Source code for sympy.simplify.cse_main

""" Tools for doing common subexpression elimination.
"""
from __future__ import print_function, division

from sympy.core import Basic, Mul, Add, Pow, sympify, Symbol, Tuple
from sympy.core.singleton import S
from sympy.core.function import _coeff_isneg
from sympy.core.exprtools import factor_terms
from sympy.core.compatibility import iterable, range
from sympy.utilities.iterables import filter_symbols, \
    numbered_symbols, sift, topological_sort, ordered

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]

# ====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


[docs]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]) >>> print(opt_subs) {x**(-2): 1/(x**2)} """ from sympy.matrices.expressions import MatAdd, MatMul, MatPow opt_subs = dict() adds = set() muls = set() seen_subexp = set() def _find_opts(expr): if not isinstance(expr, Basic): 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 _coeff_isneg(expr): neg_expr = -expr if not neg_expr.is_Atom: opt_subs[expr] = Mul(S.NegativeOne, neg_expr, evaluate=False) seen_subexp.add(neg_expr) expr = neg_expr if isinstance(expr, (Mul, MatMul)): muls.add(expr) elif isinstance(expr, (Add, MatAdd)): adds.add(expr) elif isinstance(expr, (Pow, MatPow)): if _coeff_isneg(expr.exp): opt_subs[expr] = Pow(Pow(expr.base, -expr.exp), S.NegativeOne, evaluate=False) for e in exprs: if isinstance(e, Basic): _find_opts(e) ## Process Adds and commutative Muls def _match_common_args(Func, funcs): if order != 'none': funcs = list(ordered(funcs)) else: funcs = sorted(funcs, key=lambda x: len(x.args)) func_args = [set(e.args) for e in funcs] for i in range(len(func_args)): for j in range(i + 1, len(func_args)): com_args = func_args[i].intersection(func_args[j]) if len(com_args) > 1: com_func = Func(*com_args) # for all sets, replace the common symbols by the function # over them, to allow recursive matches diff_i = func_args[i].difference(com_args) func_args[i] = diff_i | {com_func} if diff_i: opt_subs[funcs[i]] = Func(Func(*diff_i), com_func, evaluate=False) diff_j = func_args[j].difference(com_args) func_args[j] = diff_j | {com_func} opt_subs[funcs[j]] = Func(Func(*diff_j), com_func, evaluate=False) for k in range(j + 1, len(func_args)): if not com_args.difference(func_args[k]): diff_k = func_args[k].difference(com_args) func_args[k] = diff_k | {com_func} opt_subs[funcs[k]] = Func(Func(*diff_k), com_func, evaluate=False) # split muls into commutative comutative_muls = set() for m in muls: c, nc = m.args_cnc(cset=True) if c: c_mul = m.func(*c) if nc: if c_mul == 1: new_obj = m.func(*nc) else: new_obj = m.func(c_mul, m.func(*nc), evaluate=False) opt_subs[m] = new_obj if len(c) > 1: comutative_muls.add(c_mul) _match_common_args(Add, adds) _match_common_args(Mul, comutative_muls) return opt_subs
[docs]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. """ from sympy.matrices.expressions import MatrixExpr, MatrixSymbol, MatMul, MatAdd if opt_subs is None: opt_subs = dict() ## Find repeated sub-expressions to_eliminate = set() seen_subexp = set() def _find_repeated(expr): if not isinstance(expr, Basic): return if expr.is_Atom or expr.is_Order: 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 replacements = [] subs = dict() def _rebuild(expr): if not isinstance(expr, Basic): return expr if not expr.args: return expr if iterable(expr): new_args = [_rebuild(arg) for arg in expr] 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 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) # don't allow hollow nesting # e.g if p = [b + 2*d + e + f, b + 2*d + f + g, a + c + d + f + g] # and R, C = cse(p) then # R = [(x0, d + f), (x1, b + d)] # C = [e + x0 + x1, g + x0 + x1, a + c + d + f + g] # but the args of C[-1] should not be `(a + c, d + f + g)` for i in range(len(exprs)): F = reduced_exprs[i].func if not (F is Mul or F is Add): continue if any(isinstance(a, F) for a in reduced_exprs[i].args): reduced_exprs[i] = F(*reduced_exprs[i].args) return replacements, reduced_exprs
[docs]def cse(exprs, symbols=None, optimizations=None, postprocess=None, order='canonical', ignore=()): """ 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. 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]) Note that currently, y + z will not get substituted if -y - z is used. >>> cse(((w + x + y + z)*(w - y - z))/(w + x)**3) ([(x0, w + x)], [(w - y - z)*(x0 + y + z)/x0**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]) """ from sympy.matrices import (MatrixBase, Matrix, ImmutableMatrix, SparseMatrix, ImmutableSparseMatrix) # 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._mat)) elif isinstance(e, (SparseMatrix, ImmutableSparseMatrix)): temp.append(Tuple(*e._smat.items())) else: temp.append(e) exprs = temp del temp if optimizations is None: optimizations = list() 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] excluded_symbols = set().union(*[expr.atoms(Symbol) for expr in reduced_exprs]) if symbols is None: symbols = numbered_symbols() else: # In case we get passed an iterable with an __iter__ method instead of # an actual iterator. symbols = iter(symbols) symbols = filter_symbols(symbols, excluded_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)