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# This file is part of TRS (http://math.kompiler.org)
#
# TRS is free software: you can redistribute it and/or modify it under the
# terms of the GNU Affero General Public License as published by the Free
# Software Foundation, either version 3 of the License, or (at your option) any
# later version.
#
# TRS 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 Affero General Public License for more
# details.
#
# You should have received a copy of the GNU Affero General Public License
# along with TRS. If not, see <http://www.gnu.org/licenses/>.
from .utils import find_variables, first_sorted_variable
from ..node import ExpressionLeaf as L, Scope, OP_DER, OP_MUL, OP_LOG, \
OP_SIN, OP_COS, OP_TAN, OP_ADD, OP_DIV, E, sin, cos, der, ln
from ..possibilities import Possibility as P, MESSAGES
from ..translate import _
def second_arg(node):
"""
Get the second child of a node if it exists, None otherwise.
"""
return node[1] if len(node) > 1 else None
def same_der(root, new):
"""
Replace root with a new derrivative, using the same operator as root
(OP_PRIME or DXDER).
"""
return der(new, second_arg(root))
def get_derivation_variable(node, variables=None):
"""
Find the variable to derive over.
>>> print get_derivation_variable(der(L('x')))
'x'
"""
if len(node) > 1:
assert node[1].is_identifier()
return node[1].value
if not variables:
variables = find_variables(node)
if not len(variables):
return None
return first_sorted_variable(variables)
def chain_rule(root, args):
"""
Apply the chain rule:
[f(g(x)]' -> f'(g(x)) * g'(x)
f'(g(x)) is not expressable in the current syntax, so calculate it directly
using the application function in the arguments. g'(x) is simply expressed
as der(g(x), x).
"""
g, f_deriv, f_deriv_args = args
return f_deriv(root, f_deriv_args) * same_der(root, g)
MESSAGES[chain_rule] = _('Apply the chain rule to {0}.')
def match_zero_derivative(node):
"""
der(x, y) -> 0
der(n) -> 0
"""
assert node.is_op(OP_DER)
variables = find_variables(node[0])
var = get_derivation_variable(node, variables)
if not var or var not in variables:
return [P(node, zero_derivative)]
return []
def zero_derivative(root, args):
"""
der(x, y) -> 0
der(n) -> 0
"""
return L(0)
MESSAGES[zero_derivative] = _('Constant {0[0]} has derivative 0.')
def match_one_derivative(node):
"""
der(x) -> 1 # Implicit x
der(x, x) -> 1 # Explicit x
"""
assert node.is_op(OP_DER)
var = get_derivation_variable(node)
if var and node[0] == L(var):
return [P(node, one_derivative)]
return []
def one_derivative(root, args):
"""
der(x) -> 1
der(x, x) -> 1
"""
return L(1)
MESSAGES[one_derivative] = _('Variable {0[0]} has derivative 1.')
def match_const_deriv_multiplication(node):
"""
der(c * f(x), x) -> c * der(f(x), x)
"""
assert node.is_op(OP_DER)
p = []
if node[0].is_op(OP_MUL):
x = L(get_derivation_variable(node))
scope = Scope(node[0])
for n in scope:
if not n.contains(x):
p.append(P(node, const_deriv_multiplication, (scope, n, x)))
return p
def const_deriv_multiplication(root, args):
"""
der(c * f(x), x) -> c * der(f(x), x)
"""
scope, c, x = args
scope.remove(c)
return c * same_der(root, scope.as_nary_node())
MESSAGES[const_deriv_multiplication] = \
_('Bring multiplication with {2} in derivative {0} to the outside.')
def match_variable_power(node):
"""
der(x ^ n) -> n * x ^ (n - 1)
der(x ^ n, x) -> n * x ^ (n - 1)
der(f(x) ^ n) -> n * f(x) ^ (n - 1) * der(f(x)) # Chain rule
der(f(x) ^ g(x), x) -> der(e ^ ln(f(x) ^ g(x)), x)
"""
assert node.is_op(OP_DER)
if not node[0].is_power():
return []
root, exponent = node[0]
rvars = find_variables(root)
evars = find_variables(exponent)
x = get_derivation_variable(node, rvars | evars)
if x in rvars:
if x in evars:
return [P(node, power_rule)]
if root.is_variable():
return [P(node, variable_root)]
return [P(node, chain_rule, (root, variable_root, ()))]
if exponent.is_variable():
return [P(node, variable_exponent)]
return [P(node, chain_rule, (exponent, variable_exponent, ()))]
def power_rule(root, args):
"""
[f(x) ^ g(x)]' -> [e ^ ln(f(x) ^ g(x))]'
"""
return same_der(root, L(E) ** ln(root[0]))
MESSAGES[power_rule] = \
_('Write {0} as a logarithm to be able to separate root and exponent.')
def variable_root(root, args):
"""
der(x ^ n, x) -> n * x ^ (n - 1)
"""
x, n = root[0]
return (n).negate(root[0].negated) * x ** (n - 1)
MESSAGES[variable_root] = \
_('Apply standard derivative `d/dx x ^ n = n * x ^ (n - 1)` on {0}.')
def variable_exponent(root, args):
"""
der(g ^ x, x) -> g ^ x * ln(g)
Shortcut rule (because of presence on formula list):
der(e ^ x, x) -> e ^ x
"""
g, x = root[0]
if g == E:
return (g ** x).negate(root[0].negated)
return (g ** x).negate(root[0].negated) * ln(g)
MESSAGES[variable_exponent] = \
_('Apply standard derivative `d/dx g ^ x = g ^ x * ln g`.')
def match_logarithmic(node):
"""
der(log(x, g), x) -> 1 / (x * ln(g))
der(log(f(x), g), x) -> 1 / (f(x) * ln(g)) * der(f(x), x)
"""
assert node.is_op(OP_DER)
x = get_derivation_variable(node)
if x and node[0].is_op(OP_LOG):
f = node[0][0]
x = L(x)
if f == x:
return [P(node, logarithmic, ())]
if f.contains(x):
return [P(node, chain_rule, (f, logarithmic, ()))]
return []
def logarithmic(root, args):
"""
der(log(x, g), x) -> 1 / (xln(g))
Shortcut function (because of presence on formula list):
der(ln(x), x) -> 1 / x
"""
x, g = root[0]
if g == E:
return L(1) / x
return L(1) / (x * ln(g))
MESSAGES[logarithmic] = \
_('Apply standard derivative `d/dx log(x, g) = 1 / (x * ln(g))`.')
def match_goniometric(node):
"""
der(sin(x), x) -> cos(x)
der(sin(f(x)), x) -> cos(f(x)) * der(f(x), x)
der(cos(x), x) -> -sin(x)
der(cos(f(x)), x) -> -sin(f(x)) * der(f(x), x)
der(tan(x), x) -> der(sin(x) / cos(x), x)
"""
assert node.is_op(OP_DER)
x = get_derivation_variable(node)
if x and not node[0].is_leaf:
op = node[0].op
if op in (OP_SIN, OP_COS):
f = node[0][0]
x = L(x)
handler = sinus if op == OP_SIN else cosinus
if f == x:
return [P(node, handler)]
if f.contains(x):
return [P(node, chain_rule, (f, handler, ()))]
if op == OP_TAN:
return [P(node, tangens)]
return []
def sinus(root, args):
"""
der(sin(x), x) -> cos(x)
"""
return cos(root[0][0])
MESSAGES[sinus] = _('Apply standard derivative `d/dx sin(x)` is `cos(x)`.')
def cosinus(root, args):
"""
der(cos(x), x) -> -sin(x)
"""
return -sin(root[0][0])
MESSAGES[cosinus] = _('Apply standard derivative `d/dx cos(x)` is `-sin(x)`.')
def tangens(root, args):
"""
der(tan(x), x) -> der(sin(x) / cos(x), x)
"""
x = root[0][0]
return same_der(root, sin(x) / cos(x))
MESSAGES[tangens] = \
_('Convert the tanges to a division and apply the product rule.')
def match_sum_product_rule(node):
"""
[f(x) + g(x)]' -> f'(x) + g'(x)
[f(x) * g(x)]' -> f'(x) * g(x) + f(x) * g'(x)
"""
assert node.is_op(OP_DER)
x = get_derivation_variable(node)
if not x or node[0].is_leaf or node[0].op not in (OP_ADD, OP_MUL):
return []
scope = Scope(node[0])
x = L(x)
functions = [n for n in scope if n.contains(x)]
if node[0].op == OP_MUL:
if len(functions) < 2:
return []
handler = product_rule
else:
handler = sum_rule
return [P(node, handler, (scope, f)) for f in functions]
def sum_rule(root, args):
"""
[f(x) + g(x)]' -> f'(x) + g'(x)
"""
scope, f = args
scope.remove(f)
return same_der(root, f) + same_der(root, scope.as_nary_node())
MESSAGES[sum_rule] = _('Apply the sum rule to {0}.')
def product_rule(root, args):
"""
[f(x) * g(x)]' -> f'(x) * g(x) + f(x) * g'(x)
Note that implicitely:
[f(x) * g(x) * h(x)]' -> f'(x) * (g(x) * h(x)) + f(x) * [g(x) * h(x)]'
"""
scope, f = args
scope.remove(f)
gh = scope.as_nary_node()
return same_der(root, f) * gh + f * same_der(root, gh)
MESSAGES[product_rule] = _('Apply the product rule to {0}.')
def match_quotient_rule(node):
"""
[f(x) / g(x)]' -> (f'(x) * g(x) - f(x) * g'(x)) / g(x) ^ 2
"""
assert node.is_op(OP_DER)
x = get_derivation_variable(node)
if not x or not node[0].is_op(OP_DIV):
return []
f, g = node[0]
x = L(x)
if f.contains(x) and g.contains(x):
return [P(node, quotient_rule)]
return []
def quotient_rule(root, args):
"""
[f(x) / g(x)]' -> (f'(x) * g(x) - f(x) * g'(x)) / g(x) ^ 2
"""
f, g = root[0]
return (same_der(root, f) * g - f * same_der(root, g)) / g ** 2
MESSAGES[quotient_rule] = _('Apply the quotient rule to {0}.')