def diff(y, x):
"""Return the symbolic derivative, dy/dx, as an Expr.
However, you probably want to simplify the results with simp.
>>> diff(x * x, x)
((x * 1) + (x * 1))
>>> simp(diff(x * x, x))
(2 * x)
"""
if y == x: return ONE
elif not y.args: return ZERO
else:
u, op, v = y.args[0], y.op, y.args[-1]
if op == '+': return diff(u, x) + diff(v, x)
elif op == '-' and len(args) == 1: return -diff(u, x)
elif op == '-': return diff(u, x) - diff(v, x)
elif op == '*': return u * diff(v, x) + v * diff(u, x)
elif op == '/': return (v * diff(u, x) - u * diff(v, x)) / (v * v)
elif op == '**' and utils_lpw.isnumber(x.op):
return (v * u**(v - 1) * diff(u, x))
elif op == '**':
return (v * u**(v - 1) * diff(u, x) +
u**v * Expr('log')(u) * diff(v, x))
elif op == 'log':
return diff(u, x) / u
else:
raise ValueError("Unknown op: %s in diff(%s, %s)" % (op, y, x))
def expr(s):
"""Create an Expr representing a logic expression by parsing the input
string. Symbols and numbers are automatically converted to Exprs.
In addition you can use alternative spellings of these operators:
'x ==> y' parses as (x >> y) # Implication
'x <== y' parses as (x << y) # Reverse implication
'x <=> y' parses as (x % y) # Logical equivalence
'x =/= y' parses as (x ^ y) # Logical disequality (xor)
But BE CAREFUL; precedence of implication is wrong. expr('P & Q ==> R & S')
is ((P & (Q >> R)) & S); so you must use expr('(P & Q) ==> (R & S)').
>>> expr('P <=> Q(1)')
(P <=> Q(1))
>>> expr('P & Q | ~R(x, F(x))')
((P & Q) | ~R(x, F(x)))
"""
if isinstance(s, Expr): return s
if utils_lpw.isnumber(s): return Expr(s)
## Replace the alternative spellings of operators with canonical spellings
s = s.replace('==>', '>>').replace('<==', '<<')
s = s.replace('<=>', '%').replace('=/=', '^')
## Replace a symbol or number, such as 'P' with 'Expr("P")'
s = re.sub(r'([a-zA-Z0-9_.]+)', r'Expr("\1")', s)
## Now eval the string. (A security hole; do not use with an adversary.)
return eval(s, {'Expr': Expr})
def __init__(self, op, *args):
"Op is a string or number; args are Exprs (or are coerced to Exprs)."
assert isinstance(op, str) or (utils_lpw.isnumber(op) and not args)
self.op = utils_lpw.num_or_str(op)
self.args = map(expr, args) ## Coerce args to Exprs