# -*- coding: utf-8 -*-
"Some simple functions for manipulating expressions symbolically"
# Copyright (C) 2008-2010 Kristian B. Oelgaard
#
# This file is part of FFC.
#
# FFC is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# FFC is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with FFC. If not, see <http://www.gnu.org/licenses/>.
from ufl.utils.sorting import sorted_by_key
# FFC modules
from ffc.log import error
from collections import deque
[docs]def split_expression(expression, format, operator, allow_split=False):
"""Split the expression at the given operator, return list.
Do not split () or [] unless told to split (). This is to enable easy count
of double operations which can be in (), but in [] we only have integer operations."""
# Get formats
access = format["component"]("", [""])
group = format["grouping"]("")
la = access[0]
ra = access[1]
lg = group[0]
rg = group[1]
# Split with given operator
prods = deque(expression.split(operator))
new_prods = [prods.popleft()]
while prods:
# Continue while we still have list of potential products p is
# the first string in the product
p = prods.popleft()
# If the number of "[" and "]" doesn't add up in the last
# entry of the new_prods list, add p and see if it helps for
# next iteration
if new_prods[-1].count(la) != new_prods[-1].count(ra):
new_prods[-1] = operator.join([new_prods[-1], p])
# If the number of "(" and ")" doesn't add up (and we didn't
# allow a split) in the last entry of the new_prods list, add
# p and see if it helps for next iteration
elif new_prods[-1].count(lg) != new_prods[-1].count(rg) and not allow_split:
new_prods[-1] = operator.join([new_prods[-1], p])
# If everything was fine, we can start a new entry in the
# new_prods list
else:
new_prods.append(p)
return new_prods
[docs]def operation_count(expression, format):
"""This function returns the number of double operations in an
expression. We do split () but not [] as we only have unsigned
integer operations in [].
"""
# Note we do not subtract 1 for the additions, because there is
# also an assignment involved
adds = len(split_expression(expression, format, format["add"](["", ""]),
True)) - 1
mults = len(split_expression(expression, format,
format["multiply"](["", ""]), True)) - 1
return mults + adds
[docs]def get_simple_variables(expression, format):
"""This function takes as argument an expression (preferably expanded):
expression = "x*x + y*x + x*y*z"
returns a list of products and a dictionary:
prods = ["x*x", "y*x", "x*y*z"]
variables = {variable: [num_occurences, [pos_in_prods]]}
variables = {"x":[3, [0,1,2]], "y":[2, [1,2]], "z":[1, [2]]}
"""
# Get formats
add = format["add"](["", ""])
mult = format["multiply"](["", ""])
format_float = format["floating point"]
prods = split_expression(expression, format, add)
prods = [p for p in prods if p]
variables = {}
for i, p in enumerate(prods):
# Only extract unique variables
vrs = list(set(split_expression(p, format, mult)))
for v in vrs:
# Try to convert variable to floats and back (so '2' == '2.0' etc.)
try:
v = format_float(float(v))
except Exception:
pass
if v in variables:
variables[v][0] += 1
variables[v][1].append(i)
else:
variables[v] = [1, [i]]
return (prods, variables)
[docs]def group_vars(expr, format):
"""Group variables in an expression, such that:
"x + y + z + 2*y + 6*z" = "x + 3*y + 7*z"
"x*x + x*x + 2*x + 3*x + 5" = "2.0*x*x + 5.0*x + 5"
"x*y + y*x + 2*x*y + 3*x + 0*x + 5" = "5.0*x*y + 3.0*x + 5"
"(y + z)*x + 5*(y + z)*x" = "6.0*(y + z)*x"
"1/(x*x) + 2*1/(x*x) + std::sqrt(x) + 6*std::sqrt(x)" = "3*1/(x*x) + 7*std::sqrt(x)"
"""
# Get formats
format_float = format["floating point"]
add = format["add"](["", ""])
mult = format["multiply"](["", ""])
new_prods = {}
# Get list of products
prods = split_expression(expr, format, add)
# Loop products and collect factors
for p in prods:
# Get list of variables, and do a basic sort
vrs = split_expression(p, format, mult)
factor = 1
new_var = []
# Try to multiply factor with variable, else variable must be
# multiplied by factor later
# If we don't have a variable, set factor to zero and break
for v in vrs:
if v:
try:
f = float(v)
factor *= f
except Exception:
new_var.append(v)
else:
factor = 0
break
# Create new variable that must be multiplied with factor. Add
# this variable to dictionary, if it already exists add factor
# to other factors
new_var.sort()
new_var = mult.join(new_var)
if new_var in new_prods:
new_prods[new_var] += factor
else:
new_prods[new_var] = factor
# Reset products
prods = []
for prod, f in sorted_by_key(new_prods):
# If we have a product append mult of both
if prod:
# If factor is 1.0 we don't need it
if f == 1.0:
prods.append(prod)
else:
prods.append(mult.join([format_float(f), prod]))
# If we just have a factor
elif f:
prods.append(format_float(f))
prods.sort()
return add.join(prods)
[docs]def reduction_possible(variables):
"""Find the variable that occurs in the most products, if more
variables occur the same number of times and in the same products
add them to list.
"""
# Find the variable that appears in the most products
max_val = 1
max_var = ""
max_vars = []
for key, val in sorted_by_key(variables):
if max_val < val[0]:
max_val = val[0]
max_var = key
# If we found a variable that appears in products multiple times,
# check if other variables appear in the exact same products
if max_var:
for key, val in sorted_by_key(variables):
# Check if we have more variables in the same products
if max_val == val[0] and variables[max_var][1] == val[1]:
max_vars.append(key)
return max_vars
[docs]def is_constant(variable, format, constants=[], from_is_constant=False):
"""Determine if a variable is constant or not. The function accepts
an optional list of variables (loop indices) that will be regarded
as constants for the given variable. If none are supplied it is
assumed that all array accesses will result in a non-constant
variable.
v = 2.0, is constant
v = Jinv_00*det, is constant
v = w[0][1], is constant
v = 2*w[0][1], is constant
v = W0[ip], is constant if constants = ['ip'] else not
v = P_t0[ip][j], is constant if constants = ['j','ip'] else not
"""
# Get formats
access = format["array access"]("")
add = format["add"](["", ""])
mult = format["multiply"](["", ""])
l = access[0] # noqa: E741
r = access[1]
if not variable.count(l) == variable.count(r):
print("variable: ", variable)
error("Something wrong with variable")
# Be sure that we don't have a compound
variable = expand_operations(variable, format)
prods = split_expression(variable, format, add)
# Loop all products and variables and check if they're constant
for p in prods:
vrs = split_expression(p, format, mult)
for v in vrs:
# Check if each variable is constant, if just one fails
# the entire variable is considered not to be constant
const_var = False
# If variable is in constants, well....
if v in constants:
const_var = True
continue
# If we don't have any '[' or ']' we have a constant
# (unless we're dealing with a call from this funtions)
elif not v.count(l) and not from_is_constant:
const_var = True
continue
# If we have an array access variable, see if the index is
# regarded a constant
elif v.count(l):
# Check if access is OK ('[' is before ']')
if not v.index(l) < v.index(r):
print("variable: ", v)
error("Something is wrong with the array access")
# Auxiliary variables
index = ""
left = 0
inside = False
indices = []
# Loop all characters in variable and find indices
for c in v:
# If character is ']' reduce left count
if c == r:
left -= 1
# If the '[' count has returned to zero, we have a
# complete index
if left == 0 and inside:
const_index = False # Aux. var
if index in constants:
const_index = True
try:
int(index)
const_index = True
except Exception:
# Last resort, call recursively
if is_constant(index, format, constants, True):
const_index = True
pass
# Append index and reset values
if const_index:
indices.append(const_index)
else:
indices = [False]
break
index = ""
inside = False
# If we're inside an access, add character to index
if inside:
index += c
# If character is '[' increase the count, and
# we're inside an access
if c == l:
inside = True
left += 1
# If all indices were constant, the variable is constant
if all(indices):
const_var = True
continue
else:
# If it is a float, it is also constant
try:
float(v)
const_var = True
continue
except Exception:
pass
# I no tests resulted in a constant variable, there is no
# need to continue
if not const_var:
return False
# If all variables were constant return True
return True
[docs]def expand_operations(expression, format):
"""This function expands an expression and returns the value. E.g.,
((x + y)) --> x + y
2*(x + y) --> 2*x + 2*y
(x + y)*(x + y) --> x*x + y*y + 2*x*y
z*(x*(y + 3) + 2) + 1 --> 1 + 2*z + x*y*z + x*z*3
z*((y + 3)*x + 2) + 1 --> 1 + 2*z + x*y*z + x*z*3"""
# Get formats
add = format["add"](["", ""])
mult = format["multiply"](["", ""])
group = format["grouping"]("")
l = group[0] # noqa: E741
r = group[1]
# Check that we have the same number of left/right parenthesis in
# expression
if not expression.count(l) == expression.count(r):
error("Number of left/right parenthesis do not match")
# If we don't have any parenthesis, group variables and return
if expression.count(l) == 0:
return group_vars(expression, format)
# Get list of additions
adds = split_expression(expression, format, add)
new_adds = []
# Loop additions and get products
for a in adds:
prods = sorted(split_expression(a, format, mult))
new_prods = []
# FIXME: Should we use deque here?
expanded = []
for i, p in enumerate(prods):
# If we have a group, expand inner expression
if p[0] == l and p[-1] == r:
# Add remaining products to new products and multiply
# with all terms from expanded variable
expanded_var = expand_operations(p[1:-1], format)
expanded.append(split_expression(expanded_var, format, add))
# Else, just add variable to list of new products
else:
new_prods.append(p)
if expanded:
# Combine all expanded variables and multiply by factor
while len(expanded) > 1:
first = expanded.pop(0)
second = expanded.pop(0)
expanded = [[mult.join([i] + [j]) for i in first for j in second]] + expanded
new_adds += [mult.join(new_prods + [e]) for e in expanded[0]]
else:
# Else, just multiply products and add to list of products
new_adds.append(mult.join(new_prods))
# Group variables and return
return group_vars(add.join(new_adds), format)
[docs]def reduce_operations(expression, format):
"""This function reduces the number of opertions needed to compute a
given expression. It looks for the variable that appears the most
and groups terms containing this variable inside parenthesis. The
function is called recursively until no further reductions are
possible.
"x + y + x" = 2*x + y
"x*x + 2.0*x*y + y*y" = y*y + (2.0*y + x)*x, not (x + y)*(x + y) as it should be!!
z*x*y + z*x*3 + 2*z + 1" = z*(x*(y + 3) + 2) + 1
"""
# Get formats
add = format["add"](["", ""])
mult = format["multiply"](["", ""])
# Be sure that we have an expanded expression
expression = expand_operations(expression, format)
# Group variables to possibly reduce complexity
expression = group_vars(expression, format)
# Get variables and products
prods, variables = get_simple_variables(expression, format)
# Get the variables for which we can reduce the expression
max_vars = reduction_possible(variables)
new_prods = []
no_mult = []
max_vars.sort()
# If we have variables that can be moved outside
if max_vars:
for p in prods:
# Get the list of variables in current product
li = sorted(split_expression(p, format, mult))
# If the list of products is the same as what we intend of
# moving outside the parenthesis, leave it (because x +
# x*x + x*y should be x + (x + y)*x NOT (1.0 + x + y)*x)
if li == max_vars:
no_mult.append(p)
continue
else:
# Get list of all variables from max_vars that are in
# li
indices = [i for i in max_vars if i in li]
# If not all were present add to list of terms that
# shouldn't be multiplied with variables and continue
if indices != max_vars:
no_mult.append(p)
continue
# Remove variables that we are moving outside
for v in max_vars:
li.remove(v)
# Add to list of products
p = mult.join(li)
new_prods.append(p)
# Sort lists
no_mult.sort()
new_prods.sort()
else:
# No reduction possible
return expression
# Recursively reduce sums with and without reduced variable
new_prods = add.join(new_prods)
if new_prods:
new_prods = reduce_operations(new_prods, format)
if no_mult:
no_mult = [reduce_operations(add.join(no_mult), format)]
# Group new products if we have a sum
g = new_prods
len_new_prods = len(split_expression(new_prods, format, add))
if len_new_prods > 1:
g = format["grouping"](new_prods)
# The new expression is the sum of terms that couldn't be reduced
# and terms that could be reduced multiplied by the reduction
# e.g., expr = z + (x + y)*x
new_expression = add.join(no_mult + [mult.join([g, mult.join(max_vars)])])
return new_expression
[docs]def get_geo_terms(expression, geo_terms, offset, format):
"""This function returns a new expression where all geometry terms
have been substituted with geometry declarations, these
declarations are added to the geo_terms dictionary.
"""
# Get formats
add = format["add"](["", ""])
mult = format["multiply"](["", ""])
grouping = format["grouping"]
group = grouping("")
format_G = format["geometry tensor"]
gl = group[0]
gr = group[1]
# Get the number of geometry declaration, possibly offset value
num_geo = offset + len(geo_terms)
new_prods = []
# Split the expression into products
prods = split_expression(expression, format, add)
consts = []
# Loop products and check if the variables are constant
for p in prods:
vrs = split_expression(p, format, mult)
geos = []
# Generate geo code for constant coefficients e.g., w[0][5]
new_vrs = []
for v in vrs:
# If variable is a group, get the geometry terms and
# update geo number
if v[0] == gl and v[-1] == gr:
v = get_geo_terms(v[1:-1], geo_terms, offset, format)
num_geo = offset + len(geo_terms)
# If we still have a sum, regroup
if len(v.split(add)) > 1:
v = grouping(v)
# Append to new variables
new_vrs.append(v)
# If variable is constants, add to geo terms
constant = is_constant(v, format)
if constant:
geos.append(v)
# Update variable list
vrs = new_vrs
vrs.sort()
# Sort geo and create geometry term
geos.sort()
geo = mult.join(geos)
# Handle geometry term appropriately
if geo:
if geos != vrs:
if len(geos) > 1:
for g in geos:
vrs.remove(g)
if geo not in geo_terms:
geo_terms[geo] = format_G + str(num_geo)
num_geo += 1
vrs.append(geo_terms[geo])
new_prods.append(mult.join(vrs))
else:
consts.append(mult.join(vrs))
else:
new_prods.append(mult.join(vrs))
if consts:
if len(consts) > 1:
c = grouping(add.join(consts))
else:
c = add.join(consts)
if c not in geo_terms:
geo_terms[c] = format_G + str(num_geo)
num_geo += 1
consts = [geo_terms[c]]
return add.join(new_prods + consts)
[docs]def get_constants(expression, const_terms, format, constants=[]):
"""This function returns a new expression where all geometry terms
have been substituted with geometry declarations, these
declarations are added to the const_terms dictionary.
"""
# Get formats
add = format["add"](["", ""])
mult = format["multiply"](["", ""])
grouping = format["grouping"]
format_G = format["geometry tensor"] + "".join(constants) # format["geometry tensor"]
# Get the number of geometry declaration, possibly offset value
num_geo = len(const_terms)
new_prods = []
# Split the expression into products
prods = split_expression(expression, format, add)
consts = []
# Loop products and check if the variables are constant
for p in prods:
vrs = split_expression(p, format, mult)
geos = []
# Generate geo code for constant coefficients e.g., w[0][5]
new_vrs = []
for v in vrs:
# If variable is constants, add to geo terms
constant = is_constant(v, format, constants)
if constant:
geos.append(v)
# Append to new variables
new_vrs.append(v)
# Update variable list
vrs = new_vrs
vrs.sort()
# Sort geo and create geometry term
geos.sort()
geo = mult.join(geos)
if geo:
if geos != vrs:
for g in geos:
vrs.remove(g)
if geo not in const_terms:
const_terms[geo] = format_G + str(num_geo)
num_geo += 1
vrs.append(const_terms[geo])
new_prods.append(mult.join(vrs))
else:
consts.append(mult.join(vrs))
else:
new_prods.append(mult.join(vrs))
if consts:
if len(consts) > 1:
c = grouping(add.join(consts))
else:
c = add.join(consts)
if c not in const_terms:
const_terms[c] = format_G + str(num_geo)
num_geo += 1
consts = [const_terms[c]]
return add.join(new_prods + consts)
[docs]def get_indices(variable, format, from_get_indices=False):
"""This function returns the indices of a given variable. E.g.,
P[0][j], returns ['j']
P[ip][k], returns ['ip','k']
P[ip][nzc0[j] + 3], returns ['ip','j']
w[0][j + 2] , returns [j]"""
add = format["add"](["", ""])
mult = format["multiply"](["", ""])
format_access = format["array access"]
access = format_access("")
l = access[0] # noqa: E741
r = access[1]
indices = []
# If there are no '[' in variable and self is the caller
if not variable.count(l) and from_get_indices:
adds = split_expression(variable, format, add)
for a in adds:
mults = split_expression(a, format, mult)
for m in mults:
try:
float(m)
except Exception:
if m not in indices:
indices.append(m)
else:
index = ""
left = 0
inside = False
# Loop all characters in variable and find indices
for c in variable:
# If character is ']' reduce left count
if c == r:
left -= 1
# If the '[' count has returned to zero, we have a
# complete index
if left == 0 and inside:
try:
eval(index)
except Exception:
indices += get_indices(index, format, True)
index = ""
inside = False
# If we're inside an access, add character to index
if inside:
index += c
# If character is '[' increase the count, and we're inside
# an access
if c == l:
inside = True
left += 1
return indices
[docs]def get_variables(expression, variables, format, constants=[]):
"""This function returns a new expression where all geometry terms
have been substituted with geometry declarations, these
declarations are added to the const_terms dictionary.
"""
# Get formats
add = format["add"](["", ""])
mult = format["multiply"](["", ""])
format_access = format["array access"]
access = format_access("")
format_F = format["function value"]
l = access[0] # noqa: E741
# If we don't have any access operators in expression,
# we don't have any variables
if expression.count(l) == 0:
return expression
# Get the number of geometry declaration, possibly offset value
num_var = len(variables)
new_prods = []
used_vars = []
# Split the expression into products
prods = split_expression(expression, format, add)
# Loop products and check if the variables are constant
for p in prods:
vrs = split_expression(p, format, mult)
# Variables with respect to the constants in list
variables_of_interest = []
# Generate geo code for constant coefficients e.g., w[0][5]
new_vrs = []
for v in vrs:
# If we don't have any access operators, we don't have a
# variable
if v.count(l) == 0:
new_vrs.append(v)
continue
# Check if we have a variable that depends on one of the
# constants First check the easy way
is_var = False
for c in constants:
if format_access(c) in v:
is_var = True
break
if is_var:
variables_of_interest.append(v)
continue
# Then check the hard way
# Get list of indices
indices = get_indices(v, format)
depends = [True for c in constants if c in indices]
if any(depends):
variables_of_interest.append(v)
else:
new_vrs.append(v)
variables_of_interest.sort()
variables_of_interest = mult.join(variables_of_interest)
# If we have some variables, declare new variable if needed
# and add to list of variables
if variables_of_interest:
# If we didn't already declare this variable do so
if variables_of_interest not in variables:
variables[variables_of_interest] = format_F + str(num_var)
num_var += 1
# Get mapped variable
mv = variables[variables_of_interest]
new_vrs.append(mv)
if mv not in used_vars:
used_vars.append(mv)
# Sort variables and add to list of products
new_vrs.sort()
new_prods.append(mult.join(new_vrs))
# Sort list of products and return the sum
new_prods.sort()
return (add.join(new_prods), used_vars)