I am done

rl/Lib/site-packages/sympy/simplify/__init__.py

@@ -0,0 +1,60 @@
+"""The module helps converting SymPy expressions into shorter forms of them.
+
+for example:
+the expression E**(pi*I) will be converted into -1
+the expression (x+x)**2 will be converted into 4*x**2
+"""
+from .simplify import (simplify, hypersimp, hypersimilar,
+    logcombine, separatevars, posify, besselsimp, kroneckersimp,
+    signsimp, nsimplify)
+
+from .fu import FU, fu
+
+from .sqrtdenest import sqrtdenest
+
+from .cse_main import cse
+
+from .epathtools import epath, EPath
+
+from .hyperexpand import hyperexpand
+
+from .radsimp import collect, rcollect, radsimp, collect_const, fraction, numer, denom
+
+from .trigsimp import trigsimp, exptrigsimp
+
+from .powsimp import powsimp, powdenest
+
+from .combsimp import combsimp
+
+from .gammasimp import gammasimp
+
+from .ratsimp import ratsimp, ratsimpmodprime
+
+__all__ = [
+    'simplify', 'hypersimp', 'hypersimilar', 'logcombine', 'separatevars',
+    'posify', 'besselsimp', 'kroneckersimp', 'signsimp',
+    'nsimplify',
+
+    'FU', 'fu',
+
+    'sqrtdenest',
+
+    'cse',
+
+    'epath', 'EPath',
+
+    'hyperexpand',
+
+    'collect', 'rcollect', 'radsimp', 'collect_const', 'fraction', 'numer',
+    'denom',
+
+    'trigsimp', 'exptrigsimp',
+
+    'powsimp', 'powdenest',
+
+    'combsimp',
+
+    'gammasimp',
+
+    'ratsimp', 'ratsimpmodprime',
+]

rl/Lib/site-packages/sympy/simplify/combsimp.py

@@ -0,0 +1,114 @@
+from sympy.core import Mul
+from sympy.core.function import count_ops
+from sympy.core.traversal import preorder_traversal, bottom_up
+from sympy.functions.combinatorial.factorials import binomial, factorial
+from sympy.functions import gamma
+from sympy.simplify.gammasimp import gammasimp, _gammasimp
+
+from sympy.utilities.timeutils import timethis
+
+
+@timethis('combsimp')
+def combsimp(expr):
+    r"""
+    Simplify combinatorial expressions.
+
+    Explanation
+    ===========
+
+    This function takes as input an expression containing factorials,
+    binomials, Pochhammer symbol and other "combinatorial" functions,
+    and tries to minimize the number of those functions and reduce
+    the size of their arguments.
+
+    The algorithm works by rewriting all combinatorial functions as
+    gamma functions and applying gammasimp() except simplification
+    steps that may make an integer argument non-integer. See docstring
+    of gammasimp for more information.
+
+    Then it rewrites expression in terms of factorials and binomials by
+    rewriting gammas as factorials and converting (a+b)!/a!b! into
+    binomials.
+
+    If expression has gamma functions or combinatorial functions
+    with non-integer argument, it is automatically passed to gammasimp.
+
+    Examples
+    ========
+
+    >>> from sympy.simplify import combsimp
+    >>> from sympy import factorial, binomial, symbols
+    >>> n, k = symbols('n k', integer = True)
+
+    >>> combsimp(factorial(n)/factorial(n - 3))
+    n*(n - 2)*(n - 1)
+    >>> combsimp(binomial(n+1, k+1)/binomial(n, k))
+    (n + 1)/(k + 1)
+
+    """
+
+    expr = expr.rewrite(gamma, piecewise=False)
+    if any(isinstance(node, gamma) and not node.args[0].is_integer
+        for node in preorder_traversal(expr)):
+        return gammasimp(expr);
+
+    expr = _gammasimp(expr, as_comb = True)
+    expr = _gamma_as_comb(expr)
+    return expr
+
+
+def _gamma_as_comb(expr):
+    """
+    Helper function for combsimp.
+
+    Rewrites expression in terms of factorials and binomials
+    """
+
+    expr = expr.rewrite(factorial)
+
+    def f(rv):
+        if not rv.is_Mul:
+            return rv
+        rvd = rv.as_powers_dict()
+        nd_fact_args = [[], []] # numerator, denominator
+
+        for k in rvd:
+            if isinstance(k, factorial) and rvd[k].is_Integer:
+                if rvd[k].is_positive:
+                    nd_fact_args[0].extend([k.args[0]]*rvd[k])
+                else:
+                    nd_fact_args[1].extend([k.args[0]]*-rvd[k])
+                rvd[k] = 0
+        if not nd_fact_args[0] or not nd_fact_args[1]:
+            return rv
+
+        hit = False
+        for m in range(2):
+            i = 0
+            while i < len(nd_fact_args[m]):
+                ai = nd_fact_args[m][i]
+                for j in range(i + 1, len(nd_fact_args[m])):
+                    aj = nd_fact_args[m][j]
+
+                    sum = ai + aj
+                    if sum in nd_fact_args[1 - m]:
+                        hit = True
+
+                        nd_fact_args[1 - m].remove(sum)
+                        del nd_fact_args[m][j]
+                        del nd_fact_args[m][i]
+
+                        rvd[binomial(sum, ai if count_ops(ai) <
+                                count_ops(aj) else aj)] += (
+                                -1 if m == 0 else 1)
+                        break
+                else:
+                    i += 1
+
+        if hit:
+            return Mul(*([k**rvd[k] for k in rvd] + [factorial(k)
+                    for k in nd_fact_args[0]]))/Mul(*[factorial(k)
+                    for k in nd_fact_args[1]])
+        return rv
+
+    return bottom_up(expr, f)

rl/Lib/site-packages/sympy/simplify/cse_main.py

@@ -0,0 +1,946 @@
+""" Tools for doing common subexpression elimination.
+"""
+from collections import defaultdict
+
+from sympy.core import Basic, Mul, Add, Pow, sympify
+from sympy.core.containers import Tuple, OrderedSet
+from sympy.core.exprtools import factor_terms
+from sympy.core.singleton import S
+from sympy.core.sorting import ordered
+from sympy.core.symbol import symbols, Symbol
+from sympy.matrices import (MatrixBase, Matrix, ImmutableMatrix,
+                            SparseMatrix, ImmutableSparseMatrix)
+from sympy.matrices.expressions import (MatrixExpr, MatrixSymbol, MatMul,
+                                        MatAdd, MatPow, Inverse)
+from sympy.matrices.expressions.matexpr import MatrixElement
+from sympy.polys.rootoftools import RootOf
+from sympy.utilities.iterables import numbered_symbols, sift, \
+        topological_sort, iterable
+
+from . import cse_opts
+
+# (preprocessor, postprocessor) pairs which are commonly useful. They should
+# each take a SymPy expression and return a possibly transformed expression.
+# When used in the function ``cse()``, the target expressions will be transformed
+# by each of the preprocessor functions in order. After the common
+# subexpressions are eliminated, each resulting expression will have the
+# postprocessor functions transform them in *reverse* order in order to undo the
+# transformation if necessary. This allows the algorithm to operate on
+# a representation of the expressions that allows for more optimization
+# opportunities.
+# ``None`` can be used to specify no transformation for either the preprocessor or
+# postprocessor.
+
+
+basic_optimizations = [(cse_opts.sub_pre, cse_opts.sub_post),
+                       (factor_terms, None)]
+
+# sometimes we want the output in a different format; non-trivial
+# transformations can be put here for users
+# ===============================================================
+
+
+def reps_toposort(r):
+    """Sort replacements ``r`` so (k1, v1) appears before (k2, v2)
+    if k2 is in v1's free symbols. This orders items in the
+    way that cse returns its results (hence, in order to use the
+    replacements in a substitution option it would make sense
+    to reverse the order).
+
+    Examples
+    ========
+
+    >>> from sympy.simplify.cse_main import reps_toposort
+    >>> from sympy.abc import x, y
+    >>> from sympy import Eq
+    >>> for l, r in reps_toposort([(x, y + 1), (y, 2)]):
+    ...     print(Eq(l, r))
+    ...
+    Eq(y, 2)
+    Eq(x, y + 1)
+
+    """
+    r = sympify(r)
+    E = []
+    for c1, (k1, v1) in enumerate(r):
+        for c2, (k2, v2) in enumerate(r):
+            if k1 in v2.free_symbols:
+                E.append((c1, c2))
+    return [r[i] for i in topological_sort((range(len(r)), E))]
+
+
+def cse_separate(r, e):
+    """Move expressions that are in the form (symbol, expr) out of the
+    expressions and sort them into the replacements using the reps_toposort.
+
+    Examples
+    ========
+
+    >>> from sympy.simplify.cse_main import cse_separate
+    >>> from sympy.abc import x, y, z
+    >>> from sympy import cos, exp, cse, Eq, symbols
+    >>> x0, x1 = symbols('x:2')
+    >>> eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1))
+    >>> cse([eq, Eq(x, z + 1), z - 2], postprocess=cse_separate) in [
+    ... [[(x0, y + 1), (x, z + 1), (x1, x + 1)],
+    ...  [x1 + exp(x1/x0) + cos(x0), z - 2]],
+    ... [[(x1, y + 1), (x, z + 1), (x0, x + 1)],
+    ...  [x0 + exp(x0/x1) + cos(x1), z - 2]]]
+    ...
+    True
+    """
+    d = sift(e, lambda w: w.is_Equality and w.lhs.is_Symbol)
+    r = r + [w.args for w in d[True]]
+    e = d[False]
+    return [reps_toposort(r), e]
+
+
+def cse_release_variables(r, e):
+    """
+    Return tuples giving ``(a, b)`` where ``a`` is a symbol and ``b`` is
+    either an expression or None. The value of None is used when a
+    symbol is no longer needed for subsequent expressions.
+
+    Use of such output can reduce the memory footprint of lambdified
+    expressions that contain large, repeated subexpressions.
+
+    Examples
+    ========
+
+    >>> from sympy import cse
+    >>> from sympy.simplify.cse_main import cse_release_variables
+    >>> from sympy.abc import x, y
+    >>> eqs = [(x + y - 1)**2, x, x + y, (x + y)/(2*x + 1) + (x + y - 1)**2, (2*x + 1)**(x + y)]
+    >>> defs, rvs = cse_release_variables(*cse(eqs))
+    >>> for i in defs:
+    ...   print(i)
+    ...
+    (x0, x + y)
+    (x1, (x0 - 1)**2)
+    (x2, 2*x + 1)
+    (_3, x0/x2 + x1)
+    (_4, x2**x0)
+    (x2, None)
+    (_0, x1)
+    (x1, None)
+    (_2, x0)
+    (x0, None)
+    (_1, x)
+    >>> print(rvs)
+    (_0, _1, _2, _3, _4)
+    """
+    if not r:
+        return r, e
+
+    s, p = zip(*r)
+    esyms = symbols('_:%d' % len(e))
+    syms = list(esyms)
+    s = list(s)
+    in_use = set(s)
+    p = list(p)
+    # sort e so those with most sub-expressions appear first
+    e = [(e[i], syms[i]) for i in range(len(e))]
+    e, syms = zip(*sorted(e,
+        key=lambda x: -sum(p[s.index(i)].count_ops()
+        for i in x[0].free_symbols & in_use)))
+    syms = list(syms)
+    p += e
+    rv = []
+    i = len(p) - 1
+    while i >= 0:
+        _p = p.pop()
+        c = in_use & _p.free_symbols
+        if c: # sorting for canonical results
+            rv.extend([(s, None) for s in sorted(c, key=str)])
+        if i >= len(r):
+            rv.append((syms.pop(), _p))
+        else:
+            rv.append((s[i], _p))
+        in_use -= c
+        i -= 1
+    rv.reverse()
+    return rv, esyms
+
+
+# ====end of cse postprocess idioms===========================
+
+
+def preprocess_for_cse(expr, optimizations):
+    """ Preprocess an expression to optimize for common subexpression
+    elimination.
+
+    Parameters
+    ==========
+
+    expr : SymPy expression
+        The target expression to optimize.
+    optimizations : list of (callable, callable) pairs
+        The (preprocessor, postprocessor) pairs.
+
+    Returns
+    =======
+
+    expr : SymPy expression
+        The transformed expression.
+    """
+    for pre, post in optimizations:
+        if pre is not None:
+            expr = pre(expr)
+    return expr
+
+
+def postprocess_for_cse(expr, optimizations):
+    """Postprocess an expression after common subexpression elimination to
+    return the expression to canonical SymPy form.
+
+    Parameters
+    ==========
+
+    expr : SymPy expression
+        The target expression to transform.
+    optimizations : list of (callable, callable) pairs, optional
+        The (preprocessor, postprocessor) pairs.  The postprocessors will be
+        applied in reversed order to undo the effects of the preprocessors
+        correctly.
+
+    Returns
+    =======
+
+    expr : SymPy expression
+        The transformed expression.
+    """
+    for pre, post in reversed(optimizations):
+        if post is not None:
+            expr = post(expr)
+    return expr
+
+
+class FuncArgTracker:
+    """
+    A class which manages a mapping from functions to arguments and an inverse
+    mapping from arguments to functions.
+    """
+
+    def __init__(self, funcs):
+        # To minimize the number of symbolic comparisons, all function arguments
+        # get assigned a value number.
+        self.value_numbers = {}
+        self.value_number_to_value = []
+
+        # Both of these maps use integer indices for arguments / functions.
+        self.arg_to_funcset = []
+        self.func_to_argset = []
+
+        for func_i, func in enumerate(funcs):
+            func_argset = OrderedSet()
+
+            for func_arg in func.args:
+                arg_number = self.get_or_add_value_number(func_arg)
+                func_argset.add(arg_number)
+                self.arg_to_funcset[arg_number].add(func_i)
+
+            self.func_to_argset.append(func_argset)
+
+    def get_args_in_value_order(self, argset):
+        """
+        Return the list of arguments in sorted order according to their value
+        numbers.
+        """
+        return [self.value_number_to_value[argn] for argn in sorted(argset)]
+
+    def get_or_add_value_number(self, value):
+        """
+        Return the value number for the given argument.
+        """
+        nvalues = len(self.value_numbers)
+        value_number = self.value_numbers.setdefault(value, nvalues)
+        if value_number == nvalues:
+            self.value_number_to_value.append(value)
+            self.arg_to_funcset.append(OrderedSet())
+        return value_number
+
+    def stop_arg_tracking(self, func_i):
+        """
+        Remove the function func_i from the argument to function mapping.
+        """
+        for arg in self.func_to_argset[func_i]:
+            self.arg_to_funcset[arg].remove(func_i)
+
+
+    def get_common_arg_candidates(self, argset, min_func_i=0):
+        """Return a dict whose keys are function numbers. The entries of the dict are
+        the number of arguments said function has in common with
+        ``argset``. Entries have at least 2 items in common.  All keys have
+        value at least ``min_func_i``.
+        """
+        count_map = defaultdict(lambda: 0)
+        if not argset:
+            return count_map
+
+        funcsets = [self.arg_to_funcset[arg] for arg in argset]
+        # As an optimization below, we handle the largest funcset separately from
+        # the others.
+        largest_funcset = max(funcsets, key=len)
+
+        for funcset in funcsets:
+            if largest_funcset is funcset:
+                continue
+            for func_i in funcset:
+                if func_i >= min_func_i:
+                    count_map[func_i] += 1
+
+        # We pick the smaller of the two containers (count_map, largest_funcset)
+        # to iterate over to reduce the number of iterations needed.
+        (smaller_funcs_container,
+         larger_funcs_container) = sorted(
+                 [largest_funcset, count_map],
+                 key=len)
+
+        for func_i in smaller_funcs_container:
+            # Not already in count_map? It can't possibly be in the output, so
+            # skip it.
+            if count_map[func_i] < 1:
+                continue
+
+            if func_i in larger_funcs_container:
+                count_map[func_i] += 1
+
+        return {k: v for k, v in count_map.items() if v >= 2}
+
+    def get_subset_candidates(self, argset, restrict_to_funcset=None):
+        """
+        Return a set of functions each of which whose argument list contains
+        ``argset``, optionally filtered only to contain functions in
+        ``restrict_to_funcset``.
+        """
+        iarg = iter(argset)
+
+        indices = OrderedSet(
+            fi for fi in self.arg_to_funcset[next(iarg)])
+
+        if restrict_to_funcset is not None:
+            indices &= restrict_to_funcset
+
+        for arg in iarg:
+            indices &= self.arg_to_funcset[arg]
+
+        return indices
+
+    def update_func_argset(self, func_i, new_argset):
+        """
+        Update a function with a new set of arguments.
+        """
+        new_args = OrderedSet(new_argset)
+        old_args = self.func_to_argset[func_i]
+
+        for deleted_arg in old_args - new_args:
+            self.arg_to_funcset[deleted_arg].remove(func_i)
+        for added_arg in new_args - old_args:
+            self.arg_to_funcset[added_arg].add(func_i)
+
+        self.func_to_argset[func_i].clear()
+        self.func_to_argset[func_i].update(new_args)
+
+
+class Unevaluated:
+
+    def __init__(self, func, args):
+        self.func = func
+        self.args = args
+
+    def __str__(self):
+        return "Uneval<{}>({})".format(
+                self.func, ", ".join(str(a) for a in self.args))
+
+    def as_unevaluated_basic(self):
+        return self.func(*self.args, evaluate=False)
+
+    @property
+    def free_symbols(self):
+        return set().union(*[a.free_symbols for a in self.args])
+
+    __repr__ = __str__
+
+
+def match_common_args(func_class, funcs, opt_subs):
+    """
+    Recognize and extract common subexpressions of function arguments within a
+    set of function calls. For instance, for the following function calls::
+
+        x + z + y
+        sin(x + y)
+
+    this will extract a common subexpression of `x + y`::
+
+        w = x + y
+        w + z
+        sin(w)
+
+    The function we work with is assumed to be associative and commutative.
+
+    Parameters
+    ==========
+
+    func_class: class
+        The function class (e.g. Add, Mul)
+    funcs: list of functions
+        A list of function calls.
+    opt_subs: dict
+        A dictionary of substitutions which this function may update.
+    """
+
+    # Sort to ensure that whole-function subexpressions come before the items
+    # that use them.
+    funcs = sorted(funcs, key=lambda f: len(f.args))
+    arg_tracker = FuncArgTracker(funcs)
+
+    changed = OrderedSet()
+
+    for i in range(len(funcs)):
+        common_arg_candidates_counts = arg_tracker.get_common_arg_candidates(
+                arg_tracker.func_to_argset[i], min_func_i=i + 1)
+
+        # Sort the candidates in order of match size.
+        # This makes us try combining smaller matches first.
+        common_arg_candidates = OrderedSet(sorted(
+                common_arg_candidates_counts.keys(),
+                key=lambda k: (common_arg_candidates_counts[k], k)))
+
+        while common_arg_candidates:
+            j = common_arg_candidates.pop(last=False)
+
+            com_args = arg_tracker.func_to_argset[i].intersection(
+                    arg_tracker.func_to_argset[j])
+
+            if len(com_args) <= 1:
+                # This may happen if a set of common arguments was already
+                # combined in a previous iteration.
+                continue
+
+            # For all sets, replace the common symbols by the function
+            # over them, to allow recursive matches.
+
+            diff_i = arg_tracker.func_to_argset[i].difference(com_args)
+            if diff_i:
+                # com_func needs to be unevaluated to allow for recursive matches.
+                com_func = Unevaluated(
+                        func_class, arg_tracker.get_args_in_value_order(com_args))
+                com_func_number = arg_tracker.get_or_add_value_number(com_func)
+                arg_tracker.update_func_argset(i, diff_i | OrderedSet([com_func_number]))
+                changed.add(i)
+            else:
+                # Treat the whole expression as a CSE.
+                #
+                # The reason this needs to be done is somewhat subtle. Within
+                # tree_cse(), to_eliminate only contains expressions that are
+                # seen more than once. The problem is unevaluated expressions
+                # do not compare equal to the evaluated equivalent. So
+                # tree_cse() won't mark funcs[i] as a CSE if we use an
+                # unevaluated version.
+                com_func_number = arg_tracker.get_or_add_value_number(funcs[i])
+
+            diff_j = arg_tracker.func_to_argset[j].difference(com_args)
+            arg_tracker.update_func_argset(j, diff_j | OrderedSet([com_func_number]))
+            changed.add(j)
+
+            for k in arg_tracker.get_subset_candidates(
+                    com_args, common_arg_candidates):
+                diff_k = arg_tracker.func_to_argset[k].difference(com_args)
+                arg_tracker.update_func_argset(k, diff_k | OrderedSet([com_func_number]))
+                changed.add(k)
+
+        if i in changed:
+            opt_subs[funcs[i]] = Unevaluated(func_class,
+                arg_tracker.get_args_in_value_order(arg_tracker.func_to_argset[i]))
+
+        arg_tracker.stop_arg_tracking(i)
+
+
+def opt_cse(exprs, order='canonical'):
+    """Find optimization opportunities in Adds, Muls, Pows and negative
+    coefficient Muls.
+
+    Parameters
+    ==========
+
+    exprs : list of SymPy expressions
+        The expressions to optimize.
+    order : string, 'none' or 'canonical'
+        The order by which Mul and Add arguments are processed. For large
+        expressions where speed is a concern, use the setting order='none'.
+
+    Returns
+    =======
+
+    opt_subs : dictionary of expression substitutions
+        The expression substitutions which can be useful to optimize CSE.
+
+    Examples
+    ========
+
+    >>> from sympy.simplify.cse_main import opt_cse
+    >>> from sympy.abc import x
+    >>> opt_subs = opt_cse([x**-2])
+    >>> k, v = list(opt_subs.keys())[0], list(opt_subs.values())[0]
+    >>> print((k, v.as_unevaluated_basic()))
+    (x**(-2), 1/(x**2))
+    """
+    opt_subs = {}
+
+    adds = OrderedSet()
+    muls = OrderedSet()
+
+    seen_subexp = set()
+    collapsible_subexp = set()
+
+    def _find_opts(expr):
+
+        if not isinstance(expr, (Basic, Unevaluated)):
+            return
+
+        if expr.is_Atom or expr.is_Order:
+            return
+
+        if iterable(expr):
+            list(map(_find_opts, expr))
+            return
+
+        if expr in seen_subexp:
+            return expr
+        seen_subexp.add(expr)
+
+        list(map(_find_opts, expr.args))
+
+        if not isinstance(expr, MatrixExpr) and expr.could_extract_minus_sign():
+            # XXX -expr does not always work rigorously for some expressions
+            # containing UnevaluatedExpr.
+            # https://github.com/sympy/sympy/issues/24818
+            if isinstance(expr, Add):
+                neg_expr = Add(*(-i for i in expr.args))
+            else:
+                neg_expr = -expr
+
+            if not neg_expr.is_Atom:
+                opt_subs[expr] = Unevaluated(Mul, (S.NegativeOne, neg_expr))
+                seen_subexp.add(neg_expr)
+                expr = neg_expr
+
+        if isinstance(expr, (Mul, MatMul)):
+            if len(expr.args) == 1:
+                collapsible_subexp.add(expr)
+            else:
+                muls.add(expr)
+
+        elif isinstance(expr, (Add, MatAdd)):
+            if len(expr.args) == 1:
+                collapsible_subexp.add(expr)
+            else:
+                adds.add(expr)
+
+        elif isinstance(expr, Inverse):
+            # Do not want to treat `Inverse` as a `MatPow`
+            pass
+
+        elif isinstance(expr, (Pow, MatPow)):
+            base, exp = expr.base, expr.exp
+            if exp.could_extract_minus_sign():
+                opt_subs[expr] = Unevaluated(Pow, (Pow(base, -exp), -1))
+
+    for e in exprs:
+        if isinstance(e, (Basic, Unevaluated)):
+            _find_opts(e)
+
+    # Handle collapsing of multinary operations with single arguments
+    edges = [(s, s.args[0]) for s in collapsible_subexp
+             if s.args[0] in collapsible_subexp]
+    for e in reversed(topological_sort((collapsible_subexp, edges))):
+        opt_subs[e] = opt_subs.get(e.args[0], e.args[0])
+
+    # split muls into commutative
+    commutative_muls = OrderedSet()
+    for m in muls:
+        c, nc = m.args_cnc(cset=False)
+        if c:
+            c_mul = m.func(*c)
+            if nc:
+                if c_mul == 1:
+                    new_obj = m.func(*nc)
+                else:
+                    if isinstance(m, MatMul):
+                        new_obj = m.func(c_mul, *nc, evaluate=False)
+                    else:
+                        new_obj = m.func(c_mul, m.func(*nc), evaluate=False)
+                opt_subs[m] = new_obj
+            if len(c) > 1:
+                commutative_muls.add(c_mul)
+
+    match_common_args(Add, adds, opt_subs)
+    match_common_args(Mul, commutative_muls, opt_subs)
+
+    return opt_subs
+
+
+def tree_cse(exprs, symbols, opt_subs=None, order='canonical', ignore=()):
+    """Perform raw CSE on expression tree, taking opt_subs into account.
+
+    Parameters
+    ==========
+
+    exprs : list of SymPy expressions
+        The expressions to reduce.
+    symbols : infinite iterator yielding unique Symbols
+        The symbols used to label the common subexpressions which are pulled
+        out.
+    opt_subs : dictionary of expression substitutions
+        The expressions to be substituted before any CSE action is performed.
+    order : string, 'none' or 'canonical'
+        The order by which Mul and Add arguments are processed. For large
+        expressions where speed is a concern, use the setting order='none'.
+    ignore : iterable of Symbols
+        Substitutions containing any Symbol from ``ignore`` will be ignored.
+    """
+    if opt_subs is None:
+        opt_subs = {}
+
+    ## Find repeated sub-expressions
+
+    to_eliminate = set()
+
+    seen_subexp = set()
+    excluded_symbols = set()
+
+    def _find_repeated(expr):
+        if not isinstance(expr, (Basic, Unevaluated)):
+            return
+
+        if isinstance(expr, RootOf):
+            return
+
+        if isinstance(expr, Basic) and (
+                expr.is_Atom or
+                expr.is_Order or
+                isinstance(expr, (MatrixSymbol, MatrixElement))):
+            if expr.is_Symbol:
+                excluded_symbols.add(expr.name)
+            return
+
+        if iterable(expr):
+            args = expr
+
+        else:
+            if expr in seen_subexp:
+                for ign in ignore:
+                    if ign in expr.free_symbols:
+                        break
+                else:
+                    to_eliminate.add(expr)
+                    return
+
+            seen_subexp.add(expr)
+
+            if expr in opt_subs:
+                expr = opt_subs[expr]
+
+            args = expr.args
+
+        list(map(_find_repeated, args))
+
+    for e in exprs:
+        if isinstance(e, Basic):
+            _find_repeated(e)
+
+    ## Rebuild tree
+
+    # Remove symbols from the generator that conflict with names in the expressions.
+    symbols = (_ for _ in symbols if _.name not in excluded_symbols)
+
+    replacements = []
+
+    subs = {}
+
+    def _rebuild(expr):
+        if not isinstance(expr, (Basic, Unevaluated)):
+            return expr
+
+        if not expr.args:
+            return expr
+
+        if iterable(expr):
+            new_args = [_rebuild(arg) for arg in expr.args]
+            return expr.func(*new_args)
+
+        if expr in subs:
+            return subs[expr]
+
+        orig_expr = expr
+        if expr in opt_subs:
+            expr = opt_subs[expr]
+
+        # If enabled, parse Muls and Adds arguments by order to ensure
+        # replacement order independent from hashes
+        if order != 'none':
+            if isinstance(expr, (Mul, MatMul)):
+                c, nc = expr.args_cnc()
+                if c == [1]:
+                    args = nc
+                else:
+                    args = list(ordered(c)) + nc
+            elif isinstance(expr, (Add, MatAdd)):
+                args = list(ordered(expr.args))
+            else:
+                args = expr.args
+        else:
+            args = expr.args
+
+        new_args = list(map(_rebuild, args))
+        if isinstance(expr, Unevaluated) or new_args != args:
+            new_expr = expr.func(*new_args)
+        else:
+            new_expr = expr
+
+        if orig_expr in to_eliminate:
+            try:
+                sym = next(symbols)
+            except StopIteration:
+                raise ValueError("Symbols iterator ran out of symbols.")
+
+            if isinstance(orig_expr, MatrixExpr):
+                sym = MatrixSymbol(sym.name, orig_expr.rows,
+                    orig_expr.cols)
+
+            subs[orig_expr] = sym
+            replacements.append((sym, new_expr))
+            return sym
+
+        else:
+            return new_expr
+
+    reduced_exprs = []
+    for e in exprs:
+        if isinstance(e, Basic):
+            reduced_e = _rebuild(e)
+        else:
+            reduced_e = e
+        reduced_exprs.append(reduced_e)
+    return replacements, reduced_exprs
+
+
+def cse(exprs, symbols=None, optimizations=None, postprocess=None,
+        order='canonical', ignore=(), list=True):
+    """ Perform common subexpression elimination on an expression.
+
+    Parameters
+    ==========
+
+    exprs : list of SymPy expressions, or a single SymPy expression
+        The expressions to reduce.
+    symbols : infinite iterator yielding unique Symbols
+        The symbols used to label the common subexpressions which are pulled
+        out. The ``numbered_symbols`` generator is useful. The default is a
+        stream of symbols of the form "x0", "x1", etc. This must be an
+        infinite iterator.
+    optimizations : list of (callable, callable) pairs
+        The (preprocessor, postprocessor) pairs of external optimization
+        functions. Optionally 'basic' can be passed for a set of predefined
+        basic optimizations. Such 'basic' optimizations were used by default
+        in old implementation, however they can be really slow on larger
+        expressions. Now, no pre or post optimizations are made by default.
+    postprocess : a function which accepts the two return values of cse and
+        returns the desired form of output from cse, e.g. if you want the
+        replacements reversed the function might be the following lambda:
+        lambda r, e: return reversed(r), e
+    order : string, 'none' or 'canonical'
+        The order by which Mul and Add arguments are processed. If set to
+        'canonical', arguments will be canonically ordered. If set to 'none',
+        ordering will be faster but dependent on expressions hashes, thus
+        machine dependent and variable. For large expressions where speed is a
+        concern, use the setting order='none'.
+    ignore : iterable of Symbols
+        Substitutions containing any Symbol from ``ignore`` will be ignored.
+    list : bool, (default True)
+        Returns expression in list or else with same type as input (when False).
+
+    Returns
+    =======
+
+    replacements : list of (Symbol, expression) pairs
+        All of the common subexpressions that were replaced. Subexpressions
+        earlier in this list might show up in subexpressions later in this
+        list.
+    reduced_exprs : list of SymPy expressions
+        The reduced expressions with all of the replacements above.
+
+    Examples
+    ========
+
+    >>> from sympy import cse, SparseMatrix
+    >>> from sympy.abc import x, y, z, w
+    >>> cse(((w + x + y + z)*(w + y + z))/(w + x)**3)
+    ([(x0, y + z), (x1, w + x)], [(w + x0)*(x0 + x1)/x1**3])
+
+
+    List of expressions with recursive substitutions:
+
+    >>> m = SparseMatrix([x + y, x + y + z])
+    >>> cse([(x+y)**2, x + y + z, y + z, x + z + y, m])
+    ([(x0, x + y), (x1, x0 + z)], [x0**2, x1, y + z, x1, Matrix([
+    [x0],
+    [x1]])])
+
+    Note: the type and mutability of input matrices is retained.
+
+    >>> isinstance(_[1][-1], SparseMatrix)
+    True
+
+    The user may disallow substitutions containing certain symbols:
+
+    >>> cse([y**2*(x + 1), 3*y**2*(x + 1)], ignore=(y,))
+    ([(x0, x + 1)], [x0*y**2, 3*x0*y**2])
+
+    The default return value for the reduced expression(s) is a list, even if there is only
+    one expression. The `list` flag preserves the type of the input in the output:
+
+    >>> cse(x)
+    ([], [x])
+    >>> cse(x, list=False)
+    ([], x)
+    """
+    if not list:
+        return _cse_homogeneous(exprs,
+            symbols=symbols, optimizations=optimizations,
+            postprocess=postprocess, order=order, ignore=ignore)
+
+    if isinstance(exprs, (int, float)):
+        exprs = sympify(exprs)
+
+    # Handle the case if just one expression was passed.
+    if isinstance(exprs, (Basic, MatrixBase)):
+        exprs = [exprs]
+
+    copy = exprs
+    temp = []
+    for e in exprs:
+        if isinstance(e, (Matrix, ImmutableMatrix)):
+            temp.append(Tuple(*e.flat()))
+        elif isinstance(e, (SparseMatrix, ImmutableSparseMatrix)):
+            temp.append(Tuple(*e.todok().items()))
+        else:
+            temp.append(e)
+    exprs = temp
+    del temp
+
+    if optimizations is None:
+        optimizations = []
+    elif optimizations == 'basic':
+        optimizations = basic_optimizations
+
+    # Preprocess the expressions to give us better optimization opportunities.
+    reduced_exprs = [preprocess_for_cse(e, optimizations) for e in exprs]
+
+    if symbols is None:
+        symbols = numbered_symbols(cls=Symbol)
+    else:
+        # In case we get passed an iterable with an __iter__ method instead of
+        # an actual iterator.
+        symbols = iter(symbols)
+
+    # Find other optimization opportunities.
+    opt_subs = opt_cse(reduced_exprs, order)
+
+    # Main CSE algorithm.
+    replacements, reduced_exprs = tree_cse(reduced_exprs, symbols, opt_subs,
+                                           order, ignore)
+
+    # Postprocess the expressions to return the expressions to canonical form.
+    exprs = copy
+    for i, (sym, subtree) in enumerate(replacements):
+        subtree = postprocess_for_cse(subtree, optimizations)
+        replacements[i] = (sym, subtree)
+    reduced_exprs = [postprocess_for_cse(e, optimizations)
+                     for e in reduced_exprs]
+
+    # Get the matrices back
+    for i, e in enumerate(exprs):
+        if isinstance(e, (Matrix, ImmutableMatrix)):
+            reduced_exprs[i] = Matrix(e.rows, e.cols, reduced_exprs[i])
+            if isinstance(e, ImmutableMatrix):
+                reduced_exprs[i] = reduced_exprs[i].as_immutable()
+        elif isinstance(e, (SparseMatrix, ImmutableSparseMatrix)):
+            m = SparseMatrix(e.rows, e.cols, {})
+            for k, v in reduced_exprs[i]:
+                m[k] = v
+            if isinstance(e, ImmutableSparseMatrix):
+                m = m.as_immutable()
+            reduced_exprs[i] = m
+
+    if postprocess is None:
+        return replacements, reduced_exprs
+
+    return postprocess(replacements, reduced_exprs)
+
+
+def _cse_homogeneous(exprs, **kwargs):
+    """
+    Same as ``cse`` but the ``reduced_exprs`` are returned
+    with the same type as ``exprs`` or a sympified version of the same.
+
+    Parameters
+    ==========
+
+    exprs : an Expr, iterable of Expr or dictionary with Expr values
+        the expressions in which repeated subexpressions will be identified
+    kwargs : additional arguments for the ``cse`` function
+
+    Returns
+    =======
+
+    replacements : list of (Symbol, expression) pairs
+        All of the common subexpressions that were replaced. Subexpressions
+        earlier in this list might show up in subexpressions later in this
+        list.
+    reduced_exprs : list of SymPy expressions
+        The reduced expressions with all of the replacements above.
+
+    Examples
+    ========
+
+    >>> from sympy.simplify.cse_main import cse
+    >>> from sympy import cos, Tuple, Matrix
+    >>> from sympy.abc import x
+    >>> output = lambda x: type(cse(x, list=False)[1])
+    >>> output(1)
+    <class 'sympy.core.numbers.One'>
+    >>> output('cos(x)')
+    <class 'str'>
+    >>> output(cos(x))
+    cos
+    >>> output(Tuple(1, x))
+    <class 'sympy.core.containers.Tuple'>
+    >>> output(Matrix([[1,0], [0,1]]))
+    <class 'sympy.matrices.dense.MutableDenseMatrix'>
+    >>> output([1, x])
+    <class 'list'>
+    >>> output((1, x))
+    <class 'tuple'>
+    >>> output({1, x})
+    <class 'set'>
+    """
+    if isinstance(exprs, str):
+        replacements, reduced_exprs = _cse_homogeneous(
+            sympify(exprs), **kwargs)
+        return replacements, repr(reduced_exprs)
+    if isinstance(exprs, (list, tuple, set)):
+        replacements, reduced_exprs = cse(exprs, **kwargs)
+        return replacements, type(exprs)(reduced_exprs)
+    if isinstance(exprs, dict):
+        keys = list(exprs.keys()) # In order to guarantee the order of the elements.
+        replacements, values = cse([exprs[k] for k in keys], **kwargs)
+        reduced_exprs = dict(zip(keys, values))
+        return replacements, reduced_exprs
+
+    try:
+        replacements, (reduced_exprs,) = cse(exprs, **kwargs)
+    except TypeError: # For example 'mpf' objects
+        return [], exprs
+    else:
+        return replacements, reduced_exprs

rl/Lib/site-packages/sympy/simplify/cse_opts.py

@@ -0,0 +1,52 @@
+""" Optimizations of the expression tree representation for better CSE
+opportunities.
+"""
+from sympy.core import Add, Basic, Mul
+from sympy.core.singleton import S
+from sympy.core.sorting import default_sort_key
+from sympy.core.traversal import preorder_traversal
+
+
+def sub_pre(e):
+    """ Replace y - x with -(x - y) if -1 can be extracted from y - x.
+    """
+    # replacing Add, A, from which -1 can be extracted with -1*-A
+    adds = [a for a in e.atoms(Add) if a.could_extract_minus_sign()]
+    reps = {}
+    ignore = set()
+    for a in adds:
+        na = -a
+        if na.is_Mul:  # e.g. MatExpr
+            ignore.add(a)
+            continue
+        reps[a] = Mul._from_args([S.NegativeOne, na])
+
+    e = e.xreplace(reps)
+
+    # repeat again for persisting Adds but mark these with a leading 1, -1
+    # e.g. y - x -> 1*-1*(x - y)
+    if isinstance(e, Basic):
+        negs = {}
+        for a in sorted(e.atoms(Add), key=default_sort_key):
+            if a in ignore:
+                continue
+            if a in reps:
+                negs[a] = reps[a]
+            elif a.could_extract_minus_sign():
+                negs[a] = Mul._from_args([S.One, S.NegativeOne, -a])
+        e = e.xreplace(negs)
+    return e
+
+
+def sub_post(e):
+    """ Replace 1*-1*x with -x.
+    """
+    replacements = []
+    for node in preorder_traversal(e):
+        if isinstance(node, Mul) and \
+            node.args[0] is S.One and node.args[1] is S.NegativeOne:
+            replacements.append((node, -Mul._from_args(node.args[2:])))
+    for node, replacement in replacements:
+        e = e.xreplace({node: replacement})
+
+    return e

rl/Lib/site-packages/sympy/simplify/epathtools.py

@@ -0,0 +1,356 @@
+"""Tools for manipulation of expressions using paths. """
+
+from sympy.core import Basic
+
+
+class EPath:
+    r"""
+    Manipulate expressions using paths.
+
+    EPath grammar in EBNF notation::
+
+        literal   ::= /[A-Za-z_][A-Za-z_0-9]*/
+        number    ::= /-?\d+/
+        type      ::= literal
+        attribute ::= literal "?"
+        all       ::= "*"
+        slice     ::= "[" number? (":" number? (":" number?)?)? "]"
+        range     ::= all | slice
+        query     ::= (type | attribute) ("|" (type | attribute))*
+        selector  ::= range | query range?
+        path      ::= "/" selector ("/" selector)*
+
+    See the docstring of the epath() function.
+
+    """
+
+    __slots__ = ("_path", "_epath")
+
+    def __new__(cls, path):
+        """Construct new EPath. """
+        if isinstance(path, EPath):
+            return path
+
+        if not path:
+            raise ValueError("empty EPath")
+
+        _path = path
+
+        if path[0] == '/':
+            path = path[1:]
+        else:
+            raise NotImplementedError("non-root EPath")
+
+        epath = []
+
+        for selector in path.split('/'):
+            selector = selector.strip()
+
+            if not selector:
+                raise ValueError("empty selector")
+
+            index = 0
+
+            for c in selector:
+                if c.isalnum() or c in ('_', '|', '?'):
+                    index += 1
+                else:
+                    break
+
+            attrs = []
+            types = []
+
+            if index:
+                elements = selector[:index]
+                selector = selector[index:]
+
+                for element in elements.split('|'):
+                    element = element.strip()
+
+                    if not element:
+                        raise ValueError("empty element")
+
+                    if element.endswith('?'):
+                        attrs.append(element[:-1])
+                    else:
+                        types.append(element)
+
+            span = None
+
+            if selector == '*':
+                pass
+            else:
+                if selector.startswith('['):
+                    try:
+                        i = selector.index(']')
+                    except ValueError:
+                        raise ValueError("expected ']', got EOL")
+
+                    _span, span = selector[1:i], []
+
+                    if ':' not in _span:
+                        span = int(_span)
+                    else:
+                        for elt in _span.split(':', 3):
+                            if not elt:
+                                span.append(None)
+                            else:
+                                span.append(int(elt))
+
+                        span = slice(*span)
+
+                    selector = selector[i + 1:]
+
+                if selector:
+                    raise ValueError("trailing characters in selector")
+
+            epath.append((attrs, types, span))
+
+        obj = object.__new__(cls)
+
+        obj._path = _path
+        obj._epath = epath
+
+        return obj
+
+    def __repr__(self):
+        return "%s(%r)" % (self.__class__.__name__, self._path)
+
+    def _get_ordered_args(self, expr):
+        """Sort ``expr.args`` using printing order. """
+        if expr.is_Add:
+            return expr.as_ordered_terms()
+        elif expr.is_Mul:
+            return expr.as_ordered_factors()
+        else:
+            return expr.args
+
+    def _hasattrs(self, expr, attrs):
+        """Check if ``expr`` has any of ``attrs``. """
+        for attr in attrs:
+            if not hasattr(expr, attr):
+                return False
+
+        return True
+
+    def _hastypes(self, expr, types):
+        """Check if ``expr`` is any of ``types``. """
+        _types = [ cls.__name__ for cls in expr.__class__.mro() ]
+        return bool(set(_types).intersection(types))
+
+    def _has(self, expr, attrs, types):
+        """Apply ``_hasattrs`` and ``_hastypes`` to ``expr``. """
+        if not (attrs or types):
+            return True
+
+        if attrs and self._hasattrs(expr, attrs):
+            return True
+
+        if types and self._hastypes(expr, types):
+            return True
+
+        return False
+
+    def apply(self, expr, func, args=None, kwargs=None):
+        """
+        Modify parts of an expression selected by a path.
+
+        Examples
+        ========
+
+        >>> from sympy.simplify.epathtools import EPath
+        >>> from sympy import sin, cos, E
+        >>> from sympy.abc import x, y, z, t
+
+        >>> path = EPath("/*/[0]/Symbol")
+        >>> expr = [((x, 1), 2), ((3, y), z)]
+
+        >>> path.apply(expr, lambda expr: expr**2)
+        [((x**2, 1), 2), ((3, y**2), z)]
+
+        >>> path = EPath("/*/*/Symbol")
+        >>> expr = t + sin(x + 1) + cos(x + y + E)
+
+        >>> path.apply(expr, lambda expr: 2*expr)
+        t + sin(2*x + 1) + cos(2*x + 2*y + E)
+
+        """
+        def _apply(path, expr, func):
+            if not path:
+                return func(expr)
+            else:
+                selector, path = path[0], path[1:]
+                attrs, types, span = selector
+
+                if isinstance(expr, Basic):
+                    if not expr.is_Atom:
+                        args, basic = self._get_ordered_args(expr), True
+                    else:
+                        return expr
+                elif hasattr(expr, '__iter__'):
+                    args, basic = expr, False
+                else:
+                    return expr
+
+                args = list(args)
+
+                if span is not None:
+                    if isinstance(span, slice):
+                        indices = range(*span.indices(len(args)))
+                    else:
+                        indices = [span]
+                else:
+                    indices = range(len(args))
+
+                for i in indices:
+                    try:
+                        arg = args[i]
+                    except IndexError:
+                        continue
+
+                    if self._has(arg, attrs, types):
+                        args[i] = _apply(path, arg, func)
+
+                if basic:
+                    return expr.func(*args)
+                else:
+                    return expr.__class__(args)
+
+        _args, _kwargs = args or (), kwargs or {}
+        _func = lambda expr: func(expr, *_args, **_kwargs)
+
+        return _apply(self._epath, expr, _func)
+
+    def select(self, expr):
+        """
+        Retrieve parts of an expression selected by a path.
+
+        Examples
+        ========
+
+        >>> from sympy.simplify.epathtools import EPath
+        >>> from sympy import sin, cos, E
+        >>> from sympy.abc import x, y, z, t
+
+        >>> path = EPath("/*/[0]/Symbol")
+        >>> expr = [((x, 1), 2), ((3, y), z)]
+
+        >>> path.select(expr)
+        [x, y]
+
+        >>> path = EPath("/*/*/Symbol")
+        >>> expr = t + sin(x + 1) + cos(x + y + E)
+
+        >>> path.select(expr)
+        [x, x, y]
+
+        """
+        result = []
+
+        def _select(path, expr):
+            if not path:
+                result.append(expr)
+            else:
+                selector, path = path[0], path[1:]
+                attrs, types, span = selector
+
+                if isinstance(expr, Basic):
+                    args = self._get_ordered_args(expr)
+                elif hasattr(expr, '__iter__'):
+                    args = expr
+                else:
+                    return
+
+                if span is not None:
+                    if isinstance(span, slice):
+                        args = args[span]
+                    else:
+                        try:
+                            args = [args[span]]
+                        except IndexError:
+                            return
+
+                for arg in args:
+                    if self._has(arg, attrs, types):
+                        _select(path, arg)
+
+        _select(self._epath, expr)
+        return result
+
+
+def epath(path, expr=None, func=None, args=None, kwargs=None):
+    r"""
+    Manipulate parts of an expression selected by a path.
+
+    Explanation
+    ===========
+
+    This function allows to manipulate large nested expressions in single
+    line of code, utilizing techniques to those applied in XML processing
+    standards (e.g. XPath).
+
+    If ``func`` is ``None``, :func:`epath` retrieves elements selected by
+    the ``path``. Otherwise it applies ``func`` to each matching element.
+
+    Note that it is more efficient to create an EPath object and use the select
+    and apply methods of that object, since this will compile the path string
+    only once.  This function should only be used as a convenient shortcut for
+    interactive use.
+
+    This is the supported syntax:
+
+    * select all: ``/*``
+          Equivalent of ``for arg in args:``.
+    * select slice: ``/[0]`` or ``/[1:5]`` or ``/[1:5:2]``
+          Supports standard Python's slice syntax.
+    * select by type: ``/list`` or ``/list|tuple``
+          Emulates ``isinstance()``.
+    * select by attribute: ``/__iter__?``
+          Emulates ``hasattr()``.
+
+    Parameters
+    ==========
+
+    path : str | EPath
+        A path as a string or a compiled EPath.
+    expr : Basic | iterable
+        An expression or a container of expressions.
+    func : callable (optional)
+        A callable that will be applied to matching parts.
+    args : tuple (optional)
+        Additional positional arguments to ``func``.
+    kwargs : dict (optional)
+        Additional keyword arguments to ``func``.
+
+    Examples
+    ========
+
+    >>> from sympy.simplify.epathtools import epath
+    >>> from sympy import sin, cos, E
+    >>> from sympy.abc import x, y, z, t
+
+    >>> path = "/*/[0]/Symbol"
+    >>> expr = [((x, 1), 2), ((3, y), z)]
+
+    >>> epath(path, expr)
+    [x, y]
+    >>> epath(path, expr, lambda expr: expr**2)
+    [((x**2, 1), 2), ((3, y**2), z)]
+
+    >>> path = "/*/*/Symbol"
+    >>> expr = t + sin(x + 1) + cos(x + y + E)
+
+    >>> epath(path, expr)
+    [x, x, y]
+    >>> epath(path, expr, lambda expr: 2*expr)
+    t + sin(2*x + 1) + cos(2*x + 2*y + E)
+
+    """
+    _epath = EPath(path)
+
+    if expr is None:
+        return _epath
+    if func is None:
+        return _epath.select(expr)
+    else:
+        return _epath.apply(expr, func, args, kwargs)

rl/Lib/site-packages/sympy/simplify/gammasimp.py

@@ -0,0 +1,497 @@
+from sympy.core import Function, S, Mul, Pow, Add
+from sympy.core.sorting import ordered, default_sort_key
+from sympy.core.function import expand_func
+from sympy.core.symbol import Dummy
+from sympy.functions import gamma, sqrt, sin
+from sympy.polys import factor, cancel
+from sympy.utilities.iterables import sift, uniq
+
+
+def gammasimp(expr):
+    r"""
+    Simplify expressions with gamma functions.
+
+    Explanation
+    ===========
+
+    This function takes as input an expression containing gamma
+    functions or functions that can be rewritten in terms of gamma
+    functions and tries to minimize the number of those functions and
+    reduce the size of their arguments.
+
+    The algorithm works by rewriting all gamma functions as expressions
+    involving rising factorials (Pochhammer symbols) and applies
+    recurrence relations and other transformations applicable to rising
+    factorials, to reduce their arguments, possibly letting the resulting
+    rising factorial to cancel. Rising factorials with the second argument
+    being an integer are expanded into polynomial forms and finally all
+    other rising factorial are rewritten in terms of gamma functions.
+
+    Then the following two steps are performed.
+
+    1. Reduce the number of gammas by applying the reflection theorem
+       gamma(x)*gamma(1-x) == pi/sin(pi*x).
+    2. Reduce the number of gammas by applying the multiplication theorem
+       gamma(x)*gamma(x+1/n)*...*gamma(x+(n-1)/n) == C*gamma(n*x).
+
+    It then reduces the number of prefactors by absorbing them into gammas
+    where possible and expands gammas with rational argument.
+
+    All transformation rules can be found (or were derived from) here:
+
+    .. [1] https://functions.wolfram.com/GammaBetaErf/Pochhammer/17/01/02/
+    .. [2] https://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/0005/
+
+    Examples
+    ========
+
+    >>> from sympy.simplify import gammasimp
+    >>> from sympy import gamma, Symbol
+    >>> from sympy.abc import x
+    >>> n = Symbol('n', integer = True)
+
+    >>> gammasimp(gamma(x)/gamma(x - 3))
+    (x - 3)*(x - 2)*(x - 1)
+    >>> gammasimp(gamma(n + 3))
+    gamma(n + 3)
+
+    """
+
+    expr = expr.rewrite(gamma)
+
+    # compute_ST will be looking for Functions and we don't want
+    # it looking for non-gamma functions: issue 22606
+    # so we mask free, non-gamma functions
+    f = expr.atoms(Function)
+    # take out gammas
+    gammas = {i for i in f if isinstance(i, gamma)}
+    if not gammas:
+        return expr  # avoid side effects like factoring
+    f -= gammas
+    # keep only those without bound symbols
+    f = f & expr.as_dummy().atoms(Function)
+    if f:
+        dum, fun, simp = zip(*[
+            (Dummy(), fi, fi.func(*[
+                _gammasimp(a, as_comb=False) for a in fi.args]))
+            for fi in ordered(f)])
+        d = expr.xreplace(dict(zip(fun, dum)))
+        return _gammasimp(d, as_comb=False).xreplace(dict(zip(dum, simp)))
+
+    return _gammasimp(expr, as_comb=False)
+
+
+def _gammasimp(expr, as_comb):
+    """
+    Helper function for gammasimp and combsimp.
+
+    Explanation
+    ===========
+
+    Simplifies expressions written in terms of gamma function. If
+    as_comb is True, it tries to preserve integer arguments. See
+    docstring of gammasimp for more information. This was part of
+    combsimp() in combsimp.py.
+    """
+    expr = expr.replace(gamma,
+        lambda n: _rf(1, (n - 1).expand()))
+
+    if as_comb:
+        expr = expr.replace(_rf,
+            lambda a, b: gamma(b + 1))
+    else:
+        expr = expr.replace(_rf,
+            lambda a, b: gamma(a + b)/gamma(a))
+
+    def rule_gamma(expr, level=0):
+        """ Simplify products of gamma functions further. """
+
+        if expr.is_Atom:
+            return expr
+
+        def gamma_rat(x):
+            # helper to simplify ratios of gammas
+            was = x.count(gamma)
+            xx = x.replace(gamma, lambda n: _rf(1, (n - 1).expand()
+                ).replace(_rf, lambda a, b: gamma(a + b)/gamma(a)))
+            if xx.count(gamma) < was:
+                x = xx
+            return x
+
+        def gamma_factor(x):
+            # return True if there is a gamma factor in shallow args
+            if isinstance(x, gamma):
+                return True
+            if x.is_Add or x.is_Mul:
+                return any(gamma_factor(xi) for xi in x.args)
+            if x.is_Pow and (x.exp.is_integer or x.base.is_positive):
+                return gamma_factor(x.base)
+            return False
+
+        # recursion step
+        if level == 0:
+            expr = expr.func(*[rule_gamma(x, level + 1) for x in expr.args])
+            level += 1
+
+        if not expr.is_Mul:
+            return expr
+
+        # non-commutative step
+        if level == 1:
+            args, nc = expr.args_cnc()
+            if not args:
+                return expr
+            if nc:
+                return rule_gamma(Mul._from_args(args), level + 1)*Mul._from_args(nc)
+            level += 1
+
+        # pure gamma handling, not factor absorption
+        if level == 2:
+            T, F = sift(expr.args, gamma_factor, binary=True)
+            gamma_ind = Mul(*F)
+            d = Mul(*T)
+
+            nd, dd = d.as_numer_denom()
+            for ipass in range(2):
+                args = list(ordered(Mul.make_args(nd)))
+                for i, ni in enumerate(args):
+                    if ni.is_Add:
+                        ni, dd = Add(*[
+                            rule_gamma(gamma_rat(a/dd), level + 1) for a in ni.args]
+                            ).as_numer_denom()
+                        args[i] = ni
+                        if not dd.has(gamma):
+                            break
+                nd = Mul(*args)
+                if ipass ==  0 and not gamma_factor(nd):
+                    break
+                nd, dd = dd, nd  # now process in reversed order
+            expr = gamma_ind*nd/dd
+            if not (expr.is_Mul and (gamma_factor(dd) or gamma_factor(nd))):
+                return expr
+            level += 1
+
+        # iteration until constant
+        if level == 3:
+            while True:
+                was = expr
+                expr = rule_gamma(expr, 4)
+                if expr == was:
+                    return expr
+
+        numer_gammas = []
+        denom_gammas = []
+        numer_others = []
+        denom_others = []
+        def explicate(p):
+            if p is S.One:
+                return None, []
+            b, e = p.as_base_exp()
+            if e.is_Integer:
+                if isinstance(b, gamma):
+                    return True, [b.args[0]]*e
+                else:
+                    return False, [b]*e
+            else:
+                return False, [p]
+
+        newargs = list(ordered(expr.args))
+        while newargs:
+            n, d = newargs.pop().as_numer_denom()
+            isg, l = explicate(n)
+            if isg:
+                numer_gammas.extend(l)
+            elif isg is False:
+                numer_others.extend(l)
+            isg, l = explicate(d)
+            if isg:
+                denom_gammas.extend(l)
+            elif isg is False:
+                denom_others.extend(l)
+
+        # =========== level 2 work: pure gamma manipulation =========
+
+        if not as_comb:
+            # Try to reduce the number of gamma factors by applying the
+            # reflection formula gamma(x)*gamma(1-x) = pi/sin(pi*x)
+            for gammas, numer, denom in [(
+                numer_gammas, numer_others, denom_others),
+                    (denom_gammas, denom_others, numer_others)]:
+                new = []
+                while gammas:
+                    g1 = gammas.pop()
+                    if g1.is_integer:
+                        new.append(g1)
+                        continue
+                    for i, g2 in enumerate(gammas):
+                        n = g1 + g2 - 1
+                        if not n.is_Integer:
+                            continue
+                        numer.append(S.Pi)
+                        denom.append(sin(S.Pi*g1))
+                        gammas.pop(i)
+                        if n > 0:
+                            for k in range(n):
+                                numer.append(1 - g1 + k)
+                        elif n < 0:
+                            for k in range(-n):
+                                denom.append(-g1 - k)
+                        break
+                    else:
+                        new.append(g1)
+                # /!\ updating IN PLACE
+                gammas[:] = new
+
+            # Try to reduce the number of gammas by using the duplication
+            # theorem to cancel an upper and lower: gamma(2*s)/gamma(s) =
+            # 2**(2*s + 1)/(4*sqrt(pi))*gamma(s + 1/2). Although this could
+            # be done with higher argument ratios like gamma(3*x)/gamma(x),
+            # this would not reduce the number of gammas as in this case.
+            for ng, dg, no, do in [(numer_gammas, denom_gammas, numer_others,
+                                    denom_others),
+                                   (denom_gammas, numer_gammas, denom_others,
+                                    numer_others)]:
+
+                while True:
+                    for x in ng:
+                        for y in dg:
+                            n = x - 2*y
+                            if n.is_Integer:
+                                break
+                        else:
+                            continue
+                        break
+                    else:
+                        break
+                    ng.remove(x)
+                    dg.remove(y)
+                    if n > 0:
+                        for k in range(n):
+                            no.append(2*y + k)
+                    elif n < 0:
+                        for k in range(-n):
+                            do.append(2*y - 1 - k)
+                    ng.append(y + S.Half)
+                    no.append(2**(2*y - 1))
+                    do.append(sqrt(S.Pi))
+
+            # Try to reduce the number of gamma factors by applying the
+            # multiplication theorem (used when n gammas with args differing
+            # by 1/n mod 1 are encountered).
+            #
+            # run of 2 with args differing by 1/2
+            #
+            # >>> gammasimp(gamma(x)*gamma(x+S.Half))
+            # 2*sqrt(2)*2**(-2*x - 1/2)*sqrt(pi)*gamma(2*x)
+            #
+            # run of 3 args differing by 1/3 (mod 1)
+            #
+            # >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(2)/3))
+            # 6*3**(-3*x - 1/2)*pi*gamma(3*x)
+            # >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(5)/3))
+            # 2*3**(-3*x - 1/2)*pi*(3*x + 2)*gamma(3*x)
+            #
+            def _run(coeffs):
+                # find runs in coeffs such that the difference in terms (mod 1)
+                # of t1, t2, ..., tn is 1/n
+                u = list(uniq(coeffs))
+                for i in range(len(u)):
+                    dj = ([((u[j] - u[i]) % 1, j) for j in range(i + 1, len(u))])
+                    for one, j in dj:
+                        if one.p == 1 and one.q != 1:
+                            n = one.q
+                            got = [i]
+                            get = list(range(1, n))
+                            for d, j in dj:
+                                m = n*d
+                                if m.is_Integer and m in get:
+                                    get.remove(m)
+                                    got.append(j)
+                                    if not get:
+                                        break
+                            else:
+                                continue
+                            for i, j in enumerate(got):
+                                c = u[j]
+                                coeffs.remove(c)
+                                got[i] = c
+                            return one.q, got[0], got[1:]
+
+            def _mult_thm(gammas, numer, denom):
+                # pull off and analyze the leading coefficient from each gamma arg
+                # looking for runs in those Rationals
+
+                # expr -> coeff + resid -> rats[resid] = coeff
+                rats = {}
+                for g in gammas:
+                    c, resid = g.as_coeff_Add()
+                    rats.setdefault(resid, []).append(c)
+
+                # look for runs in Rationals for each resid
+                keys = sorted(rats, key=default_sort_key)
+                for resid in keys:
+                    coeffs = sorted(rats[resid])
+                    new = []
+                    while True:
+                        run = _run(coeffs)
+                        if run is None:
+                            break
+
+                        # process the sequence that was found:
+                        # 1) convert all the gamma functions to have the right
+                        #    argument (could be off by an integer)
+                        # 2) append the factors corresponding to the theorem
+                        # 3) append the new gamma function
+
+                        n, ui, other = run
+
+                        # (1)
+                        for u in other:
+                            con = resid + u - 1
+                            for k in range(int(u - ui)):
+                                numer.append(con - k)
+
+                        con = n*(resid + ui)  # for (2) and (3)
+
+                        # (2)
+                        numer.append((2*S.Pi)**(S(n - 1)/2)*
+                                     n**(S.Half - con))
+                        # (3)
+                        new.append(con)
+
+                    # restore resid to coeffs
+                    rats[resid] = [resid + c for c in coeffs] + new
+
+                # rebuild the gamma arguments
+                g = []
+                for resid in keys:
+                    g += rats[resid]
+                # /!\ updating IN PLACE
+                gammas[:] = g
+
+            for l, numer, denom in [(numer_gammas, numer_others, denom_others),
+                                    (denom_gammas, denom_others, numer_others)]:
+                _mult_thm(l, numer, denom)
+
+        # =========== level >= 2 work: factor absorption =========
+
+        if level >= 2:
+            # Try to absorb factors into the gammas: x*gamma(x) -> gamma(x + 1)
+            # and gamma(x)/(x - 1) -> gamma(x - 1)
+            # This code (in particular repeated calls to find_fuzzy) can be very
+            # slow.
+            def find_fuzzy(l, x):
+                if not l:
+                    return
+                S1, T1 = compute_ST(x)
+                for y in l:
+                    S2, T2 = inv[y]
+                    if T1 != T2 or (not S1.intersection(S2) and
+                                    (S1 != set() or S2 != set())):
+                        continue
+                    # XXX we want some simplification (e.g. cancel or
+                    # simplify) but no matter what it's slow.
+                    a = len(cancel(x/y).free_symbols)
+                    b = len(x.free_symbols)
+                    c = len(y.free_symbols)
+                    # TODO is there a better heuristic?
+                    if a == 0 and (b > 0 or c > 0):
+                        return y
+
+            # We thus try to avoid expensive calls by building the following
+            # "invariants": For every factor or gamma function argument
+            #   - the set of free symbols S
+            #   - the set of functional components T
+            # We will only try to absorb if T1==T2 and (S1 intersect S2 != emptyset
+            # or S1 == S2 == emptyset)
+            inv = {}
+
+            def compute_ST(expr):
+                if expr in inv:
+                    return inv[expr]
+                return (expr.free_symbols, expr.atoms(Function).union(
+                        {e.exp for e in expr.atoms(Pow)}))
+
+            def update_ST(expr):
+                inv[expr] = compute_ST(expr)
+            for expr in numer_gammas + denom_gammas + numer_others + denom_others:
+                update_ST(expr)
+
+            for gammas, numer, denom in [(
+                numer_gammas, numer_others, denom_others),
+                    (denom_gammas, denom_others, numer_others)]:
+                new = []
+                while gammas:
+                    g = gammas.pop()
+                    cont = True
+                    while cont:
+                        cont = False
+                        y = find_fuzzy(numer, g)
+                        if y is not None:
+                            numer.remove(y)
+                            if y != g:
+                                numer.append(y/g)
+                                update_ST(y/g)
+                            g += 1
+                            cont = True
+                        y = find_fuzzy(denom, g - 1)
+                        if y is not None:
+                            denom.remove(y)
+                            if y != g - 1:
+                                numer.append((g - 1)/y)
+                                update_ST((g - 1)/y)
+                            g -= 1
+                            cont = True
+                    new.append(g)
+                # /!\ updating IN PLACE
+                gammas[:] = new
+
+        # =========== rebuild expr ==================================
+
+        return Mul(*[gamma(g) for g in numer_gammas]) \
+            / Mul(*[gamma(g) for g in denom_gammas]) \
+            * Mul(*numer_others) / Mul(*denom_others)
+
+    was = factor(expr)
+    # (for some reason we cannot use Basic.replace in this case)
+    expr = rule_gamma(was)
+    if expr != was:
+        expr = factor(expr)
+
+    expr = expr.replace(gamma,
+        lambda n: expand_func(gamma(n)) if n.is_Rational else gamma(n))
+
+    return expr
+
+
+class _rf(Function):
+    @classmethod
+    def eval(cls, a, b):
+        if b.is_Integer:
+            if not b:
+                return S.One
+
+            n = int(b)
+
+            if n > 0:
+                return Mul(*[a + i for i in range(n)])
+            elif n < 0:
+                return 1/Mul(*[a - i for i in range(1, -n + 1)])
+        else:
+            if b.is_Add:
+                c, _b = b.as_coeff_Add()
+
+                if c.is_Integer:
+                    if c > 0:
+                        return _rf(a, _b)*_rf(a + _b, c)
+                    elif c < 0:
+                        return _rf(a, _b)/_rf(a + _b + c, -c)
+
+            if a.is_Add:
+                c, _a = a.as_coeff_Add()
+
+                if c.is_Integer:
+                    if c > 0:
+                        return _rf(_a, b)*_rf(_a + b, c)/_rf(_a, c)
+                    elif c < 0:
+                        return _rf(_a, b)*_rf(_a + c, -c)/_rf(_a + b + c, -c)

rl/Lib/site-packages/sympy/simplify/hyperexpand_doc.py

@@ -0,0 +1,18 @@
+""" This module cooks up a docstring when imported. Its only purpose is to
+    be displayed in the sphinx documentation. """
+
+from sympy.core.relational import Eq
+from sympy.functions.special.hyper import hyper
+from sympy.printing.latex import latex
+from sympy.simplify.hyperexpand import FormulaCollection
+
+c = FormulaCollection()
+
+doc = ""
+
+for f in c.formulae:
+    obj = Eq(hyper(f.func.ap, f.func.bq, f.z),
+             f.closed_form.rewrite('nonrepsmall'))
+    doc += ".. math::\n  %s\n" % latex(obj)
+
+__doc__ = doc

rl/Lib/site-packages/sympy/simplify/powsimp.py

@@ -0,0 +1,714 @@
+from collections import defaultdict
+from functools import reduce
+from math import prod
+
+from sympy.core.function import expand_log, count_ops, _coeff_isneg
+from sympy.core import sympify, Basic, Dummy, S, Add, Mul, Pow, expand_mul, factor_terms
+from sympy.core.sorting import ordered, default_sort_key
+from sympy.core.numbers import Integer, Rational
+from sympy.core.mul import _keep_coeff
+from sympy.core.rules import Transform
+from sympy.functions import exp_polar, exp, log, root, polarify, unpolarify
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.polys import lcm, gcd
+from sympy.ntheory.factor_ import multiplicity
+
+
+
+def powsimp(expr, deep=False, combine='all', force=False, measure=count_ops):
+    """
+    Reduce expression by combining powers with similar bases and exponents.
+
+    Explanation
+    ===========
+
+    If ``deep`` is ``True`` then powsimp() will also simplify arguments of
+    functions. By default ``deep`` is set to ``False``.
+
+    If ``force`` is ``True`` then bases will be combined without checking for
+    assumptions, e.g. sqrt(x)*sqrt(y) -> sqrt(x*y) which is not true
+    if x and y are both negative.
+
+    You can make powsimp() only combine bases or only combine exponents by
+    changing combine='base' or combine='exp'.  By default, combine='all',
+    which does both.  combine='base' will only combine::
+
+         a   a          a                          2x      x
+        x * y  =>  (x*y)   as well as things like 2   =>  4
+
+    and combine='exp' will only combine
+    ::
+
+         a   b      (a + b)
+        x * x  =>  x
+
+    combine='exp' will strictly only combine exponents in the way that used
+    to be automatic.  Also use deep=True if you need the old behavior.
+
+    When combine='all', 'exp' is evaluated first.  Consider the first
+    example below for when there could be an ambiguity relating to this.
+    This is done so things like the second example can be completely
+    combined.  If you want 'base' combined first, do something like
+    powsimp(powsimp(expr, combine='base'), combine='exp').
+
+    Examples
+    ========
+
+    >>> from sympy import powsimp, exp, log, symbols
+    >>> from sympy.abc import x, y, z, n
+    >>> powsimp(x**y*x**z*y**z, combine='all')
+    x**(y + z)*y**z
+    >>> powsimp(x**y*x**z*y**z, combine='exp')
+    x**(y + z)*y**z
+    >>> powsimp(x**y*x**z*y**z, combine='base', force=True)
+    x**y*(x*y)**z
+
+    >>> powsimp(x**z*x**y*n**z*n**y, combine='all', force=True)
+    (n*x)**(y + z)
+    >>> powsimp(x**z*x**y*n**z*n**y, combine='exp')
+    n**(y + z)*x**(y + z)
+    >>> powsimp(x**z*x**y*n**z*n**y, combine='base', force=True)
+    (n*x)**y*(n*x)**z
+
+    >>> x, y = symbols('x y', positive=True)
+    >>> powsimp(log(exp(x)*exp(y)))
+    log(exp(x)*exp(y))
+    >>> powsimp(log(exp(x)*exp(y)), deep=True)
+    x + y
+
+    Radicals with Mul bases will be combined if combine='exp'
+
+    >>> from sympy import sqrt
+    >>> x, y = symbols('x y')
+
+    Two radicals are automatically joined through Mul:
+
+    >>> a=sqrt(x*sqrt(y))
+    >>> a*a**3 == a**4
+    True
+
+    But if an integer power of that radical has been
+    autoexpanded then Mul does not join the resulting factors:
+
+    >>> a**4 # auto expands to a Mul, no longer a Pow
+    x**2*y
+    >>> _*a # so Mul doesn't combine them
+    x**2*y*sqrt(x*sqrt(y))
+    >>> powsimp(_) # but powsimp will
+    (x*sqrt(y))**(5/2)
+    >>> powsimp(x*y*a) # but won't when doing so would violate assumptions
+    x*y*sqrt(x*sqrt(y))
+
+    """
+    def recurse(arg, **kwargs):
+        _deep = kwargs.get('deep', deep)
+        _combine = kwargs.get('combine', combine)
+        _force = kwargs.get('force', force)
+        _measure = kwargs.get('measure', measure)
+        return powsimp(arg, _deep, _combine, _force, _measure)
+
+    expr = sympify(expr)
+
+    if (not isinstance(expr, Basic) or isinstance(expr, MatrixSymbol) or (
+            expr.is_Atom or expr in (exp_polar(0), exp_polar(1)))):
+        return expr
+
+    if deep or expr.is_Add or expr.is_Mul and _y not in expr.args:
+        expr = expr.func(*[recurse(w) for w in expr.args])
+
+    if expr.is_Pow:
+        return recurse(expr*_y, deep=False)/_y
+
+    if not expr.is_Mul:
+        return expr
+
+    # handle the Mul
+    if combine in ('exp', 'all'):
+        # Collect base/exp data, while maintaining order in the
+        # non-commutative parts of the product
+        c_powers = defaultdict(list)
+        nc_part = []
+        newexpr = []
+        coeff = S.One
+        for term in expr.args:
+            if term.is_Rational:
+                coeff *= term
+                continue
+            if term.is_Pow:
+                term = _denest_pow(term)
+            if term.is_commutative:
+                b, e = term.as_base_exp()
+                if deep:
+                    b, e = [recurse(i) for i in [b, e]]
+                if b.is_Pow or isinstance(b, exp):
+                    # don't let smthg like sqrt(x**a) split into x**a, 1/2
+                    # or else it will be joined as x**(a/2) later
+                    b, e = b**e, S.One
+                c_powers[b].append(e)
+            else:
+                # This is the logic that combines exponents for equal,
+                # but non-commutative bases: A**x*A**y == A**(x+y).
+                if nc_part:
+                    b1, e1 = nc_part[-1].as_base_exp()
+                    b2, e2 = term.as_base_exp()
+                    if (b1 == b2 and
+                            e1.is_commutative and e2.is_commutative):
+                        nc_part[-1] = Pow(b1, Add(e1, e2))
+                        continue
+                nc_part.append(term)
+
+        # add up exponents of common bases
+        for b, e in ordered(iter(c_powers.items())):
+            # allow 2**x/4 -> 2**(x - 2); don't do this when b and e are
+            # Numbers since autoevaluation will undo it, e.g.
+            # 2**(1/3)/4 -> 2**(1/3 - 2) -> 2**(1/3)/4
+            if (b and b.is_Rational and not all(ei.is_Number for ei in e) and \
+                    coeff is not S.One and
+                    b not in (S.One, S.NegativeOne)):
+                m = multiplicity(abs(b), abs(coeff))
+                if m:
+                    e.append(m)
+                    coeff /= b**m
+            c_powers[b] = Add(*e)
+        if coeff is not S.One:
+            if coeff in c_powers:
+                c_powers[coeff] += S.One
+            else:
+                c_powers[coeff] = S.One
+
+        # convert to plain dictionary
+        c_powers = dict(c_powers)
+
+        # check for base and inverted base pairs
+        be = list(c_powers.items())
+        skip = set()  # skip if we already saw them
+        for b, e in be:
+            if b in skip:
+                continue
+            bpos = b.is_positive or b.is_polar
+            if bpos:
+                binv = 1/b
+                if b != binv and binv in c_powers:
+                    if b.as_numer_denom()[0] is S.One:
+                        c_powers.pop(b)
+                        c_powers[binv] -= e
+                    else:
+                        skip.add(binv)
+                        e = c_powers.pop(binv)
+                        c_powers[b] -= e
+
+        # check for base and negated base pairs
+        be = list(c_powers.items())
+        _n = S.NegativeOne
+        for b, e in be:
+            if (b.is_Symbol or b.is_Add) and -b in c_powers and b in c_powers:
+                if (b.is_positive is not None or e.is_integer):
+                    if e.is_integer or b.is_negative:
+                        c_powers[-b] += c_powers.pop(b)
+                    else:  # (-b).is_positive so use its e
+                        e = c_powers.pop(-b)
+                        c_powers[b] += e
+                    if _n in c_powers:
+                        c_powers[_n] += e
+                    else:
+                        c_powers[_n] = e
+
+        # filter c_powers and convert to a list
+        c_powers = [(b, e) for b, e in c_powers.items() if e]
+
+        # ==============================================================
+        # check for Mul bases of Rational powers that can be combined with
+        # separated bases, e.g. x*sqrt(x*y)*sqrt(x*sqrt(x*y)) ->
+        # (x*sqrt(x*y))**(3/2)
+        # ---------------- helper functions
+
+        def ratq(x):
+            '''Return Rational part of x's exponent as it appears in the bkey.
+            '''
+            return bkey(x)[0][1]
+
+        def bkey(b, e=None):
+            '''Return (b**s, c.q), c.p where e -> c*s. If e is not given then
+            it will be taken by using as_base_exp() on the input b.
+            e.g.
+                x**3/2 -> (x, 2), 3
+                x**y -> (x**y, 1), 1
+                x**(2*y/3) -> (x**y, 3), 2
+                exp(x/2) -> (exp(a), 2), 1
+
+            '''
+            if e is not None:  # coming from c_powers or from below
+                if e.is_Integer:
+                    return (b, S.One), e
+                elif e.is_Rational:
+                    return (b, Integer(e.q)), Integer(e.p)
+                else:
+                    c, m = e.as_coeff_Mul(rational=True)
+                    if c is not S.One:
+                        if m.is_integer:
+                            return (b, Integer(c.q)), m*Integer(c.p)
+                        return (b**m, Integer(c.q)), Integer(c.p)
+                    else:
+                        return (b**e, S.One), S.One
+            else:
+                return bkey(*b.as_base_exp())
+
+        def update(b):
+            '''Decide what to do with base, b. If its exponent is now an
+            integer multiple of the Rational denominator, then remove it
+            and put the factors of its base in the common_b dictionary or
+            update the existing bases if necessary. If it has been zeroed
+            out, simply remove the base.
+            '''
+            newe, r = divmod(common_b[b], b[1])
+            if not r:
+                common_b.pop(b)
+                if newe:
+                    for m in Mul.make_args(b[0]**newe):
+                        b, e = bkey(m)
+                        if b not in common_b:
+                            common_b[b] = 0
+                        common_b[b] += e
+                        if b[1] != 1:
+                            bases.append(b)
+        # ---------------- end of helper functions
+
+        # assemble a dictionary of the factors having a Rational power
+        common_b = {}
+        done = []
+        bases = []
+        for b, e in c_powers:
+            b, e = bkey(b, e)
+            if b in common_b:
+                common_b[b] = common_b[b] + e
+            else:
+                common_b[b] = e
+            if b[1] != 1 and b[0].is_Mul:
+                bases.append(b)
+        bases.sort(key=default_sort_key)  # this makes tie-breaking canonical
+        bases.sort(key=measure, reverse=True)  # handle longest first
+        for base in bases:
+            if base not in common_b:  # it may have been removed already
+                continue
+            b, exponent = base
+            last = False  # True when no factor of base is a radical
+            qlcm = 1  # the lcm of the radical denominators
+            while True:
+                bstart = b
+                qstart = qlcm
+
+                bb = []  # list of factors
+                ee = []  # (factor's expo. and it's current value in common_b)
+                for bi in Mul.make_args(b):
+                    bib, bie = bkey(bi)
+                    if bib not in common_b or common_b[bib] < bie:
+                        ee = bb = []  # failed
+                        break
+                    ee.append([bie, common_b[bib]])
+                    bb.append(bib)
+                if ee:
+                    # find the number of integral extractions possible
+                    # e.g. [(1, 2), (2, 2)] -> min(2/1, 2/2) -> 1
+                    min1 = ee[0][1]//ee[0][0]
+                    for i in range(1, len(ee)):
+                        rat = ee[i][1]//ee[i][0]
+                        if rat < 1:
+                            break
+                        min1 = min(min1, rat)
+                    else:
+                        # update base factor counts
+                        # e.g. if ee = [(2, 5), (3, 6)] then min1 = 2
+                        # and the new base counts will be 5-2*2 and 6-2*3
+                        for i in range(len(bb)):
+                            common_b[bb[i]] -= min1*ee[i][0]
+                            update(bb[i])
+                        # update the count of the base
+                        # e.g. x**2*y*sqrt(x*sqrt(y)) the count of x*sqrt(y)
+                        # will increase by 4 to give bkey (x*sqrt(y), 2, 5)
+                        common_b[base] += min1*qstart*exponent
+                if (last  # no more radicals in base
+                    or len(common_b) == 1  # nothing left to join with
+                    or all(k[1] == 1 for k in common_b)  # no rad's in common_b
+                        ):
+                    break
+                # see what we can exponentiate base by to remove any radicals
+                # so we know what to search for
+                # e.g. if base were x**(1/2)*y**(1/3) then we should
+                # exponentiate by 6 and look for powers of x and y in the ratio
+                # of 2 to 3
+                qlcm = lcm([ratq(bi) for bi in Mul.make_args(bstart)])
+                if qlcm == 1:
+                    break  # we are done
+                b = bstart**qlcm
+                qlcm *= qstart
+                if all(ratq(bi) == 1 for bi in Mul.make_args(b)):
+                    last = True  # we are going to be done after this next pass
+            # this base no longer can find anything to join with and
+            # since it was longer than any other we are done with it
+            b, q = base
+            done.append((b, common_b.pop(base)*Rational(1, q)))
+
+        # update c_powers and get ready to continue with powsimp
+        c_powers = done
+        # there may be terms still in common_b that were bases that were
+        # identified as needing processing, so remove those, too
+        for (b, q), e in common_b.items():
+            if (b.is_Pow or isinstance(b, exp)) and \
+                    q is not S.One and not b.exp.is_Rational:
+                b, be = b.as_base_exp()
+                b = b**(be/q)
+            else:
+                b = root(b, q)
+            c_powers.append((b, e))
+        check = len(c_powers)
+        c_powers = dict(c_powers)
+        assert len(c_powers) == check  # there should have been no duplicates
+        # ==============================================================
+
+        # rebuild the expression
+        newexpr = expr.func(*(newexpr + [Pow(b, e) for b, e in c_powers.items()]))
+        if combine == 'exp':
+            return expr.func(newexpr, expr.func(*nc_part))
+        else:
+            return recurse(expr.func(*nc_part), combine='base') * \
+                recurse(newexpr, combine='base')
+
+    elif combine == 'base':
+
+        # Build c_powers and nc_part.  These must both be lists not
+        # dicts because exp's are not combined.
+        c_powers = []
+        nc_part = []
+        for term in expr.args:
+            if term.is_commutative:
+                c_powers.append(list(term.as_base_exp()))
+            else:
+                nc_part.append(term)
+
+        # Pull out numerical coefficients from exponent if assumptions allow
+        # e.g., 2**(2*x) => 4**x
+        for i in range(len(c_powers)):
+            b, e = c_powers[i]
+            if not (all(x.is_nonnegative for x in b.as_numer_denom()) or e.is_integer or force or b.is_polar):
+                continue
+            exp_c, exp_t = e.as_coeff_Mul(rational=True)
+            if exp_c is not S.One and exp_t is not S.One:
+                c_powers[i] = [Pow(b, exp_c), exp_t]
+
+        # Combine bases whenever they have the same exponent and
+        # assumptions allow
+        # first gather the potential bases under the common exponent
+        c_exp = defaultdict(list)
+        for b, e in c_powers:
+            if deep:
+                e = recurse(e)
+            if e.is_Add and (b.is_positive or e.is_integer):
+                e = factor_terms(e)
+                if _coeff_isneg(e):
+                    e = -e
+                    b = 1/b
+            c_exp[e].append(b)
+        del c_powers
+
+        # Merge back in the results of the above to form a new product
+        c_powers = defaultdict(list)
+        for e in c_exp:
+            bases = c_exp[e]
+
+            # calculate the new base for e
+
+            if len(bases) == 1:
+                new_base = bases[0]
+            elif e.is_integer or force:
+                new_base = expr.func(*bases)
+            else:
+                # see which ones can be joined
+                unk = []
+                nonneg = []
+                neg = []
+                for bi in bases:
+                    if bi.is_negative:
+                        neg.append(bi)
+                    elif bi.is_nonnegative:
+                        nonneg.append(bi)
+                    elif bi.is_polar:
+                        nonneg.append(
+                            bi)  # polar can be treated like non-negative
+                    else:
+                        unk.append(bi)
+                if len(unk) == 1 and not neg or len(neg) == 1 and not unk:
+                    # a single neg or a single unk can join the rest
+                    nonneg.extend(unk + neg)
+                    unk = neg = []
+                elif neg:
+                    # their negative signs cancel in groups of 2*q if we know
+                    # that e = p/q else we have to treat them as unknown
+                    israt = False
+                    if e.is_Rational:
+                        israt = True
+                    else:
+                        p, d = e.as_numer_denom()
+                        if p.is_integer and d.is_integer:
+                            israt = True
+                    if israt:
+                        neg = [-w for w in neg]
+                        unk.extend([S.NegativeOne]*len(neg))
+                    else:
+                        unk.extend(neg)
+                        neg = []
+                    del israt
+
+                # these shouldn't be joined
+                for b in unk:
+                    c_powers[b].append(e)
+                # here is a new joined base
+                new_base = expr.func(*(nonneg + neg))
+                # if there are positive parts they will just get separated
+                # again unless some change is made
+
+                def _terms(e):
+                    # return the number of terms of this expression
+                    # when multiplied out -- assuming no joining of terms
+                    if e.is_Add:
+                        return sum(_terms(ai) for ai in e.args)
+                    if e.is_Mul:
+                        return prod([_terms(mi) for mi in e.args])
+                    return 1
+                xnew_base = expand_mul(new_base, deep=False)
+                if len(Add.make_args(xnew_base)) < _terms(new_base):
+                    new_base = factor_terms(xnew_base)
+
+            c_powers[new_base].append(e)
+
+        # break out the powers from c_powers now
+        c_part = [Pow(b, ei) for b, e in c_powers.items() for ei in e]
+
+        # we're done
+        return expr.func(*(c_part + nc_part))
+
+    else:
+        raise ValueError("combine must be one of ('all', 'exp', 'base').")
+
+
+def powdenest(eq, force=False, polar=False):
+    r"""
+    Collect exponents on powers as assumptions allow.
+
+    Explanation
+    ===========
+
+    Given ``(bb**be)**e``, this can be simplified as follows:
+        * if ``bb`` is positive, or
+        * ``e`` is an integer, or
+        * ``|be| < 1`` then this simplifies to ``bb**(be*e)``
+
+    Given a product of powers raised to a power, ``(bb1**be1 *
+    bb2**be2...)**e``, simplification can be done as follows:
+
+    - if e is positive, the gcd of all bei can be joined with e;
+    - all non-negative bb can be separated from those that are negative
+      and their gcd can be joined with e; autosimplification already
+      handles this separation.
+    - integer factors from powers that have integers in the denominator
+      of the exponent can be removed from any term and the gcd of such
+      integers can be joined with e
+
+    Setting ``force`` to ``True`` will make symbols that are not explicitly
+    negative behave as though they are positive, resulting in more
+    denesting.
+
+    Setting ``polar`` to ``True`` will do simplifications on the Riemann surface of
+    the logarithm, also resulting in more denestings.
+
+    When there are sums of logs in exp() then a product of powers may be
+    obtained e.g. ``exp(3*(log(a) + 2*log(b)))`` - > ``a**3*b**6``.
+
+    Examples
+    ========
+
+    >>> from sympy.abc import a, b, x, y, z
+    >>> from sympy import Symbol, exp, log, sqrt, symbols, powdenest
+
+    >>> powdenest((x**(2*a/3))**(3*x))
+    (x**(2*a/3))**(3*x)
+    >>> powdenest(exp(3*x*log(2)))
+    2**(3*x)
+
+    Assumptions may prevent expansion:
+
+    >>> powdenest(sqrt(x**2))
+    sqrt(x**2)
+
+    >>> p = symbols('p', positive=True)
+    >>> powdenest(sqrt(p**2))
+    p
+
+    No other expansion is done.
+
+    >>> i, j = symbols('i,j', integer=True)
+    >>> powdenest((x**x)**(i + j)) # -X-> (x**x)**i*(x**x)**j
+    x**(x*(i + j))
+
+    But exp() will be denested by moving all non-log terms outside of
+    the function; this may result in the collapsing of the exp to a power
+    with a different base:
+
+    >>> powdenest(exp(3*y*log(x)))
+    x**(3*y)
+    >>> powdenest(exp(y*(log(a) + log(b))))
+    (a*b)**y
+    >>> powdenest(exp(3*(log(a) + log(b))))
+    a**3*b**3
+
+    If assumptions allow, symbols can also be moved to the outermost exponent:
+
+    >>> i = Symbol('i', integer=True)
+    >>> powdenest(((x**(2*i))**(3*y))**x)
+    ((x**(2*i))**(3*y))**x
+    >>> powdenest(((x**(2*i))**(3*y))**x, force=True)
+    x**(6*i*x*y)
+
+    >>> powdenest(((x**(2*a/3))**(3*y/i))**x)
+    ((x**(2*a/3))**(3*y/i))**x
+    >>> powdenest((x**(2*i)*y**(4*i))**z, force=True)
+    (x*y**2)**(2*i*z)
+
+    >>> n = Symbol('n', negative=True)
+
+    >>> powdenest((x**i)**y, force=True)
+    x**(i*y)
+    >>> powdenest((n**i)**x, force=True)
+    (n**i)**x
+
+    """
+    from sympy.simplify.simplify import posify
+
+    if force:
+        def _denest(b, e):
+            if not isinstance(b, (Pow, exp)):
+                return b.is_positive, Pow(b, e, evaluate=False)
+            return _denest(b.base, b.exp*e)
+        reps = []
+        for p in eq.atoms(Pow, exp):
+            if isinstance(p.base, (Pow, exp)):
+                ok, dp = _denest(*p.args)
+                if ok is not False:
+                    reps.append((p, dp))
+        if reps:
+            eq = eq.subs(reps)
+        eq, reps = posify(eq)
+        return powdenest(eq, force=False, polar=polar).xreplace(reps)
+
+    if polar:
+        eq, rep = polarify(eq)
+        return unpolarify(powdenest(unpolarify(eq, exponents_only=True)), rep)
+
+    new = powsimp(eq)
+    return new.xreplace(Transform(
+        _denest_pow, filter=lambda m: m.is_Pow or isinstance(m, exp)))
+
+_y = Dummy('y')
+
+
+def _denest_pow(eq):
+    """
+    Denest powers.
+
+    This is a helper function for powdenest that performs the actual
+    transformation.
+    """
+    from sympy.simplify.simplify import logcombine
+
+    b, e = eq.as_base_exp()
+    if b.is_Pow or isinstance(b, exp) and e != 1:
+        new = b._eval_power(e)
+        if new is not None:
+            eq = new
+            b, e = new.as_base_exp()
+
+    # denest exp with log terms in exponent
+    if b is S.Exp1 and e.is_Mul:
+        logs = []
+        other = []
+        for ei in e.args:
+            if any(isinstance(ai, log) for ai in Add.make_args(ei)):
+                logs.append(ei)
+            else:
+                other.append(ei)
+        logs = logcombine(Mul(*logs))
+        return Pow(exp(logs), Mul(*other))
+
+    _, be = b.as_base_exp()
+    if be is S.One and not (b.is_Mul or
+                            b.is_Rational and b.q != 1 or
+                            b.is_positive):
+        return eq
+
+    # denest eq which is either pos**e or Pow**e or Mul**e or
+    # Mul(b1**e1, b2**e2)
+
+    # handle polar numbers specially
+    polars, nonpolars = [], []
+    for bb in Mul.make_args(b):
+        if bb.is_polar:
+            polars.append(bb.as_base_exp())
+        else:
+            nonpolars.append(bb)
+    if len(polars) == 1 and not polars[0][0].is_Mul:
+        return Pow(polars[0][0], polars[0][1]*e)*powdenest(Mul(*nonpolars)**e)
+    elif polars:
+        return Mul(*[powdenest(bb**(ee*e)) for (bb, ee) in polars]) \
+            *powdenest(Mul(*nonpolars)**e)
+
+    if b.is_Integer:
+        # use log to see if there is a power here
+        logb = expand_log(log(b))
+        if logb.is_Mul:
+            c, logb = logb.args
+            e *= c
+            base = logb.args[0]
+            return Pow(base, e)
+
+    # if b is not a Mul or any factor is an atom then there is nothing to do
+    if not b.is_Mul or any(s.is_Atom for s in Mul.make_args(b)):
+        return eq
+
+    # let log handle the case of the base of the argument being a Mul, e.g.
+    # sqrt(x**(2*i)*y**(6*i)) -> x**i*y**(3**i) if x and y are positive; we
+    # will take the log, expand it, and then factor out the common powers that
+    # now appear as coefficient. We do this manually since terms_gcd pulls out
+    # fractions, terms_gcd(x+x*y/2) -> x*(y + 2)/2 and we don't want the 1/2;
+    # gcd won't pull out numerators from a fraction: gcd(3*x, 9*x/2) -> x but
+    # we want 3*x. Neither work with noncommutatives.
+
+    def nc_gcd(aa, bb):
+        a, b = [i.as_coeff_Mul() for i in [aa, bb]]
+        c = gcd(a[0], b[0]).as_numer_denom()[0]
+        g = Mul(*(a[1].args_cnc(cset=True)[0] & b[1].args_cnc(cset=True)[0]))
+        return _keep_coeff(c, g)
+
+    glogb = expand_log(log(b))
+    if glogb.is_Add:
+        args = glogb.args
+        g = reduce(nc_gcd, args)
+        if g != 1:
+            cg, rg = g.as_coeff_Mul()
+            glogb = _keep_coeff(cg, rg*Add(*[a/g for a in args]))
+
+    # now put the log back together again
+    if isinstance(glogb, log) or not glogb.is_Mul:
+        if glogb.args[0].is_Pow or isinstance(glogb.args[0], exp):
+            glogb = _denest_pow(glogb.args[0])
+            if (abs(glogb.exp) < 1) == True:
+                return Pow(glogb.base, glogb.exp*e)
+        return eq
+
+    # the log(b) was a Mul so join any adds with logcombine
+    add = []
+    other = []
+    for a in glogb.args:
+        if a.is_Add:
+            add.append(a)
+        else:
+            other.append(a)
+    return Pow(exp(logcombine(Mul(*add))), e*Mul(*other))

rl/Lib/site-packages/sympy/simplify/ratsimp.py

@@ -0,0 +1,222 @@
+from itertools import combinations_with_replacement
+from sympy.core import symbols, Add, Dummy
+from sympy.core.numbers import Rational
+from sympy.polys import cancel, ComputationFailed, parallel_poly_from_expr, reduced, Poly
+from sympy.polys.monomials import Monomial, monomial_div
+from sympy.polys.polyerrors import DomainError, PolificationFailed
+from sympy.utilities.misc import debug, debugf
+
+def ratsimp(expr):
+    """
+    Put an expression over a common denominator, cancel and reduce.
+
+    Examples
+    ========
+
+    >>> from sympy import ratsimp
+    >>> from sympy.abc import x, y
+    >>> ratsimp(1/x + 1/y)
+    (x + y)/(x*y)
+    """
+
+    f, g = cancel(expr).as_numer_denom()
+    try:
+        Q, r = reduced(f, [g], field=True, expand=False)
+    except ComputationFailed:
+        return f/g
+
+    return Add(*Q) + cancel(r/g)
+
+
+def ratsimpmodprime(expr, G, *gens, quick=True, polynomial=False, **args):
+    """
+    Simplifies a rational expression ``expr`` modulo the prime ideal
+    generated by ``G``.  ``G`` should be a Groebner basis of the
+    ideal.
+
+    Examples
+    ========
+
+    >>> from sympy.simplify.ratsimp import ratsimpmodprime
+    >>> from sympy.abc import x, y
+    >>> eq = (x + y**5 + y)/(x - y)
+    >>> ratsimpmodprime(eq, [x*y**5 - x - y], x, y, order='lex')
+    (-x**2 - x*y - x - y)/(-x**2 + x*y)
+
+    If ``polynomial`` is ``False``, the algorithm computes a rational
+    simplification which minimizes the sum of the total degrees of
+    the numerator and the denominator.
+
+    If ``polynomial`` is ``True``, this function just brings numerator and
+    denominator into a canonical form. This is much faster, but has
+    potentially worse results.
+
+    References
+    ==========
+
+    .. [1] M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial
+        Ideal, https://dl.acm.org/doi/pdf/10.1145/1145768.1145809
+        (specifically, the second algorithm)
+    """
+    from sympy.solvers.solvers import solve
+
+    debug('ratsimpmodprime', expr)
+
+    # usual preparation of polynomials:
+
+    num, denom = cancel(expr).as_numer_denom()
+
+    try:
+        polys, opt = parallel_poly_from_expr([num, denom] + G, *gens, **args)
+    except PolificationFailed:
+        return expr
+
+    domain = opt.domain
+
+    if domain.has_assoc_Field:
+        opt.domain = domain.get_field()
+    else:
+        raise DomainError(
+            "Cannot compute rational simplification over %s" % domain)
+
+    # compute only once
+    leading_monomials = [g.LM(opt.order) for g in polys[2:]]
+    tested = set()
+
+    def staircase(n):
+        """
+        Compute all monomials with degree less than ``n`` that are
+        not divisible by any element of ``leading_monomials``.
+        """
+        if n == 0:
+            return [1]
+        S = []
+        for mi in combinations_with_replacement(range(len(opt.gens)), n):
+            m = [0]*len(opt.gens)
+            for i in mi:
+                m[i] += 1
+            if all(monomial_div(m, lmg) is None for lmg in
+                   leading_monomials):
+                S.append(m)
+
+        return [Monomial(s).as_expr(*opt.gens) for s in S] + staircase(n - 1)
+
+    def _ratsimpmodprime(a, b, allsol, N=0, D=0):
+        r"""
+        Computes a rational simplification of ``a/b`` which minimizes
+        the sum of the total degrees of the numerator and the denominator.
+
+        Explanation
+        ===========
+
+        The algorithm proceeds by looking at ``a * d - b * c`` modulo
+        the ideal generated by ``G`` for some ``c`` and ``d`` with degree
+        less than ``a`` and ``b`` respectively.
+        The coefficients of ``c`` and ``d`` are indeterminates and thus
+        the coefficients of the normalform of ``a * d - b * c`` are
+        linear polynomials in these indeterminates.
+        If these linear polynomials, considered as system of
+        equations, have a nontrivial solution, then `\frac{a}{b}
+        \equiv \frac{c}{d}` modulo the ideal generated by ``G``. So,
+        by construction, the degree of ``c`` and ``d`` is less than
+        the degree of ``a`` and ``b``, so a simpler representation
+        has been found.
+        After a simpler representation has been found, the algorithm
+        tries to reduce the degree of the numerator and denominator
+        and returns the result afterwards.
+
+        As an extension, if quick=False, we look at all possible degrees such
+        that the total degree is less than *or equal to* the best current
+        solution. We retain a list of all solutions of minimal degree, and try
+        to find the best one at the end.
+        """
+        c, d = a, b
+        steps = 0
+
+        maxdeg = a.total_degree() + b.total_degree()
+        if quick:
+            bound = maxdeg - 1
+        else:
+            bound = maxdeg
+        while N + D <= bound:
+            if (N, D) in tested:
+                break
+            tested.add((N, D))
+
+            M1 = staircase(N)
+            M2 = staircase(D)
+            debugf('%s / %s: %s, %s', (N, D, M1, M2))
+
+            Cs = symbols("c:%d" % len(M1), cls=Dummy)
+            Ds = symbols("d:%d" % len(M2), cls=Dummy)
+            ng = Cs + Ds
+
+            c_hat = Poly(
+                sum(Cs[i] * M1[i] for i in range(len(M1))), opt.gens + ng)
+            d_hat = Poly(
+                sum(Ds[i] * M2[i] for i in range(len(M2))), opt.gens + ng)
+
+            r = reduced(a * d_hat - b * c_hat, G, opt.gens + ng,
+                        order=opt.order, polys=True)[1]
+
+            S = Poly(r, gens=opt.gens).coeffs()
+            sol = solve(S, Cs + Ds, particular=True, quick=True)
+
+            if sol and not all(s == 0 for s in sol.values()):
+                c = c_hat.subs(sol)
+                d = d_hat.subs(sol)
+
+                # The "free" variables occurring before as parameters
+                # might still be in the substituted c, d, so set them
+                # to the value chosen before:
+                c = c.subs(dict(list(zip(Cs + Ds, [1] * (len(Cs) + len(Ds))))))
+                d = d.subs(dict(list(zip(Cs + Ds, [1] * (len(Cs) + len(Ds))))))
+
+                c = Poly(c, opt.gens)
+                d = Poly(d, opt.gens)
+                if d == 0:
+                    raise ValueError('Ideal not prime?')
+
+                allsol.append((c_hat, d_hat, S, Cs + Ds))
+                if N + D != maxdeg:
+                    allsol = [allsol[-1]]
+
+                break
+
+            steps += 1
+            N += 1
+            D += 1
+
+        if steps > 0:
+            c, d, allsol = _ratsimpmodprime(c, d, allsol, N, D - steps)
+            c, d, allsol = _ratsimpmodprime(c, d, allsol, N - steps, D)
+
+        return c, d, allsol
+
+    # preprocessing. this improves performance a bit when deg(num)
+    # and deg(denom) are large:
+    num = reduced(num, G, opt.gens, order=opt.order)[1]
+    denom = reduced(denom, G, opt.gens, order=opt.order)[1]
+
+    if polynomial:
+        return (num/denom).cancel()
+
+    c, d, allsol = _ratsimpmodprime(
+        Poly(num, opt.gens, domain=opt.domain), Poly(denom, opt.gens, domain=opt.domain), [])
+    if not quick and allsol:
+        debugf('Looking for best minimal solution. Got: %s', len(allsol))
+        newsol = []
+        for c_hat, d_hat, S, ng in allsol:
+            sol = solve(S, ng, particular=True, quick=False)
+            # all values of sol should be numbers; if not, solve is broken
+            newsol.append((c_hat.subs(sol), d_hat.subs(sol)))
+        c, d = min(newsol, key=lambda x: len(x[0].terms()) + len(x[1].terms()))
+
+    if not domain.is_Field:
+        cn, c = c.clear_denoms(convert=True)
+        dn, d = d.clear_denoms(convert=True)
+        r = Rational(cn, dn)
+    else:
+        r = Rational(1)
+
+    return (c*r.q)/(d*r.p)

rl/Lib/site-packages/sympy/simplify/sqrtdenest.py

@@ -0,0 +1,678 @@
+from sympy.core import Add, Expr, Mul, S, sympify
+from sympy.core.function import _mexpand, count_ops, expand_mul
+from sympy.core.sorting import default_sort_key
+from sympy.core.symbol import Dummy
+from sympy.functions import root, sign, sqrt
+from sympy.polys import Poly, PolynomialError
+
+
+def is_sqrt(expr):
+    """Return True if expr is a sqrt, otherwise False."""
+
+    return expr.is_Pow and expr.exp.is_Rational and abs(expr.exp) is S.Half
+
+
+def sqrt_depth(p) -> int:
+    """Return the maximum depth of any square root argument of p.
+
+    >>> from sympy.functions.elementary.miscellaneous import sqrt
+    >>> from sympy.simplify.sqrtdenest import sqrt_depth
+
+    Neither of these square roots contains any other square roots
+    so the depth is 1:
+
+    >>> sqrt_depth(1 + sqrt(2)*(1 + sqrt(3)))
+    1
+
+    The sqrt(3) is contained within a square root so the depth is
+    2:
+
+    >>> sqrt_depth(1 + sqrt(2)*sqrt(1 + sqrt(3)))
+    2
+    """
+    if p is S.ImaginaryUnit:
+        return 1
+    if p.is_Atom:
+        return 0
+    if p.is_Add or p.is_Mul:
+        return max(sqrt_depth(x) for x in p.args)
+    if is_sqrt(p):
+        return sqrt_depth(p.base) + 1
+    return 0
+
+
+def is_algebraic(p):
+    """Return True if p is comprised of only Rationals or square roots
+    of Rationals and algebraic operations.
+
+    Examples
+    ========
+
+    >>> from sympy.functions.elementary.miscellaneous import sqrt
+    >>> from sympy.simplify.sqrtdenest import is_algebraic
+    >>> from sympy import cos
+    >>> is_algebraic(sqrt(2)*(3/(sqrt(7) + sqrt(5)*sqrt(2))))
+    True
+    >>> is_algebraic(sqrt(2)*(3/(sqrt(7) + sqrt(5)*cos(2))))
+    False
+    """
+
+    if p.is_Rational:
+        return True
+    elif p.is_Atom:
+        return False
+    elif is_sqrt(p) or p.is_Pow and p.exp.is_Integer:
+        return is_algebraic(p.base)
+    elif p.is_Add or p.is_Mul:
+        return all(is_algebraic(x) for x in p.args)
+    else:
+        return False
+
+
+def _subsets(n):
+    """
+    Returns all possible subsets of the set (0, 1, ..., n-1) except the
+    empty set, listed in reversed lexicographical order according to binary
+    representation, so that the case of the fourth root is treated last.
+
+    Examples
+    ========
+
+    >>> from sympy.simplify.sqrtdenest import _subsets
+    >>> _subsets(2)
+    [[1, 0], [0, 1], [1, 1]]
+
+    """
+    if n == 1:
+        a = [[1]]
+    elif n == 2:
+        a = [[1, 0], [0, 1], [1, 1]]
+    elif n == 3:
+        a = [[1, 0, 0], [0, 1, 0], [1, 1, 0],
+             [0, 0, 1], [1, 0, 1], [0, 1, 1], [1, 1, 1]]
+    else:
+        b = _subsets(n - 1)
+        a0 = [x + [0] for x in b]
+        a1 = [x + [1] for x in b]
+        a = a0 + [[0]*(n - 1) + [1]] + a1
+    return a
+
+
+def sqrtdenest(expr, max_iter=3):
+    """Denests sqrts in an expression that contain other square roots
+    if possible, otherwise returns the expr unchanged. This is based on the
+    algorithms of [1].
+
+    Examples
+    ========
+
+    >>> from sympy.simplify.sqrtdenest import sqrtdenest
+    >>> from sympy import sqrt
+    >>> sqrtdenest(sqrt(5 + 2 * sqrt(6)))
+    sqrt(2) + sqrt(3)
+
+    See Also
+    ========
+
+    sympy.solvers.solvers.unrad
+
+    References
+    ==========
+
+    .. [1] https://web.archive.org/web/20210806201615/https://researcher.watson.ibm.com/researcher/files/us-fagin/symb85.pdf
+
+    .. [2] D. J. Jeffrey and A. D. Rich, 'Symplifying Square Roots of Square Roots
+           by Denesting' (available at https://www.cybertester.com/data/denest.pdf)
+
+    """
+    expr = expand_mul(expr)
+    for i in range(max_iter):
+        z = _sqrtdenest0(expr)
+        if expr == z:
+            return expr
+        expr = z
+    return expr
+
+
+def _sqrt_match(p):
+    """Return [a, b, r] for p.match(a + b*sqrt(r)) where, in addition to
+    matching, sqrt(r) also has then maximal sqrt_depth among addends of p.
+
+    Examples
+    ========
+
+    >>> from sympy.functions.elementary.miscellaneous import sqrt
+    >>> from sympy.simplify.sqrtdenest import _sqrt_match
+    >>> _sqrt_match(1 + sqrt(2) + sqrt(2)*sqrt(3) +  2*sqrt(1+sqrt(5)))
+    [1 + sqrt(2) + sqrt(6), 2, 1 + sqrt(5)]
+    """
+    from sympy.simplify.radsimp import split_surds
+
+    p = _mexpand(p)
+    if p.is_Number:
+        res = (p, S.Zero, S.Zero)
+    elif p.is_Add:
+        pargs = sorted(p.args, key=default_sort_key)
+        sqargs = [x**2 for x in pargs]
+        if all(sq.is_Rational and sq.is_positive for sq in sqargs):
+            r, b, a = split_surds(p)
+            res = a, b, r
+            return list(res)
+        # to make the process canonical, the argument is included in the tuple
+        # so when the max is selected, it will be the largest arg having a
+        # given depth
+        v = [(sqrt_depth(x), x, i) for i, x in enumerate(pargs)]
+        nmax = max(v, key=default_sort_key)
+        if nmax[0] == 0:
+            res = []
+        else:
+            # select r
+            depth, _, i = nmax
+            r = pargs.pop(i)
+            v.pop(i)
+            b = S.One
+            if r.is_Mul:
+                bv = []
+                rv = []
+                for x in r.args:
+                    if sqrt_depth(x) < depth:
+                        bv.append(x)
+                    else:
+                        rv.append(x)
+                b = Mul._from_args(bv)
+                r = Mul._from_args(rv)
+            # collect terms comtaining r
+            a1 = []
+            b1 = [b]
+            for x in v:
+                if x[0] < depth:
+                    a1.append(x[1])
+                else:
+                    x1 = x[1]
+                    if x1 == r:
+                        b1.append(1)
+                    else:
+                        if x1.is_Mul:
+                            x1args = list(x1.args)
+                            if r in x1args:
+                                x1args.remove(r)
+                                b1.append(Mul(*x1args))
+                            else:
+                                a1.append(x[1])
+                        else:
+                            a1.append(x[1])
+            a = Add(*a1)
+            b = Add(*b1)
+            res = (a, b, r**2)
+    else:
+        b, r = p.as_coeff_Mul()
+        if is_sqrt(r):
+            res = (S.Zero, b, r**2)
+        else:
+            res = []
+    return list(res)
+
+
+class SqrtdenestStopIteration(StopIteration):
+    pass
+
+
+def _sqrtdenest0(expr):
+    """Returns expr after denesting its arguments."""
+
+    if is_sqrt(expr):
+        n, d = expr.as_numer_denom()
+        if d is S.One:  # n is a square root
+            if n.base.is_Add:
+                args = sorted(n.base.args, key=default_sort_key)
+                if len(args) > 2 and all((x**2).is_Integer for x in args):
+                    try:
+                        return _sqrtdenest_rec(n)
+                    except SqrtdenestStopIteration:
+                        pass
+                expr = sqrt(_mexpand(Add(*[_sqrtdenest0(x) for x in args])))
+            return _sqrtdenest1(expr)
+        else:
+            n, d = [_sqrtdenest0(i) for i in (n, d)]
+            return n/d
+
+    if isinstance(expr, Add):
+        cs = []
+        args = []
+        for arg in expr.args:
+            c, a = arg.as_coeff_Mul()
+            cs.append(c)
+            args.append(a)
+
+        if all(c.is_Rational for c in cs) and all(is_sqrt(arg) for arg in args):
+            return _sqrt_ratcomb(cs, args)
+
+    if isinstance(expr, Expr):
+        args = expr.args
+        if args:
+            return expr.func(*[_sqrtdenest0(a) for a in args])
+    return expr
+
+
+def _sqrtdenest_rec(expr):
+    """Helper that denests the square root of three or more surds.
+
+    Explanation
+    ===========
+
+    It returns the denested expression; if it cannot be denested it
+    throws SqrtdenestStopIteration
+
+    Algorithm: expr.base is in the extension Q_m = Q(sqrt(r_1),..,sqrt(r_k));
+    split expr.base = a + b*sqrt(r_k), where `a` and `b` are on
+    Q_(m-1) = Q(sqrt(r_1),..,sqrt(r_(k-1))); then a**2 - b**2*r_k is
+    on Q_(m-1); denest sqrt(a**2 - b**2*r_k) and so on.
+    See [1], section 6.
+
+    Examples
+    ========
+
+    >>> from sympy import sqrt
+    >>> from sympy.simplify.sqrtdenest import _sqrtdenest_rec
+    >>> _sqrtdenest_rec(sqrt(-72*sqrt(2) + 158*sqrt(5) + 498))
+    -sqrt(10) + sqrt(2) + 9 + 9*sqrt(5)
+    >>> w=-6*sqrt(55)-6*sqrt(35)-2*sqrt(22)-2*sqrt(14)+2*sqrt(77)+6*sqrt(10)+65
+    >>> _sqrtdenest_rec(sqrt(w))
+    -sqrt(11) - sqrt(7) + sqrt(2) + 3*sqrt(5)
+    """
+    from sympy.simplify.radsimp import radsimp, rad_rationalize, split_surds
+    if not expr.is_Pow:
+        return sqrtdenest(expr)
+    if expr.base < 0:
+        return sqrt(-1)*_sqrtdenest_rec(sqrt(-expr.base))
+    g, a, b = split_surds(expr.base)
+    a = a*sqrt(g)
+    if a < b:
+        a, b = b, a
+    c2 = _mexpand(a**2 - b**2)
+    if len(c2.args) > 2:
+        g, a1, b1 = split_surds(c2)
+        a1 = a1*sqrt(g)
+        if a1 < b1:
+            a1, b1 = b1, a1
+        c2_1 = _mexpand(a1**2 - b1**2)
+        c_1 = _sqrtdenest_rec(sqrt(c2_1))
+        d_1 = _sqrtdenest_rec(sqrt(a1 + c_1))
+        num, den = rad_rationalize(b1, d_1)
+        c = _mexpand(d_1/sqrt(2) + num/(den*sqrt(2)))
+    else:
+        c = _sqrtdenest1(sqrt(c2))
+
+    if sqrt_depth(c) > 1:
+        raise SqrtdenestStopIteration
+    ac = a + c
+    if len(ac.args) >= len(expr.args):
+        if count_ops(ac) >= count_ops(expr.base):
+            raise SqrtdenestStopIteration
+    d = sqrtdenest(sqrt(ac))
+    if sqrt_depth(d) > 1:
+        raise SqrtdenestStopIteration
+    num, den = rad_rationalize(b, d)
+    r = d/sqrt(2) + num/(den*sqrt(2))
+    r = radsimp(r)
+    return _mexpand(r)
+
+
+def _sqrtdenest1(expr, denester=True):
+    """Return denested expr after denesting with simpler methods or, that
+    failing, using the denester."""
+
+    from sympy.simplify.simplify import radsimp
+
+    if not is_sqrt(expr):
+        return expr
+
+    a = expr.base
+    if a.is_Atom:
+        return expr
+    val = _sqrt_match(a)
+    if not val:
+        return expr
+
+    a, b, r = val
+    # try a quick numeric denesting
+    d2 = _mexpand(a**2 - b**2*r)
+    if d2.is_Rational:
+        if d2.is_positive:
+            z = _sqrt_numeric_denest(a, b, r, d2)
+            if z is not None:
+                return z
+        else:
+            # fourth root case
+            # sqrtdenest(sqrt(3 + 2*sqrt(3))) =
+            # sqrt(2)*3**(1/4)/2 + sqrt(2)*3**(3/4)/2
+            dr2 = _mexpand(-d2*r)
+            dr = sqrt(dr2)
+            if dr.is_Rational:
+                z = _sqrt_numeric_denest(_mexpand(b*r), a, r, dr2)
+                if z is not None:
+                    return z/root(r, 4)
+
+    else:
+        z = _sqrt_symbolic_denest(a, b, r)
+        if z is not None:
+            return z
+
+    if not denester or not is_algebraic(expr):
+        return expr
+
+    res = sqrt_biquadratic_denest(expr, a, b, r, d2)
+    if res:
+        return res
+
+    # now call to the denester
+    av0 = [a, b, r, d2]
+    z = _denester([radsimp(expr**2)], av0, 0, sqrt_depth(expr))[0]
+    if av0[1] is None:
+        return expr
+    if z is not None:
+        if sqrt_depth(z) == sqrt_depth(expr) and count_ops(z) > count_ops(expr):
+            return expr
+        return z
+    return expr
+
+
+def _sqrt_symbolic_denest(a, b, r):
+    """Given an expression, sqrt(a + b*sqrt(b)), return the denested
+    expression or None.
+
+    Explanation
+    ===========
+
+    If r = ra + rb*sqrt(rr), try replacing sqrt(rr) in ``a`` with
+    (y**2 - ra)/rb, and if the result is a quadratic, ca*y**2 + cb*y + cc, and
+    (cb + b)**2 - 4*ca*cc is 0, then sqrt(a + b*sqrt(r)) can be rewritten as
+    sqrt(ca*(sqrt(r) + (cb + b)/(2*ca))**2).
+
+    Examples
+    ========
+
+    >>> from sympy.simplify.sqrtdenest import _sqrt_symbolic_denest, sqrtdenest
+    >>> from sympy import sqrt, Symbol
+    >>> from sympy.abc import x
+
+    >>> a, b, r = 16 - 2*sqrt(29), 2, -10*sqrt(29) + 55
+    >>> _sqrt_symbolic_denest(a, b, r)
+    sqrt(11 - 2*sqrt(29)) + sqrt(5)
+
+    If the expression is numeric, it will be simplified:
+
+    >>> w = sqrt(sqrt(sqrt(3) + 1) + 1) + 1 + sqrt(2)
+    >>> sqrtdenest(sqrt((w**2).expand()))
+    1 + sqrt(2) + sqrt(1 + sqrt(1 + sqrt(3)))
+
+    Otherwise, it will only be simplified if assumptions allow:
+
+    >>> w = w.subs(sqrt(3), sqrt(x + 3))
+    >>> sqrtdenest(sqrt((w**2).expand()))
+    sqrt((sqrt(sqrt(sqrt(x + 3) + 1) + 1) + 1 + sqrt(2))**2)
+
+    Notice that the argument of the sqrt is a square. If x is made positive
+    then the sqrt of the square is resolved:
+
+    >>> _.subs(x, Symbol('x', positive=True))
+    sqrt(sqrt(sqrt(x + 3) + 1) + 1) + 1 + sqrt(2)
+    """
+
+    a, b, r = map(sympify, (a, b, r))
+    rval = _sqrt_match(r)
+    if not rval:
+        return None
+    ra, rb, rr = rval
+    if rb:
+        y = Dummy('y', positive=True)
+        try:
+            newa = Poly(a.subs(sqrt(rr), (y**2 - ra)/rb), y)
+        except PolynomialError:
+            return None
+        if newa.degree() == 2:
+            ca, cb, cc = newa.all_coeffs()
+            cb += b
+            if _mexpand(cb**2 - 4*ca*cc).equals(0):
+                z = sqrt(ca*(sqrt(r) + cb/(2*ca))**2)
+                if z.is_number:
+                    z = _mexpand(Mul._from_args(z.as_content_primitive()))
+                return z
+
+
+def _sqrt_numeric_denest(a, b, r, d2):
+    r"""Helper that denest
+    $\sqrt{a + b \sqrt{r}}, d^2 = a^2 - b^2 r > 0$
+
+    If it cannot be denested, it returns ``None``.
+    """
+    d = sqrt(d2)
+    s = a + d
+    # sqrt_depth(res) <= sqrt_depth(s) + 1
+    # sqrt_depth(expr) = sqrt_depth(r) + 2
+    # there is denesting if sqrt_depth(s) + 1 < sqrt_depth(r) + 2
+    # if s**2 is Number there is a fourth root
+    if sqrt_depth(s) < sqrt_depth(r) + 1 or (s**2).is_Rational:
+        s1, s2 = sign(s), sign(b)
+        if s1 == s2 == -1:
+            s1 = s2 = 1
+        res = (s1 * sqrt(a + d) + s2 * sqrt(a - d)) * sqrt(2) / 2
+        return res.expand()
+
+
+def sqrt_biquadratic_denest(expr, a, b, r, d2):
+    """denest expr = sqrt(a + b*sqrt(r))
+    where a, b, r are linear combinations of square roots of
+    positive rationals on the rationals (SQRR) and r > 0, b != 0,
+    d2 = a**2 - b**2*r > 0
+
+    If it cannot denest it returns None.
+
+    Explanation
+    ===========
+
+    Search for a solution A of type SQRR of the biquadratic equation
+    4*A**4 - 4*a*A**2 + b**2*r = 0                               (1)
+    sqd = sqrt(a**2 - b**2*r)
+    Choosing the sqrt to be positive, the possible solutions are
+    A = sqrt(a/2 +/- sqd/2)
+    Since a, b, r are SQRR, then a**2 - b**2*r is a SQRR,
+    so if sqd can be denested, it is done by
+    _sqrtdenest_rec, and the result is a SQRR.
+    Similarly for A.
+    Examples of solutions (in both cases a and sqd are positive):
+
+      Example of expr with solution sqrt(a/2 + sqd/2) but not
+      solution sqrt(a/2 - sqd/2):
+      expr = sqrt(-sqrt(15) - sqrt(2)*sqrt(-sqrt(5) + 5) - sqrt(3) + 8)
+      a = -sqrt(15) - sqrt(3) + 8; sqd = -2*sqrt(5) - 2 + 4*sqrt(3)
+
+      Example of expr with solution sqrt(a/2 - sqd/2) but not
+      solution sqrt(a/2 + sqd/2):
+      w = 2 + r2 + r3 + (1 + r3)*sqrt(2 + r2 + 5*r3)
+      expr = sqrt((w**2).expand())
+      a = 4*sqrt(6) + 8*sqrt(2) + 47 + 28*sqrt(3)
+      sqd = 29 + 20*sqrt(3)
+
+    Define B = b/2*A; eq.(1) implies a = A**2 + B**2*r; then
+    expr**2 = a + b*sqrt(r) = (A + B*sqrt(r))**2
+
+    Examples
+    ========
+
+    >>> from sympy import sqrt
+    >>> from sympy.simplify.sqrtdenest import _sqrt_match, sqrt_biquadratic_denest
+    >>> z = sqrt((2*sqrt(2) + 4)*sqrt(2 + sqrt(2)) + 5*sqrt(2) + 8)
+    >>> a, b, r = _sqrt_match(z**2)
+    >>> d2 = a**2 - b**2*r
+    >>> sqrt_biquadratic_denest(z, a, b, r, d2)
+    sqrt(2) + sqrt(sqrt(2) + 2) + 2
+    """
+    from sympy.simplify.radsimp import radsimp, rad_rationalize
+    if r <= 0 or d2 < 0 or not b or sqrt_depth(expr.base) < 2:
+        return None
+    for x in (a, b, r):
+        for y in x.args:
+            y2 = y**2
+            if not y2.is_Integer or not y2.is_positive:
+                return None
+    sqd = _mexpand(sqrtdenest(sqrt(radsimp(d2))))
+    if sqrt_depth(sqd) > 1:
+        return None
+    x1, x2 = [a/2 + sqd/2, a/2 - sqd/2]
+    # look for a solution A with depth 1
+    for x in (x1, x2):
+        A = sqrtdenest(sqrt(x))
+        if sqrt_depth(A) > 1:
+            continue
+        Bn, Bd = rad_rationalize(b, _mexpand(2*A))
+        B = Bn/Bd
+        z = A + B*sqrt(r)
+        if z < 0:
+            z = -z
+        return _mexpand(z)
+    return None
+
+
+def _denester(nested, av0, h, max_depth_level):
+    """Denests a list of expressions that contain nested square roots.
+
+    Explanation
+    ===========
+
+    Algorithm based on <http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf>.
+
+    It is assumed that all of the elements of 'nested' share the same
+    bottom-level radicand. (This is stated in the paper, on page 177, in
+    the paragraph immediately preceding the algorithm.)
+
+    When evaluating all of the arguments in parallel, the bottom-level
+    radicand only needs to be denested once. This means that calling
+    _denester with x arguments results in a recursive invocation with x+1
+    arguments; hence _denester has polynomial complexity.
+
+    However, if the arguments were evaluated separately, each call would
+    result in two recursive invocations, and the algorithm would have
+    exponential complexity.
+
+    This is discussed in the paper in the middle paragraph of page 179.
+    """
+    from sympy.simplify.simplify import radsimp
+    if h > max_depth_level:
+        return None, None
+    if av0[1] is None:
+        return None, None
+    if (av0[0] is None and
+            all(n.is_Number for n in nested)):  # no arguments are nested
+        for f in _subsets(len(nested)):  # test subset 'f' of nested
+            p = _mexpand(Mul(*[nested[i] for i in range(len(f)) if f[i]]))
+            if f.count(1) > 1 and f[-1]:
+                p = -p
+            sqp = sqrt(p)
+            if sqp.is_Rational:
+                return sqp, f  # got a perfect square so return its square root.
+        # Otherwise, return the radicand from the previous invocation.
+        return sqrt(nested[-1]), [0]*len(nested)
+    else:
+        R = None
+        if av0[0] is not None:
+            values = [av0[:2]]
+            R = av0[2]
+            nested2 = [av0[3], R]
+            av0[0] = None
+        else:
+            values = list(filter(None, [_sqrt_match(expr) for expr in nested]))
+            for v in values:
+                if v[2]:  # Since if b=0, r is not defined
+                    if R is not None:
+                        if R != v[2]:
+                            av0[1] = None
+                            return None, None
+                    else:
+                        R = v[2]
+            if R is None:
+                # return the radicand from the previous invocation
+                return sqrt(nested[-1]), [0]*len(nested)
+            nested2 = [_mexpand(v[0]**2) -
+                       _mexpand(R*v[1]**2) for v in values] + [R]
+        d, f = _denester(nested2, av0, h + 1, max_depth_level)
+        if not f:
+            return None, None
+        if not any(f[i] for i in range(len(nested))):
+            v = values[-1]
+            return sqrt(v[0] + _mexpand(v[1]*d)), f
+        else:
+            p = Mul(*[nested[i] for i in range(len(nested)) if f[i]])
+            v = _sqrt_match(p)
+            if 1 in f and f.index(1) < len(nested) - 1 and f[len(nested) - 1]:
+                v[0] = -v[0]
+                v[1] = -v[1]
+            if not f[len(nested)]:  # Solution denests with square roots
+                vad = _mexpand(v[0] + d)
+                if vad <= 0:
+                    # return the radicand from the previous invocation.
+                    return sqrt(nested[-1]), [0]*len(nested)
+                if not(sqrt_depth(vad) <= sqrt_depth(R) + 1 or
+                       (vad**2).is_Number):
+                    av0[1] = None
+                    return None, None
+
+                sqvad = _sqrtdenest1(sqrt(vad), denester=False)
+                if not (sqrt_depth(sqvad) <= sqrt_depth(R) + 1):
+                    av0[1] = None
+                    return None, None
+                sqvad1 = radsimp(1/sqvad)
+                res = _mexpand(sqvad/sqrt(2) + (v[1]*sqrt(R)*sqvad1/sqrt(2)))
+                return res, f
+
+                      #          sign(v[1])*sqrt(_mexpand(v[1]**2*R*vad1/2))), f
+            else:  # Solution requires a fourth root
+                s2 = _mexpand(v[1]*R) + d
+                if s2 <= 0:
+                    return sqrt(nested[-1]), [0]*len(nested)
+                FR, s = root(_mexpand(R), 4), sqrt(s2)
+                return _mexpand(s/(sqrt(2)*FR) + v[0]*FR/(sqrt(2)*s)), f
+
+
+def _sqrt_ratcomb(cs, args):
+    """Denest rational combinations of radicals.
+
+    Based on section 5 of [1].
+
+    Examples
+    ========
+
+    >>> from sympy import sqrt
+    >>> from sympy.simplify.sqrtdenest import sqrtdenest
+    >>> z = sqrt(1+sqrt(3)) + sqrt(3+3*sqrt(3)) - sqrt(10+6*sqrt(3))
+    >>> sqrtdenest(z)
+    0
+    """
+    from sympy.simplify.radsimp import radsimp
+
+    # check if there exists a pair of sqrt that can be denested
+    def find(a):
+        n = len(a)
+        for i in range(n - 1):
+            for j in range(i + 1, n):
+                s1 = a[i].base
+                s2 = a[j].base
+                p = _mexpand(s1 * s2)
+                s = sqrtdenest(sqrt(p))
+                if s != sqrt(p):
+                    return s, i, j
+
+    indices = find(args)
+    if indices is None:
+        return Add(*[c * arg for c, arg in zip(cs, args)])
+
+    s, i1, i2 = indices
+
+    c2 = cs.pop(i2)
+    args.pop(i2)
+    a1 = args[i1]
+
+    # replace a2 by s/a1
+    cs[i1] += radsimp(c2 * s / a1.base)
+
+    return _sqrt_ratcomb(cs, args)

rl/Lib/site-packages/sympy/simplify/tests/test_combsimp.py

@@ -0,0 +1,75 @@
+from sympy.core.numbers import Rational
+from sympy.core.symbol import symbols
+from sympy.functions.combinatorial.factorials import (FallingFactorial, RisingFactorial, binomial, factorial)
+from sympy.functions.special.gamma_functions import gamma
+from sympy.simplify.combsimp import combsimp
+from sympy.abc import x
+
+
+def test_combsimp():
+    k, m, n = symbols('k m n', integer = True)
+
+    assert combsimp(factorial(n)) == factorial(n)
+    assert combsimp(binomial(n, k)) == binomial(n, k)
+
+    assert combsimp(factorial(n)/factorial(n - 3)) == n*(-1 + n)*(-2 + n)
+    assert combsimp(binomial(n + 1, k + 1)/binomial(n, k)) == (1 + n)/(1 + k)
+
+    assert combsimp(binomial(3*n + 4, n + 1)/binomial(3*n + 1, n)) == \
+        Rational(3, 2)*((3*n + 2)*(3*n + 4)/((n + 1)*(2*n + 3)))
+
+    assert combsimp(factorial(n)**2/factorial(n - 3)) == \
+        factorial(n)*n*(-1 + n)*(-2 + n)
+    assert combsimp(factorial(n)*binomial(n + 1, k + 1)/binomial(n, k)) == \
+        factorial(n + 1)/(1 + k)
+
+    assert combsimp(gamma(n + 3)) == factorial(n + 2)
+
+    assert combsimp(factorial(x)) == gamma(x + 1)
+
+    # issue 9699
+    assert combsimp((n + 1)*factorial(n)) == factorial(n + 1)
+    assert combsimp(factorial(n)/n) == factorial(n-1)
+
+    # issue 6658
+    assert combsimp(binomial(n, n - k)) == binomial(n, k)
+
+    # issue 6341, 7135
+    assert combsimp(factorial(n)/(factorial(k)*factorial(n - k))) == \
+        binomial(n, k)
+    assert combsimp(factorial(k)*factorial(n - k)/factorial(n)) == \
+        1/binomial(n, k)
+    assert combsimp(factorial(2*n)/factorial(n)**2) == binomial(2*n, n)
+    assert combsimp(factorial(2*n)*factorial(k)*factorial(n - k)/
+        factorial(n)**3) == binomial(2*n, n)/binomial(n, k)
+
+    assert combsimp(factorial(n*(1 + n) - n**2 - n)) == 1
+
+    assert combsimp(6*FallingFactorial(-4, n)/factorial(n)) == \
+        (-1)**n*(n + 1)*(n + 2)*(n + 3)
+    assert combsimp(6*FallingFactorial(-4, n - 1)/factorial(n - 1)) == \
+        (-1)**(n - 1)*n*(n + 1)*(n + 2)
+    assert combsimp(6*FallingFactorial(-4, n - 3)/factorial(n - 3)) == \
+        (-1)**(n - 3)*n*(n - 1)*(n - 2)
+    assert combsimp(6*FallingFactorial(-4, -n - 1)/factorial(-n - 1)) == \
+        -(-1)**(-n - 1)*n*(n - 1)*(n - 2)
+
+    assert combsimp(6*RisingFactorial(4, n)/factorial(n)) == \
+        (n + 1)*(n + 2)*(n + 3)
+    assert combsimp(6*RisingFactorial(4, n - 1)/factorial(n - 1)) == \
+        n*(n + 1)*(n + 2)
+    assert combsimp(6*RisingFactorial(4, n - 3)/factorial(n - 3)) == \
+        n*(n - 1)*(n - 2)
+    assert combsimp(6*RisingFactorial(4, -n - 1)/factorial(-n - 1)) == \
+        -n*(n - 1)*(n - 2)
+
+
+def test_issue_6878():
+    n = symbols('n', integer=True)
+    assert combsimp(RisingFactorial(-10, n)) == 3628800*(-1)**n/factorial(10 - n)
+
+
+def test_issue_14528():
+    p = symbols("p", integer=True, positive=True)
+    assert combsimp(binomial(1,p)) == 1/(factorial(p)*factorial(1-p))
+    assert combsimp(factorial(2-p)) == factorial(2-p)

rl/Lib/site-packages/sympy/simplify/tests/test_cse.py

@@ -0,0 +1,758 @@
+from functools import reduce
+import itertools
+from operator import add
+
+from sympy.codegen.matrix_nodes import MatrixSolve
+from sympy.core.add import Add
+from sympy.core.containers import Tuple
+from sympy.core.expr import UnevaluatedExpr
+from sympy.core.function import Function
+from sympy.core.mul import Mul
+from sympy.core.power import Pow
+from sympy.core.relational import Eq
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.core.sympify import sympify
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.matrices.dense import Matrix
+from sympy.matrices.expressions import Inverse, MatAdd, MatMul, Transpose
+from sympy.polys.rootoftools import CRootOf
+from sympy.series.order import O
+from sympy.simplify.cse_main import cse
+from sympy.simplify.simplify import signsimp
+from sympy.tensor.indexed import (Idx, IndexedBase)
+
+from sympy.core.function import count_ops
+from sympy.simplify.cse_opts import sub_pre, sub_post
+from sympy.functions.special.hyper import meijerg
+from sympy.simplify import cse_main, cse_opts
+from sympy.utilities.iterables import subsets
+from sympy.testing.pytest import XFAIL, raises
+from sympy.matrices import (MutableDenseMatrix, MutableSparseMatrix,
+        ImmutableDenseMatrix, ImmutableSparseMatrix)
+from sympy.matrices.expressions import MatrixSymbol
+
+
+w, x, y, z = symbols('w,x,y,z')
+x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12 = symbols('x:13')
+
+
+def test_numbered_symbols():
+    ns = cse_main.numbered_symbols(prefix='y')
+    assert list(itertools.islice(
+        ns, 0, 10)) == [Symbol('y%s' % i) for i in range(0, 10)]
+    ns = cse_main.numbered_symbols(prefix='y')
+    assert list(itertools.islice(
+        ns, 10, 20)) == [Symbol('y%s' % i) for i in range(10, 20)]
+    ns = cse_main.numbered_symbols()
+    assert list(itertools.islice(
+        ns, 0, 10)) == [Symbol('x%s' % i) for i in range(0, 10)]
+
+# Dummy "optimization" functions for testing.
+
+
+def opt1(expr):
+    return expr + y
+
+
+def opt2(expr):
+    return expr*z
+
+
+def test_preprocess_for_cse():
+    assert cse_main.preprocess_for_cse(x, [(opt1, None)]) == x + y
+    assert cse_main.preprocess_for_cse(x, [(None, opt1)]) == x
+    assert cse_main.preprocess_for_cse(x, [(None, None)]) == x
+    assert cse_main.preprocess_for_cse(x, [(opt1, opt2)]) == x + y
+    assert cse_main.preprocess_for_cse(
+        x, [(opt1, None), (opt2, None)]) == (x + y)*z
+
+
+def test_postprocess_for_cse():
+    assert cse_main.postprocess_for_cse(x, [(opt1, None)]) == x
+    assert cse_main.postprocess_for_cse(x, [(None, opt1)]) == x + y
+    assert cse_main.postprocess_for_cse(x, [(None, None)]) == x
+    assert cse_main.postprocess_for_cse(x, [(opt1, opt2)]) == x*z
+    # Note the reverse order of application.
+    assert cse_main.postprocess_for_cse(
+        x, [(None, opt1), (None, opt2)]) == x*z + y
+
+
+def test_cse_single():
+    # Simple substitution.
+    e = Add(Pow(x + y, 2), sqrt(x + y))
+    substs, reduced = cse([e])
+    assert substs == [(x0, x + y)]
+    assert reduced == [sqrt(x0) + x0**2]
+
+    subst42, (red42,) = cse([42])  # issue_15082
+    assert len(subst42) == 0 and red42 == 42
+    subst_half, (red_half,) = cse([0.5])
+    assert len(subst_half) == 0 and red_half == 0.5
+
+
+def test_cse_single2():
+    # Simple substitution, test for being able to pass the expression directly
+    e = Add(Pow(x + y, 2), sqrt(x + y))
+    substs, reduced = cse(e)
+    assert substs == [(x0, x + y)]
+    assert reduced == [sqrt(x0) + x0**2]
+    substs, reduced = cse(Matrix([[1]]))
+    assert isinstance(reduced[0], Matrix)
+
+    subst42, (red42,) = cse(42)  # issue 15082
+    assert len(subst42) == 0 and red42 == 42
+    subst_half, (red_half,) = cse(0.5)  # issue 15082
+    assert len(subst_half) == 0 and red_half == 0.5
+
+
+def test_cse_not_possible():
+    # No substitution possible.
+    e = Add(x, y)
+    substs, reduced = cse([e])
+    assert substs == []
+    assert reduced == [x + y]
+    # issue 6329
+    eq = (meijerg((1, 2), (y, 4), (5,), [], x) +
+          meijerg((1, 3), (y, 4), (5,), [], x))
+    assert cse(eq) == ([], [eq])
+
+
+def test_nested_substitution():
+    # Substitution within a substitution.
+    e = Add(Pow(w*x + y, 2), sqrt(w*x + y))
+    substs, reduced = cse([e])
+    assert substs == [(x0, w*x + y)]
+    assert reduced == [sqrt(x0) + x0**2]
+
+
+def test_subtraction_opt():
+    # Make sure subtraction is optimized.
+    e = (x - y)*(z - y) + exp((x - y)*(z - y))
+    substs, reduced = cse(
+        [e], optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)])
+    assert substs == [(x0, (x - y)*(y - z))]
+    assert reduced == [-x0 + exp(-x0)]
+    e = -(x - y)*(z - y) + exp(-(x - y)*(z - y))
+    substs, reduced = cse(
+        [e], optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)])
+    assert substs == [(x0, (x - y)*(y - z))]
+    assert reduced == [x0 + exp(x0)]
+    # issue 4077
+    n = -1 + 1/x
+    e = n/x/(-n)**2 - 1/n/x
+    assert cse(e, optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)]) == \
+        ([], [0])
+    assert cse(((w + x + y + z)*(w - y - z))/(w + x)**3) == \
+        ([(x0, w + x), (x1, y + z)], [(w - x1)*(x0 + x1)/x0**3])
+
+
+def test_multiple_expressions():
+    e1 = (x + y)*z
+    e2 = (x + y)*w
+    substs, reduced = cse([e1, e2])
+    assert substs == [(x0, x + y)]
+    assert reduced == [x0*z, x0*w]
+    l = [w*x*y + z, w*y]
+    substs, reduced = cse(l)
+    rsubsts, _ = cse(reversed(l))
+    assert substs == rsubsts
+    assert reduced == [z + x*x0, x0]
+    l = [w*x*y, w*x*y + z, w*y]
+    substs, reduced = cse(l)
+    rsubsts, _ = cse(reversed(l))
+    assert substs == rsubsts
+    assert reduced == [x1, x1 + z, x0]
+    l = [(x - z)*(y - z), x - z, y - z]
+    substs, reduced = cse(l)
+    rsubsts, _ = cse(reversed(l))
+    assert substs == [(x0, -z), (x1, x + x0), (x2, x0 + y)]
+    assert rsubsts == [(x0, -z), (x1, x0 + y), (x2, x + x0)]
+    assert reduced == [x1*x2, x1, x2]
+    l = [w*y + w + x + y + z, w*x*y]
+    assert cse(l) == ([(x0, w*y)], [w + x + x0 + y + z, x*x0])
+    assert cse([x + y, x + y + z]) == ([(x0, x + y)], [x0, z + x0])
+    assert cse([x + y, x + z]) == ([], [x + y, x + z])
+    assert cse([x*y, z + x*y, x*y*z + 3]) == \
+        ([(x0, x*y)], [x0, z + x0, 3 + x0*z])
+
+
+@XFAIL # CSE of non-commutative Mul terms is disabled
+def test_non_commutative_cse():
+    A, B, C = symbols('A B C', commutative=False)
+    l = [A*B*C, A*C]
+    assert cse(l) == ([], l)
+    l = [A*B*C, A*B]
+    assert cse(l) == ([(x0, A*B)], [x0*C, x0])
+
+
+# Test if CSE of non-commutative Mul terms is disabled
+def test_bypass_non_commutatives():
+    A, B, C = symbols('A B C', commutative=False)
+    l = [A*B*C, A*C]
+    assert cse(l) == ([], l)
+    l = [A*B*C, A*B]
+    assert cse(l) == ([], l)
+    l = [B*C, A*B*C]
+    assert cse(l) == ([], l)
+
+
+@XFAIL # CSE fails when replacing non-commutative sub-expressions
+def test_non_commutative_order():
+    A, B, C = symbols('A B C', commutative=False)
+    x0 = symbols('x0', commutative=False)
+    l = [B+C, A*(B+C)]
+    assert cse(l) == ([(x0, B+C)], [x0, A*x0])
+
+
+@XFAIL # Worked in gh-11232, but was reverted due to performance considerations
+def test_issue_10228():
+    assert cse([x*y**2 + x*y]) == ([(x0, x*y)], [x0*y + x0])
+    assert cse([x + y, 2*x + y]) == ([(x0, x + y)], [x0, x + x0])
+    assert cse((w + 2*x + y + z, w + x + 1)) == (
+        [(x0, w + x)], [x0 + x + y + z, x0 + 1])
+    assert cse(((w + x + y + z)*(w - x))/(w + x)) == (
+        [(x0, w + x)], [(x0 + y + z)*(w - x)/x0])
+    a, b, c, d, f, g, j, m = symbols('a, b, c, d, f, g, j, m')
+    exprs = (d*g**2*j*m, 4*a*f*g*m, a*b*c*f**2)
+    assert cse(exprs) == (
+        [(x0, g*m), (x1, a*f)], [d*g*j*x0, 4*x0*x1, b*c*f*x1]
+)
+
+@XFAIL
+def test_powers():
+    assert cse(x*y**2 + x*y) == ([(x0, x*y)], [x0*y + x0])
+
+
+def test_issue_4498():
+    assert cse(w/(x - y) + z/(y - x), optimizations='basic') == \
+        ([], [(w - z)/(x - y)])
+
+
+def test_issue_4020():
+    assert cse(x**5 + x**4 + x**3 + x**2, optimizations='basic') \
+        == ([(x0, x**2)], [x0*(x**3 + x + x0 + 1)])
+
+
+def test_issue_4203():
+    assert cse(sin(x**x)/x**x) == ([(x0, x**x)], [sin(x0)/x0])
+
+
+def test_issue_6263():
+    e = Eq(x*(-x + 1) + x*(x - 1), 0)
+    assert cse(e, optimizations='basic') == ([], [True])
+
+
+def test_issue_25043():
+    c = symbols("c")
+    x = symbols("x0", real=True)
+    cse_expr = cse(c*x**2 + c*(x**4 - x**2))[-1][-1]
+    free = cse_expr.free_symbols
+    assert len(free) == len({i.name for i in free})
+
+
+def test_dont_cse_tuples():
+    from sympy.core.function import Subs
+    f = Function("f")
+    g = Function("g")
+
+    name_val, (expr,) = cse(
+        Subs(f(x, y), (x, y), (0, 1))
+        + Subs(g(x, y), (x, y), (0, 1)))
+
+    assert name_val == []
+    assert expr == (Subs(f(x, y), (x, y), (0, 1))
+            + Subs(g(x, y), (x, y), (0, 1)))
+
+    name_val, (expr,) = cse(
+        Subs(f(x, y), (x, y), (0, x + y))
+        + Subs(g(x, y), (x, y), (0, x + y)))
+
+    assert name_val == [(x0, x + y)]
+    assert expr == Subs(f(x, y), (x, y), (0, x0)) + \
+        Subs(g(x, y), (x, y), (0, x0))
+
+
+def test_pow_invpow():
+    assert cse(1/x**2 + x**2) == \
+        ([(x0, x**2)], [x0 + 1/x0])
+    assert cse(x**2 + (1 + 1/x**2)/x**2) == \
+        ([(x0, x**2), (x1, 1/x0)], [x0 + x1*(x1 + 1)])
+    assert cse(1/x**2 + (1 + 1/x**2)*x**2) == \
+        ([(x0, x**2), (x1, 1/x0)], [x0*(x1 + 1) + x1])
+    assert cse(cos(1/x**2) + sin(1/x**2)) == \
+        ([(x0, x**(-2))], [sin(x0) + cos(x0)])
+    assert cse(cos(x**2) + sin(x**2)) == \
+        ([(x0, x**2)], [sin(x0) + cos(x0)])
+    assert cse(y/(2 + x**2) + z/x**2/y) == \
+        ([(x0, x**2)], [y/(x0 + 2) + z/(x0*y)])
+    assert cse(exp(x**2) + x**2*cos(1/x**2)) == \
+        ([(x0, x**2)], [x0*cos(1/x0) + exp(x0)])
+    assert cse((1 + 1/x**2)/x**2) == \
+        ([(x0, x**(-2))], [x0*(x0 + 1)])
+    assert cse(x**(2*y) + x**(-2*y)) == \
+        ([(x0, x**(2*y))], [x0 + 1/x0])
+
+
+def test_postprocess():
+    eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1))
+    assert cse([eq, Eq(x, z + 1), z - 2, (z + 1)*(x + 1)],
+        postprocess=cse_main.cse_separate) == \
+        [[(x0, y + 1), (x2, z + 1), (x, x2), (x1, x + 1)],
+        [x1 + exp(x1/x0) + cos(x0), z - 2, x1*x2]]
+
+
+def test_issue_4499():
+    # previously, this gave 16 constants
+    from sympy.abc import a, b
+    B = Function('B')
+    G = Function('G')
+    t = Tuple(*
+        (a, a + S.Half, 2*a, b, 2*a - b + 1, (sqrt(z)/2)**(-2*a + 1)*B(2*a -
+        b, sqrt(z))*B(b - 1, sqrt(z))*G(b)*G(2*a - b + 1),
+        sqrt(z)*(sqrt(z)/2)**(-2*a + 1)*B(b, sqrt(z))*B(2*a - b,
+        sqrt(z))*G(b)*G(2*a - b + 1), sqrt(z)*(sqrt(z)/2)**(-2*a + 1)*B(b - 1,
+        sqrt(z))*B(2*a - b + 1, sqrt(z))*G(b)*G(2*a - b + 1),
+        (sqrt(z)/2)**(-2*a + 1)*B(b, sqrt(z))*B(2*a - b + 1,
+        sqrt(z))*G(b)*G(2*a - b + 1), 1, 0, S.Half, z/2, -b + 1, -2*a + b,
+        -2*a))
+    c = cse(t)
+    ans = (
+        [(x0, 2*a), (x1, -b + x0), (x2, x1 + 1), (x3, b - 1), (x4, sqrt(z)),
+         (x5, B(x3, x4)), (x6, (x4/2)**(1 - x0)*G(b)*G(x2)), (x7, x6*B(x1, x4)),
+         (x8, B(b, x4)), (x9, x6*B(x2, x4))],
+        [(a, a + S.Half, x0, b, x2, x5*x7, x4*x7*x8, x4*x5*x9, x8*x9,
+          1, 0, S.Half, z/2, -x3, -x1, -x0)])
+    assert ans == c
+
+
+def test_issue_6169():
+    r = CRootOf(x**6 - 4*x**5 - 2, 1)
+    assert cse(r) == ([], [r])
+    # and a check that the right thing is done with the new
+    # mechanism
+    assert sub_post(sub_pre((-x - y)*z - x - y)) == -z*(x + y) - x - y
+
+
+def test_cse_Indexed():
+    len_y = 5
+    y = IndexedBase('y', shape=(len_y,))
+    x = IndexedBase('x', shape=(len_y,))
+    i = Idx('i', len_y-1)
+
+    expr1 = (y[i+1]-y[i])/(x[i+1]-x[i])
+    expr2 = 1/(x[i+1]-x[i])
+    replacements, reduced_exprs = cse([expr1, expr2])
+    assert len(replacements) > 0
+
+
+def test_cse_MatrixSymbol():
+    # MatrixSymbols have non-Basic args, so make sure that works
+    A = MatrixSymbol("A", 3, 3)
+    assert cse(A) == ([], [A])
+
+    n = symbols('n', integer=True)
+    B = MatrixSymbol("B", n, n)
+    assert cse(B) == ([], [B])
+
+    assert cse(A[0] * A[0]) == ([], [A[0]*A[0]])
+
+    assert cse(A[0,0]*A[0,1] + A[0,0]*A[0,1]*A[0,2]) == ([(x0, A[0, 0]*A[0, 1])], [x0*A[0, 2] + x0])
+
+def test_cse_MatrixExpr():
+    A = MatrixSymbol('A', 3, 3)
+    y = MatrixSymbol('y', 3, 1)
+
+    expr1 = (A.T*A).I * A * y
+    expr2 = (A.T*A) * A * y
+    replacements, reduced_exprs = cse([expr1, expr2])
+    assert len(replacements) > 0
+
+    replacements, reduced_exprs = cse([expr1 + expr2, expr1])
+    assert replacements
+
+    replacements, reduced_exprs = cse([A**2, A + A**2])
+    assert replacements
+
+
+def test_Piecewise():
+    f = Piecewise((-z + x*y, Eq(y, 0)), (-z - x*y, True))
+    ans = cse(f)
+    actual_ans = ([(x0, x*y)],
+        [Piecewise((x0 - z, Eq(y, 0)), (-z - x0, True))])
+    assert ans == actual_ans
+
+
+def test_ignore_order_terms():
+    eq = exp(x).series(x,0,3) + sin(y+x**3) - 1
+    assert cse(eq) == ([], [sin(x**3 + y) + x + x**2/2 + O(x**3)])
+
+
+def test_name_conflict():
+    z1 = x0 + y
+    z2 = x2 + x3
+    l = [cos(z1) + z1, cos(z2) + z2, x0 + x2]
+    substs, reduced = cse(l)
+    assert [e.subs(reversed(substs)) for e in reduced] == l
+
+
+def test_name_conflict_cust_symbols():
+    z1 = x0 + y
+    z2 = x2 + x3
+    l = [cos(z1) + z1, cos(z2) + z2, x0 + x2]
+    substs, reduced = cse(l, symbols("x:10"))
+    assert [e.subs(reversed(substs)) for e in reduced] == l
+
+
+def test_symbols_exhausted_error():
+    l = cos(x+y)+x+y+cos(w+y)+sin(w+y)
+    sym = [x, y, z]
+    with raises(ValueError):
+        cse(l, symbols=sym)
+
+
+def test_issue_7840():
+    # daveknippers' example
+    C393 = sympify( \
+        'Piecewise((C391 - 1.65, C390 < 0.5), (Piecewise((C391 - 1.65, \
+        C391 > 2.35), (C392, True)), True))'
+    )
+    C391 = sympify( \
+        'Piecewise((2.05*C390**(-1.03), C390 < 0.5), (2.5*C390**(-0.625), True))'
+    )
+    C393 = C393.subs('C391',C391)
+    # simple substitution
+    sub = {}
+    sub['C390'] = 0.703451854
+    sub['C392'] = 1.01417794
+    ss_answer = C393.subs(sub)
+    # cse
+    substitutions,new_eqn = cse(C393)
+    for pair in substitutions:
+        sub[pair[0].name] = pair[1].subs(sub)
+    cse_answer = new_eqn[0].subs(sub)
+    # both methods should be the same
+    assert ss_answer == cse_answer
+
+    # GitRay's example
+    expr = sympify(
+        "Piecewise((Symbol('ON'), Equality(Symbol('mode'), Symbol('ON'))), \
+        (Piecewise((Piecewise((Symbol('OFF'), StrictLessThan(Symbol('x'), \
+        Symbol('threshold'))), (Symbol('ON'), true)), Equality(Symbol('mode'), \
+        Symbol('AUTO'))), (Symbol('OFF'), true)), true))"
+    )
+    substitutions, new_eqn = cse(expr)
+    # this Piecewise should be exactly the same
+    assert new_eqn[0] == expr
+    # there should not be any replacements
+    assert len(substitutions) < 1
+
+
+def test_issue_8891():
+    for cls in (MutableDenseMatrix, MutableSparseMatrix,
+            ImmutableDenseMatrix, ImmutableSparseMatrix):
+        m = cls(2, 2, [x + y, 0, 0, 0])
+        res = cse([x + y, m])
+        ans = ([(x0, x + y)], [x0, cls([[x0, 0], [0, 0]])])
+        assert res == ans
+        assert isinstance(res[1][-1], cls)
+
+
+def test_issue_11230():
+    # a specific test that always failed
+    a, b, f, k, l, i = symbols('a b f k l i')
+    p = [a*b*f*k*l, a*i*k**2*l, f*i*k**2*l]
+    R, C = cse(p)
+    assert not any(i.is_Mul for a in C for i in a.args)
+
+    # random tests for the issue
+    from sympy.core.random import choice
+    from sympy.core.function import expand_mul
+    s = symbols('a:m')
+    # 35 Mul tests, none of which should ever fail
+    ex = [Mul(*[choice(s) for i in range(5)]) for i in range(7)]
+    for p in subsets(ex, 3):
+        p = list(p)
+        R, C = cse(p)
+        assert not any(i.is_Mul for a in C for i in a.args)
+        for ri in reversed(R):
+            for i in range(len(C)):
+                C[i] = C[i].subs(*ri)
+        assert p == C
+    # 35 Add tests, none of which should ever fail
+    ex = [Add(*[choice(s[:7]) for i in range(5)]) for i in range(7)]
+    for p in subsets(ex, 3):
+        p = list(p)
+        R, C = cse(p)
+        assert not any(i.is_Add for a in C for i in a.args)
+        for ri in reversed(R):
+            for i in range(len(C)):
+                C[i] = C[i].subs(*ri)
+        # use expand_mul to handle cases like this:
+        # p = [a + 2*b + 2*e, 2*b + c + 2*e, b + 2*c + 2*g]
+        # x0 = 2*(b + e) is identified giving a rebuilt p that
+        # is now `[a + 2*(b + e), c + 2*(b + e), b + 2*c + 2*g]`
+        assert p == [expand_mul(i) for i in C]
+
+
+@XFAIL
+def test_issue_11577():
+    def check(eq):
+        r, c = cse(eq)
+        assert eq.count_ops() >= \
+            len(r) + sum(i[1].count_ops() for i in r) + \
+            count_ops(c)
+
+    eq = x**5*y**2 + x**5*y + x**5
+    assert cse(eq) == (
+        [(x0, x**4), (x1, x*y)], [x**5 + x0*x1*y + x0*x1])
+        # ([(x0, x**5*y)], [x0*y + x0 + x**5]) or
+        # ([(x0, x**5)], [x0*y**2 + x0*y + x0])
+    check(eq)
+
+    eq = x**2/(y + 1)**2 + x/(y + 1)
+    assert cse(eq) == (
+        [(x0, y + 1)], [x**2/x0**2 + x/x0])
+        # ([(x0, x/(y + 1))], [x0**2 + x0])
+    check(eq)
+
+
+def test_hollow_rejection():
+    eq = [x + 3, x + 4]
+    assert cse(eq) == ([], eq)
+
+
+def test_cse_ignore():
+    exprs = [exp(y)*(3*y + 3*sqrt(x+1)), exp(y)*(5*y + 5*sqrt(x+1))]
+    subst1, red1 = cse(exprs)
+    assert any(y in sub.free_symbols for _, sub in subst1), "cse failed to identify any term with y"
+
+    subst2, red2 = cse(exprs, ignore=(y,))  # y is not allowed in substitutions
+    assert not any(y in sub.free_symbols for _, sub in subst2), "Sub-expressions containing y must be ignored"
+    assert any(sub - sqrt(x + 1) == 0 for _, sub in subst2), "cse failed to identify sqrt(x + 1) as sub-expression"
+
+def test_cse_ignore_issue_15002():
+    l = [
+        w*exp(x)*exp(-z),
+        exp(y)*exp(x)*exp(-z)
+    ]
+    substs, reduced = cse(l, ignore=(x,))
+    rl = [e.subs(reversed(substs)) for e in reduced]
+    assert rl == l
+
+
+def test_cse_unevaluated():
+    xp1 = UnevaluatedExpr(x + 1)
+    # This used to cause RecursionError
+    [(x0, ue)], [red] = cse([(-1 - xp1) / (1 - xp1)])
+    if ue == xp1:
+        assert red == (-1 - x0) / (1 - x0)
+    elif ue == -xp1:
+        assert red == (-1 + x0) / (1 + x0)
+    else:
+        msg = f'Expected common subexpression {xp1} or {-xp1}, instead got {ue}'
+        assert False, msg
+
+
+def test_cse__performance():
+    nexprs, nterms = 3, 20
+    x = symbols('x:%d' % nterms)
+    exprs = [
+        reduce(add, [x[j]*(-1)**(i+j) for j in range(nterms)])
+        for i in range(nexprs)
+    ]
+    assert (exprs[0] + exprs[1]).simplify() == 0
+    subst, red = cse(exprs)
+    assert len(subst) > 0, "exprs[0] == -exprs[2], i.e. a CSE"
+    for i, e in enumerate(red):
+        assert (e.subs(reversed(subst)) - exprs[i]).simplify() == 0
+
+
+def test_issue_12070():
+    exprs = [x + y, 2 + x + y, x + y + z, 3 + x + y + z]
+    subst, red = cse(exprs)
+    assert 6 >= (len(subst) + sum(v.count_ops() for k, v in subst) +
+                 count_ops(red))
+
+
+def test_issue_13000():
+    eq = x/(-4*x**2 + y**2)
+    cse_eq = cse(eq)[1][0]
+    assert cse_eq == eq
+
+
+def test_issue_18203():
+    eq = CRootOf(x**5 + 11*x - 2, 0) + CRootOf(x**5 + 11*x - 2, 1)
+    assert cse(eq) == ([], [eq])
+
+
+def test_unevaluated_mul():
+    eq = Mul(x + y, x + y, evaluate=False)
+    assert cse(eq) == ([(x0, x + y)], [x0**2])
+
+def test_cse_release_variables():
+    from sympy.simplify.cse_main import cse_release_variables
+    _0, _1, _2, _3, _4 = symbols('_:5')
+    eqs = [(x + y - 1)**2, x,
+        x + y, (x + y)/(2*x + 1) + (x + y - 1)**2,
+        (2*x + 1)**(x + y)]
+    r, e = cse(eqs, postprocess=cse_release_variables)
+    # this can change in keeping with the intention of the function
+    assert r, e == ([
+    (x0, x + y), (x1, (x0 - 1)**2), (x2, 2*x + 1),
+    (_3, x0/x2 + x1), (_4, x2**x0), (x2, None), (_0, x1),
+    (x1, None), (_2, x0), (x0, None), (_1, x)], (_0, _1, _2, _3, _4))
+    r.reverse()
+    r = [(s, v) for s, v in r if v is not None]
+    assert eqs == [i.subs(r) for i in e]
+
+def test_cse_list():
+    _cse = lambda x: cse(x, list=False)
+    assert _cse(x) == ([], x)
+    assert _cse('x') == ([], 'x')
+    it = [x]
+    for c in (list, tuple, set):
+        assert _cse(c(it)) == ([], c(it))
+    #Tuple works different from tuple:
+    assert _cse(Tuple(*it)) == ([], Tuple(*it))
+    d = {x: 1}
+    assert _cse(d) == ([], d)
+
+def test_issue_18991():
+    A = MatrixSymbol('A', 2, 2)
+    assert signsimp(-A * A - A) == -A * A - A
+
+
+def test_unevaluated_Mul():
+    m = [Mul(1, 2, evaluate=False)]
+    assert cse(m) == ([], m)
+
+
+def test_cse_matrix_expression_inverse():
+    A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2)
+    x = Inverse(A)
+    cse_expr = cse(x)
+    assert cse_expr == ([], [Inverse(A)])
+
+
+def test_cse_matrix_expression_matmul_inverse():
+    A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2)
+    b = ImmutableDenseMatrix(symbols('b:2'))
+    x = MatMul(Inverse(A), b)
+    cse_expr = cse(x)
+    assert cse_expr == ([], [x])
+
+
+def test_cse_matrix_negate_matrix():
+    A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2)
+    x = MatMul(S.NegativeOne, A)
+    cse_expr = cse(x)
+    assert cse_expr == ([], [x])
+
+
+def test_cse_matrix_negate_matmul_not_extracted():
+    A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2)
+    B = ImmutableDenseMatrix(symbols('B:4')).reshape(2, 2)
+    x = MatMul(S.NegativeOne, A, B)
+    cse_expr = cse(x)
+    assert cse_expr == ([], [x])
+
+
+@XFAIL  # No simplification rule for nested associative operations
+def test_cse_matrix_nested_matmul_collapsed():
+    A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2)
+    B = ImmutableDenseMatrix(symbols('B:4')).reshape(2, 2)
+    x = MatMul(S.NegativeOne, MatMul(A, B))
+    cse_expr = cse(x)
+    assert cse_expr == ([], [MatMul(S.NegativeOne, A, B)])
+
+
+def test_cse_matrix_optimize_out_single_argument_mul():
+    A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2)
+    x = MatMul(MatMul(MatMul(A)))
+    cse_expr = cse(x)
+    assert cse_expr == ([], [A])
+
+
+@XFAIL  # Multiple simplification passed not supported in CSE
+def test_cse_matrix_optimize_out_single_argument_mul_combined():
+    A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2)
+    x = MatAdd(MatMul(MatMul(MatMul(A))), MatMul(MatMul(A)), MatMul(A), A)
+    cse_expr = cse(x)
+    assert cse_expr == ([], [MatMul(4, A)])
+
+
+def test_cse_matrix_optimize_out_single_argument_add():
+    A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2)
+    x = MatAdd(MatAdd(MatAdd(MatAdd(A))))
+    cse_expr = cse(x)
+    assert cse_expr == ([], [A])
+
+
+@XFAIL  # Multiple simplification passed not supported in CSE
+def test_cse_matrix_optimize_out_single_argument_add_combined():
+    A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2)
+    x = MatMul(MatAdd(MatAdd(MatAdd(A))), MatAdd(MatAdd(A)), MatAdd(A), A)
+    cse_expr = cse(x)
+    assert cse_expr == ([], [MatMul(4, A)])
+
+
+def test_cse_matrix_expression_matrix_solve():
+    A = ImmutableDenseMatrix(symbols('A:4')).reshape(2, 2)
+    b = ImmutableDenseMatrix(symbols('b:2'))
+    x = MatrixSolve(A, b)
+    cse_expr = cse(x)
+    assert cse_expr == ([], [x])
+
+
+def test_cse_matrix_matrix_expression():
+    X = ImmutableDenseMatrix(symbols('X:4')).reshape(2, 2)
+    y = ImmutableDenseMatrix(symbols('y:2'))
+    b = MatMul(Inverse(MatMul(Transpose(X), X)), Transpose(X), y)
+    cse_expr = cse(b)
+    x0 = MatrixSymbol('x0', 2, 2)
+    reduced_expr_expected = MatMul(Inverse(MatMul(x0, X)), x0, y)
+    assert cse_expr == ([(x0, Transpose(X))], [reduced_expr_expected])
+
+
+def test_cse_matrix_kalman_filter():
+    """Kalman Filter example from Matthew Rocklin's SciPy 2013 talk.
+
+    Talk titled: "Matrix Expressions and BLAS/LAPACK; SciPy 2013 Presentation"
+
+    Video: https://pyvideo.org/scipy-2013/matrix-expressions-and-blaslapack-scipy-2013-pr.html
+
+    Notes
+    =====
+
+    Equations are:
+
+    new_mu = mu + Sigma*H.T * (R + H*Sigma*H.T).I * (H*mu - data)
+           = MatAdd(mu, MatMul(Sigma, Transpose(H), Inverse(MatAdd(R, MatMul(H, Sigma, Transpose(H)))), MatAdd(MatMul(H, mu), MatMul(S.NegativeOne, data))))
+    new_Sigma = Sigma - Sigma*H.T * (R + H*Sigma*H.T).I * H * Sigma
+              = MatAdd(Sigma, MatMul(S.NegativeOne, Sigma, Transpose(H)), Inverse(MatAdd(R, MatMul(H*Sigma*Transpose(H)))), H, Sigma))
+
+    """
+    N = 2
+    mu = ImmutableDenseMatrix(symbols(f'mu:{N}'))
+    Sigma = ImmutableDenseMatrix(symbols(f'Sigma:{N * N}')).reshape(N, N)
+    H = ImmutableDenseMatrix(symbols(f'H:{N * N}')).reshape(N, N)
+    R = ImmutableDenseMatrix(symbols(f'R:{N * N}')).reshape(N, N)
+    data = ImmutableDenseMatrix(symbols(f'data:{N}'))
+    new_mu = MatAdd(mu, MatMul(Sigma, Transpose(H), Inverse(MatAdd(R, MatMul(H, Sigma, Transpose(H)))), MatAdd(MatMul(H, mu), MatMul(S.NegativeOne, data))))
+    new_Sigma = MatAdd(Sigma, MatMul(S.NegativeOne, Sigma, Transpose(H), Inverse(MatAdd(R, MatMul(H, Sigma, Transpose(H)))), H, Sigma))
+    cse_expr = cse([new_mu, new_Sigma])
+    x0 = MatrixSymbol('x0', N, N)
+    x1 = MatrixSymbol('x1', N, N)
+    replacements_expected = [
+        (x0, Transpose(H)),
+        (x1, Inverse(MatAdd(R, MatMul(H, Sigma, x0)))),
+    ]
+    reduced_exprs_expected = [
+        MatAdd(mu, MatMul(Sigma, x0, x1, MatAdd(MatMul(H, mu), MatMul(S.NegativeOne, data)))),
+        MatAdd(Sigma, MatMul(S.NegativeOne, Sigma, x0, x1, H, Sigma)),
+    ]
+    assert cse_expr == (replacements_expected, reduced_exprs_expected)

rl/Lib/site-packages/sympy/simplify/tests/test_epathtools.py

@@ -0,0 +1,90 @@
+"""Tests for tools for manipulation of expressions using paths. """
+
+from sympy.simplify.epathtools import epath, EPath
+from sympy.testing.pytest import raises
+
+from sympy.core.numbers import E
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.abc import x, y, z, t
+
+
+def test_epath_select():
+    expr = [((x, 1, t), 2), ((3, y, 4), z)]
+
+    assert epath("/*", expr) == [((x, 1, t), 2), ((3, y, 4), z)]
+    assert epath("/*/*", expr) == [(x, 1, t), 2, (3, y, 4), z]
+    assert epath("/*/*/*", expr) == [x, 1, t, 3, y, 4]
+    assert epath("/*/*/*/*", expr) == []
+
+    assert epath("/[:]", expr) == [((x, 1, t), 2), ((3, y, 4), z)]
+    assert epath("/[:]/[:]", expr) == [(x, 1, t), 2, (3, y, 4), z]
+    assert epath("/[:]/[:]/[:]", expr) == [x, 1, t, 3, y, 4]
+    assert epath("/[:]/[:]/[:]/[:]", expr) == []
+
+    assert epath("/*/[:]", expr) == [(x, 1, t), 2, (3, y, 4), z]
+
+    assert epath("/*/[0]", expr) == [(x, 1, t), (3, y, 4)]
+    assert epath("/*/[1]", expr) == [2, z]
+    assert epath("/*/[2]", expr) == []
+
+    assert epath("/*/int", expr) == [2]
+    assert epath("/*/Symbol", expr) == [z]
+    assert epath("/*/tuple", expr) == [(x, 1, t), (3, y, 4)]
+    assert epath("/*/__iter__?", expr) == [(x, 1, t), (3, y, 4)]
+
+    assert epath("/*/int|tuple", expr) == [(x, 1, t), 2, (3, y, 4)]
+    assert epath("/*/Symbol|tuple", expr) == [(x, 1, t), (3, y, 4), z]
+    assert epath("/*/int|Symbol|tuple", expr) == [(x, 1, t), 2, (3, y, 4), z]
+
+    assert epath("/*/int|__iter__?", expr) == [(x, 1, t), 2, (3, y, 4)]
+    assert epath("/*/Symbol|__iter__?", expr) == [(x, 1, t), (3, y, 4), z]
+    assert epath(
+        "/*/int|Symbol|__iter__?", expr) == [(x, 1, t), 2, (3, y, 4), z]
+
+    assert epath("/*/[0]/int", expr) == [1, 3, 4]
+    assert epath("/*/[0]/Symbol", expr) == [x, t, y]
+
+    assert epath("/*/[0]/int[1:]", expr) == [1, 4]
+    assert epath("/*/[0]/Symbol[1:]", expr) == [t, y]
+
+    assert epath("/Symbol", x + y + z + 1) == [x, y, z]
+    assert epath("/*/*/Symbol", t + sin(x + 1) + cos(x + y + E)) == [x, x, y]
+
+
+def test_epath_apply():
+    expr = [((x, 1, t), 2), ((3, y, 4), z)]
+    func = lambda expr: expr**2
+
+    assert epath("/*", expr, list) == [[(x, 1, t), 2], [(3, y, 4), z]]
+
+    assert epath("/*/[0]", expr, list) == [([x, 1, t], 2), ([3, y, 4], z)]
+    assert epath("/*/[1]", expr, func) == [((x, 1, t), 4), ((3, y, 4), z**2)]
+    assert epath("/*/[2]", expr, list) == expr
+
+    assert epath("/*/[0]/int", expr, func) == [((x, 1, t), 2), ((9, y, 16), z)]
+    assert epath("/*/[0]/Symbol", expr, func) == [((x**2, 1, t**2), 2),
+                 ((3, y**2, 4), z)]
+    assert epath(
+        "/*/[0]/int[1:]", expr, func) == [((x, 1, t), 2), ((3, y, 16), z)]
+    assert epath("/*/[0]/Symbol[1:]", expr, func) == [((x, 1, t**2),
+                 2), ((3, y**2, 4), z)]
+
+    assert epath("/Symbol", x + y + z + 1, func) == x**2 + y**2 + z**2 + 1
+    assert epath("/*/*/Symbol", t + sin(x + 1) + cos(x + y + E), func) == \
+        t + sin(x**2 + 1) + cos(x**2 + y**2 + E)
+
+
+def test_EPath():
+    assert EPath("/*/[0]")._path == "/*/[0]"
+    assert EPath(EPath("/*/[0]"))._path == "/*/[0]"
+    assert isinstance(epath("/*/[0]"), EPath) is True
+
+    assert repr(EPath("/*/[0]")) == "EPath('/*/[0]')"
+
+    raises(ValueError, lambda: EPath(""))
+    raises(ValueError, lambda: EPath("/"))
+    raises(ValueError, lambda: EPath("/|x"))
+    raises(ValueError, lambda: EPath("/["))
+    raises(ValueError, lambda: EPath("/[0]%"))
+
+    raises(NotImplementedError, lambda: EPath("Symbol"))

rl/Lib/site-packages/sympy/simplify/tests/test_fu.py

@@ -0,0 +1,492 @@
+from sympy.core.add import Add
+from sympy.core.mul import Mul
+from sympy.core.numbers import (I, Rational, pi)
+from sympy.core.parameters import evaluate
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.functions.elementary.hyperbolic import (cosh, coth, csch, sech, sinh, tanh)
+from sympy.functions.elementary.miscellaneous import (root, sqrt)
+from sympy.functions.elementary.trigonometric import (cos, cot, csc, sec, sin, tan)
+from sympy.simplify.powsimp import powsimp
+from sympy.simplify.fu import (
+    L, TR1, TR10, TR10i, TR11, _TR11, TR12, TR12i, TR13, TR14, TR15, TR16,
+    TR111, TR2, TR2i, TR3, TR4, TR5, TR6, TR7, TR8, TR9, TRmorrie, _TR56 as T,
+    TRpower, hyper_as_trig, fu, process_common_addends, trig_split,
+    as_f_sign_1)
+from sympy.core.random import verify_numerically
+from sympy.abc import a, b, c, x, y, z
+
+
+def test_TR1():
+    assert TR1(2*csc(x) + sec(x)) == 1/cos(x) + 2/sin(x)
+
+
+def test_TR2():
+    assert TR2(tan(x)) == sin(x)/cos(x)
+    assert TR2(cot(x)) == cos(x)/sin(x)
+    assert TR2(tan(tan(x) - sin(x)/cos(x))) == 0
+
+
+def test_TR2i():
+    # just a reminder that ratios of powers only simplify if both
+    # numerator and denominator satisfy the condition that each
+    # has a positive base or an integer exponent; e.g. the following,
+    # at y=-1, x=1/2 gives sqrt(2)*I != -sqrt(2)*I
+    assert powsimp(2**x/y**x) != (2/y)**x
+
+    assert TR2i(sin(x)/cos(x)) == tan(x)
+    assert TR2i(sin(x)*sin(y)/cos(x)) == tan(x)*sin(y)
+    assert TR2i(1/(sin(x)/cos(x))) == 1/tan(x)
+    assert TR2i(1/(sin(x)*sin(y)/cos(x))) == 1/tan(x)/sin(y)
+    assert TR2i(sin(x)/2/(cos(x) + 1)) == sin(x)/(cos(x) + 1)/2
+
+    assert TR2i(sin(x)/2/(cos(x) + 1), half=True) == tan(x/2)/2
+    assert TR2i(sin(1)/(cos(1) + 1), half=True) == tan(S.Half)
+    assert TR2i(sin(2)/(cos(2) + 1), half=True) == tan(1)
+    assert TR2i(sin(4)/(cos(4) + 1), half=True) == tan(2)
+    assert TR2i(sin(5)/(cos(5) + 1), half=True) == tan(5*S.Half)
+    assert TR2i((cos(1) + 1)/sin(1), half=True) == 1/tan(S.Half)
+    assert TR2i((cos(2) + 1)/sin(2), half=True) == 1/tan(1)
+    assert TR2i((cos(4) + 1)/sin(4), half=True) == 1/tan(2)
+    assert TR2i((cos(5) + 1)/sin(5), half=True) == 1/tan(5*S.Half)
+    assert TR2i((cos(1) + 1)**(-a)*sin(1)**a, half=True) == tan(S.Half)**a
+    assert TR2i((cos(2) + 1)**(-a)*sin(2)**a, half=True) == tan(1)**a
+    assert TR2i((cos(4) + 1)**(-a)*sin(4)**a, half=True) == (cos(4) + 1)**(-a)*sin(4)**a
+    assert TR2i((cos(5) + 1)**(-a)*sin(5)**a, half=True) == (cos(5) + 1)**(-a)*sin(5)**a
+    assert TR2i((cos(1) + 1)**a*sin(1)**(-a), half=True) == tan(S.Half)**(-a)
+    assert TR2i((cos(2) + 1)**a*sin(2)**(-a), half=True) == tan(1)**(-a)
+    assert TR2i((cos(4) + 1)**a*sin(4)**(-a), half=True) == (cos(4) + 1)**a*sin(4)**(-a)
+    assert TR2i((cos(5) + 1)**a*sin(5)**(-a), half=True) == (cos(5) + 1)**a*sin(5)**(-a)
+
+    i = symbols('i', integer=True)
+    assert TR2i(((cos(5) + 1)**i*sin(5)**(-i)), half=True) == tan(5*S.Half)**(-i)
+    assert TR2i(1/((cos(5) + 1)**i*sin(5)**(-i)), half=True) == tan(5*S.Half)**i
+
+
+def test_TR3():
+    assert TR3(cos(y - x*(y - x))) == cos(x*(x - y) + y)
+    assert cos(pi/2 + x) == -sin(x)
+    assert cos(30*pi/2 + x) == -cos(x)
+
+    for f in (cos, sin, tan, cot, csc, sec):
+        i = f(pi*Rational(3, 7))
+        j = TR3(i)
+        assert verify_numerically(i, j) and i.func != j.func
+
+    with evaluate(False):
+        eq = cos(9*pi/22)
+    assert eq.has(9*pi) and TR3(eq) == sin(pi/11)
+
+
+def test_TR4():
+    for i in [0, pi/6, pi/4, pi/3, pi/2]:
+        with evaluate(False):
+            eq = cos(i)
+        assert isinstance(eq, cos) and TR4(eq) == cos(i)
+
+
+def test__TR56():
+    h = lambda x: 1 - x
+    assert T(sin(x)**3, sin, cos, h, 4, False) == sin(x)*(-cos(x)**2 + 1)
+    assert T(sin(x)**10, sin, cos, h, 4, False) == sin(x)**10
+    assert T(sin(x)**6, sin, cos, h, 6, False) == (-cos(x)**2 + 1)**3
+    assert T(sin(x)**6, sin, cos, h, 6, True) == sin(x)**6
+    assert T(sin(x)**8, sin, cos, h, 10, True) == (-cos(x)**2 + 1)**4
+
+    # issue 17137
+    assert T(sin(x)**I, sin, cos, h, 4, True) == sin(x)**I
+    assert T(sin(x)**(2*I + 1), sin, cos, h, 4, True) == sin(x)**(2*I + 1)
+
+
+def test_TR5():
+    assert TR5(sin(x)**2) == -cos(x)**2 + 1
+    assert TR5(sin(x)**-2) == sin(x)**(-2)
+    assert TR5(sin(x)**4) == (-cos(x)**2 + 1)**2
+
+
+def test_TR6():
+    assert TR6(cos(x)**2) == -sin(x)**2 + 1
+    assert TR6(cos(x)**-2) == cos(x)**(-2)
+    assert TR6(cos(x)**4) == (-sin(x)**2 + 1)**2
+
+
+def test_TR7():
+    assert TR7(cos(x)**2) == cos(2*x)/2 + S.Half
+    assert TR7(cos(x)**2 + 1) == cos(2*x)/2 + Rational(3, 2)
+
+
+def test_TR8():
+    assert TR8(cos(2)*cos(3)) == cos(5)/2 + cos(1)/2
+    assert TR8(cos(2)*sin(3)) == sin(5)/2 + sin(1)/2
+    assert TR8(sin(2)*sin(3)) == -cos(5)/2 + cos(1)/2
+    assert TR8(sin(1)*sin(2)*sin(3)) == sin(4)/4 - sin(6)/4 + sin(2)/4
+    assert TR8(cos(2)*cos(3)*cos(4)*cos(5)) == \
+        cos(4)/4 + cos(10)/8 + cos(2)/8 + cos(8)/8 + cos(14)/8 + \
+        cos(6)/8 + Rational(1, 8)
+    assert TR8(cos(2)*cos(3)*cos(4)*cos(5)*cos(6)) == \
+        cos(10)/8 + cos(4)/8 + 3*cos(2)/16 + cos(16)/16 + cos(8)/8 + \
+        cos(14)/16 + cos(20)/16 + cos(12)/16 + Rational(1, 16) + cos(6)/8
+    assert TR8(sin(pi*Rational(3, 7))**2*cos(pi*Rational(3, 7))**2/(16*sin(pi/7)**2)) == Rational(1, 64)
+
+def test_TR9():
+    a = S.Half
+    b = 3*a
+    assert TR9(a) == a
+    assert TR9(cos(1) + cos(2)) == 2*cos(a)*cos(b)
+    assert TR9(cos(1) - cos(2)) == 2*sin(a)*sin(b)
+    assert TR9(sin(1) - sin(2)) == -2*sin(a)*cos(b)
+    assert TR9(sin(1) + sin(2)) == 2*sin(b)*cos(a)
+    assert TR9(cos(1) + 2*sin(1) + 2*sin(2)) == cos(1) + 4*sin(b)*cos(a)
+    assert TR9(cos(4) + cos(2) + 2*cos(1)*cos(3)) == 4*cos(1)*cos(3)
+    assert TR9((cos(4) + cos(2))/cos(3)/2 + cos(3)) == 2*cos(1)*cos(2)
+    assert TR9(cos(3) + cos(4) + cos(5) + cos(6)) == \
+        4*cos(S.Half)*cos(1)*cos(Rational(9, 2))
+    assert TR9(cos(3) + cos(3)*cos(2)) == cos(3) + cos(2)*cos(3)
+    assert TR9(-cos(y) + cos(x*y)) == -2*sin(x*y/2 - y/2)*sin(x*y/2 + y/2)
+    assert TR9(-sin(y) + sin(x*y)) == 2*sin(x*y/2 - y/2)*cos(x*y/2 + y/2)
+    c = cos(x)
+    s = sin(x)
+    for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)):
+        for a in ((c, s), (s, c), (cos(x), cos(x*y)), (sin(x), sin(x*y))):
+            args = zip(si, a)
+            ex = Add(*[Mul(*ai) for ai in args])
+            t = TR9(ex)
+            assert not (a[0].func == a[1].func and (
+                not verify_numerically(ex, t.expand(trig=True)) or t.is_Add)
+                or a[1].func != a[0].func and ex != t)
+
+
+def test_TR10():
+    assert TR10(cos(a + b)) == -sin(a)*sin(b) + cos(a)*cos(b)
+    assert TR10(sin(a + b)) == sin(a)*cos(b) + sin(b)*cos(a)
+    assert TR10(sin(a + b + c)) == \
+        (-sin(a)*sin(b) + cos(a)*cos(b))*sin(c) + \
+        (sin(a)*cos(b) + sin(b)*cos(a))*cos(c)
+    assert TR10(cos(a + b + c)) == \
+        (-sin(a)*sin(b) + cos(a)*cos(b))*cos(c) - \
+        (sin(a)*cos(b) + sin(b)*cos(a))*sin(c)
+
+
+def test_TR10i():
+    assert TR10i(cos(1)*cos(3) + sin(1)*sin(3)) == cos(2)
+    assert TR10i(cos(1)*cos(3) - sin(1)*sin(3)) == cos(4)
+    assert TR10i(cos(1)*sin(3) - sin(1)*cos(3)) == sin(2)
+    assert TR10i(cos(1)*sin(3) + sin(1)*cos(3)) == sin(4)
+    assert TR10i(cos(1)*sin(3) + sin(1)*cos(3) + 7) == sin(4) + 7
+    assert TR10i(cos(1)*sin(3) + sin(1)*cos(3) + cos(3)) == cos(3) + sin(4)
+    assert TR10i(2*cos(1)*sin(3) + 2*sin(1)*cos(3) + cos(3)) == \
+        2*sin(4) + cos(3)
+    assert TR10i(cos(2)*cos(3) + sin(2)*(cos(1)*sin(2) + cos(2)*sin(1))) == \
+        cos(1)
+    eq = (cos(2)*cos(3) + sin(2)*(
+        cos(1)*sin(2) + cos(2)*sin(1)))*cos(5) + sin(1)*sin(5)
+    assert TR10i(eq) == TR10i(eq.expand()) == cos(4)
+    assert TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x) == \
+        2*sqrt(2)*x*sin(x + pi/6)
+    assert TR10i(cos(x)/sqrt(6) + sin(x)/sqrt(2) +
+            cos(x)/sqrt(6)/3 + sin(x)/sqrt(2)/3) == 4*sqrt(6)*sin(x + pi/6)/9
+    assert TR10i(cos(x)/sqrt(6) + sin(x)/sqrt(2) +
+            cos(y)/sqrt(6)/3 + sin(y)/sqrt(2)/3) == \
+        sqrt(6)*sin(x + pi/6)/3 + sqrt(6)*sin(y + pi/6)/9
+    assert TR10i(cos(x) + sqrt(3)*sin(x) + 2*sqrt(3)*cos(x + pi/6)) == 4*cos(x)
+    assert TR10i(cos(x) + sqrt(3)*sin(x) +
+            2*sqrt(3)*cos(x + pi/6) + 4*sin(x)) == 4*sqrt(2)*sin(x + pi/4)
+    assert TR10i(cos(2)*sin(3) + sin(2)*cos(4)) == \
+        sin(2)*cos(4) + sin(3)*cos(2)
+
+    A = Symbol('A', commutative=False)
+    assert TR10i(sqrt(2)*cos(x)*A + sqrt(6)*sin(x)*A) == \
+        2*sqrt(2)*sin(x + pi/6)*A
+
+
+    c = cos(x)
+    s = sin(x)
+    h = sin(y)
+    r = cos(y)
+    for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)):
+        for argsi in ((c*r, s*h), (c*h, s*r)): # explicit 2-args
+            args = zip(si, argsi)
+            ex = Add(*[Mul(*ai) for ai in args])
+            t = TR10i(ex)
+            assert not (ex - t.expand(trig=True) or t.is_Add)
+
+    c = cos(x)
+    s = sin(x)
+    h = sin(pi/6)
+    r = cos(pi/6)
+    for si in ((1, 1), (1, -1), (-1, 1), (-1, -1)):
+        for argsi in ((c*r, s*h), (c*h, s*r)): # induced
+            args = zip(si, argsi)
+            ex = Add(*[Mul(*ai) for ai in args])
+            t = TR10i(ex)
+            assert not (ex - t.expand(trig=True) or t.is_Add)
+
+
+def test_TR11():
+
+    assert TR11(sin(2*x)) == 2*sin(x)*cos(x)
+    assert TR11(sin(4*x)) == 4*((-sin(x)**2 + cos(x)**2)*sin(x)*cos(x))
+    assert TR11(sin(x*Rational(4, 3))) == \
+        4*((-sin(x/3)**2 + cos(x/3)**2)*sin(x/3)*cos(x/3))
+
+    assert TR11(cos(2*x)) == -sin(x)**2 + cos(x)**2
+    assert TR11(cos(4*x)) == \
+        (-sin(x)**2 + cos(x)**2)**2 - 4*sin(x)**2*cos(x)**2
+
+    assert TR11(cos(2)) == cos(2)
+
+    assert TR11(cos(pi*Rational(3, 7)), pi*Rational(2, 7)) == -cos(pi*Rational(2, 7))**2 + sin(pi*Rational(2, 7))**2
+    assert TR11(cos(4), 2) == -sin(2)**2 + cos(2)**2
+    assert TR11(cos(6), 2) == cos(6)
+    assert TR11(sin(x)/cos(x/2), x/2) == 2*sin(x/2)
+
+def test__TR11():
+
+    assert _TR11(sin(x/3)*sin(2*x)*sin(x/4)/(cos(x/6)*cos(x/8))) == \
+        4*sin(x/8)*sin(x/6)*sin(2*x),_TR11(sin(x/3)*sin(2*x)*sin(x/4)/(cos(x/6)*cos(x/8)))
+    assert _TR11(sin(x/3)/cos(x/6)) == 2*sin(x/6)
+
+    assert _TR11(cos(x/6)/sin(x/3)) == 1/(2*sin(x/6))
+    assert _TR11(sin(2*x)*cos(x/8)/sin(x/4)) == sin(2*x)/(2*sin(x/8)), _TR11(sin(2*x)*cos(x/8)/sin(x/4))
+    assert _TR11(sin(x)/sin(x/2)) == 2*cos(x/2)
+
+
+def test_TR12():
+    assert TR12(tan(x + y)) == (tan(x) + tan(y))/(-tan(x)*tan(y) + 1)
+    assert TR12(tan(x + y + z)) ==\
+        (tan(z) + (tan(x) + tan(y))/(-tan(x)*tan(y) + 1))/(
+        1 - (tan(x) + tan(y))*tan(z)/(-tan(x)*tan(y) + 1))
+    assert TR12(tan(x*y)) == tan(x*y)
+
+
+def test_TR13():
+    assert TR13(tan(3)*tan(2)) == -tan(2)/tan(5) - tan(3)/tan(5) + 1
+    assert TR13(cot(3)*cot(2)) == 1 + cot(3)*cot(5) + cot(2)*cot(5)
+    assert TR13(tan(1)*tan(2)*tan(3)) == \
+        (-tan(2)/tan(5) - tan(3)/tan(5) + 1)*tan(1)
+    assert TR13(tan(1)*tan(2)*cot(3)) == \
+        (-tan(2)/tan(3) + 1 - tan(1)/tan(3))*cot(3)
+
+
+def test_L():
+    assert L(cos(x) + sin(x)) == 2
+
+
+def test_fu():
+
+    assert fu(sin(50)**2 + cos(50)**2 + sin(pi/6)) == Rational(3, 2)
+    assert fu(sqrt(6)*cos(x) + sqrt(2)*sin(x)) == 2*sqrt(2)*sin(x + pi/3)
+
+
+    eq = sin(x)**4 - cos(y)**2 + sin(y)**2 + 2*cos(x)**2
+    assert fu(eq) == cos(x)**4 - 2*cos(y)**2 + 2
+
+    assert fu(S.Half - cos(2*x)/2) == sin(x)**2
+
+    assert fu(sin(a)*(cos(b) - sin(b)) + cos(a)*(sin(b) + cos(b))) == \
+        sqrt(2)*sin(a + b + pi/4)
+
+    assert fu(sqrt(3)*cos(x)/2 + sin(x)/2) == sin(x + pi/3)
+
+    assert fu(1 - sin(2*x)**2/4 - sin(y)**2 - cos(x)**4) == \
+        -cos(x)**2 + cos(y)**2
+
+    assert fu(cos(pi*Rational(4, 9))) == sin(pi/18)
+    assert fu(cos(pi/9)*cos(pi*Rational(2, 9))*cos(pi*Rational(3, 9))*cos(pi*Rational(4, 9))) == Rational(1, 16)
+
+    assert fu(
+        tan(pi*Rational(7, 18)) + tan(pi*Rational(5, 18)) - sqrt(3)*tan(pi*Rational(5, 18))*tan(pi*Rational(7, 18))) == \
+        -sqrt(3)
+
+    assert fu(tan(1)*tan(2)) == tan(1)*tan(2)
+
+    expr = Mul(*[cos(2**i) for i in range(10)])
+    assert fu(expr) == sin(1024)/(1024*sin(1))
+
+    # issue #18059:
+    assert fu(cos(x) + sqrt(sin(x)**2)) == cos(x) + sqrt(sin(x)**2)
+
+    assert fu((-14*sin(x)**3 + 35*sin(x) + 6*sqrt(3)*cos(x)**3 + 9*sqrt(3)*cos(x))/((cos(2*x) + 4))) == \
+        7*sin(x) + 3*sqrt(3)*cos(x)
+
+
+def test_objective():
+    assert fu(sin(x)/cos(x), measure=lambda x: x.count_ops()) == \
+            tan(x)
+    assert fu(sin(x)/cos(x), measure=lambda x: -x.count_ops()) == \
+            sin(x)/cos(x)
+
+
+def test_process_common_addends():
+    # this tests that the args are not evaluated as they are given to do
+    # and that key2 works when key1 is False
+    do = lambda x: Add(*[i**(i%2) for i in x.args])
+    assert process_common_addends(Add(*[1, 2, 3, 4], evaluate=False), do,
+        key2=lambda x: x%2, key1=False) == 1**1 + 3**1 + 2**0 + 4**0
+
+
+def test_trig_split():
+    assert trig_split(cos(x), cos(y)) == (1, 1, 1, x, y, True)
+    assert trig_split(2*cos(x), -2*cos(y)) == (2, 1, -1, x, y, True)
+    assert trig_split(cos(x)*sin(y), cos(y)*sin(y)) == \
+        (sin(y), 1, 1, x, y, True)
+
+    assert trig_split(cos(x), -sqrt(3)*sin(x), two=True) == \
+        (2, 1, -1, x, pi/6, False)
+    assert trig_split(cos(x), sin(x), two=True) == \
+        (sqrt(2), 1, 1, x, pi/4, False)
+    assert trig_split(cos(x), -sin(x), two=True) == \
+        (sqrt(2), 1, -1, x, pi/4, False)
+    assert trig_split(sqrt(2)*cos(x), -sqrt(6)*sin(x), two=True) == \
+        (2*sqrt(2), 1, -1, x, pi/6, False)
+    assert trig_split(-sqrt(6)*cos(x), -sqrt(2)*sin(x), two=True) == \
+        (-2*sqrt(2), 1, 1, x, pi/3, False)
+    assert trig_split(cos(x)/sqrt(6), sin(x)/sqrt(2), two=True) == \
+        (sqrt(6)/3, 1, 1, x, pi/6, False)
+    assert trig_split(-sqrt(6)*cos(x)*sin(y),
+            -sqrt(2)*sin(x)*sin(y), two=True) == \
+        (-2*sqrt(2)*sin(y), 1, 1, x, pi/3, False)
+
+    assert trig_split(cos(x), sin(x)) is None
+    assert trig_split(cos(x), sin(z)) is None
+    assert trig_split(2*cos(x), -sin(x)) is None
+    assert trig_split(cos(x), -sqrt(3)*sin(x)) is None
+    assert trig_split(cos(x)*cos(y), sin(x)*sin(z)) is None
+    assert trig_split(cos(x)*cos(y), sin(x)*sin(y)) is None
+    assert trig_split(-sqrt(6)*cos(x), sqrt(2)*sin(x)*sin(y), two=True) is \
+        None
+
+    assert trig_split(sqrt(3)*sqrt(x), cos(3), two=True) is None
+    assert trig_split(sqrt(3)*root(x, 3), sin(3)*cos(2), two=True) is None
+    assert trig_split(cos(5)*cos(6), cos(7)*sin(5), two=True) is None
+
+
+def test_TRmorrie():
+    assert TRmorrie(7*Mul(*[cos(i) for i in range(10)])) == \
+        7*sin(12)*sin(16)*cos(5)*cos(7)*cos(9)/(64*sin(1)*sin(3))
+    assert TRmorrie(x) == x
+    assert TRmorrie(2*x) == 2*x
+    e = cos(pi/7)*cos(pi*Rational(2, 7))*cos(pi*Rational(4, 7))
+    assert TR8(TRmorrie(e)) == Rational(-1, 8)
+    e = Mul(*[cos(2**i*pi/17) for i in range(1, 17)])
+    assert TR8(TR3(TRmorrie(e))) == Rational(1, 65536)
+    # issue 17063
+    eq = cos(x)/cos(x/2)
+    assert TRmorrie(eq) == eq
+    # issue #20430
+    eq = cos(x/2)*sin(x/2)*cos(x)**3
+    assert TRmorrie(eq) == sin(2*x)*cos(x)**2/4
+
+
+def test_TRpower():
+    assert TRpower(1/sin(x)**2) == 1/sin(x)**2
+    assert TRpower(cos(x)**3*sin(x/2)**4) == \
+        (3*cos(x)/4 + cos(3*x)/4)*(-cos(x)/2 + cos(2*x)/8 + Rational(3, 8))
+    for k in range(2, 8):
+        assert verify_numerically(sin(x)**k, TRpower(sin(x)**k))
+        assert verify_numerically(cos(x)**k, TRpower(cos(x)**k))
+
+
+def test_hyper_as_trig():
+    from sympy.simplify.fu import _osborne, _osbornei
+
+    eq = sinh(x)**2 + cosh(x)**2
+    t, f = hyper_as_trig(eq)
+    assert f(fu(t)) == cosh(2*x)
+    e, f = hyper_as_trig(tanh(x + y))
+    assert f(TR12(e)) == (tanh(x) + tanh(y))/(tanh(x)*tanh(y) + 1)
+
+    d = Dummy()
+    assert _osborne(sinh(x), d) == I*sin(x*d)
+    assert _osborne(tanh(x), d) == I*tan(x*d)
+    assert _osborne(coth(x), d) == cot(x*d)/I
+    assert _osborne(cosh(x), d) == cos(x*d)
+    assert _osborne(sech(x), d) == sec(x*d)
+    assert _osborne(csch(x), d) == csc(x*d)/I
+    for func in (sinh, cosh, tanh, coth, sech, csch):
+        h = func(pi)
+        assert _osbornei(_osborne(h, d), d) == h
+    # /!\ the _osborne functions are not meant to work
+    # in the o(i(trig, d), d) direction so we just check
+    # that they work as they are supposed to work
+    assert _osbornei(cos(x*y + z), y) == cosh(x + z*I)
+    assert _osbornei(sin(x*y + z), y) == sinh(x + z*I)/I
+    assert _osbornei(tan(x*y + z), y) == tanh(x + z*I)/I
+    assert _osbornei(cot(x*y + z), y) == coth(x + z*I)*I
+    assert _osbornei(sec(x*y + z), y) == sech(x + z*I)
+    assert _osbornei(csc(x*y + z), y) == csch(x + z*I)*I
+
+
+def test_TR12i():
+    ta, tb, tc = [tan(i) for i in (a, b, c)]
+    assert TR12i((ta + tb)/(-ta*tb + 1)) == tan(a + b)
+    assert TR12i((ta + tb)/(ta*tb - 1)) == -tan(a + b)
+    assert TR12i((-ta - tb)/(ta*tb - 1)) == tan(a + b)
+    eq = (ta + tb)/(-ta*tb + 1)**2*(-3*ta - 3*tc)/(2*(ta*tc - 1))
+    assert TR12i(eq.expand()) == \
+        -3*tan(a + b)*tan(a + c)/(tan(a) + tan(b) - 1)/2
+    assert TR12i(tan(x)/sin(x)) == tan(x)/sin(x)
+    eq = (ta + cos(2))/(-ta*tb + 1)
+    assert TR12i(eq) == eq
+    eq = (ta + tb + 2)**2/(-ta*tb + 1)
+    assert TR12i(eq) == eq
+    eq = ta/(-ta*tb + 1)
+    assert TR12i(eq) == eq
+    eq = (((ta + tb)*(a + 1)).expand())**2/(ta*tb - 1)
+    assert TR12i(eq) == -(a + 1)**2*tan(a + b)
+
+
+def test_TR14():
+    eq = (cos(x) - 1)*(cos(x) + 1)
+    ans = -sin(x)**2
+    assert TR14(eq) == ans
+    assert TR14(1/eq) == 1/ans
+    assert TR14((cos(x) - 1)**2*(cos(x) + 1)**2) == ans**2
+    assert TR14((cos(x) - 1)**2*(cos(x) + 1)**3) == ans**2*(cos(x) + 1)
+    assert TR14((cos(x) - 1)**3*(cos(x) + 1)**2) == ans**2*(cos(x) - 1)
+    eq = (cos(x) - 1)**y*(cos(x) + 1)**y
+    assert TR14(eq) == eq
+    eq = (cos(x) - 2)**y*(cos(x) + 1)
+    assert TR14(eq) == eq
+    eq = (tan(x) - 2)**2*(cos(x) + 1)
+    assert TR14(eq) == eq
+    i = symbols('i', integer=True)
+    assert TR14((cos(x) - 1)**i*(cos(x) + 1)**i) == ans**i
+    assert TR14((sin(x) - 1)**i*(sin(x) + 1)**i) == (-cos(x)**2)**i
+    # could use extraction in this case
+    eq = (cos(x) - 1)**(i + 1)*(cos(x) + 1)**i
+    assert TR14(eq) in [(cos(x) - 1)*ans**i, eq]
+
+    assert TR14((sin(x) - 1)*(sin(x) + 1)) == -cos(x)**2
+    p1 = (cos(x) + 1)*(cos(x) - 1)
+    p2 = (cos(y) - 1)*2*(cos(y) + 1)
+    p3 = (3*(cos(y) - 1))*(3*(cos(y) + 1))
+    assert TR14(p1*p2*p3*(x - 1)) == -18*((x - 1)*sin(x)**2*sin(y)**4)
+
+
+def test_TR15_16_17():
+    assert TR15(1 - 1/sin(x)**2) == -cot(x)**2
+    assert TR16(1 - 1/cos(x)**2) == -tan(x)**2
+    assert TR111(1 - 1/tan(x)**2) == 1 - cot(x)**2
+
+
+def test_as_f_sign_1():
+    assert as_f_sign_1(x + 1) == (1, x, 1)
+    assert as_f_sign_1(x - 1) == (1, x, -1)
+    assert as_f_sign_1(-x + 1) == (-1, x, -1)
+    assert as_f_sign_1(-x - 1) == (-1, x, 1)
+    assert as_f_sign_1(2*x + 2) == (2, x, 1)
+    assert as_f_sign_1(x*y - y) == (y, x, -1)
+    assert as_f_sign_1(-x*y + y) == (-y, x, -1)
+
+
+def test_issue_25590():
+    A = Symbol('A', commutative=False)
+    B = Symbol('B', commutative=False)
+
+    assert TR8(2*cos(x)*sin(x)*B*A) == sin(2*x)*B*A
+    assert TR13(tan(2)*tan(3)*B*A) == (-tan(2)/tan(5) - tan(3)/tan(5) + 1)*B*A
+
+    # XXX The result may not be optimal than
+    # sin(2*x)*B*A + cos(x)**2 and may change in the future
+    assert (2*cos(x)*sin(x)*B*A + cos(x)**2).simplify() == sin(2*x)*B*A + cos(2*x)/2 + S.One/2

rl/Lib/site-packages/sympy/simplify/tests/test_function.py

@@ -0,0 +1,54 @@
+""" Unit tests for Hyper_Function"""
+from sympy.core import symbols, Dummy, Tuple, S, Rational
+from sympy.functions import hyper
+
+from sympy.simplify.hyperexpand import Hyper_Function
+
+def test_attrs():
+    a, b = symbols('a, b', cls=Dummy)
+    f = Hyper_Function([2, a], [b])
+    assert f.ap == Tuple(2, a)
+    assert f.bq == Tuple(b)
+    assert f.args == (Tuple(2, a), Tuple(b))
+    assert f.sizes == (2, 1)
+
+def test_call():
+    a, b, x = symbols('a, b, x', cls=Dummy)
+    f = Hyper_Function([2, a], [b])
+    assert f(x) == hyper([2, a], [b], x)
+
+def test_has():
+    a, b, c = symbols('a, b, c', cls=Dummy)
+    f = Hyper_Function([2, -a], [b])
+    assert f.has(a)
+    assert f.has(Tuple(b))
+    assert not f.has(c)
+
+def test_eq():
+    assert Hyper_Function([1], []) == Hyper_Function([1], [])
+    assert (Hyper_Function([1], []) != Hyper_Function([1], [])) is False
+    assert Hyper_Function([1], []) != Hyper_Function([2], [])
+    assert Hyper_Function([1], []) != Hyper_Function([1, 2], [])
+    assert Hyper_Function([1], []) != Hyper_Function([1], [2])
+
+def test_gamma():
+    assert Hyper_Function([2, 3], [-1]).gamma == 0
+    assert Hyper_Function([-2, -3], [-1]).gamma == 2
+    n = Dummy(integer=True)
+    assert Hyper_Function([-1, n, 1], []).gamma == 1
+    assert Hyper_Function([-1, -n, 1], []).gamma == 1
+    p = Dummy(integer=True, positive=True)
+    assert Hyper_Function([-1, p, 1], []).gamma == 1
+    assert Hyper_Function([-1, -p, 1], []).gamma == 2
+
+def test_suitable_origin():
+    assert Hyper_Function((S.Half,), (Rational(3, 2),))._is_suitable_origin() is True
+    assert Hyper_Function((S.Half,), (S.Half,))._is_suitable_origin() is False
+    assert Hyper_Function((S.Half,), (Rational(-1, 2),))._is_suitable_origin() is False
+    assert Hyper_Function((S.Half,), (0,))._is_suitable_origin() is False
+    assert Hyper_Function((S.Half,), (-1, 1,))._is_suitable_origin() is False
+    assert Hyper_Function((S.Half, 0), (1,))._is_suitable_origin() is False
+    assert Hyper_Function((S.Half, 1),
+            (2, Rational(-2, 3)))._is_suitable_origin() is True
+    assert Hyper_Function((S.Half, 1),
+            (2, Rational(-2, 3), Rational(3, 2)))._is_suitable_origin() is True

rl/Lib/site-packages/sympy/simplify/tests/test_gammasimp.py

@@ -0,0 +1,127 @@
+from sympy.core.function import Function
+from sympy.core.numbers import (Rational, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import symbols
+from sympy.functions.combinatorial.factorials import (rf, binomial, factorial)
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.functions.special.gamma_functions import gamma
+from sympy.simplify.gammasimp import gammasimp
+from sympy.simplify.powsimp import powsimp
+from sympy.simplify.simplify import simplify
+
+from sympy.abc import x, y, n, k
+
+
+def test_gammasimp():
+    R = Rational
+
+    # was part of test_combsimp_gamma() in test_combsimp.py
+    assert gammasimp(gamma(x)) == gamma(x)
+    assert gammasimp(gamma(x + 1)/x) == gamma(x)
+    assert gammasimp(gamma(x)/(x - 1)) == gamma(x - 1)
+    assert gammasimp(x*gamma(x)) == gamma(x + 1)
+    assert gammasimp((x + 1)*gamma(x + 1)) == gamma(x + 2)
+    assert gammasimp(gamma(x + y)*(x + y)) == gamma(x + y + 1)
+    assert gammasimp(x/gamma(x + 1)) == 1/gamma(x)
+    assert gammasimp((x + 1)**2/gamma(x + 2)) == (x + 1)/gamma(x + 1)
+    assert gammasimp(x*gamma(x) + gamma(x + 3)/(x + 2)) == \
+        (x + 2)*gamma(x + 1)
+
+    assert gammasimp(gamma(2*x)*x) == gamma(2*x + 1)/2
+    assert gammasimp(gamma(2*x)/(x - S.Half)) == 2*gamma(2*x - 1)
+
+    assert gammasimp(gamma(x)*gamma(1 - x)) == pi/sin(pi*x)
+    assert gammasimp(gamma(x)*gamma(-x)) == -pi/(x*sin(pi*x))
+    assert gammasimp(1/gamma(x + 3)/gamma(1 - x)) == \
+        sin(pi*x)/(pi*x*(x + 1)*(x + 2))
+
+    assert gammasimp(factorial(n + 2)) == gamma(n + 3)
+    assert gammasimp(binomial(n, k)) == \
+        gamma(n + 1)/(gamma(k + 1)*gamma(-k + n + 1))
+
+    assert powsimp(gammasimp(
+        gamma(x)*gamma(x + S.Half)*gamma(y)/gamma(x + y))) == \
+        2**(-2*x + 1)*sqrt(pi)*gamma(2*x)*gamma(y)/gamma(x + y)
+    assert gammasimp(1/gamma(x)/gamma(x - Rational(1, 3))/gamma(x + Rational(1, 3))) == \
+        3**(3*x - Rational(3, 2))/(2*pi*gamma(3*x - 1))
+    assert simplify(
+        gamma(S.Half + x/2)*gamma(1 + x/2)/gamma(1 + x)/sqrt(pi)*2**x) == 1
+    assert gammasimp(gamma(Rational(-1, 4))*gamma(Rational(-3, 4))) == 16*sqrt(2)*pi/3
+
+    assert powsimp(gammasimp(gamma(2*x)/gamma(x))) == \
+        2**(2*x - 1)*gamma(x + S.Half)/sqrt(pi)
+
+    # issue 6792
+    e = (-gamma(k)*gamma(k + 2) + gamma(k + 1)**2)/gamma(k)**2
+    assert gammasimp(e) == -k
+    assert gammasimp(1/e) == -1/k
+    e = (gamma(x) + gamma(x + 1))/gamma(x)
+    assert gammasimp(e) == x + 1
+    assert gammasimp(1/e) == 1/(x + 1)
+    e = (gamma(x) + gamma(x + 2))*(gamma(x - 1) + gamma(x))/gamma(x)
+    assert gammasimp(e) == (x**2 + x + 1)*gamma(x + 1)/(x - 1)
+    e = (-gamma(k)*gamma(k + 2) + gamma(k + 1)**2)/gamma(k)**2
+    assert gammasimp(e**2) == k**2
+    assert gammasimp(e**2/gamma(k + 1)) == k/gamma(k)
+    a = R(1, 2) + R(1, 3)
+    b = a + R(1, 3)
+    assert gammasimp(gamma(2*k)/gamma(k)*gamma(k + a)*gamma(k + b)
+        ) == 3*2**(2*k + 1)*3**(-3*k - 2)*sqrt(pi)*gamma(3*k + R(3, 2))/2
+
+    # issue 9699
+    assert gammasimp((x + 1)*factorial(x)/gamma(y)) == gamma(x + 2)/gamma(y)
+    assert gammasimp(rf(x + n, k)*binomial(n, k)).simplify() == Piecewise(
+        (gamma(n + 1)*gamma(k + n + x)/(gamma(k + 1)*gamma(n + x)*gamma(-k + n + 1)), n > -x),
+        ((-1)**k*gamma(n + 1)*gamma(-n - x + 1)/(gamma(k + 1)*gamma(-k + n + 1)*gamma(-k - n - x + 1)), True))
+
+    A, B = symbols('A B', commutative=False)
+    assert gammasimp(e*B*A) == gammasimp(e)*B*A
+
+    # check iteration
+    assert gammasimp(gamma(2*k)/gamma(k)*gamma(-k - R(1, 2))) == (
+        -2**(2*k + 1)*sqrt(pi)/(2*((2*k + 1)*cos(pi*k))))
+    assert gammasimp(
+        gamma(k)*gamma(k + R(1, 3))*gamma(k + R(2, 3))/gamma(k*R(3, 2))) == (
+        3*2**(3*k + 1)*3**(-3*k - S.Half)*sqrt(pi)*gamma(k*R(3, 2) + S.Half)/2)
+
+    # issue 6153
+    assert gammasimp(gamma(Rational(1, 4))/gamma(Rational(5, 4))) == 4
+
+    # was part of test_combsimp() in test_combsimp.py
+    assert gammasimp(binomial(n + 2, k + S.Half)) == gamma(n + 3)/ \
+        (gamma(k + R(3, 2))*gamma(-k + n + R(5, 2)))
+    assert gammasimp(binomial(n + 2, k + 2.0)) == \
+        gamma(n + 3)/(gamma(k + 3.0)*gamma(-k + n + 1))
+
+    # issue 11548
+    assert gammasimp(binomial(0, x)) == sin(pi*x)/(pi*x)
+
+    e = gamma(n + Rational(1, 3))*gamma(n + R(2, 3))
+    assert gammasimp(e) == e
+    assert gammasimp(gamma(4*n + S.Half)/gamma(2*n - R(3, 4))) == \
+        2**(4*n - R(5, 2))*(8*n - 3)*gamma(2*n + R(3, 4))/sqrt(pi)
+
+    i, m = symbols('i m', integer = True)
+    e = gamma(exp(i))
+    assert gammasimp(e) == e
+    e = gamma(m + 3)
+    assert gammasimp(e) == e
+    e = gamma(m + 1)/(gamma(i + 1)*gamma(-i + m + 1))
+    assert gammasimp(e) == e
+
+    p = symbols("p", integer=True, positive=True)
+    assert gammasimp(gamma(-p + 4)) == gamma(-p + 4)
+
+
+def test_issue_22606():
+    fx = Function('f')(x)
+    eq = x + gamma(y)
+    # seems like ans should be `eq`, not `(x*y + gamma(y + 1))/y`
+    ans = gammasimp(eq)
+    assert gammasimp(eq.subs(x, fx)).subs(fx, x) == ans
+    assert gammasimp(eq.subs(x, cos(x))).subs(cos(x), x) == ans
+    assert 1/gammasimp(1/eq) == ans
+    assert gammasimp(fx.subs(x, eq)).args[0] == ans

rl/Lib/site-packages/sympy/simplify/tests/test_powsimp.py

@@ -0,0 +1,366 @@
+from sympy.core.function import Function
+from sympy.core.mul import Mul
+from sympy.core.numbers import (E, I, Rational, oo, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Dummy, Symbol, symbols)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import (root, sqrt)
+from sympy.functions.elementary.trigonometric import sin
+from sympy.functions.special.gamma_functions import gamma
+from sympy.functions.special.hyper import hyper
+from sympy.matrices.expressions.matexpr import MatrixSymbol
+from sympy.simplify.powsimp import (powdenest, powsimp)
+from sympy.simplify.simplify import (signsimp, simplify)
+from sympy.core.symbol import Str
+
+from sympy.abc import x, y, z, a, b
+
+
+def test_powsimp():
+    x, y, z, n = symbols('x,y,z,n')
+    f = Function('f')
+    assert powsimp( 4**x * 2**(-x) * 2**(-x) ) == 1
+    assert powsimp( (-4)**x * (-2)**(-x) * 2**(-x) ) == 1
+
+    assert powsimp(
+        f(4**x * 2**(-x) * 2**(-x)) ) == f(4**x * 2**(-x) * 2**(-x))
+    assert powsimp( f(4**x * 2**(-x) * 2**(-x)), deep=True ) == f(1)
+    assert exp(x)*exp(y) == exp(x)*exp(y)
+    assert powsimp(exp(x)*exp(y)) == exp(x + y)
+    assert powsimp(exp(x)*exp(y)*2**x*2**y) == (2*E)**(x + y)
+    assert powsimp(exp(x)*exp(y)*2**x*2**y, combine='exp') == \
+        exp(x + y)*2**(x + y)
+    assert powsimp(exp(x)*exp(y)*exp(2)*sin(x) + sin(y) + 2**x*2**y) == \
+        exp(2 + x + y)*sin(x) + sin(y) + 2**(x + y)
+    assert powsimp(sin(exp(x)*exp(y))) == sin(exp(x)*exp(y))
+    assert powsimp(sin(exp(x)*exp(y)), deep=True) == sin(exp(x + y))
+    assert powsimp(x**2*x**y) == x**(2 + y)
+    # This should remain factored, because 'exp' with deep=True is supposed
+    # to act like old automatic exponent combining.
+    assert powsimp((1 + E*exp(E))*exp(-E), combine='exp', deep=True) == \
+        (1 + exp(1 + E))*exp(-E)
+    assert powsimp((1 + E*exp(E))*exp(-E), deep=True) == \
+        (1 + exp(1 + E))*exp(-E)
+    assert powsimp((1 + E*exp(E))*exp(-E)) == (1 + exp(1 + E))*exp(-E)
+    assert powsimp((1 + E*exp(E))*exp(-E), combine='exp') == \
+        (1 + exp(1 + E))*exp(-E)
+    assert powsimp((1 + E*exp(E))*exp(-E), combine='base') == \
+        (1 + E*exp(E))*exp(-E)
+    x, y = symbols('x,y', nonnegative=True)
+    n = Symbol('n', real=True)
+    assert powsimp(y**n * (y/x)**(-n)) == x**n
+    assert powsimp(x**(x**(x*y)*y**(x*y))*y**(x**(x*y)*y**(x*y)), deep=True) \
+        == (x*y)**(x*y)**(x*y)
+    assert powsimp(2**(2**(2*x)*x), deep=False) == 2**(2**(2*x)*x)
+    assert powsimp(2**(2**(2*x)*x), deep=True) == 2**(x*4**x)
+    assert powsimp(
+        exp(-x + exp(-x)*exp(-x*log(x))), deep=False, combine='exp') == \
+        exp(-x + exp(-x)*exp(-x*log(x)))
+    assert powsimp(
+        exp(-x + exp(-x)*exp(-x*log(x))), deep=False, combine='exp') == \
+        exp(-x + exp(-x)*exp(-x*log(x)))
+    assert powsimp((x + y)/(3*z), deep=False, combine='exp') == (x + y)/(3*z)
+    assert powsimp((x/3 + y/3)/z, deep=True, combine='exp') == (x/3 + y/3)/z
+    assert powsimp(exp(x)/(1 + exp(x)*exp(y)), deep=True) == \
+        exp(x)/(1 + exp(x + y))
+    assert powsimp(x*y**(z**x*z**y), deep=True) == x*y**(z**(x + y))
+    assert powsimp((z**x*z**y)**x, deep=True) == (z**(x + y))**x
+    assert powsimp(x*(z**x*z**y)**x, deep=True) == x*(z**(x + y))**x
+    p = symbols('p', positive=True)
+    assert powsimp((1/x)**log(2)/x) == (1/x)**(1 + log(2))
+    assert powsimp((1/p)**log(2)/p) == p**(-1 - log(2))
+
+    # coefficient of exponent can only be simplified for positive bases
+    assert powsimp(2**(2*x)) == 4**x
+    assert powsimp((-1)**(2*x)) == (-1)**(2*x)
+    i = symbols('i', integer=True)
+    assert powsimp((-1)**(2*i)) == 1
+    assert powsimp((-1)**(-x)) != (-1)**x  # could be 1/((-1)**x), but is not
+    # force=True overrides assumptions
+    assert powsimp((-1)**(2*x), force=True) == 1
+
+    # rational exponents allow combining of negative terms
+    w, n, m = symbols('w n m', negative=True)
+    e = i/a  # not a rational exponent if `a` is unknown
+    ex = w**e*n**e*m**e
+    assert powsimp(ex) == m**(i/a)*n**(i/a)*w**(i/a)
+    e = i/3
+    ex = w**e*n**e*m**e
+    assert powsimp(ex) == (-1)**i*(-m*n*w)**(i/3)
+    e = (3 + i)/i
+    ex = w**e*n**e*m**e
+    assert powsimp(ex) == (-1)**(3*e)*(-m*n*w)**e
+
+    eq = x**(a*Rational(2, 3))
+    # eq != (x**a)**(2/3) (try x = -1 and a = 3 to see)
+    assert powsimp(eq).exp == eq.exp == a*Rational(2, 3)
+    # powdenest goes the other direction
+    assert powsimp(2**(2*x)) == 4**x
+
+    assert powsimp(exp(p/2)) == exp(p/2)
+
+    # issue 6368
+    eq = Mul(*[sqrt(Dummy(imaginary=True)) for i in range(3)])
+    assert powsimp(eq) == eq and eq.is_Mul
+
+    assert all(powsimp(e) == e for e in (sqrt(x**a), sqrt(x**2)))
+
+    # issue 8836
+    assert str( powsimp(exp(I*pi/3)*root(-1,3)) ) == '(-1)**(2/3)'
+
+    # issue 9183
+    assert powsimp(-0.1**x) == -0.1**x
+
+    # issue 10095
+    assert powsimp((1/(2*E))**oo) == (exp(-1)/2)**oo
+
+    # PR 13131
+    eq = sin(2*x)**2*sin(2.0*x)**2
+    assert powsimp(eq) == eq
+
+    # issue 14615
+    assert powsimp(x**2*y**3*(x*y**2)**Rational(3, 2)
+        ) == x*y*(x*y**2)**Rational(5, 2)
+
+
+def test_powsimp_negated_base():
+    assert powsimp((-x + y)/sqrt(x - y)) == -sqrt(x - y)
+    assert powsimp((-x + y)*(-z + y)/sqrt(x - y)/sqrt(z - y)) == sqrt(x - y)*sqrt(z - y)
+    p = symbols('p', positive=True)
+    reps = {p: 2, a: S.Half}
+    assert powsimp((-p)**a/p**a).subs(reps) == ((-1)**a).subs(reps)
+    assert powsimp((-p)**a*p**a).subs(reps) == ((-p**2)**a).subs(reps)
+    n = symbols('n', negative=True)
+    reps = {p: -2, a: S.Half}
+    assert powsimp((-n)**a/n**a).subs(reps) == (-1)**(-a).subs(a, S.Half)
+    assert powsimp((-n)**a*n**a).subs(reps) == ((-n**2)**a).subs(reps)
+    # if x is 0 then the lhs is 0**a*oo**a which is not (-1)**a
+    eq = (-x)**a/x**a
+    assert powsimp(eq) == eq
+
+
+def test_powsimp_nc():
+    x, y, z = symbols('x,y,z')
+    A, B, C = symbols('A B C', commutative=False)
+
+    assert powsimp(A**x*A**y, combine='all') == A**(x + y)
+    assert powsimp(A**x*A**y, combine='base') == A**x*A**y
+    assert powsimp(A**x*A**y, combine='exp') == A**(x + y)
+
+    assert powsimp(A**x*B**x, combine='all') == A**x*B**x
+    assert powsimp(A**x*B**x, combine='base') == A**x*B**x
+    assert powsimp(A**x*B**x, combine='exp') == A**x*B**x
+
+    assert powsimp(B**x*A**x, combine='all') == B**x*A**x
+    assert powsimp(B**x*A**x, combine='base') == B**x*A**x
+    assert powsimp(B**x*A**x, combine='exp') == B**x*A**x
+
+    assert powsimp(A**x*A**y*A**z, combine='all') == A**(x + y + z)
+    assert powsimp(A**x*A**y*A**z, combine='base') == A**x*A**y*A**z
+    assert powsimp(A**x*A**y*A**z, combine='exp') == A**(x + y + z)
+
+    assert powsimp(A**x*B**x*C**x, combine='all') == A**x*B**x*C**x
+    assert powsimp(A**x*B**x*C**x, combine='base') == A**x*B**x*C**x
+    assert powsimp(A**x*B**x*C**x, combine='exp') == A**x*B**x*C**x
+
+    assert powsimp(B**x*A**x*C**x, combine='all') == B**x*A**x*C**x
+    assert powsimp(B**x*A**x*C**x, combine='base') == B**x*A**x*C**x
+    assert powsimp(B**x*A**x*C**x, combine='exp') == B**x*A**x*C**x
+
+
+def test_issue_6440():
+    assert powsimp(16*2**a*8**b) == 2**(a + 3*b + 4)
+
+
+def test_powdenest():
+    x, y = symbols('x,y')
+    p, q = symbols('p q', positive=True)
+    i, j = symbols('i,j', integer=True)
+
+    assert powdenest(x) == x
+    assert powdenest(x + 2*(x**(a*Rational(2, 3)))**(3*x)) == (x + 2*(x**(a*Rational(2, 3)))**(3*x))
+    assert powdenest((exp(a*Rational(2, 3)))**(3*x))  # -X-> (exp(a/3))**(6*x)
+    assert powdenest((x**(a*Rational(2, 3)))**(3*x)) == ((x**(a*Rational(2, 3)))**(3*x))
+    assert powdenest(exp(3*x*log(2))) == 2**(3*x)
+    assert powdenest(sqrt(p**2)) == p
+    eq = p**(2*i)*q**(4*i)
+    assert powdenest(eq) == (p*q**2)**(2*i)
+    # -X-> (x**x)**i*(x**x)**j == x**(x*(i + j))
+    assert powdenest((x**x)**(i + j))
+    assert powdenest(exp(3*y*log(x))) == x**(3*y)
+    assert powdenest(exp(y*(log(a) + log(b)))) == (a*b)**y
+    assert powdenest(exp(3*(log(a) + log(b)))) == a**3*b**3
+    assert powdenest(((x**(2*i))**(3*y))**x) == ((x**(2*i))**(3*y))**x
+    assert powdenest(((x**(2*i))**(3*y))**x, force=True) == x**(6*i*x*y)
+    assert powdenest(((x**(a*Rational(2, 3)))**(3*y/i))**x) == \
+        (((x**(a*Rational(2, 3)))**(3*y/i))**x)
+    assert powdenest((x**(2*i)*y**(4*i))**z, force=True) == (x*y**2)**(2*i*z)
+    assert powdenest((p**(2*i)*q**(4*i))**j) == (p*q**2)**(2*i*j)
+    e = ((p**(2*a))**(3*y))**x
+    assert powdenest(e) == e
+    e = ((x**2*y**4)**a)**(x*y)
+    assert powdenest(e) == e
+    e = (((x**2*y**4)**a)**(x*y))**3
+    assert powdenest(e) == ((x**2*y**4)**a)**(3*x*y)
+    assert powdenest((((x**2*y**4)**a)**(x*y)), force=True) == \
+        (x*y**2)**(2*a*x*y)
+    assert powdenest((((x**2*y**4)**a)**(x*y))**3, force=True) == \
+        (x*y**2)**(6*a*x*y)
+    assert powdenest((x**2*y**6)**i) != (x*y**3)**(2*i)
+    x, y = symbols('x,y', positive=True)
+    assert powdenest((x**2*y**6)**i) == (x*y**3)**(2*i)
+
+    assert powdenest((x**(i*Rational(2, 3))*y**(i/2))**(2*i)) == (x**Rational(4, 3)*y)**(i**2)
+    assert powdenest(sqrt(x**(2*i)*y**(6*i))) == (x*y**3)**i
+
+    assert powdenest(4**x) == 2**(2*x)
+    assert powdenest((4**x)**y) == 2**(2*x*y)
+    assert powdenest(4**x*y) == 2**(2*x)*y
+
+
+def test_powdenest_polar():
+    x, y, z = symbols('x y z', polar=True)
+    a, b, c = symbols('a b c')
+    assert powdenest((x*y*z)**a) == x**a*y**a*z**a
+    assert powdenest((x**a*y**b)**c) == x**(a*c)*y**(b*c)
+    assert powdenest(((x**a)**b*y**c)**c) == x**(a*b*c)*y**(c**2)
+
+
+def test_issue_5805():
+    arg = ((gamma(x)*hyper((), (), x))*pi)**2
+    assert powdenest(arg) == (pi*gamma(x)*hyper((), (), x))**2
+    assert arg.is_positive is None
+
+
+def test_issue_9324_powsimp_on_matrix_symbol():
+    M = MatrixSymbol('M', 10, 10)
+    expr = powsimp(M, deep=True)
+    assert expr == M
+    assert expr.args[0] == Str('M')
+
+
+def test_issue_6367():
+    z = -5*sqrt(2)/(2*sqrt(2*sqrt(29) + 29)) + sqrt(-sqrt(29)/29 + S.Half)
+    assert Mul(*[powsimp(a) for a in Mul.make_args(z.normal())]) == 0
+    assert powsimp(z.normal()) == 0
+    assert simplify(z) == 0
+    assert powsimp(sqrt(2 + sqrt(3))*sqrt(2 - sqrt(3)) + 1) == 2
+    assert powsimp(z) != 0
+
+
+def test_powsimp_polar():
+    from sympy.functions.elementary.complexes import polar_lift
+    from sympy.functions.elementary.exponential import exp_polar
+    x, y, z = symbols('x y z')
+    p, q, r = symbols('p q r', polar=True)
+
+    assert (polar_lift(-1))**(2*x) == exp_polar(2*pi*I*x)
+    assert powsimp(p**x * q**x) == (p*q)**x
+    assert p**x * (1/p)**x == 1
+    assert (1/p)**x == p**(-x)
+
+    assert exp_polar(x)*exp_polar(y) == exp_polar(x)*exp_polar(y)
+    assert powsimp(exp_polar(x)*exp_polar(y)) == exp_polar(x + y)
+    assert powsimp(exp_polar(x)*exp_polar(y)*p**x*p**y) == \
+        (p*exp_polar(1))**(x + y)
+    assert powsimp(exp_polar(x)*exp_polar(y)*p**x*p**y, combine='exp') == \
+        exp_polar(x + y)*p**(x + y)
+    assert powsimp(
+        exp_polar(x)*exp_polar(y)*exp_polar(2)*sin(x) + sin(y) + p**x*p**y) \
+        == p**(x + y) + sin(x)*exp_polar(2 + x + y) + sin(y)
+    assert powsimp(sin(exp_polar(x)*exp_polar(y))) == \
+        sin(exp_polar(x)*exp_polar(y))
+    assert powsimp(sin(exp_polar(x)*exp_polar(y)), deep=True) == \
+        sin(exp_polar(x + y))
+
+
+def test_issue_5728():
+    b = x*sqrt(y)
+    a = sqrt(b)
+    c = sqrt(sqrt(x)*y)
+    assert powsimp(a*b) == sqrt(b)**3
+    assert powsimp(a*b**2*sqrt(y)) == sqrt(y)*a**5
+    assert powsimp(a*x**2*c**3*y) == c**3*a**5
+    assert powsimp(a*x*c**3*y**2) == c**7*a
+    assert powsimp(x*c**3*y**2) == c**7
+    assert powsimp(x*c**3*y) == x*y*c**3
+    assert powsimp(sqrt(x)*c**3*y) == c**5
+    assert powsimp(sqrt(x)*a**3*sqrt(y)) == sqrt(x)*sqrt(y)*a**3
+    assert powsimp(Mul(sqrt(x)*c**3*sqrt(y), y, evaluate=False)) == \
+        sqrt(x)*sqrt(y)**3*c**3
+    assert powsimp(a**2*a*x**2*y) == a**7
+
+    # symbolic powers work, too
+    b = x**y*y
+    a = b*sqrt(b)
+    assert a.is_Mul is True
+    assert powsimp(a) == sqrt(b)**3
+
+    # as does exp
+    a = x*exp(y*Rational(2, 3))
+    assert powsimp(a*sqrt(a)) == sqrt(a)**3
+    assert powsimp(a**2*sqrt(a)) == sqrt(a)**5
+    assert powsimp(a**2*sqrt(sqrt(a))) == sqrt(sqrt(a))**9
+
+
+def test_issue_from_PR1599():
+    n1, n2, n3, n4 = symbols('n1 n2 n3 n4', negative=True)
+    assert (powsimp(sqrt(n1)*sqrt(n2)*sqrt(n3)) ==
+        -I*sqrt(-n1)*sqrt(-n2)*sqrt(-n3))
+    assert (powsimp(root(n1, 3)*root(n2, 3)*root(n3, 3)*root(n4, 3)) ==
+        -(-1)**Rational(1, 3)*
+        (-n1)**Rational(1, 3)*(-n2)**Rational(1, 3)*(-n3)**Rational(1, 3)*(-n4)**Rational(1, 3))
+
+
+def test_issue_10195():
+    a = Symbol('a', integer=True)
+    l = Symbol('l', even=True, nonzero=True)
+    n = Symbol('n', odd=True)
+    e_x = (-1)**(n/2 - S.Half) - (-1)**(n*Rational(3, 2) - S.Half)
+    assert powsimp((-1)**(l/2)) == I**l
+    assert powsimp((-1)**(n/2)) == I**n
+    assert powsimp((-1)**(n*Rational(3, 2))) == -I**n
+    assert powsimp(e_x) == (-1)**(n/2 - S.Half) + (-1)**(n*Rational(3, 2) +
+            S.Half)
+    assert powsimp((-1)**(a*Rational(3, 2))) == (-I)**a
+
+def test_issue_15709():
+    assert powsimp(3**x*Rational(2, 3)) == 2*3**(x-1)
+    assert powsimp(2*3**x/3) == 2*3**(x-1)
+
+
+def test_issue_11981():
+    x, y = symbols('x y', commutative=False)
+    assert powsimp((x*y)**2 * (y*x)**2) == (x*y)**2 * (y*x)**2
+
+
+def test_issue_17524():
+    a = symbols("a", real=True)
+    e = (-1 - a**2)*sqrt(1 + a**2)
+    assert signsimp(powsimp(e)) == signsimp(e) == -(a**2 + 1)**(S(3)/2)
+
+
+def test_issue_19627():
+    # if you use force the user must verify
+    assert powdenest(sqrt(sin(x)**2), force=True) == sin(x)
+    assert powdenest((x**(S.Half/y))**(2*y), force=True) == x
+    from sympy.core.function import expand_power_base
+    e = 1 - a
+    expr = (exp(z/e)*x**(b/e)*y**((1 - b)/e))**e
+    assert powdenest(expand_power_base(expr, force=True), force=True
+        ) == x**b*y**(1 - b)*exp(z)
+
+
+def test_issue_22546():
+    p1, p2 = symbols('p1, p2', positive=True)
+    ref = powsimp(p1**z/p2**z)
+    e = z + 1
+    ans = ref.subs(z, e)
+    assert ans.is_Pow
+    assert powsimp(p1**e/p2**e) == ans
+    i = symbols('i', integer=True)
+    ref = powsimp(x**i/y**i)
+    e = i + 1
+    ans = ref.subs(i, e)
+    assert ans.is_Pow
+    assert powsimp(x**e/y**e) == ans

rl/Lib/site-packages/sympy/simplify/tests/test_radsimp.py

@@ -0,0 +1,490 @@
+from sympy.core.add import Add
+from sympy.core.function import (Derivative, Function, diff)
+from sympy.core.mul import Mul
+from sympy.core.numbers import (I, Rational)
+from sympy.core.power import Pow
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, Wild, symbols)
+from sympy.functions.elementary.complexes import Abs
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.miscellaneous import (root, sqrt)
+from sympy.functions.elementary.trigonometric import (cos, sin)
+from sympy.polys.polytools import factor
+from sympy.series.order import O
+from sympy.simplify.radsimp import (collect, collect_const, fraction, radsimp, rcollect)
+
+from sympy.core.expr import unchanged
+from sympy.core.mul import _unevaluated_Mul as umul
+from sympy.simplify.radsimp import (_unevaluated_Add,
+    collect_sqrt, fraction_expand, collect_abs)
+from sympy.testing.pytest import raises
+
+from sympy.abc import x, y, z, a, b, c, d
+
+
+def test_radsimp():
+    r2 = sqrt(2)
+    r3 = sqrt(3)
+    r5 = sqrt(5)
+    r7 = sqrt(7)
+    assert fraction(radsimp(1/r2)) == (sqrt(2), 2)
+    assert radsimp(1/(1 + r2)) == \
+        -1 + sqrt(2)
+    assert radsimp(1/(r2 + r3)) == \
+        -sqrt(2) + sqrt(3)
+    assert fraction(radsimp(1/(1 + r2 + r3))) == \
+        (-sqrt(6) + sqrt(2) + 2, 4)
+    assert fraction(radsimp(1/(r2 + r3 + r5))) == \
+        (-sqrt(30) + 2*sqrt(3) + 3*sqrt(2), 12)
+    assert fraction(radsimp(1/(1 + r2 + r3 + r5))) == (
+        (-34*sqrt(10) - 26*sqrt(15) - 55*sqrt(3) - 61*sqrt(2) + 14*sqrt(30) +
+        93 + 46*sqrt(6) + 53*sqrt(5), 71))
+    assert fraction(radsimp(1/(r2 + r3 + r5 + r7))) == (
+        (-50*sqrt(42) - 133*sqrt(5) - 34*sqrt(70) - 145*sqrt(3) + 22*sqrt(105)
+        + 185*sqrt(2) + 62*sqrt(30) + 135*sqrt(7), 215))
+    z = radsimp(1/(1 + r2/3 + r3/5 + r5 + r7))
+    assert len((3616791619821680643598*z).args) == 16
+    assert radsimp(1/z) == 1/z
+    assert radsimp(1/z, max_terms=20).expand() == 1 + r2/3 + r3/5 + r5 + r7
+    assert radsimp(1/(r2*3)) == \
+        sqrt(2)/6
+    assert radsimp(1/(r2*a + r3 + r5 + r7)) == (
+        (8*sqrt(2)*a**7 - 8*sqrt(7)*a**6 - 8*sqrt(5)*a**6 - 8*sqrt(3)*a**6 -
+        180*sqrt(2)*a**5 + 8*sqrt(30)*a**5 + 8*sqrt(42)*a**5 + 8*sqrt(70)*a**5
+        - 24*sqrt(105)*a**4 + 84*sqrt(3)*a**4 + 100*sqrt(5)*a**4 +
+        116*sqrt(7)*a**4 - 72*sqrt(70)*a**3 - 40*sqrt(42)*a**3 -
+        8*sqrt(30)*a**3 + 782*sqrt(2)*a**3 - 462*sqrt(3)*a**2 -
+        302*sqrt(7)*a**2 - 254*sqrt(5)*a**2 + 120*sqrt(105)*a**2 -
+        795*sqrt(2)*a - 62*sqrt(30)*a + 82*sqrt(42)*a + 98*sqrt(70)*a -
+        118*sqrt(105) + 59*sqrt(7) + 295*sqrt(5) + 531*sqrt(3))/(16*a**8 -
+        480*a**6 + 3128*a**4 - 6360*a**2 + 3481))
+    assert radsimp(1/(r2*a + r2*b + r3 + r7)) == (
+        (sqrt(2)*a*(a + b)**2 - 5*sqrt(2)*a + sqrt(42)*a + sqrt(2)*b*(a +
+        b)**2 - 5*sqrt(2)*b + sqrt(42)*b - sqrt(7)*(a + b)**2 - sqrt(3)*(a +
+        b)**2 - 2*sqrt(3) + 2*sqrt(7))/(2*a**4 + 8*a**3*b + 12*a**2*b**2 -
+        20*a**2 + 8*a*b**3 - 40*a*b + 2*b**4 - 20*b**2 + 8))
+    assert radsimp(1/(r2*a + r2*b + r2*c + r2*d)) == \
+        sqrt(2)/(2*a + 2*b + 2*c + 2*d)
+    assert radsimp(1/(1 + r2*a + r2*b + r2*c + r2*d)) == (
+        (sqrt(2)*a + sqrt(2)*b + sqrt(2)*c + sqrt(2)*d - 1)/(2*a**2 + 4*a*b +
+        4*a*c + 4*a*d + 2*b**2 + 4*b*c + 4*b*d + 2*c**2 + 4*c*d + 2*d**2 - 1))
+    assert radsimp((y**2 - x)/(y - sqrt(x))) == \
+        sqrt(x) + y
+    assert radsimp(-(y**2 - x)/(y - sqrt(x))) == \
+        -(sqrt(x) + y)
+    assert radsimp(1/(1 - I + a*I)) == \
+        (-I*a + 1 + I)/(a**2 - 2*a + 2)
+    assert radsimp(1/((-x + y)*(x - sqrt(y)))) == \
+        (-x - sqrt(y))/((x - y)*(x**2 - y))
+    e = (3 + 3*sqrt(2))*x*(3*x - 3*sqrt(y))
+    assert radsimp(e) == x*(3 + 3*sqrt(2))*(3*x - 3*sqrt(y))
+    assert radsimp(1/e) == (
+        (-9*x + 9*sqrt(2)*x - 9*sqrt(y) + 9*sqrt(2)*sqrt(y))/(9*x*(9*x**2 -
+        9*y)))
+    assert radsimp(1 + 1/(1 + sqrt(3))) == \
+        Mul(S.Half, -1 + sqrt(3), evaluate=False) + 1
+    A = symbols("A", commutative=False)
+    assert radsimp(x**2 + sqrt(2)*x**2 - sqrt(2)*x*A) == \
+        x**2 + sqrt(2)*x**2 - sqrt(2)*x*A
+    assert radsimp(1/sqrt(5 + 2 * sqrt(6))) == -sqrt(2) + sqrt(3)
+    assert radsimp(1/sqrt(5 + 2 * sqrt(6))**3) == -(-sqrt(3) + sqrt(2))**3
+
+    # issue 6532
+    assert fraction(radsimp(1/sqrt(x))) == (sqrt(x), x)
+    assert fraction(radsimp(1/sqrt(2*x + 3))) == (sqrt(2*x + 3), 2*x + 3)
+    assert fraction(radsimp(1/sqrt(2*(x + 3)))) == (sqrt(2*x + 6), 2*x + 6)
+
+    # issue 5994
+    e = S('-(2 + 2*sqrt(2) + 4*2**(1/4))/'
+        '(1 + 2**(3/4) + 3*2**(1/4) + 3*sqrt(2))')
+    assert radsimp(e).expand() == -2*2**Rational(3, 4) - 2*2**Rational(1, 4) + 2 + 2*sqrt(2)
+
+    # issue 5986 (modifications to radimp didn't initially recognize this so
+    # the test is included here)
+    assert radsimp(1/(-sqrt(5)/2 - S.Half + (-sqrt(5)/2 - S.Half)**2)) == 1
+
+    # from issue 5934
+    eq = (
+        (-240*sqrt(2)*sqrt(sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) -
+        360*sqrt(2)*sqrt(-8*sqrt(5) + 40)*sqrt(-sqrt(5) + 5) -
+        120*sqrt(10)*sqrt(-8*sqrt(5) + 40)*sqrt(-sqrt(5) + 5) +
+        120*sqrt(2)*sqrt(-sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) +
+        120*sqrt(2)*sqrt(-8*sqrt(5) + 40)*sqrt(sqrt(5) + 5) +
+        120*sqrt(10)*sqrt(-sqrt(5) + 5)*sqrt(8*sqrt(5) + 40) +
+        120*sqrt(10)*sqrt(-8*sqrt(5) + 40)*sqrt(sqrt(5) + 5))/(-36000 -
+        7200*sqrt(5) + (12*sqrt(10)*sqrt(sqrt(5) + 5) +
+        24*sqrt(10)*sqrt(-sqrt(5) + 5))**2))
+    assert radsimp(eq) is S.NaN  # it's 0/0
+
+    # work with normal form
+    e = 1/sqrt(sqrt(7)/7 + 2*sqrt(2) + 3*sqrt(3) + 5*sqrt(5)) + 3
+    assert radsimp(e) == (
+        -sqrt(sqrt(7) + 14*sqrt(2) + 21*sqrt(3) +
+        35*sqrt(5))*(-11654899*sqrt(35) - 1577436*sqrt(210) - 1278438*sqrt(15)
+        - 1346996*sqrt(10) + 1635060*sqrt(6) + 5709765 + 7539830*sqrt(14) +
+        8291415*sqrt(21))/1300423175 + 3)
+
+    # obey power rules
+    base = sqrt(3) - sqrt(2)
+    assert radsimp(1/base**3) == (sqrt(3) + sqrt(2))**3
+    assert radsimp(1/(-base)**3) == -(sqrt(2) + sqrt(3))**3
+    assert radsimp(1/(-base)**x) == (-base)**(-x)
+    assert radsimp(1/base**x) == (sqrt(2) + sqrt(3))**x
+    assert radsimp(root(1/(-1 - sqrt(2)), -x)) == (-1)**(-1/x)*(1 + sqrt(2))**(1/x)
+
+    # recurse
+    e = cos(1/(1 + sqrt(2)))
+    assert radsimp(e) == cos(-sqrt(2) + 1)
+    assert radsimp(e/2) == cos(-sqrt(2) + 1)/2
+    assert radsimp(1/e) == 1/cos(-sqrt(2) + 1)
+    assert radsimp(2/e) == 2/cos(-sqrt(2) + 1)
+    assert fraction(radsimp(e/sqrt(x))) == (sqrt(x)*cos(-sqrt(2)+1), x)
+
+    # test that symbolic denominators are not processed
+    r = 1 + sqrt(2)
+    assert radsimp(x/r, symbolic=False) == -x*(-sqrt(2) + 1)
+    assert radsimp(x/(y + r), symbolic=False) == x/(y + 1 + sqrt(2))
+    assert radsimp(x/(y + r)/r, symbolic=False) == \
+        -x*(-sqrt(2) + 1)/(y + 1 + sqrt(2))
+
+    # issue 7408
+    eq = sqrt(x)/sqrt(y)
+    assert radsimp(eq) == umul(sqrt(x), sqrt(y), 1/y)
+    assert radsimp(eq, symbolic=False) == eq
+
+    # issue 7498
+    assert radsimp(sqrt(x)/sqrt(y)**3) == umul(sqrt(x), sqrt(y**3), 1/y**3)
+
+    # for coverage
+    eq = sqrt(x)/y**2
+    assert radsimp(eq) == eq
+
+
+def test_radsimp_issue_3214():
+    c, p = symbols('c p', positive=True)
+    s = sqrt(c**2 - p**2)
+    b = (c + I*p - s)/(c + I*p + s)
+    assert radsimp(b) == -I*(c + I*p - sqrt(c**2 - p**2))**2/(2*c*p)
+
+
+def test_collect_1():
+    """Collect with respect to Symbol"""
+    x, y, z, n = symbols('x,y,z,n')
+    assert collect(1, x) == 1
+    assert collect( x + y*x, x ) == x * (1 + y)
+    assert collect( x + x**2, x ) == x + x**2
+    assert collect( x**2 + y*x**2, x ) == (x**2)*(1 + y)
+    assert collect( x**2 + y*x, x ) == x*y + x**2
+    assert collect( 2*x**2 + y*x**2 + 3*x*y, [x] ) == x**2*(2 + y) + 3*x*y
+    assert collect( 2*x**2 + y*x**2 + 3*x*y, [y] ) == 2*x**2 + y*(x**2 + 3*x)
+
+    assert collect( ((1 + y + x)**4).expand(), x) == ((1 + y)**4).expand() + \
+        x*(4*(1 + y)**3).expand() + x**2*(6*(1 + y)**2).expand() + \
+        x**3*(4*(1 + y)).expand() + x**4
+    # symbols can be given as any iterable
+    expr = x + y
+    assert collect(expr, expr.free_symbols) == expr
+    assert collect(x*exp(x) + sin(x)*y + sin(x)*2 + 3*x, x, exact=None
+        ) == x*exp(x) + 3*x + (y + 2)*sin(x)
+    assert collect(x*exp(x) + sin(x)*y + sin(x)*2 + 3*x + y*x +
+        y*x*exp(x), x, exact=None
+        ) == x*exp(x)*(y + 1) + (3 + y)*x + (y + 2)*sin(x)
+
+
+def test_collect_2():
+    """Collect with respect to a sum"""
+    a, b, x = symbols('a,b,x')
+    assert collect(a*(cos(x) + sin(x)) + b*(cos(x) + sin(x)),
+        sin(x) + cos(x)) == (a + b)*(cos(x) + sin(x))
+
+
+def test_collect_3():
+    """Collect with respect to a product"""
+    a, b, c = symbols('a,b,c')
+    f = Function('f')
+    x, y, z, n = symbols('x,y,z,n')
+
+    assert collect(-x/8 + x*y, -x) == x*(y - Rational(1, 8))
+
+    assert collect( 1 + x*(y**2), x*y ) == 1 + x*(y**2)
+    assert collect( x*y + a*x*y, x*y) == x*y*(1 + a)
+    assert collect( 1 + x*y + a*x*y, x*y) == 1 + x*y*(1 + a)
+    assert collect(a*x*f(x) + b*(x*f(x)), x*f(x)) == x*(a + b)*f(x)
+
+    assert collect(a*x*log(x) + b*(x*log(x)), x*log(x)) == x*(a + b)*log(x)
+    assert collect(a*x**2*log(x)**2 + b*(x*log(x))**2, x*log(x)) == \
+        x**2*log(x)**2*(a + b)
+
+    # with respect to a product of three symbols
+    assert collect(y*x*z + a*x*y*z, x*y*z) == (1 + a)*x*y*z
+
+
+def test_collect_4():
+    """Collect with respect to a power"""
+    a, b, c, x = symbols('a,b,c,x')
+
+    assert collect(a*x**c + b*x**c, x**c) == x**c*(a + b)
+    # issue 6096: 2 stays with c (unless c is integer or x is positive0
+    assert collect(a*x**(2*c) + b*x**(2*c), x**c) == x**(2*c)*(a + b)
+
+
+def test_collect_5():
+    """Collect with respect to a tuple"""
+    a, x, y, z, n = symbols('a,x,y,z,n')
+    assert collect(x**2*y**4 + z*(x*y**2)**2 + z + a*z, [x*y**2, z]) in [
+        z*(1 + a + x**2*y**4) + x**2*y**4,
+        z*(1 + a) + x**2*y**4*(1 + z) ]
+    assert collect((1 + (x + y) + (x + y)**2).expand(),
+                   [x, y]) == 1 + y + x*(1 + 2*y) + x**2 + y**2
+
+
+def test_collect_pr19431():
+    """Unevaluated collect with respect to a product"""
+    a = symbols('a')
+    assert collect(a**2*(a**2 + 1), a**2, evaluate=False)[a**2] == (a**2 + 1)
+
+
+def test_collect_D():
+    D = Derivative
+    f = Function('f')
+    x, a, b = symbols('x,a,b')
+    fx = D(f(x), x)
+    fxx = D(f(x), x, x)
+
+    assert collect(a*fx + b*fx, fx) == (a + b)*fx
+    assert collect(a*D(fx, x) + b*D(fx, x), fx) == (a + b)*D(fx, x)
+    assert collect(a*fxx + b*fxx, fx) == (a + b)*D(fx, x)
+    # issue 4784
+    assert collect(5*f(x) + 3*fx, fx) == 5*f(x) + 3*fx
+    assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x)) == \
+        (x*f(x) + f(x))*D(f(x), x) + f(x)
+    assert collect(f(x) + f(x)*diff(f(x), x) + x*diff(f(x), x)*f(x), f(x).diff(x), exact=True) == \
+        (x*f(x) + f(x))*D(f(x), x) + f(x)
+    assert collect(1/f(x) + 1/f(x)*diff(f(x), x) + x*diff(f(x), x)/f(x), f(x).diff(x), exact=True) == \
+        (1/f(x) + x/f(x))*D(f(x), x) + 1/f(x)
+    e = (1 + x*fx + fx)/f(x)
+    assert collect(e.expand(), fx) == fx*(x/f(x) + 1/f(x)) + 1/f(x)
+
+
+def test_collect_func():
+    f = ((x + a + 1)**3).expand()
+
+    assert collect(f, x) == a**3 + 3*a**2 + 3*a + x**3 + x**2*(3*a + 3) + \
+        x*(3*a**2 + 6*a + 3) + 1
+    assert collect(f, x, factor) == x**3 + 3*x**2*(a + 1) + 3*x*(a + 1)**2 + \
+        (a + 1)**3
+
+    assert collect(f, x, evaluate=False) == {
+        S.One: a**3 + 3*a**2 + 3*a + 1,
+        x: 3*a**2 + 6*a + 3, x**2: 3*a + 3,
+        x**3: 1
+    }
+
+    assert collect(f, x, factor, evaluate=False) == {
+        S.One: (a + 1)**3, x: 3*(a + 1)**2,
+        x**2: umul(S(3), a + 1), x**3: 1}
+
+
+def test_collect_order():
+    a, b, x, t = symbols('a,b,x,t')
+
+    assert collect(t + t*x + t*x**2 + O(x**3), t) == t*(1 + x + x**2 + O(x**3))
+    assert collect(t + t*x + x**2 + O(x**3), t) == \
+        t*(1 + x + O(x**3)) + x**2 + O(x**3)
+
+    f = a*x + b*x + c*x**2 + d*x**2 + O(x**3)
+    g = x*(a + b) + x**2*(c + d) + O(x**3)
+
+    assert collect(f, x) == g
+    assert collect(f, x, distribute_order_term=False) == g
+
+    f = sin(a + b).series(b, 0, 10)
+
+    assert collect(f, [sin(a), cos(a)]) == \
+        sin(a)*cos(b).series(b, 0, 10) + cos(a)*sin(b).series(b, 0, 10)
+    assert collect(f, [sin(a), cos(a)], distribute_order_term=False) == \
+        sin(a)*cos(b).series(b, 0, 10).removeO() + \
+        cos(a)*sin(b).series(b, 0, 10).removeO() + O(b**10)
+
+
+def test_rcollect():
+    assert rcollect((x**2*y + x*y + x + y)/(x + y), y) == \
+        (x + y*(1 + x + x**2))/(x + y)
+    assert rcollect(sqrt(-((x + 1)*(y + 1))), z) == sqrt(-((x + 1)*(y + 1)))
+
+
+def test_collect_D_0():
+    D = Derivative
+    f = Function('f')
+    x, a, b = symbols('x,a,b')
+    fxx = D(f(x), x, x)
+
+    assert collect(a*fxx + b*fxx, fxx) == (a + b)*fxx
+
+
+def test_collect_Wild():
+    """Collect with respect to functions with Wild argument"""
+    a, b, x, y = symbols('a b x y')
+    f = Function('f')
+    w1 = Wild('.1')
+    w2 = Wild('.2')
+    assert collect(f(x) + a*f(x), f(w1)) == (1 + a)*f(x)
+    assert collect(f(x, y) + a*f(x, y), f(w1)) == f(x, y) + a*f(x, y)
+    assert collect(f(x, y) + a*f(x, y), f(w1, w2)) == (1 + a)*f(x, y)
+    assert collect(f(x, y) + a*f(x, y), f(w1, w1)) == f(x, y) + a*f(x, y)
+    assert collect(f(x, x) + a*f(x, x), f(w1, w1)) == (1 + a)*f(x, x)
+    assert collect(a*(x + 1)**y + (x + 1)**y, w1**y) == (1 + a)*(x + 1)**y
+    assert collect(a*(x + 1)**y + (x + 1)**y, w1**b) == \
+        a*(x + 1)**y + (x + 1)**y
+    assert collect(a*(x + 1)**y + (x + 1)**y, (x + 1)**w2) == \
+        (1 + a)*(x + 1)**y
+    assert collect(a*(x + 1)**y + (x + 1)**y, w1**w2) == (1 + a)*(x + 1)**y
+
+
+def test_collect_const():
+    # coverage not provided by above tests
+    assert collect_const(2*sqrt(3) + 4*a*sqrt(5)) == \
+        2*(2*sqrt(5)*a + sqrt(3))  # let the primitive reabsorb
+    assert collect_const(2*sqrt(3) + 4*a*sqrt(5), sqrt(3)) == \
+        2*sqrt(3) + 4*a*sqrt(5)
+    assert collect_const(sqrt(2)*(1 + sqrt(2)) + sqrt(3) + x*sqrt(2)) == \
+        sqrt(2)*(x + 1 + sqrt(2)) + sqrt(3)
+
+    # issue 5290
+    assert collect_const(2*x + 2*y + 1, 2) == \
+        collect_const(2*x + 2*y + 1) == \
+        Add(S.One, Mul(2, x + y, evaluate=False), evaluate=False)
+    assert collect_const(-y - z) == Mul(-1, y + z, evaluate=False)
+    assert collect_const(2*x - 2*y - 2*z, 2) == \
+        Mul(2, x - y - z, evaluate=False)
+    assert collect_const(2*x - 2*y - 2*z, -2) == \
+        _unevaluated_Add(2*x, Mul(-2, y + z, evaluate=False))
+
+    # this is why the content_primitive is used
+    eq = (sqrt(15 + 5*sqrt(2))*x + sqrt(3 + sqrt(2))*y)*2
+    assert collect_sqrt(eq + 2) == \
+        2*sqrt(sqrt(2) + 3)*(sqrt(5)*x + y) + 2
+
+    # issue 16296
+    assert collect_const(a + b + x/2 + y/2) == a + b + Mul(S.Half, x + y, evaluate=False)
+
+
+def test_issue_13143():
+    f = Function('f')
+    fx = f(x).diff(x)
+    e = f(x) + fx + f(x)*fx
+    # collect function before derivative
+    assert collect(e, Wild('w')) == f(x)*(fx + 1) + fx
+    e = f(x) + f(x)*fx + x*fx*f(x)
+    assert collect(e, fx) == (x*f(x) + f(x))*fx + f(x)
+    assert collect(e, f(x)) == (x*fx + fx + 1)*f(x)
+    e = f(x) + fx + f(x)*fx
+    assert collect(e, [f(x), fx]) == f(x)*(1 + fx) + fx
+    assert collect(e, [fx, f(x)]) == fx*(1 + f(x)) + f(x)
+
+
+def test_issue_6097():
+    assert collect(a*y**(2.0*x) + b*y**(2.0*x), y**x) == (a + b)*(y**x)**2.0
+    assert collect(a*2**(2.0*x) + b*2**(2.0*x), 2**x) == (a + b)*(2**x)**2.0
+
+
+def test_fraction_expand():
+    eq = (x + y)*y/x
+    assert eq.expand(frac=True) == fraction_expand(eq) == (x*y + y**2)/x
+    assert eq.expand() == y + y**2/x
+
+
+def test_fraction():
+    x, y, z = map(Symbol, 'xyz')
+    A = Symbol('A', commutative=False)
+
+    assert fraction(S.Half) == (1, 2)
+
+    assert fraction(x) == (x, 1)
+    assert fraction(1/x) == (1, x)
+    assert fraction(x/y) == (x, y)
+    assert fraction(x/2) == (x, 2)
+
+    assert fraction(x*y/z) == (x*y, z)
+    assert fraction(x/(y*z)) == (x, y*z)
+
+    assert fraction(1/y**2) == (1, y**2)
+    assert fraction(x/y**2) == (x, y**2)
+
+    assert fraction((x**2 + 1)/y) == (x**2 + 1, y)
+    assert fraction(x*(y + 1)/y**7) == (x*(y + 1), y**7)
+
+    assert fraction(exp(-x), exact=True) == (exp(-x), 1)
+    assert fraction((1/(x + y))/2, exact=True) == (1, Mul(2,(x + y), evaluate=False))
+
+    assert fraction(x*A/y) == (x*A, y)
+    assert fraction(x*A**-1/y) == (x*A**-1, y)
+
+    n = symbols('n', negative=True)
+    assert fraction(exp(n)) == (1, exp(-n))
+    assert fraction(exp(-n)) == (exp(-n), 1)
+
+    p = symbols('p', positive=True)
+    assert fraction(exp(-p)*log(p), exact=True) == (exp(-p)*log(p), 1)
+
+    m = Mul(1, 1, S.Half, evaluate=False)
+    assert fraction(m) == (1, 2)
+    assert fraction(m, exact=True) == (Mul(1, 1, evaluate=False), 2)
+
+    m = Mul(1, 1, S.Half, S.Half, Pow(1, -1, evaluate=False), evaluate=False)
+    assert fraction(m) == (1, 4)
+    assert fraction(m, exact=True) == \
+            (Mul(1, 1, evaluate=False), Mul(2, 2, 1, evaluate=False))
+
+
+def test_issue_5615():
+    aA, Re, a, b, D = symbols('aA Re a b D')
+    e = ((D**3*a + b*aA**3)/Re).expand()
+    assert collect(e, [aA**3/Re, a]) == e
+
+
+def test_issue_5933():
+    from sympy.geometry.polygon import (Polygon, RegularPolygon)
+    from sympy.simplify.radsimp import denom
+    x = Polygon(*RegularPolygon((0, 0), 1, 5).vertices).centroid.x
+    assert abs(denom(x).n()) > 1e-12
+    assert abs(denom(radsimp(x))) > 1e-12  # in case simplify didn't handle it
+
+
+def test_issue_14608():
+    a, b = symbols('a b', commutative=False)
+    x, y = symbols('x y')
+    raises(AttributeError, lambda: collect(a*b + b*a, a))
+    assert collect(x*y + y*(x+1), a) == x*y + y*(x+1)
+    assert collect(x*y + y*(x+1) + a*b + b*a, y) == y*(2*x + 1) + a*b + b*a
+
+
+def test_collect_abs():
+    s = abs(x) + abs(y)
+    assert collect_abs(s) == s
+    assert unchanged(Mul, abs(x), abs(y))
+    ans = Abs(x*y)
+    assert isinstance(ans, Abs)
+    assert collect_abs(abs(x)*abs(y)) == ans
+    assert collect_abs(1 + exp(abs(x)*abs(y))) == 1 + exp(ans)
+
+    # See https://github.com/sympy/sympy/issues/12910
+    p = Symbol('p', positive=True)
+    assert collect_abs(p/abs(1-p)).is_commutative is True
+
+
+def test_issue_19149():
+    eq = exp(3*x/4)
+    assert collect(eq, exp(x)) == eq
+
+def test_issue_19719():
+    a, b = symbols('a, b')
+    expr = a**2 * (b + 1) + (7 + 1/b)/a
+    collected = collect(expr, (a**2, 1/a), evaluate=False)
+    # Would return {_Dummy_20**(-2): b + 1, 1/a: 7 + 1/b} without xreplace
+    assert collected == {a**2: b + 1, 1/a: 7 + 1/b}
+
+
+def test_issue_21355():
+    assert radsimp(1/(x + sqrt(x**2))) == 1/(x + sqrt(x**2))
+    assert radsimp(1/(x - sqrt(x**2))) == 1/(x - sqrt(x**2))

rl/Lib/site-packages/sympy/simplify/tests/test_ratsimp.py

@@ -0,0 +1,78 @@
+from sympy.core.numbers import (Rational, pi)
+from sympy.functions.elementary.exponential import log
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.special.error_functions import erf
+from sympy.polys.domains import GF
+from sympy.simplify.ratsimp import (ratsimp, ratsimpmodprime)
+
+from sympy.abc import x, y, z, t, a, b, c, d, e
+
+
+def test_ratsimp():
+    f, g = 1/x + 1/y, (x + y)/(x*y)
+
+    assert f != g and ratsimp(f) == g
+
+    f, g = 1/(1 + 1/x), 1 - 1/(x + 1)
+
+    assert f != g and ratsimp(f) == g
+
+    f, g = x/(x + y) + y/(x + y), 1
+
+    assert f != g and ratsimp(f) == g
+
+    f, g = -x - y - y**2/(x + y) + x**2/(x + y), -2*y
+
+    assert f != g and ratsimp(f) == g
+
+    f = (a*c*x*y + a*c*z - b*d*x*y - b*d*z - b*t*x*y - b*t*x - b*t*z +
+         e*x)/(x*y + z)
+    G = [a*c - b*d - b*t + (-b*t*x + e*x)/(x*y + z),
+         a*c - b*d - b*t - ( b*t*x - e*x)/(x*y + z)]
+
+    assert f != g and ratsimp(f) in G
+
+    A = sqrt(pi)
+
+    B = log(erf(x) - 1)
+    C = log(erf(x) + 1)
+
+    D = 8 - 8*erf(x)
+
+    f = A*B/D - A*C/D + A*C*erf(x)/D - A*B*erf(x)/D + 2*A/D
+
+    assert ratsimp(f) == A*B/8 - A*C/8 - A/(4*erf(x) - 4)
+
+
+def test_ratsimpmodprime():
+    a = y**5 + x + y
+    b = x - y
+    F = [x*y**5 - x - y]
+    assert ratsimpmodprime(a/b, F, x, y, order='lex') == \
+        (-x**2 - x*y - x - y) / (-x**2 + x*y)
+
+    a = x + y**2 - 2
+    b = x + y**2 - y - 1
+    F = [x*y - 1]
+    assert ratsimpmodprime(a/b, F, x, y, order='lex') == \
+        (1 + y - x)/(y - x)
+
+    a = 5*x**3 + 21*x**2 + 4*x*y + 23*x + 12*y + 15
+    b = 7*x**3 - y*x**2 + 31*x**2 + 2*x*y + 15*y + 37*x + 21
+    F = [x**2 + y**2 - 1]
+    assert ratsimpmodprime(a/b, F, x, y, order='lex') == \
+        (1 + 5*y - 5*x)/(8*y - 6*x)
+
+    a = x*y - x - 2*y + 4
+    b = x + y**2 - 2*y
+    F = [x - 2, y - 3]
+    assert ratsimpmodprime(a/b, F, x, y, order='lex') == \
+        Rational(2, 5)
+
+    # Test a bug where denominators would be dropped
+    assert ratsimpmodprime(x, [y - 2*x], order='lex') == \
+        y/2
+
+    a = (x**5 + 2*x**4 + 2*x**3 + 2*x**2 + x + 2/x + x**(-2))
+    assert ratsimpmodprime(a, [x + 1], domain=GF(2)) == 1
+    assert ratsimpmodprime(a, [x + 1], domain=GF(3)) == -1

rl/Lib/site-packages/sympy/simplify/tests/test_rewrite.py

@@ -0,0 +1,31 @@
+from sympy.core.numbers import I
+from sympy.core.symbol import symbols
+from sympy.functions.elementary.exponential import exp
+from sympy.functions.elementary.trigonometric import (cos, cot, sin)
+from sympy.testing.pytest import _both_exp_pow
+
+x, y, z, n = symbols('x,y,z,n')
+
+
+@_both_exp_pow
+def test_has():
+    assert cot(x).has(x)
+    assert cot(x).has(cot)
+    assert not cot(x).has(sin)
+    assert sin(x).has(x)
+    assert sin(x).has(sin)
+    assert not sin(x).has(cot)
+    assert exp(x).has(exp)
+
+
+@_both_exp_pow
+def test_sin_exp_rewrite():
+    assert sin(x).rewrite(sin, exp) == -I/2*(exp(I*x) - exp(-I*x))
+    assert sin(x).rewrite(sin, exp).rewrite(exp, sin) == sin(x)
+    assert cos(x).rewrite(cos, exp).rewrite(exp, cos) == cos(x)
+    assert (sin(5*y) - sin(
+        2*x)).rewrite(sin, exp).rewrite(exp, sin) == sin(5*y) - sin(2*x)
+    assert sin(x + y).rewrite(sin, exp).rewrite(exp, sin) == sin(x + y)
+    assert cos(x + y).rewrite(cos, exp).rewrite(exp, cos) == cos(x + y)
+    # This next test currently passes... not clear whether it should or not?
+    assert cos(x).rewrite(cos, exp).rewrite(exp, sin) == cos(x)

rl/Lib/site-packages/sympy/simplify/tests/test_sqrtdenest.py

@@ -0,0 +1,204 @@
+from sympy.core.mul import Mul
+from sympy.core.numbers import (I, Integer, Rational)
+from sympy.core.symbol import Symbol
+from sympy.functions.elementary.miscellaneous import (root, sqrt)
+from sympy.functions.elementary.trigonometric import cos
+from sympy.integrals.integrals import Integral
+from sympy.simplify.sqrtdenest import sqrtdenest
+from sympy.simplify.sqrtdenest import (
+    _subsets as subsets, _sqrt_numeric_denest)
+
+r2, r3, r5, r6, r7, r10, r15, r29 = [sqrt(x) for x in (2, 3, 5, 6, 7, 10,
+                                          15, 29)]
+
+
+def test_sqrtdenest():
+    d = {sqrt(5 + 2 * r6): r2 + r3,
+        sqrt(5. + 2 * r6): sqrt(5. + 2 * r6),
+        sqrt(5. + 4*sqrt(5 + 2 * r6)): sqrt(5.0 + 4*r2 + 4*r3),
+        sqrt(r2): sqrt(r2),
+        sqrt(5 + r7): sqrt(5 + r7),
+        sqrt(3 + sqrt(5 + 2*r7)):
+         3*r2*(5 + 2*r7)**Rational(1, 4)/(2*sqrt(6 + 3*r7)) +
+         r2*sqrt(6 + 3*r7)/(2*(5 + 2*r7)**Rational(1, 4)),
+        sqrt(3 + 2*r3): 3**Rational(3, 4)*(r6/2 + 3*r2/2)/3}
+    for i in d:
+        assert sqrtdenest(i) == d[i], i
+
+
+def test_sqrtdenest2():
+    assert sqrtdenest(sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))) == \
+        r5 + sqrt(11 - 2*r29)
+    e = sqrt(-r5 + sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16))
+    assert sqrtdenest(e) == root(-2*r29 + 11, 4)
+    r = sqrt(1 + r7)
+    assert sqrtdenest(sqrt(1 + r)) == sqrt(1 + r)
+    e = sqrt(((1 + sqrt(1 + 2*sqrt(3 + r2 + r5)))**2).expand())
+    assert sqrtdenest(e) == 1 + sqrt(1 + 2*sqrt(r2 + r5 + 3))
+
+    assert sqrtdenest(sqrt(5*r3 + 6*r2)) == \
+        sqrt(2)*root(3, 4) + root(3, 4)**3
+
+    assert sqrtdenest(sqrt(((1 + r5 + sqrt(1 + r3))**2).expand())) == \
+        1 + r5 + sqrt(1 + r3)
+
+    assert sqrtdenest(sqrt(((1 + r5 + r7 + sqrt(1 + r3))**2).expand())) == \
+        1 + sqrt(1 + r3) + r5 + r7
+
+    e = sqrt(((1 + cos(2) + cos(3) + sqrt(1 + r3))**2).expand())
+    assert sqrtdenest(e) == cos(3) + cos(2) + 1 + sqrt(1 + r3)
+
+    e = sqrt(-2*r10 + 2*r2*sqrt(-2*r10 + 11) + 14)
+    assert sqrtdenest(e) == sqrt(-2*r10 - 2*r2 + 4*r5 + 14)
+
+    # check that the result is not more complicated than the input
+    z = sqrt(-2*r29 + cos(2) + 2*sqrt(-10*r29 + 55) + 16)
+    assert sqrtdenest(z) == z
+
+    assert sqrtdenest(sqrt(r6 + sqrt(15))) == sqrt(r6 + sqrt(15))
+
+    z = sqrt(15 - 2*sqrt(31) + 2*sqrt(55 - 10*r29))
+    assert sqrtdenest(z) == z
+
+
+def test_sqrtdenest_rec():
+    assert sqrtdenest(sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 33)) == \
+        -r2 + r3 + 2*r7
+    assert sqrtdenest(sqrt(-28*r7 - 14*r5 + 4*sqrt(35) + 82)) == \
+        -7 + r5 + 2*r7
+    assert sqrtdenest(sqrt(6*r2/11 + 2*sqrt(22)/11 + 6*sqrt(11)/11 + 2)) == \
+        sqrt(11)*(r2 + 3 + sqrt(11))/11
+    assert sqrtdenest(sqrt(468*r3 + 3024*r2 + 2912*r6 + 19735)) == \
+        9*r3 + 26 + 56*r6
+    z = sqrt(-490*r3 - 98*sqrt(115) - 98*sqrt(345) - 2107)
+    assert sqrtdenest(z) == sqrt(-1)*(7*r5 + 7*r15 + 7*sqrt(23))
+    z = sqrt(-4*sqrt(14) - 2*r6 + 4*sqrt(21) + 34)
+    assert sqrtdenest(z) == z
+    assert sqrtdenest(sqrt(-8*r2 - 2*r5 + 18)) == -r10 + 1 + r2 + r5
+    assert sqrtdenest(sqrt(8*r2 + 2*r5 - 18)) == \
+        sqrt(-1)*(-r10 + 1 + r2 + r5)
+    assert sqrtdenest(sqrt(8*r2/3 + 14*r5/3 + Rational(154, 9))) == \
+        -r10/3 + r2 + r5 + 3
+    assert sqrtdenest(sqrt(sqrt(2*r6 + 5) + sqrt(2*r7 + 8))) == \
+        sqrt(1 + r2 + r3 + r7)
+    assert sqrtdenest(sqrt(4*r15 + 8*r5 + 12*r3 + 24)) == 1 + r3 + r5 + r15
+
+    w = 1 + r2 + r3 + r5 + r7
+    assert sqrtdenest(sqrt((w**2).expand())) == w
+    z = sqrt((w**2).expand() + 1)
+    assert sqrtdenest(z) == z
+
+    z = sqrt(2*r10 + 6*r2 + 4*r5 + 12 + 10*r15 + 30*r3)
+    assert sqrtdenest(z) == z
+
+
+def test_issue_6241():
+    z = sqrt( -320 + 32*sqrt(5) + 64*r15)
+    assert sqrtdenest(z) == z
+
+
+def test_sqrtdenest3():
+    z = sqrt(13 - 2*r10 + 2*r2*sqrt(-2*r10 + 11))
+    assert sqrtdenest(z) == -1 + r2 + r10
+    assert sqrtdenest(z, max_iter=1) == -1 + sqrt(2) + sqrt(10)
+    z = sqrt(sqrt(r2 + 2) + 2)
+    assert sqrtdenest(z) == z
+    assert sqrtdenest(sqrt(-2*r10 + 4*r2*sqrt(-2*r10 + 11) + 20)) == \
+        sqrt(-2*r10 - 4*r2 + 8*r5 + 20)
+    assert sqrtdenest(sqrt((112 + 70*r2) + (46 + 34*r2)*r5)) == \
+        r10 + 5 + 4*r2 + 3*r5
+    z = sqrt(5 + sqrt(2*r6 + 5)*sqrt(-2*r29 + 2*sqrt(-10*r29 + 55) + 16))
+    r = sqrt(-2*r29 + 11)
+    assert sqrtdenest(z) == sqrt(r2*r + r3*r + r10 + r15 + 5)
+
+    n = sqrt(2*r6/7 + 2*r7/7 + 2*sqrt(42)/7 + 2)
+    d = sqrt(16 - 2*r29 + 2*sqrt(55 - 10*r29))
+    assert sqrtdenest(n/d) == r7*(1 + r6 + r7)/(Mul(7, (sqrt(-2*r29 + 11) + r5),
+                                                    evaluate=False))
+
+
+def test_sqrtdenest4():
+    # see Denest_en.pdf in https://github.com/sympy/sympy/issues/3192
+    z = sqrt(8 - r2*sqrt(5 - r5) - sqrt(3)*(1 + r5))
+    z1 = sqrtdenest(z)
+    c = sqrt(-r5 + 5)
+    z1 = ((-r15*c - r3*c + c + r5*c - r6 - r2 + r10 + sqrt(30))/4).expand()
+    assert sqrtdenest(z) == z1
+
+    z = sqrt(2*r2*sqrt(r2 + 2) + 5*r2 + 4*sqrt(r2 + 2) + 8)
+    assert sqrtdenest(z) == r2 + sqrt(r2 + 2) + 2
+
+    w = 2 + r2 + r3 + (1 + r3)*sqrt(2 + r2 + 5*r3)
+    z = sqrt((w**2).expand())
+    assert sqrtdenest(z) == w.expand()
+
+
+def test_sqrt_symbolic_denest():
+    x = Symbol('x')
+    z = sqrt(((1 + sqrt(sqrt(2 + x) + 3))**2).expand())
+    assert sqrtdenest(z) == sqrt((1 + sqrt(sqrt(2 + x) + 3))**2)
+    z = sqrt(((1 + sqrt(sqrt(2 + cos(1)) + 3))**2).expand())
+    assert sqrtdenest(z) == 1 + sqrt(sqrt(2 + cos(1)) + 3)
+    z = ((1 + cos(2))**4 + 1).expand()
+    assert sqrtdenest(z) == z
+    z = sqrt(((1 + sqrt(sqrt(2 + cos(3*x)) + 3))**2 + 1).expand())
+    assert sqrtdenest(z) == z
+    c = cos(3)
+    c2 = c**2
+    assert sqrtdenest(sqrt(2*sqrt(1 + r3)*c + c2 + 1 + r3*c2)) == \
+        -1 - sqrt(1 + r3)*c
+    ra = sqrt(1 + r3)
+    z = sqrt(20*ra*sqrt(3 + 3*r3) + 12*r3*ra*sqrt(3 + 3*r3) + 64*r3 + 112)
+    assert sqrtdenest(z) == z
+
+
+def test_issue_5857():
+    from sympy.abc import x, y
+    z = sqrt(1/(4*r3 + 7) + 1)
+    ans = (r2 + r6)/(r3 + 2)
+    assert sqrtdenest(z) == ans
+    assert sqrtdenest(1 + z) == 1 + ans
+    assert sqrtdenest(Integral(z + 1, (x, 1, 2))) == \
+        Integral(1 + ans, (x, 1, 2))
+    assert sqrtdenest(x + sqrt(y)) == x + sqrt(y)
+    ans = (r2 + r6)/(r3 + 2)
+    assert sqrtdenest(z) == ans
+    assert sqrtdenest(1 + z) == 1 + ans
+    assert sqrtdenest(Integral(z + 1, (x, 1, 2))) == \
+        Integral(1 + ans, (x, 1, 2))
+    assert sqrtdenest(x + sqrt(y)) == x + sqrt(y)
+
+
+def test_subsets():
+    assert subsets(1) == [[1]]
+    assert subsets(4) == [
+        [1, 0, 0, 0], [0, 1, 0, 0], [1, 1, 0, 0], [0, 0, 1, 0], [1, 0, 1, 0],
+        [0, 1, 1, 0], [1, 1, 1, 0], [0, 0, 0, 1], [1, 0, 0, 1], [0, 1, 0, 1],
+        [1, 1, 0, 1], [0, 0, 1, 1], [1, 0, 1, 1], [0, 1, 1, 1], [1, 1, 1, 1]]
+
+
+def test_issue_5653():
+    assert sqrtdenest(
+        sqrt(2 + sqrt(2 + sqrt(2)))) == sqrt(2 + sqrt(2 + sqrt(2)))
+
+def test_issue_12420():
+    assert sqrtdenest((3 - sqrt(2)*sqrt(4 + 3*I) + 3*I)/2) == I
+    e = 3 - sqrt(2)*sqrt(4 + I) + 3*I
+    assert sqrtdenest(e) == e
+
+def test_sqrt_ratcomb():
+    assert sqrtdenest(sqrt(1 + r3) + sqrt(3 + 3*r3) - sqrt(10 + 6*r3)) == 0
+
+def test_issue_18041():
+    e = -sqrt(-2 + 2*sqrt(3)*I)
+    assert sqrtdenest(e) == -1 - sqrt(3)*I
+
+def test_issue_19914():
+    a = Integer(-8)
+    b = Integer(-1)
+    r = Integer(63)
+    d2 = a*a - b*b*r
+
+    assert _sqrt_numeric_denest(a, b, r, d2) == \
+        sqrt(14)*I/2 + 3*sqrt(2)*I/2
+    assert sqrtdenest(sqrt(-8-sqrt(63))) == sqrt(14)*I/2 + 3*sqrt(2)*I/2

rl/Lib/site-packages/sympy/simplify/tests/test_trigsimp.py

@@ -0,0 +1,520 @@
+from itertools import product
+from sympy.core.function import (Subs, count_ops, diff, expand)
+from sympy.core.numbers import (E, I, Rational, pi)
+from sympy.core.singleton import S
+from sympy.core.symbol import (Symbol, symbols)
+from sympy.functions.elementary.exponential import (exp, log)
+from sympy.functions.elementary.hyperbolic import (cosh, coth, sinh, tanh)
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.functions.elementary.piecewise import Piecewise
+from sympy.functions.elementary.trigonometric import (cos, cot, sin, tan)
+from sympy.functions.elementary.trigonometric import (acos, asin, atan2)
+from sympy.functions.elementary.trigonometric import (asec, acsc)
+from sympy.functions.elementary.trigonometric import (acot, atan)
+from sympy.integrals.integrals import integrate
+from sympy.matrices.dense import Matrix
+from sympy.simplify.simplify import simplify
+from sympy.simplify.trigsimp import (exptrigsimp, trigsimp)
+
+from sympy.testing.pytest import XFAIL
+
+from sympy.abc import x, y
+
+
+
+def test_trigsimp1():
+    x, y = symbols('x,y')
+
+    assert trigsimp(1 - sin(x)**2) == cos(x)**2
+    assert trigsimp(1 - cos(x)**2) == sin(x)**2
+    assert trigsimp(sin(x)**2 + cos(x)**2) == 1
+    assert trigsimp(1 + tan(x)**2) == 1/cos(x)**2
+    assert trigsimp(1/cos(x)**2 - 1) == tan(x)**2
+    assert trigsimp(1/cos(x)**2 - tan(x)**2) == 1
+    assert trigsimp(1 + cot(x)**2) == 1/sin(x)**2
+    assert trigsimp(1/sin(x)**2 - 1) == 1/tan(x)**2
+    assert trigsimp(1/sin(x)**2 - cot(x)**2) == 1
+
+    assert trigsimp(5*cos(x)**2 + 5*sin(x)**2) == 5
+    assert trigsimp(5*cos(x/2)**2 + 2*sin(x/2)**2) == 3*cos(x)/2 + Rational(7, 2)
+
+    assert trigsimp(sin(x)/cos(x)) == tan(x)
+    assert trigsimp(2*tan(x)*cos(x)) == 2*sin(x)
+    assert trigsimp(cot(x)**3*sin(x)**3) == cos(x)**3
+    assert trigsimp(y*tan(x)**2/sin(x)**2) == y/cos(x)**2
+    assert trigsimp(cot(x)/cos(x)) == 1/sin(x)
+
+    assert trigsimp(sin(x + y) + sin(x - y)) == 2*sin(x)*cos(y)
+    assert trigsimp(sin(x + y) - sin(x - y)) == 2*sin(y)*cos(x)
+    assert trigsimp(cos(x + y) + cos(x - y)) == 2*cos(x)*cos(y)
+    assert trigsimp(cos(x + y) - cos(x - y)) == -2*sin(x)*sin(y)
+    assert trigsimp(tan(x + y) - tan(x)/(1 - tan(x)*tan(y))) == \
+        sin(y)/(-sin(y)*tan(x) + cos(y))  # -tan(y)/(tan(x)*tan(y) - 1)
+
+    assert trigsimp(sinh(x + y) + sinh(x - y)) == 2*sinh(x)*cosh(y)
+    assert trigsimp(sinh(x + y) - sinh(x - y)) == 2*sinh(y)*cosh(x)
+    assert trigsimp(cosh(x + y) + cosh(x - y)) == 2*cosh(x)*cosh(y)
+    assert trigsimp(cosh(x + y) - cosh(x - y)) == 2*sinh(x)*sinh(y)
+    assert trigsimp(tanh(x + y) - tanh(x)/(1 + tanh(x)*tanh(y))) == \
+        sinh(y)/(sinh(y)*tanh(x) + cosh(y))
+
+    assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2) == 1.0
+    e = 2*sin(x)**2 + 2*cos(x)**2
+    assert trigsimp(log(e)) == log(2)
+
+
+def test_trigsimp1a():
+    assert trigsimp(sin(2)**2*cos(3)*exp(2)/cos(2)**2) == tan(2)**2*cos(3)*exp(2)
+    assert trigsimp(tan(2)**2*cos(3)*exp(2)*cos(2)**2) == sin(2)**2*cos(3)*exp(2)
+    assert trigsimp(cot(2)*cos(3)*exp(2)*sin(2)) == cos(3)*exp(2)*cos(2)
+    assert trigsimp(tan(2)*cos(3)*exp(2)/sin(2)) == cos(3)*exp(2)/cos(2)
+    assert trigsimp(cot(2)*cos(3)*exp(2)/cos(2)) == cos(3)*exp(2)/sin(2)
+    assert trigsimp(cot(2)*cos(3)*exp(2)*tan(2)) == cos(3)*exp(2)
+    assert trigsimp(sinh(2)*cos(3)*exp(2)/cosh(2)) == tanh(2)*cos(3)*exp(2)
+    assert trigsimp(tanh(2)*cos(3)*exp(2)*cosh(2)) == sinh(2)*cos(3)*exp(2)
+    assert trigsimp(coth(2)*cos(3)*exp(2)*sinh(2)) == cosh(2)*cos(3)*exp(2)
+    assert trigsimp(tanh(2)*cos(3)*exp(2)/sinh(2)) == cos(3)*exp(2)/cosh(2)
+    assert trigsimp(coth(2)*cos(3)*exp(2)/cosh(2)) == cos(3)*exp(2)/sinh(2)
+    assert trigsimp(coth(2)*cos(3)*exp(2)*tanh(2)) == cos(3)*exp(2)
+
+
+def test_trigsimp2():
+    x, y = symbols('x,y')
+    assert trigsimp(cos(x)**2*sin(y)**2 + cos(x)**2*cos(y)**2 + sin(x)**2,
+            recursive=True) == 1
+    assert trigsimp(sin(x)**2*sin(y)**2 + sin(x)**2*cos(y)**2 + cos(x)**2,
+            recursive=True) == 1
+    assert trigsimp(
+        Subs(x, x, sin(y)**2 + cos(y)**2)) == Subs(x, x, 1)
+
+
+def test_issue_4373():
+    x = Symbol("x")
+    assert abs(trigsimp(2.0*sin(x)**2 + 2.0*cos(x)**2) - 2.0) < 1e-10
+
+
+def test_trigsimp3():
+    x, y = symbols('x,y')
+    assert trigsimp(sin(x)/cos(x)) == tan(x)
+    assert trigsimp(sin(x)**2/cos(x)**2) == tan(x)**2
+    assert trigsimp(sin(x)**3/cos(x)**3) == tan(x)**3
+    assert trigsimp(sin(x)**10/cos(x)**10) == tan(x)**10
+
+    assert trigsimp(cos(x)/sin(x)) == 1/tan(x)
+    assert trigsimp(cos(x)**2/sin(x)**2) == 1/tan(x)**2
+    assert trigsimp(cos(x)**10/sin(x)**10) == 1/tan(x)**10
+
+    assert trigsimp(tan(x)) == trigsimp(sin(x)/cos(x))
+
+
+def test_issue_4661():
+    a, x, y = symbols('a x y')
+    eq = -4*sin(x)**4 + 4*cos(x)**4 - 8*cos(x)**2
+    assert trigsimp(eq) == -4
+    n = sin(x)**6 + 4*sin(x)**4*cos(x)**2 + 5*sin(x)**2*cos(x)**4 + 2*cos(x)**6
+    d = -sin(x)**2 - 2*cos(x)**2
+    assert simplify(n/d) == -1
+    assert trigsimp(-2*cos(x)**2 + cos(x)**4 - sin(x)**4) == -1
+    eq = (- sin(x)**3/4)*cos(x) + (cos(x)**3/4)*sin(x) - sin(2*x)*cos(2*x)/8
+    assert trigsimp(eq) == 0
+
+
+def test_issue_4494():
+    a, b = symbols('a b')
+    eq = sin(a)**2*sin(b)**2 + cos(a)**2*cos(b)**2*tan(a)**2 + cos(a)**2
+    assert trigsimp(eq) == 1
+
+
+def test_issue_5948():
+    a, x, y = symbols('a x y')
+    assert trigsimp(diff(integrate(cos(x)/sin(x)**7, x), x)) == \
+           cos(x)/sin(x)**7
+
+
+def test_issue_4775():
+    a, x, y = symbols('a x y')
+    assert trigsimp(sin(x)*cos(y)+cos(x)*sin(y)) == sin(x + y)
+    assert trigsimp(sin(x)*cos(y)+cos(x)*sin(y)+3) == sin(x + y) + 3
+
+
+def test_issue_4280():
+    a, x, y = symbols('a x y')
+    assert trigsimp(cos(x)**2 + cos(y)**2*sin(x)**2 + sin(y)**2*sin(x)**2) == 1
+    assert trigsimp(a**2*sin(x)**2 + a**2*cos(y)**2*cos(x)**2 + a**2*cos(x)**2*sin(y)**2) == a**2
+    assert trigsimp(a**2*cos(y)**2*sin(x)**2 + a**2*sin(y)**2*sin(x)**2) == a**2*sin(x)**2
+
+
+def test_issue_3210():
+    eqs = (sin(2)*cos(3) + sin(3)*cos(2),
+        -sin(2)*sin(3) + cos(2)*cos(3),
+        sin(2)*cos(3) - sin(3)*cos(2),
+        sin(2)*sin(3) + cos(2)*cos(3),
+        sin(2)*sin(3) + cos(2)*cos(3) + cos(2),
+        sinh(2)*cosh(3) + sinh(3)*cosh(2),
+        sinh(2)*sinh(3) + cosh(2)*cosh(3),
+        )
+    assert [trigsimp(e) for e in eqs] == [
+        sin(5),
+        cos(5),
+        -sin(1),
+        cos(1),
+        cos(1) + cos(2),
+        sinh(5),
+        cosh(5),
+        ]
+
+
+def test_trigsimp_issues():
+    a, x, y = symbols('a x y')
+
+    # issue 4625 - factor_terms works, too
+    assert trigsimp(sin(x)**3 + cos(x)**2*sin(x)) == sin(x)
+
+    # issue 5948
+    assert trigsimp(diff(integrate(cos(x)/sin(x)**3, x), x)) == \
+        cos(x)/sin(x)**3
+    assert trigsimp(diff(integrate(sin(x)/cos(x)**3, x), x)) == \
+        sin(x)/cos(x)**3
+
+    # check integer exponents
+    e = sin(x)**y/cos(x)**y
+    assert trigsimp(e) == e
+    assert trigsimp(e.subs(y, 2)) == tan(x)**2
+    assert trigsimp(e.subs(x, 1)) == tan(1)**y
+
+    # check for multiple patterns
+    assert (cos(x)**2/sin(x)**2*cos(y)**2/sin(y)**2).trigsimp() == \
+        1/tan(x)**2/tan(y)**2
+    assert trigsimp(cos(x)/sin(x)*cos(x+y)/sin(x+y)) == \
+        1/(tan(x)*tan(x + y))
+
+    eq = cos(2)*(cos(3) + 1)**2/(cos(3) - 1)**2
+    assert trigsimp(eq) == eq.factor()  # factor makes denom (-1 + cos(3))**2
+    assert trigsimp(cos(2)*(cos(3) + 1)**2*(cos(3) - 1)**2) == \
+        cos(2)*sin(3)**4
+
+    # issue 6789; this generates an expression that formerly caused
+    # trigsimp to hang
+    assert cot(x).equals(tan(x)) is False
+
+    # nan or the unchanged expression is ok, but not sin(1)
+    z = cos(x)**2 + sin(x)**2 - 1
+    z1 = tan(x)**2 - 1/cot(x)**2
+    n = (1 + z1/z)
+    assert trigsimp(sin(n)) != sin(1)
+    eq = x*(n - 1) - x*n
+    assert trigsimp(eq) is S.NaN
+    assert trigsimp(eq, recursive=True) is S.NaN
+    assert trigsimp(1).is_Integer
+
+    assert trigsimp(-sin(x)**4 - 2*sin(x)**2*cos(x)**2 - cos(x)**4) == -1
+
+
+def test_trigsimp_issue_2515():
+    x = Symbol('x')
+    assert trigsimp(x*cos(x)*tan(x)) == x*sin(x)
+    assert trigsimp(-sin(x) + cos(x)*tan(x)) == 0
+
+
+def test_trigsimp_issue_3826():
+    assert trigsimp(tan(2*x).expand(trig=True)) == tan(2*x)
+
+
+def test_trigsimp_issue_4032():
+    n = Symbol('n', integer=True, positive=True)
+    assert trigsimp(2**(n/2)*cos(pi*n/4)/2 + 2**(n - 1)/2) == \
+        2**(n/2)*cos(pi*n/4)/2 + 2**n/4
+
+
+def test_trigsimp_issue_7761():
+    assert trigsimp(cosh(pi/4)) == cosh(pi/4)
+
+
+def test_trigsimp_noncommutative():
+    x, y = symbols('x,y')
+    A, B = symbols('A,B', commutative=False)
+
+    assert trigsimp(A - A*sin(x)**2) == A*cos(x)**2
+    assert trigsimp(A - A*cos(x)**2) == A*sin(x)**2
+    assert trigsimp(A*sin(x)**2 + A*cos(x)**2) == A
+    assert trigsimp(A + A*tan(x)**2) == A/cos(x)**2
+    assert trigsimp(A/cos(x)**2 - A) == A*tan(x)**2
+    assert trigsimp(A/cos(x)**2 - A*tan(x)**2) == A
+    assert trigsimp(A + A*cot(x)**2) == A/sin(x)**2
+    assert trigsimp(A/sin(x)**2 - A) == A/tan(x)**2
+    assert trigsimp(A/sin(x)**2 - A*cot(x)**2) == A
+
+    assert trigsimp(y*A*cos(x)**2 + y*A*sin(x)**2) == y*A
+
+    assert trigsimp(A*sin(x)/cos(x)) == A*tan(x)
+    assert trigsimp(A*tan(x)*cos(x)) == A*sin(x)
+    assert trigsimp(A*cot(x)**3*sin(x)**3) == A*cos(x)**3
+    assert trigsimp(y*A*tan(x)**2/sin(x)**2) == y*A/cos(x)**2
+    assert trigsimp(A*cot(x)/cos(x)) == A/sin(x)
+
+    assert trigsimp(A*sin(x + y) + A*sin(x - y)) == 2*A*sin(x)*cos(y)
+    assert trigsimp(A*sin(x + y) - A*sin(x - y)) == 2*A*sin(y)*cos(x)
+    assert trigsimp(A*cos(x + y) + A*cos(x - y)) == 2*A*cos(x)*cos(y)
+    assert trigsimp(A*cos(x + y) - A*cos(x - y)) == -2*A*sin(x)*sin(y)
+
+    assert trigsimp(A*sinh(x + y) + A*sinh(x - y)) == 2*A*sinh(x)*cosh(y)
+    assert trigsimp(A*sinh(x + y) - A*sinh(x - y)) == 2*A*sinh(y)*cosh(x)
+    assert trigsimp(A*cosh(x + y) + A*cosh(x - y)) == 2*A*cosh(x)*cosh(y)
+    assert trigsimp(A*cosh(x + y) - A*cosh(x - y)) == 2*A*sinh(x)*sinh(y)
+
+    assert trigsimp(A*cos(0.12345)**2 + A*sin(0.12345)**2) == 1.0*A
+
+
+def test_hyperbolic_simp():
+    x, y = symbols('x,y')
+
+    assert trigsimp(sinh(x)**2 + 1) == cosh(x)**2
+    assert trigsimp(cosh(x)**2 - 1) == sinh(x)**2
+    assert trigsimp(cosh(x)**2 - sinh(x)**2) == 1
+    assert trigsimp(1 - tanh(x)**2) == 1/cosh(x)**2
+    assert trigsimp(1 - 1/cosh(x)**2) == tanh(x)**2
+    assert trigsimp(tanh(x)**2 + 1/cosh(x)**2) == 1
+    assert trigsimp(coth(x)**2 - 1) == 1/sinh(x)**2
+    assert trigsimp(1/sinh(x)**2 + 1) == 1/tanh(x)**2
+    assert trigsimp(coth(x)**2 - 1/sinh(x)**2) == 1
+
+    assert trigsimp(5*cosh(x)**2 - 5*sinh(x)**2) == 5
+    assert trigsimp(5*cosh(x/2)**2 - 2*sinh(x/2)**2) == 3*cosh(x)/2 + Rational(7, 2)
+
+    assert trigsimp(sinh(x)/cosh(x)) == tanh(x)
+    assert trigsimp(tanh(x)) == trigsimp(sinh(x)/cosh(x))
+    assert trigsimp(cosh(x)/sinh(x)) == 1/tanh(x)
+    assert trigsimp(2*tanh(x)*cosh(x)) == 2*sinh(x)
+    assert trigsimp(coth(x)**3*sinh(x)**3) == cosh(x)**3
+    assert trigsimp(y*tanh(x)**2/sinh(x)**2) == y/cosh(x)**2
+    assert trigsimp(coth(x)/cosh(x)) == 1/sinh(x)
+
+    for a in (pi/6*I, pi/4*I, pi/3*I):
+        assert trigsimp(sinh(a)*cosh(x) + cosh(a)*sinh(x)) == sinh(x + a)
+        assert trigsimp(-sinh(a)*cosh(x) + cosh(a)*sinh(x)) == sinh(x - a)
+
+    e = 2*cosh(x)**2 - 2*sinh(x)**2
+    assert trigsimp(log(e)) == log(2)
+
+    # issue 19535:
+    assert trigsimp(sqrt(cosh(x)**2 - 1)) == sqrt(sinh(x)**2)
+
+    assert trigsimp(cosh(x)**2*cosh(y)**2 - cosh(x)**2*sinh(y)**2 - sinh(x)**2,
+            recursive=True) == 1
+    assert trigsimp(sinh(x)**2*sinh(y)**2 - sinh(x)**2*cosh(y)**2 + cosh(x)**2,
+            recursive=True) == 1
+
+    assert abs(trigsimp(2.0*cosh(x)**2 - 2.0*sinh(x)**2) - 2.0) < 1e-10
+
+    assert trigsimp(sinh(x)**2/cosh(x)**2) == tanh(x)**2
+    assert trigsimp(sinh(x)**3/cosh(x)**3) == tanh(x)**3
+    assert trigsimp(sinh(x)**10/cosh(x)**10) == tanh(x)**10
+    assert trigsimp(cosh(x)**3/sinh(x)**3) == 1/tanh(x)**3
+
+    assert trigsimp(cosh(x)/sinh(x)) == 1/tanh(x)
+    assert trigsimp(cosh(x)**2/sinh(x)**2) == 1/tanh(x)**2
+    assert trigsimp(cosh(x)**10/sinh(x)**10) == 1/tanh(x)**10
+
+    assert trigsimp(x*cosh(x)*tanh(x)) == x*sinh(x)
+    assert trigsimp(-sinh(x) + cosh(x)*tanh(x)) == 0
+
+    assert tan(x) != 1/cot(x)  # cot doesn't auto-simplify
+
+    assert trigsimp(tan(x) - 1/cot(x)) == 0
+    assert trigsimp(3*tanh(x)**7 - 2/coth(x)**7) == tanh(x)**7
+
+
+def test_trigsimp_groebner():
+    from sympy.simplify.trigsimp import trigsimp_groebner
+
+    c = cos(x)
+    s = sin(x)
+    ex = (4*s*c + 12*s + 5*c**3 + 21*c**2 + 23*c + 15)/(
+        -s*c**2 + 2*s*c + 15*s + 7*c**3 + 31*c**2 + 37*c + 21)
+    resnum = (5*s - 5*c + 1)
+    resdenom = (8*s - 6*c)
+    results = [resnum/resdenom, (-resnum)/(-resdenom)]
+    assert trigsimp_groebner(ex) in results
+    assert trigsimp_groebner(s/c, hints=[tan]) == tan(x)
+    assert trigsimp_groebner(c*s) == c*s
+    assert trigsimp((-s + 1)/c + c/(-s + 1),
+                    method='groebner') == 2/c
+    assert trigsimp((-s + 1)/c + c/(-s + 1),
+                    method='groebner', polynomial=True) == 2/c
+
+    # Test quick=False works
+    assert trigsimp_groebner(ex, hints=[2]) in results
+    assert trigsimp_groebner(ex, hints=[int(2)]) in results
+
+    # test "I"
+    assert trigsimp_groebner(sin(I*x)/cos(I*x), hints=[tanh]) == I*tanh(x)
+
+    # test hyperbolic / sums
+    assert trigsimp_groebner((tanh(x)+tanh(y))/(1+tanh(x)*tanh(y)),
+                             hints=[(tanh, x, y)]) == tanh(x + y)
+
+
+def test_issue_2827_trigsimp_methods():
+    measure1 = lambda expr: len(str(expr))
+    measure2 = lambda expr: -count_ops(expr)
+                                       # Return the most complicated result
+    expr = (x + 1)/(x + sin(x)**2 + cos(x)**2)
+    ans = Matrix([1])
+    M = Matrix([expr])
+    assert trigsimp(M, method='fu', measure=measure1) == ans
+    assert trigsimp(M, method='fu', measure=measure2) != ans
+    # all methods should work with Basic expressions even if they
+    # aren't Expr
+    M = Matrix.eye(1)
+    assert all(trigsimp(M, method=m) == M for m in
+        'fu matching groebner old'.split())
+    # watch for E in exptrigsimp, not only exp()
+    eq = 1/sqrt(E) + E
+    assert exptrigsimp(eq) == eq
+
+def test_issue_15129_trigsimp_methods():
+    t1 = Matrix([sin(Rational(1, 50)), cos(Rational(1, 50)), 0])
+    t2 = Matrix([sin(Rational(1, 25)), cos(Rational(1, 25)), 0])
+    t3 = Matrix([cos(Rational(1, 25)), sin(Rational(1, 25)), 0])
+    r1 = t1.dot(t2)
+    r2 = t1.dot(t3)
+    assert trigsimp(r1) == cos(Rational(1, 50))
+    assert trigsimp(r2) == sin(Rational(3, 50))
+
+def test_exptrigsimp():
+    def valid(a, b):
+        from sympy.core.random import verify_numerically as tn
+        if not (tn(a, b) and a == b):
+            return False
+        return True
+
+    assert exptrigsimp(exp(x) + exp(-x)) == 2*cosh(x)
+    assert exptrigsimp(exp(x) - exp(-x)) == 2*sinh(x)
+    assert exptrigsimp((2*exp(x)-2*exp(-x))/(exp(x)+exp(-x))) == 2*tanh(x)
+    assert exptrigsimp((2*exp(2*x)-2)/(exp(2*x)+1)) == 2*tanh(x)
+    e = [cos(x) + I*sin(x), cos(x) - I*sin(x),
+         cosh(x) - sinh(x), cosh(x) + sinh(x)]
+    ok = [exp(I*x), exp(-I*x), exp(-x), exp(x)]
+    assert all(valid(i, j) for i, j in zip(
+        [exptrigsimp(ei) for ei in e], ok))
+
+    ue = [cos(x) + sin(x), cos(x) - sin(x),
+          cosh(x) + I*sinh(x), cosh(x) - I*sinh(x)]
+    assert [exptrigsimp(ei) == ei for ei in ue]
+
+    res = []
+    ok = [y*tanh(1), 1/(y*tanh(1)), I*y*tan(1), -I/(y*tan(1)),
+        y*tanh(x), 1/(y*tanh(x)), I*y*tan(x), -I/(y*tan(x)),
+        y*tanh(1 + I), 1/(y*tanh(1 + I))]
+    for a in (1, I, x, I*x, 1 + I):
+        w = exp(a)
+        eq = y*(w - 1/w)/(w + 1/w)
+        res.append(simplify(eq))
+        res.append(simplify(1/eq))
+    assert all(valid(i, j) for i, j in zip(res, ok))
+
+    for a in range(1, 3):
+        w = exp(a)
+        e = w + 1/w
+        s = simplify(e)
+        assert s == exptrigsimp(e)
+        assert valid(s, 2*cosh(a))
+        e = w - 1/w
+        s = simplify(e)
+        assert s == exptrigsimp(e)
+        assert valid(s, 2*sinh(a))
+
+def test_exptrigsimp_noncommutative():
+    a,b = symbols('a b', commutative=False)
+    x = Symbol('x', commutative=True)
+    assert exp(a + x) == exptrigsimp(exp(a)*exp(x))
+    p = exp(a)*exp(b) - exp(b)*exp(a)
+    assert p == exptrigsimp(p) != 0
+
+def test_powsimp_on_numbers():
+    assert 2**(Rational(1, 3) - 2) == 2**Rational(1, 3)/4
+
+
+@XFAIL
+def test_issue_6811_fail():
+    # from doc/src/modules/physics/mechanics/examples.rst, the current `eq`
+    # at Line 576 (in different variables) was formerly the equivalent and
+    # shorter expression given below...it would be nice to get the short one
+    # back again
+    xp, y, x, z = symbols('xp, y, x, z')
+    eq = 4*(-19*sin(x)*y + 5*sin(3*x)*y + 15*cos(2*x)*z - 21*z)*xp/(9*cos(x) - 5*cos(3*x))
+    assert trigsimp(eq) == -2*(2*cos(x)*tan(x)*y + 3*z)*xp/cos(x)
+
+
+def test_Piecewise():
+    e1 = x*(x + y) - y*(x + y)
+    e2 = sin(x)**2 + cos(x)**2
+    e3 = expand((x + y)*y/x)
+    # s1 = simplify(e1)
+    s2 = simplify(e2)
+    # s3 = simplify(e3)
+
+    # trigsimp tries not to touch non-trig containing args
+    assert trigsimp(Piecewise((e1, e3 < e2), (e3, True))) == \
+        Piecewise((e1, e3 < s2), (e3, True))
+
+
+def test_issue_21594():
+    assert simplify(exp(Rational(1,2)) + exp(Rational(-1,2))) == cosh(S.Half)*2
+
+
+def test_trigsimp_old():
+    x, y = symbols('x,y')
+
+    assert trigsimp(1 - sin(x)**2, old=True) == cos(x)**2
+    assert trigsimp(1 - cos(x)**2, old=True) == sin(x)**2
+    assert trigsimp(sin(x)**2 + cos(x)**2, old=True) == 1
+    assert trigsimp(1 + tan(x)**2, old=True) == 1/cos(x)**2
+    assert trigsimp(1/cos(x)**2 - 1, old=True) == tan(x)**2
+    assert trigsimp(1/cos(x)**2 - tan(x)**2, old=True) == 1
+    assert trigsimp(1 + cot(x)**2, old=True) == 1/sin(x)**2
+    assert trigsimp(1/sin(x)**2 - cot(x)**2, old=True) == 1
+
+    assert trigsimp(5*cos(x)**2 + 5*sin(x)**2, old=True) == 5
+
+    assert trigsimp(sin(x)/cos(x), old=True) == tan(x)
+    assert trigsimp(2*tan(x)*cos(x), old=True) == 2*sin(x)
+    assert trigsimp(cot(x)**3*sin(x)**3, old=True) == cos(x)**3
+    assert trigsimp(y*tan(x)**2/sin(x)**2, old=True) == y/cos(x)**2
+    assert trigsimp(cot(x)/cos(x), old=True) == 1/sin(x)
+
+    assert trigsimp(sin(x + y) + sin(x - y), old=True) == 2*sin(x)*cos(y)
+    assert trigsimp(sin(x + y) - sin(x - y), old=True) == 2*sin(y)*cos(x)
+    assert trigsimp(cos(x + y) + cos(x - y), old=True) == 2*cos(x)*cos(y)
+    assert trigsimp(cos(x + y) - cos(x - y), old=True) == -2*sin(x)*sin(y)
+
+    assert trigsimp(sinh(x + y) + sinh(x - y), old=True) == 2*sinh(x)*cosh(y)
+    assert trigsimp(sinh(x + y) - sinh(x - y), old=True) == 2*sinh(y)*cosh(x)
+    assert trigsimp(cosh(x + y) + cosh(x - y), old=True) == 2*cosh(x)*cosh(y)
+    assert trigsimp(cosh(x + y) - cosh(x - y), old=True) == 2*sinh(x)*sinh(y)
+
+    assert trigsimp(cos(0.12345)**2 + sin(0.12345)**2, old=True) == 1.0
+
+    assert trigsimp(sin(x)/cos(x), old=True, method='combined') == tan(x)
+    assert trigsimp(sin(x)/cos(x), old=True, method='groebner') == sin(x)/cos(x)
+    assert trigsimp(sin(x)/cos(x), old=True, method='groebner', hints=[tan]) == tan(x)
+
+    assert trigsimp(1-sin(sin(x)**2+cos(x)**2)**2, old=True, deep=True) == cos(1)**2
+
+
+def test_trigsimp_inverse():
+    alpha = symbols('alpha')
+    s, c = sin(alpha), cos(alpha)
+
+    for finv in [asin, acos, asec, acsc, atan, acot]:
+        f = finv.inverse(None)
+        assert alpha == trigsimp(finv(f(alpha)), inverse=True)
+
+    # test atan2(cos, sin), atan2(sin, cos), etc...
+    for a, b in [[c, s], [s, c]]:
+        for i, j in product([-1, 1], repeat=2):
+            angle = atan2(i*b, j*a)
+            angle_inverted = trigsimp(angle, inverse=True)
+            assert angle_inverted != angle  # assures simplification happened
+            assert sin(angle_inverted) == trigsimp(sin(angle))
+            assert cos(angle_inverted) == trigsimp(cos(angle))

rl/Lib/site-packages/sympy/simplify/traversaltools.py

@@ -0,0 +1,15 @@
+from sympy.core.traversal import use as _use
+from sympy.utilities.decorator import deprecated
+
+use = deprecated(
+    """
+    Using use from the sympy.simplify.traversaltools submodule is
+    deprecated.
+
+    Instead, use use from the top-level sympy namespace, like
+
+        sympy.use
+    """,
+    deprecated_since_version="1.10",
+    active_deprecations_target="deprecated-traversal-functions-moved"
+)(_use)