def _product_deriv(terms, wrt):
"""
Return an AST expressing the derivative of the product of all the terms.
"""
if len(terms) == 1:
return _diff_ast(terms[0], wrt)
deriv_terms = []
for ii, term in enumerate(terms):
term_d = _diff_ast(term, wrt)
other_terms = terms[:ii] + terms[ii + 1:]
deriv_terms.append(AST._make_product(other_terms + [term_d]))
sum = deriv_terms[0]
for term in deriv_terms[1:]:
sum = BinOp(left=term, op=Add(), right=sum)
return sum
def _diff_ast(ast, wrt):
"""
Return an AST that is the derivative of ast with respect the variable with
name 'wrt'.
"""
# For now, the strategy is to return the most general forms, and let
# the simplifier take care of the special cases.
if isinstance(ast, Name):
if ast.id == wrt:
return _ONE
else:
return _ZERO
elif isinstance(ast, Constant):
return _ZERO
elif isinstance(ast, BinOp) and (isinstance(ast.op, Add)
or isinstance(ast.op, Sub)):
# Just take the derivative of the arguments. The call to ast.__class__
# lets us use the same code from Add and Sub.
return (BinOp(left=_diff_ast(ast.left, wrt),
op=ast.op,
right=_diff_ast(ast.right, wrt)))
elif isinstance(ast, BinOp) and (isinstance(ast.op, Mult)
or isinstance(ast.op, Div)):
# Collect all the numerators and denominators together
nums, denoms = [], []
AST._collect_num_denom(ast, nums, denoms)
# Collect the numerator terms into a single AST
num = AST._make_product(nums)
# Take the derivative of the numerator terms as a product
num_d = _product_deriv(nums, wrt)
if not denoms:
# If there is no denominator
return num_d
denom = AST._make_product(denoms)
denom_d = _product_deriv(denoms, wrt)
# Derivative of x/y is x'/y + -x*y'/y**2
term1 = BinOp(left=num_d, op=Div(), right=denom)
term2 = BinOp(left=BinOp(left=UnaryOp(op=USub(), operand=num),
op=Mult(),
right=denom_d),
op=Div(),
right=BinOp(left=denom,
op=Pow(),
right=Constant(value=2)))
return BinOp(left=term1, op=Add(), right=term2)
elif isinstance(ast, BinOp) and isinstance(ast.op, Pow):
# Use the derivative of the 'pow' function
ast = Call(func=Name(id='pow', ctx=Load()), args=[ast.left, ast.right])
return _diff_ast(ast, wrt)
elif isinstance(ast, Call):
func_name = AST.ast2str(ast.func)
args = ast.args
args_d = [_diff_ast(arg, wrt) for arg in args]
if (func_name, len(args)) in _KNOWN_FUNCS:
form = copy.deepcopy(_KNOWN_FUNCS[(func_name, len(args))])
else:
# If this isn't a known function, our form is
# (f_0(args), f_1(args), ...)
args_expr = [
Name(id='arg%i' % ii, ctx=Load()) for ii in range(len(args))
]
form = [
Call(func=Name(id='%s_%i' % (func_name, ii), ctx=Load()),
args=args_expr,
keywords=[]) for ii in range(len(args))
]
# We build up the terms in our derivative
# f_0(x,y)*x' + f_1(x,y)*y', etc.
outs = []
for arg_d, arg_form_d in zip(args_d, form):
# We skip arguments with 0 derivative
if arg_d == _ZERO:
continue
for ii, arg in enumerate(args):
Substitution._sub_subtrees_for_vars(arg_form_d,
{'arg%i' % ii: arg})
outs.append(BinOp(left=arg_form_d, op=Mult(), right=arg_d))
# If all arguments had zero deriviative
if not outs:
return _ZERO
else:
# We add up all our terms
ret = outs[0]
for term in outs[1:]:
ret = BinOp(left=ret, op=Add(), right=term)
return ret
elif isinstance(ast, UnaryOp) and isinstance(ast.op, USub):
return UnaryOp(op=USub(), operand=_diff_ast(ast.operand, wrt))
elif isinstance(ast, UnaryOp) and isinstance(ast.op, UAdd):
return UnaryOp(op=UAdd(), operand=_diff_ast(ast.operand, wrt))
def _simplify_ast(ast):
"""
Return a simplified ast.
Current simplifications:
Special cases for zeros and ones, and combining of constants, in
addition, subtraction, multiplication, division.
Note that at present we only handle constants applied left to right.
1+1+x -> 2+x, but x+1+1 -> x+1+1.
x - x = 0
--x = x
"""
if isinstance(ast, Name) or isinstance(ast, Constant):
return ast
elif isinstance(ast, BinOp) and (isinstance(ast.op, Add)
or isinstance(ast.op, Sub)):
# We collect positive and negative terms and simplify each of them
pos, neg = [], []
AST._collect_pos_neg(ast, pos, neg)
pos = [_simplify_ast(term) for term in pos]
neg = [_simplify_ast(term) for term in neg]
# We collect and sum the constant values
values = [term.value for term in pos if isinstance(term, Constant)] +\
[-term.value for term in neg if isinstance(term, Constant)]
value = sum(values)
# Remove the constants from our pos and neg lists
pos = [term for term in pos if not isinstance(term, Constant)]
neg = [term for term in neg if not isinstance(term, Constant)]
new_pos, new_neg = [], []
for term in pos:
if isinstance(term, UnaryOp):
if isinstance(term.op, USub):
new_neg.append(term.operand)
else:
new_pos.append(term)
for term in neg:
if isinstance(term, UnaryOp):
if isinstance(term.op, USub):
new_pos.append(term.operand)
else:
new_neg.append(term)
pos, neg = new_pos, new_neg
# Append the constant value sum to pos or neg
if value > 0:
pos.append(Constant(value=value))
elif value < 0:
neg.append(Constant(value=abs(value)))
# Count the number of occurances of each term.
term_counts = [
(term,
get_count_from_ast(pos, term) - get_count_from_ast(neg, term))
for term in pos + neg
]
# Tricky: We use the str(term) as the key for the dictionary to ensure
# that each entry represents a unique term. We also drop terms
# that have a total count of 0.
term_counts = dict([(AST.ast2str(term), (term, count))
for term, count in term_counts])
# We find the first term with non-zero count.
ii = 0
for ii, term in enumerate(pos + neg):
ast_out, count = term_counts[AST.ast2str(term)]
if count != 0:
break
else:
# We get here if we don't break out of the loop, implying that
# all our terms had count of 0
return _ZERO
term_counts[AST.ast2str(term)] = (ast_out, 0)
if abs(count) != 1:
ast_out = BinOp(left=Constant(value=abs(count)),
op=Mult(),
right=ast_out)
if count < 0:
ast_out = UnaryOp(op=USub(), operand=ast_out)
# And add in all the rest
for term in (pos + neg)[ii:]:
term, count = term_counts[AST.ast2str(term)]
term_counts[AST.ast2str(term)] = (term, 0)
if abs(count) != 1:
term = BinOp(left=Constant(value=abs(count)),
op=Mult(),
right=term)
if count > 0:
ast_out = BinOp(left=ast_out, op=Add(), right=term)
elif count < 0:
ast_out = BinOp(left=ast_out, op=Sub(), right=term)
return ast_out
elif isinstance(ast, BinOp) and (isinstance(ast.op, Mult)
or isinstance(ast.op, Div)):
# We collect numerator and denominator terms and simplify each of them
num, denom = [], []
AST._collect_num_denom(ast, num, denom)
num = [_simplify_ast(term) for term in num]
denom = [_simplify_ast(term) for term in denom]
# We collect and sum the constant values
values = [term.value for term in num if isinstance(term, Constant)] +\
[1./term.value for term in denom if isinstance(term, Constant)]
# This takes the product of all our values
value = functools.reduce(operator.mul, values + [1])
# If our value is 0, the expression is 0
if not value:
return _ZERO
# Remove the constants from our pos and neg lists
num = [term for term in num if not isinstance(term, Constant)]
denom = [term for term in denom if not isinstance(term, Constant)]
# Here we count all the negative (UnarySub) elements of our expression.
# We also remove the UnarySubs from their arguments. We'll correct
# for it at the end.
num_neg = 0
for list_of_terms in [num, denom]:
for ii, term in enumerate(list_of_terms):
if isinstance(term, UnaryOp) and isinstance(term.op, USub):
list_of_terms[ii] = term.operand
num_neg += 1
# Append the constant value sum to pos or neg
if abs(value) != 1:
num.append(Constant(value=abs(value)))
if value < 0:
num_neg += 1
make_neg = num_neg % 2
# Count the number of occurances of each term.
term_counts = [
(term,
get_count_from_ast(num, term) - get_count_from_ast(denom, term))
for term in num + denom
]
# Tricky: We use the str(term) as the key for the dictionary to ensure
# that each entry represents a unique term. We also drop terms
# that have a total count of 0.
term_counts = dict([(AST.ast2str(term), (term, count))
for term, count in term_counts])
nums, denoms = [], []
# We walk through terms in num+denom in order, so we rearrange a little
# as possible.
for term in num + denom:
term, count = term_counts[AST.ast2str(term)]
# Once a term has been done, we set its term_counts to 0, so it
# doesn't get done again.
term_counts[AST.ast2str(term)] = (term, 0)
if abs(count) > 1:
term = BinOp(left=term,
op=Pow(),
right=Constant(value=abs(count)))
if count > 0:
nums.append(term)
elif count < 0:
denoms.append(term)
# We return the product of the numerator terms over the product of the
# denominator terms
out = AST._make_product(nums)
if denoms:
denom = AST._make_product(denoms)
out = BinOp(left=out, op=Div(), right=denom)
if make_neg:
out = UnaryOp(op=USub(), operand=out)
return out
elif isinstance(ast, BinOp) and isinstance(ast.op, Pow):
# These cases all have a left and a right, so we group them just to
# avoid some code duplication.
power = _simplify_ast(ast.right)
base = _simplify_ast(ast.left)
if power == _ZERO:
# Anything, including 0, to the 0th power is 1, so this
# test should come first
return _ONE
if base == _ZERO or base == _ONE or power == _ONE:
return base
elif isinstance(base, Constant) and\
isinstance(power, Constant):
return Constant(value=base.value**power.value)
# Getting here implies that no simplifications are possible, so just
# return with simplified arguments
return BinOp(left=base, op=Pow(), right=power)
elif isinstance(ast, UnaryOp) and isinstance(ast.op, USub):
simple_expr = _simplify_ast(ast.operand)
if isinstance(simple_expr, UnaryOp) and isinstance(
simple_expr.op, USub):
# Case --x
return _simplify_ast(simple_expr.operand)
elif isinstance(simple_expr, Constant):
if simple_expr.value == 0:
return Constant(value=0)
else:
return Constant(value=-simple_expr.value)
else:
return UnaryOp(op=USub(), operand=simple_expr)
elif isinstance(ast, UnaryOp) and isinstance(ast.op, UAdd):
simple_expr = _simplify_ast(ast.operand)
return simple_expr
elif isinstance(ast, list):
simple_list = [_simplify_ast(elem) for elem in ast]
return simple_list
elif isinstance(ast, tuple):
return tuple(_simplify_ast(list(ast)))
elif ast.__class__ in AST._node_attrs:
# Handle node types with no special cases.
for attr_name in AST._node_attrs[ast.__class__]:
attr = getattr(ast, attr_name)
if isinstance(attr, list):
for ii, elem in enumerate(attr):
attr[ii] = _simplify_ast(elem)
else:
setattr(ast, attr_name, _simplify_ast(attr))
return ast
else:
return ast