3, 0), f2],
                ....:                [(0, infinity), f3]])
                sage: f.integral()
                piecewise(y|-->1/3*y^3 + 3*y^2 + 9*y + 9 on (-oo, -3),
                          y|-->1/2*y^2 + 3*y + 9/2 on (-3, 0),
                          y|-->3*y + 9/2 on (0, +oo); y)

            ::

                sage: f1(x) = e^(-abs(x))
                sage: f = piecewise([[(-infinity, infinity), f1]])
                sage: result = f.integral(definite=True)
                ...
                sage: result
                2
                sage: f.integral()
                piecewise(x|-->-integrate(e^(-abs(x)), x, x, +Infinity) on (-oo, +oo); x)

            ::

                sage: f = piecewise([((0, 5), cos(x))])
                sage: f.integral()
                piecewise(x|-->sin(x) on (0, 5); x)

            TESTS:

            Verify that piecewise integrals of zero work (:issue:`10841`)::

                sage: f0(x) = 0
                sage: f = piecewise([[[0,1], f0]])
                sage: f.integral(x,0,1)
                0
                sage: f = piecewise([[[0,1], 0]])
                sage: f.integral(x,0,1)
                0
                sage: f = piecewise([[[0,1], SR(0)]])
                sage: f.integral(x,0,1)
                0

            Check that the algorithm keyword can be used::

                sage: ex = piecewise([([0, 1], 1), ((1, oo), 1/x**2)])
                sage: integral(ex, x, 0, 100, algorithm='sympy')
                199/100
                sage: integral(ex, x, algorithm='sympy')
                piecewise(x|-->x on [0, 1], x|-->-1/x + 2 on (1, +oo); x)
            """
            if a is not None and b is not None:
                F = self.integral(x, **kwds)
                return F(b) - F(a)

            if a is not None or b is not None:
                raise TypeError('only one endpoint given')

            area = 0
            new_pieces = []

            if x is None:
                x = self.default_variable()

            # The integral is computed by iterating over the pieces in order.
            # The definite integral for each piece is calculated and accumulated in `area`.
            # The indefinite integral of each piece is also calculated,
            # and the `area` before each piece is added to the piece.
            #
            # If a definite integral is requested, `area` is returned.
            # Otherwise, a piecewise function is constructed from the indefinite integrals
            # and returned.
            #
            # An exception is made if integral is called on a piecewise function
            # that starts at -infinity. In this case, we do not try to calculate the
            # definite integral of the first piece, and the value of `area` remains 0
            # after the first piece.

            from sage.symbolic.assumptions import assume, forget
            for domain, fun in parameters:
                for interval in domain:
                    start = interval.lower()
                    end = interval.upper()
                    if start == -infinity and not definite:
                        fun_integrated = fun.integral(x, end, x, **kwds)
                    else:
                        try:
                            assume(start < x)
                        except ValueError:  # Assumption is redundant
                            pass
                        fun_integrated = fun.integral(x, start, x, **kwds) + area
                        forget(start < x)
                        if definite or end != infinity:
                            area += fun.integral(x, start, end, **kwds)
                    new_pieces.append([interval, SR(fun_integrated).function(x)])

            if definite:
                return SR(area)
            else:
                return piecewise(new_pieces)

        def critical_points(self, parameters, variable):
            """
            Return the critical points of this piecewise function.

            EXAMPLES::

                sage: R.<x> = QQ[]
                sage: f1 = x^0
                sage: f2 = 10*x - x^2
                sage: f3 = 3*x^4 - 156*x^3 + 3036*x^2 - 26208*x
                sage: f = piecewise([[(0,3), f1], [(3,10), f2], [(10,20), f3]])
                sage: expected = [5, 12, 13, 14]
                sage: all(abs(e-a) < 0.001 for e,a in zip(expected, f.critical_points()))
                True

            TESTS:

            Use variables other than x (:issue:`13836`)::

                sage: R.<y> = QQ[]
                sage: f1 = y^0
                sage: f2 = 10*y - y^2
                sage: f3 = 3*y^4 - 156*y^3 + 3036*y^2 - 26208*y
                sage: f = piecewise([[(0,3), f1], [(3,10), f2], [(10,20), f3]])
                sage: expected = [5, 12, 13, 14]
                sage: all(abs(e-a) < 0.001 for e,a in zip(expected, f.critical_points()))
                True
            """
            from sage.calculus.calculus import maxima
            x = self.default_variable()
            crit_pts = []
            for domain, f in parameters:
                for interval in domain:
                    a = interval.lower()
                    b = interval.upper()
                    for root in maxima.allroots(SR(f).diff(x) == 0):
                        root = float(root.rhs())
                        if a < root < b:
                            crit_pts.append(root)
            return crit_pts

        def convolution(self, parameters, variable, other):
            r"""
            Return the convolution function,
            `f*g(t)=\int_{-\infty}^\infty f(u)g(t-u)du`, for compactly
            supported `f,g`.

            EXAMPLES::

                sage: x = PolynomialRing(QQ, 'x').gen()

            Example 0::

                sage: f = piecewise([[[0,1], 1]])
                sage: g = f.convolution(f); g
                piecewise(x|-->x on (0, 1],
                          x|-->-x + 2 on (1, 2]; x)
                sage: h = f.convolution(g); h
                piecewise(x|-->1/2*x^2 on (0, 1],
                          x|-->-x^2 + 3*x - 3/2 on (1, 2],
                          x|-->1/2*x^2 - 3*x + 9/2 on (2, 3]; x)

            Example 1::

                sage: f = piecewise([[(0,1), 1], [(1,2), 2], [(2,3), 1]])
                sage: g = f.convolution(f)
                sage: h = f.convolution(g); h
                piecewise(x|-->1/2*x^2 on (0, 1],
                          x|-->2*x^2 - 3*x + 3/2 on (1, 3],
                          x|-->-2*x^2 + 21*x - 69/2 on (3, 4],
                          x|-->-5*x^2 + 45*x - 165/2 on (4, 5],
                          x|-->-2*x^2 + 15*x - 15/2 on (5, 6],
                          x|-->2*x^2 - 33*x + 273/2 on (6, 8],
                          x|-->1/2*x^2 - 9*x + 81/2 on (8, 9]; x)

            Example 2::

                sage: f = piecewise([[(-1,1), 1]])
                sage: g = piecewise([[(0,3), x]])
                sage: f.convolution(g)
                piecewise(x|-->1/2*x^2 + x + 1/2 on (-1, 1],
                          x|-->2*x on (1, 2],
                          x|-->-1/2*x^2 + x + 4 on (2, 4]; x)
                sage: g = piecewise([[(0,3), 1], [(3,4), 2]])
                sage: f.convolution(g)
                piecewise(x|-->x + 1 on (-1, 1],
                          x|-->2 on (1, 2],
                          x|-->x on (2, 3],
                          x|-->-x + 6 on (3, 4],
                          x|-->-2*x + 10 on (4, 5]; x)

            Some unbounded but convergent cases now work::

                sage: p = piecewise([[(2,oo),exp(-x)]])
                sage: q = piecewise([[[2,3],x]])
                sage: p.convolution(q)
                piecewise(x|-->(x - 3)*e^(-2) - e^(-x + 2) on (4, 5]; x)
                sage: q.convolution(p)
                piecewise(x|-->(x - 3)*e^(-2) - e^(-x + 2) on (4, 5]; x)

            TESTS:

            Check that the bugs raised in :issue:`12123` are fixed::

                sage: f = piecewise([[(-2, 2), 2]])
                sage: g = piecewise([[(0, 2), 3/4]])
                sage: f.convolution(g)
                piecewise(x|-->3/2*x + 3 on (-2, 0],
                          x|-->3 on (0, 2],
                          x|-->-3/2*x + 6 on (2, 4]; x)
                sage: f = piecewise([[(-1, 1), 1]])
                sage: g = piecewise([[(0, 1), x], [(1, 2), -x + 2]])
                sage: f.convolution(g)
                piecewise(x|-->1/2*x^2 + x + 1/2 on (-1, 0],
                          x|-->-1/2*x^2 + x + 1/2 on (0, 2],
                          x|-->1/2*x^2 - 3*x + 9/2 on (2, 3]; x)
            """
            from sage.symbolic.integration.integral import definite_integral
            f = self
            g = other
            if not f.end_points() or not g.end_points():
                raise ValueError('one of the piecewise functions is nowhere defined')
            fd, f0 = parameters[0]
            gd, g0 = next(other.items())
            if len(f) == 1 == len(g):
                a1 = fd[0].lower()
                a2 = fd[0].upper()
                b1 = gd[0].lower()
                b2 = gd[0].upper()
                a1b1 = a1 + b1
                a2b2 = a2 + b2
                delta_a = a2 - a1
                delta_b = b2 - b1

                # this fails in some unbounded cases:
                a1b2 = a1 + b2
                a2b1 = a2 + b1

                todo = []
                if delta_a > delta_b:
                    if a1b2 is not minus_infinity:
                        todo.append((a1b1, a1b2, a1, variable - b1))
                    todo.append((a1b2, a2b1, variable - b2, variable - b1))
                    if a2b1 is not infinity:
                        todo.append((a2b1, a2b2, variable - b2, a2))
                elif delta_a < delta_b:
                    if a2b1 is not minus_infinity:
                        todo.append((a1b1, a2b1, a1, variable - b1))
                    todo.append((a2b1, a1b2, a1, a2))
                    if a1b2 is not infinity:
                        todo.append((a1b2, a2b2, variable - b2, a2))
                else:
                    if a2b1 is not minus_infinity:
                        todo.append((a1b1, a2b1, a1, variable - b1))
                        todo.append((a2b1, a2b2, variable - b2, a2))

                if not todo:
                    raise ValueError("no domain of integration")

                with SR.temp_var() as uu:
                    i1 = f0.subs({variable: uu})
                    i2 = g0.subs({variable: variable - uu})
                    expr = i1 * i2
                    h = piecewise([[(start, stop),
                                    definite_integral(expr, uu, mini, maxi)]
                                   for start, stop, mini, maxi in todo])
                flat_zero = piecewise([[(minus_infinity, infinity), 0]])
                return (flat_zero.piecewise_add(h)).unextend_zero()  # why ?

            z = piecewise([[(0, 0), 0]])
            for fpiece in f.pieces():
                for gpiece in g.pieces():
                    h = gpiece.convolution(fpiece)
                    z = z.piecewise_add(h)
            return z.unextend_zero()

        def trapezoid(self, parameters, variable, N):
            """
            Return the piecewise line function defined by the trapezoid rule
            for numerical integration based on a subdivision of each domain
            interval into N subintervals.

            EXAMPLES::

                sage: f = piecewise([[[0,1], x^2],
                ....:                [RealSet.open_closed(1,2), 5 - x^2]])
                sage: f.trapezoid(2)
                piecewise(x|-->1/2*x on (0, 1/2),
                          x|-->3/2*x - 1/2 on (1/2, 1),
                          x|-->7/2*x - 5/2 on (1, 3/2),
                          x|-->-7/2*x + 8 on (3/2, 2); x)
                sage: f = piecewise([[[-1,1], 1 - x^2]])
                sage: f.trapezoid(4).integral(definite=True)
                5/4
                sage: f = piecewise([[[-1,1], 1/2 + x - x^3]])          ## example 3
                sage: f.trapezoid(6).integral(definite=True)
                1

            TESTS:

            Use variables or rings other than x (:issue:`13836`)::

                sage: R.<y> = QQ[]
                sage: f1 = y^2
                sage: f2 = 5 - y^2
                sage: f = piecewise([[[0,1], f1], [RealSet.open_closed(1,2), f2]])
                sage: f.trapezoid(2)
                piecewise(y|-->1/2*y on (0, 1/2),
                          y|-->3/2*y - 1/2 on (1/2, 1),
                          y|-->7/2*y - 5/2 on (1, 3/2),
                          y|-->-7/2*y + 8 on (3/2, 2); y)
            """
            def func(x0, x1):
                f0, f1 = self(x0), self(x1)
                return [[(x0, x1), f0 + (f1-f0) * (x1-x0)**(-1)
                         * (self.default_variable()-x0)]]
            rsum = []
            for domain, f in parameters:
                for interval in domain:
                    a = interval.lower()
                    b = interval.upper()
                    h = (b-a)/N
                    for i in range(N):
                        x0 = a+i*h
                        x1 = a+(i+1)*h
                        rsum += func(x0, x1)
            return piecewise(rsum)

        def laplace(self, parameters, variable, x='x', s='t'):
            r"""
            Return the Laplace transform of ``self`` with respect to the variable
            var.

            INPUT:

            - ``x`` -- variable of ``self``

            - ``s`` -- variable of Laplace transform

            We assume that a piecewise function is 0 outside of its domain and
            that the left-most endpoint of the domain is 0.

            EXAMPLES::

                sage: x, s, w = var('x, s, w')
                sage: f = piecewise([[(0,1), 1], [[1,2], 1 - x]])
                sage: f.laplace(x, s)
                -e^(-s)/s + (s + 1)*e^(-2*s)/s^2 + 1/s - e^(-s)/s^2
                sage: f.laplace(x, w)
                -e^(-w)/w + (w + 1)*e^(-2*w)/w^2 + 1/w - e^(-w)/w^2

            ::

                sage: y, t = var('y, t')
                sage: f = piecewise([[[1,2], 1 - y]])
                sage: f.laplace(y, t)
                (t + 1)*e^(-2*t)/t^2 - e^(-t)/t^2

            ::

                sage: s = var('s')
                sage: t = var('t')
                sage: f1(t) = -t
                sage: f2(t) = 2
                sage: f = piecewise([[[0,1], f1], [(1,infinity), f2]])
                sage: f.laplace(t, s)
                (s + 1)*e^(-s)/s^2 + 2*e^(-s)/s - 1/s^2
            """
            from sage.symbolic.assumptions import assume, forget

            x = SR.var(x)
            s = SR.var(s)
            assume(s > 0)
            exp_sx = (-s * x).exp()
            result = 0
            for domain, f in parameters:
                for interval in domain:
                    a = interval.lower()
                    b = interval.upper()
                    result += (SR(f) * exp_sx).integral(x, a, b)
            forget(s > 0)
            return result

        def fourier_series_cosine_coefficient(self, parameters,
                                              variable, n, L=None):
            r"""
            Return the `n`-th cosine coefficient of the Fourier series of
            the periodic function `f` extending the piecewise-defined
            function ``self``.

            Given an integer `n\geq 0`, the `n`-th cosine coefficient of
            the Fourier series of `f` is defined by

            .. MATH::

                a_n = \frac{1}{L}\int_{-L}^L
                        f(x)\cos\left(\frac{n\pi x}{L}\right) dx,

            where `L` is the half-period of `f`. For `n\geq 1`, `a_n` is
            the coefficient of `\cos(n\pi x/L)` in the Fourier series of
            `f`, while `a_0` is twice the coefficient of the constant
            term `\cos(0 x)`, i.e. twice the mean value of `f` over one
            period (cf. :meth:`fourier_series_partial_sum`).

            INPUT:

            - ``n`` -- nonnegative integer

            - ``L`` -- (default: ``None``) the half-period of `f`; if none
              is provided, `L` is assumed to be the half-width of the domain
              of ``self``

            OUTPUT: the Fourier coefficient `a_n`, as defined above

            EXAMPLES:

            A triangle wave function of period 2::

                sage: f = piecewise([((0,1), x), ((1,2), 2 - x)])
                sage: f.fourier_series_cosine_coefficient(0)
                1
                sage: f.fourier_series_cosine_coefficient(3)
                -4/9/pi^2

            If the domain of the piecewise-defined function encompasses
            more than one period, the half-period must be passed as the
            second argument; for instance::

                sage: f2 = piecewise([((0,1), x), ((1,2), 2 - x),
                ....:                 ((2,3), x - 2), ((3,4), 2 - (x-2))])
                sage: bool(f2.restriction((0,2)) == f)  # f2 extends f on (0,4)
                True
                sage: f2.fourier_series_cosine_coefficient(3, 1)  # half-period = 1
                -4/9/pi^2

            The default half-period is 2 and one has::

                sage: f2.fourier_series_cosine_coefficient(3)     # half-period = 2
                0

            The Fourier coefficient `-4/(9\pi^2)` obtained above is actually
            recovered for `n=6`::

                sage: f2.fourier_series_cosine_coefficient(6)
                -4/9/pi^2

            Other examples::

                sage: f(x) = x^2
                sage: f = piecewise([[(-1,1), f]])
                sage: f.fourier_series_cosine_coefficient(2)
                pi^(-2)
                sage: f1(x) = -1
                sage: f2(x) = 2
                sage: f = piecewise([[(-pi, pi/2), f1], [(pi/2, pi), f2]])
                sage: f.fourier_series_cosine_coefficient(5, pi)
                -3/5/pi
            """
            from sage.functions.trig import cos
            from sage.symbolic.constants import pi
            L0 = (self.domain().sup() - self.domain().inf()) / 2
            if not L:
                L = L0
            else:
                m = L0 / L
                if not (m.is_integer() and m > 0):
                    raise ValueError("the width of the domain of " +
                                     "{} is not a multiple ".format(self) +
                                     "of the given period")
            result = 0
            for domain, f in parameters:
                for interval in domain:
                    a = interval.lower()
                    b = interval.upper()
                    result += (f*cos(pi*variable*n/L)).integrate(variable, a, b)
            return SR(result / L0).simplify_trig()

        def fourier_series_sine_coefficient(self, parameters, variable,
                                            n, L=None):
            r"""
            Return the `n`-th sine coefficient of the Fourier series of
            the periodic function `f` extending the piecewise-defined
            function ``self``.

            Given an integer `n\geq 0`, the `n`-th sine coefficient of
            the Fourier series of `f` is defined by

            .. MATH::

                b_n = \frac{1}{L}\int_{-L}^L
                        f(x)\sin\left(\frac{n\pi x}{L}\right) dx,

            where `L` is the half-period of `f`. The number `b_n` is
            the coefficient of `\sin(n\pi x/L)` in the Fourier
            series of `f` (cf. :meth:`fourier_series_partial_sum`).

            INPUT:

            - ``n`` -- nonnegative integer

            - ``L`` -- (default: ``None``) the half-period of `f`; if none
              is provided, `L` is assumed to be the half-width of the domain
              of ``self``

            OUTPUT: the Fourier coefficient `b_n`, as defined above

            EXAMPLES:

            A square wave function of period 2::

                sage: f = piecewise([((-1,0), -1), ((0,1), 1)])
                sage: f.fourier_series_sine_coefficient(1)
                4/pi
                sage: f.fourier_series_sine_coefficient(2)
                0
                sage: f.fourier_series_sine_coefficient(3)
                4/3/pi

            If the domain of the piecewise-defined function encompasses
            more than one period, the half-period must be passed as the
            second argument; for instance::

                sage: f2 = piecewise([((-1,0), -1), ((0,1), 1),
                ....:                 ((1,2), -1), ((2,3), 1)])
                sage: bool(f2.restriction((-1,1)) == f)  # f2 extends f on (-1,3)
                True
                sage: f2.fourier_series_sine_coefficient(1, 1)  # half-period = 1
                4/pi
                sage: f2.fourier_series_sine_coefficient(3, 1)  # half-period = 1
                4/3/pi

            The default half-period is 2 and one has::

                sage: f2.fourier_series_sine_coefficient(1)  # half-period = 2
                0
                sage: f2.fourier_series_sine_coefficient(3)  # half-period = 2
                0

            The Fourier coefficients obtained from ``f`` are actually
            recovered for `n=2` and `n=6` respectively::

                sage: f2.fourier_series_sine_coefficient(2)
                4/pi
                sage: f2.fourier_series_sine_coefficient(6)
                4/3/pi
            """
            from sage.functions.trig import sin
            from sage.symbolic.constants import pi
            L0 = (self.domain().sup() - self.domain().inf()) / 2
            if not L:
                L = L0
            else:
                m = L0 / L
                if not (m.is_integer() and m > 0):
                    raise ValueError("the width of the domain of " +
                                     "{} is not a multiple ".format(self) +
                                     "of the given period")
            result = 0
            for domain, f in parameters:
                for interval in domain:
                    a = interval.lower()
                    b = interval.upper()
                    result += (f*sin(pi*variable*n/L)).integrate(variable, a, b)
            return SR(result/L0).simplify_trig()

        def fourier_series_partial_sum(self, parameters, variable, N,
                                       L=None):
            r"""
            Return the partial sum up to a given order of the Fourier series
            of the periodic function `f` extending the piecewise-defined
            function ``self``.

            The Fourier partial sum of order `N` is defined as

            .. MATH::

                S_{N}(x) = \frac{a_0}{2} + \sum_{n=1}^{N} \left[
                      a_n\cos\left(\frac{n\pi x}{L}\right)
                    + b_n\sin\left(\frac{n\pi x}{L}\right)\right],

            where `L` is the half-period of `f` and the `a_n`'s and `b_n`'s
            are respectively the cosine coefficients and sine coefficients
            of the Fourier series of `f` (cf.
            :meth:`fourier_series_cosine_coefficient` and
            :meth:`fourier_series_sine_coefficient`).

            INPUT:

            - ``N`` -- positive integer; the order of the partial sum

            - ``L`` -- (default: ``None``) the half-period of `f`; if none
              is provided, `L` is assumed to be the half-width of the domain
              of ``self``

            OUTPUT:

            - the partial sum `S_{N}(x)`, as a symbolic expression

            EXAMPLES:

            A square wave function of period 2::

                sage: f = piecewise([((-1,0), -1), ((0,1), 1)])
                sage: f.fourier_series_partial_sum(5)
                4/5*sin(5*pi*x)/pi + 4/3*sin(3*pi*x)/pi + 4*sin(pi*x)/pi

            If the domain of the piecewise-defined function encompasses
            more than one period, the half-period must be passed as the
            second argument; for instance::

                sage: f2 = piecewise([((-1,0), -1), ((0,1), 1),
                ....:                 ((1,2), -1), ((2,3), 1)])
                sage: bool(f2.restriction((-1,1)) == f)  # f2 extends f on (-1,3)
                True
                sage: f2.fourier_series_partial_sum(5, 1)  # half-period = 1
                4/5*sin(5*pi*x)/pi + 4/3*sin(3*pi*x)/pi + 4*sin(pi*x)/pi
                sage: bool(f2.fourier_series_partial_sum(5, 1) ==
                ....:      f.fourier_series_partial_sum(5))
                True

            The default half-period is 2, so that skipping the second
            argument yields a different result::

                sage: f2.fourier_series_partial_sum(5)  # half-period = 2
                4*sin(pi*x)/pi

            An example of partial sum involving both cosine and sine terms::

                sage: f = piecewise([((-1,0), 0), ((0,1/2), 2*x),
                ....:                ((1/2,1), 2*(1-x))])
                sage: f.fourier_series_partial_sum(5)
                -2*cos(2*pi*x)/pi^2 + 4/25*sin(5*pi*x)/pi^2
                 - 4/9*sin(3*pi*x)/pi^2 + 4*sin(pi*x)/pi^2 + 1/4
            """
            from sage.symbolic.constants import pi
            from sage.functions.trig import cos, sin
            from sage.arith.srange import srange

            if not L:
                L = (self.domain().sup() - self.domain().inf()) / 2
            x = self.default_variable()
            a0 = self.fourier_series_cosine_coefficient(0, L)
            result = a0/2 + sum([(self.fourier_series_cosine_coefficient(n, L)*cos(n*pi*x/L) +
                                  self.fourier_series_sine_coefficient(n, L)*sin(n*pi*x/L))
                                 for n in srange(1, N+1)])
            return SR(result).expand()

        def _sympy_(self, parameters, variable):
            """
            Convert this piecewise expression to its SymPy equivalent.

            EXAMPLES::

                sage: ex = piecewise([((0, 1), pi), ([1, 2], x)])
                sage: f = ex._sympy_(); f                                               # needs sympy
                Piecewise((pi, (x > 0) & (x < 1)), (x, (x >= 1) & (x <= 2)))
                sage: f.diff()                                                          # needs sympy
                Piecewise((0, (x > 0) & (x < 1)), (1, (x >= 1) & (x <= 2)))

                sage: ex = piecewise([((-100, -2), 1/x), ((1, +oo), cos(x))])
                sage: g = ex._sympy_(); g                                               # needs sympy
                Piecewise((1/x, (x > -100) & (x < -2)), (cos(x), x > 1))
                sage: g.diff()                                                          # needs sympy
                Piecewise((-1/x**2, (x > -100) & (x < -2)), (-sin(x), x > 1))
            """
            from sympy import Piecewise as pw
            args = [(func._sympy_(),
                     domain._sympy_condition_(variable))
                    for domain, func in parameters]
            return pw(*args)

        def _giac_init_(self, parameters, variable):
            """
            Convert this piecewise expression to its Giac equivalent.

            Backward conversion is not yet implemented.

            EXAMPLES::

                sage: # needs giac
                sage: ex = piecewise([((0, 1), pi), ([1, 2], x)])
                sage: f = ex._giac_(); f
                piecewise(((sageVARx>0) and (1>sageVARx)),pi,((sageVARx>=1) and (2>=sageVARx)),sageVARx)
                sage: f.diff(x)
                piecewise(((sageVARx>0) and (1>sageVARx)),0,((sageVARx>=1) and (2>=sageVARx)),1)

                sage: # needs giac
                sage: ex = piecewise([((-100, -2), 1/x), ((1, +oo), cos(x))])
                sage: g = ex._giac_(); g
                piecewise(((sageVARx>-100) and ((-2)>sageVARx)),1/sageVARx,sageVARx>1,cos(sageVARx))
                sage: g.diff(x)
                piecewise(((sageVARx>-100) and ((-2)>sageVARx)),-1/sageVARx^2,sageVARx>1,-sin(sageVARx))

            TESTS::

                sage: f = piecewise([([0,1], x), ((1,2), 3*x)])
                sage: a = libgiac(f)  # random because verbose                          # needs sage.libs.giac
                sage: a                                                                 # needs sage.libs.giac
                piecewise(((sageVARx>=0) and (1>=sageVARx)),sageVARx,((sageVARx>1) and (2>sageVARx)),sageVARx*3)
            """
            from sage.misc.flatten import flatten
            args = [(domain._giac_condition_(variable),
                     func._giac_init_())
                    for domain, func in parameters]
            args = ",".join(flatten(args))
            return f"piecewise({args})"


piecewise = PiecewiseFunction()