644 lines
22 KiB
Python
644 lines
22 KiB
Python
""" rewrite of lambdify - This stuff is not stable at all.
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It is for internal use in the new plotting module.
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It may (will! see the Q'n'A in the source) be rewritten.
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It's completely self contained. Especially it does not use lambdarepr.
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It does not aim to replace the current lambdify. Most importantly it will never
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ever support anything else than sympy expressions (no Matrices, dictionaries
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and so on).
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"""
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import re
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from sympy import Symbol, NumberSymbol, I, zoo, oo
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from sympy.utilities.iterables import numbered_symbols
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# We parse the expression string into a tree that identifies functions. Then
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# we translate the names of the functions and we translate also some strings
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# that are not names of functions (all this according to translation
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# dictionaries).
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# If the translation goes to another module (like numpy) the
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# module is imported and 'func' is translated to 'module.func'.
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# If a function can not be translated, the inner nodes of that part of the
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# tree are not translated. So if we have Integral(sqrt(x)), sqrt is not
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# translated to np.sqrt and the Integral does not crash.
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# A namespace for all this is generated by crawling the (func, args) tree of
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# the expression. The creation of this namespace involves many ugly
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# workarounds.
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# The namespace consists of all the names needed for the sympy expression and
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# all the name of modules used for translation. Those modules are imported only
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# as a name (import numpy as np) in order to keep the namespace small and
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# manageable.
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# Please, if there is a bug, do not try to fix it here! Rewrite this by using
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# the method proposed in the last Q'n'A below. That way the new function will
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# work just as well, be just as simple, but it wont need any new workarounds.
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# If you insist on fixing it here, look at the workarounds in the function
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# sympy_expression_namespace and in lambdify.
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# Q: Why are you not using python abstract syntax tree?
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# A: Because it is more complicated and not much more powerful in this case.
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# Q: What if I have Symbol('sin') or g=Function('f')?
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# A: You will break the algorithm. We should use srepr to defend against this?
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# The problem with Symbol('sin') is that it will be printed as 'sin'. The
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# parser will distinguish it from the function 'sin' because functions are
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# detected thanks to the opening parenthesis, but the lambda expression won't
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# understand the difference if we have also the sin function.
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# The solution (complicated) is to use srepr and maybe ast.
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# The problem with the g=Function('f') is that it will be printed as 'f' but in
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# the global namespace we have only 'g'. But as the same printer is used in the
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# constructor of the namespace there will be no problem.
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# Q: What if some of the printers are not printing as expected?
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# A: The algorithm wont work. You must use srepr for those cases. But even
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# srepr may not print well. All problems with printers should be considered
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# bugs.
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# Q: What about _imp_ functions?
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# A: Those are taken care for by evalf. A special case treatment will work
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# faster but it's not worth the code complexity.
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# Q: Will ast fix all possible problems?
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# A: No. You will always have to use some printer. Even srepr may not work in
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# some cases. But if the printer does not work, that should be considered a
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# bug.
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# Q: Is there same way to fix all possible problems?
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# A: Probably by constructing our strings ourself by traversing the (func,
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# args) tree and creating the namespace at the same time. That actually sounds
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# good.
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from sympy.external import import_module
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import warnings
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#TODO debugging output
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class vectorized_lambdify:
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""" Return a sufficiently smart, vectorized and lambdified function.
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Returns only reals.
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Explanation
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===========
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This function uses experimental_lambdify to created a lambdified
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expression ready to be used with numpy. Many of the functions in sympy
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are not implemented in numpy so in some cases we resort to python cmath or
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even to evalf.
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The following translations are tried:
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only numpy complex
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- on errors raised by sympy trying to work with ndarray:
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only python cmath and then vectorize complex128
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When using python cmath there is no need for evalf or float/complex
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because python cmath calls those.
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This function never tries to mix numpy directly with evalf because numpy
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does not understand sympy Float. If this is needed one can use the
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float_wrap_evalf/complex_wrap_evalf options of experimental_lambdify or
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better one can be explicit about the dtypes that numpy works with.
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Check numpy bug http://projects.scipy.org/numpy/ticket/1013 to know what
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types of errors to expect.
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"""
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def __init__(self, args, expr):
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self.args = args
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self.expr = expr
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self.np = import_module('numpy')
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self.lambda_func_1 = experimental_lambdify(
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args, expr, use_np=True)
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self.vector_func_1 = self.lambda_func_1
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self.lambda_func_2 = experimental_lambdify(
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args, expr, use_python_cmath=True)
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self.vector_func_2 = self.np.vectorize(
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self.lambda_func_2, otypes=[complex])
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self.vector_func = self.vector_func_1
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self.failure = False
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def __call__(self, *args):
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np = self.np
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try:
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temp_args = (np.array(a, dtype=complex) for a in args)
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results = self.vector_func(*temp_args)
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results = np.ma.masked_where(
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np.abs(results.imag) > 1e-7 * np.abs(results),
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results.real, copy=False)
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return results
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except ValueError:
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if self.failure:
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raise
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self.failure = True
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self.vector_func = self.vector_func_2
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warnings.warn(
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'The evaluation of the expression is problematic. '
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'We are trying a failback method that may still work. '
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'Please report this as a bug.')
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return self.__call__(*args)
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class lambdify:
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"""Returns the lambdified function.
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Explanation
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===========
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This function uses experimental_lambdify to create a lambdified
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expression. It uses cmath to lambdify the expression. If the function
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is not implemented in python cmath, python cmath calls evalf on those
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functions.
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"""
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def __init__(self, args, expr):
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self.args = args
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self.expr = expr
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self.lambda_func_1 = experimental_lambdify(
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args, expr, use_python_cmath=True, use_evalf=True)
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self.lambda_func_2 = experimental_lambdify(
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args, expr, use_python_math=True, use_evalf=True)
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self.lambda_func_3 = experimental_lambdify(
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args, expr, use_evalf=True, complex_wrap_evalf=True)
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self.lambda_func = self.lambda_func_1
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self.failure = False
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def __call__(self, args):
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try:
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#The result can be sympy.Float. Hence wrap it with complex type.
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result = complex(self.lambda_func(args))
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if abs(result.imag) > 1e-7 * abs(result):
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return None
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return result.real
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except (ZeroDivisionError, OverflowError, TypeError) as e:
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if isinstance(e, ZeroDivisionError) or isinstance(e, OverflowError):
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return None
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if self.failure:
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raise e
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if self.lambda_func == self.lambda_func_1:
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self.lambda_func = self.lambda_func_2
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return self.__call__(args)
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self.failure = True
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self.lambda_func = self.lambda_func_3
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warnings.warn(
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'The evaluation of the expression is problematic. '
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'We are trying a failback method that may still work. '
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'Please report this as a bug.')
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return self.__call__(args)
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def experimental_lambdify(*args, **kwargs):
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l = Lambdifier(*args, **kwargs)
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return l
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class Lambdifier:
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def __init__(self, args, expr, print_lambda=False, use_evalf=False,
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float_wrap_evalf=False, complex_wrap_evalf=False,
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use_np=False, use_python_math=False, use_python_cmath=False,
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use_interval=False):
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self.print_lambda = print_lambda
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self.use_evalf = use_evalf
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self.float_wrap_evalf = float_wrap_evalf
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self.complex_wrap_evalf = complex_wrap_evalf
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self.use_np = use_np
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self.use_python_math = use_python_math
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self.use_python_cmath = use_python_cmath
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self.use_interval = use_interval
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# Constructing the argument string
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# - check
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if not all([isinstance(a, Symbol) for a in args]):
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raise ValueError('The arguments must be Symbols.')
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# - use numbered symbols
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syms = numbered_symbols(exclude=expr.free_symbols)
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newargs = [next(syms) for _ in args]
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expr = expr.xreplace(dict(zip(args, newargs)))
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argstr = ', '.join([str(a) for a in newargs])
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del syms, newargs, args
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# Constructing the translation dictionaries and making the translation
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self.dict_str = self.get_dict_str()
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self.dict_fun = self.get_dict_fun()
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exprstr = str(expr)
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newexpr = self.tree2str_translate(self.str2tree(exprstr))
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# Constructing the namespaces
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namespace = {}
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namespace.update(self.sympy_atoms_namespace(expr))
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namespace.update(self.sympy_expression_namespace(expr))
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# XXX Workaround
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# Ugly workaround because Pow(a,Half) prints as sqrt(a)
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# and sympy_expression_namespace can not catch it.
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from sympy import sqrt
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namespace.update({'sqrt': sqrt})
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namespace.update({'Eq': lambda x, y: x == y})
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namespace.update({'Ne': lambda x, y: x != y})
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# End workaround.
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if use_python_math:
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namespace.update({'math': __import__('math')})
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if use_python_cmath:
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namespace.update({'cmath': __import__('cmath')})
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if use_np:
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try:
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namespace.update({'np': __import__('numpy')})
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except ImportError:
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raise ImportError(
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'experimental_lambdify failed to import numpy.')
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if use_interval:
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namespace.update({'imath': __import__(
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'sympy.plotting.intervalmath', fromlist=['intervalmath'])})
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namespace.update({'math': __import__('math')})
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# Construct the lambda
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if self.print_lambda:
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print(newexpr)
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eval_str = 'lambda %s : ( %s )' % (argstr, newexpr)
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self.eval_str = eval_str
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exec("from __future__ import division; MYNEWLAMBDA = %s" % eval_str, namespace)
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self.lambda_func = namespace['MYNEWLAMBDA']
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def __call__(self, *args, **kwargs):
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return self.lambda_func(*args, **kwargs)
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##############################################################################
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# Dicts for translating from sympy to other modules
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##############################################################################
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###
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# builtins
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###
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# Functions with different names in builtins
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builtin_functions_different = {
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'Min': 'min',
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'Max': 'max',
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'Abs': 'abs',
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}
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# Strings that should be translated
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builtin_not_functions = {
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'I': '1j',
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# 'oo': '1e400',
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}
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###
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# numpy
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###
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# Functions that are the same in numpy
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numpy_functions_same = [
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'sin', 'cos', 'tan', 'sinh', 'cosh', 'tanh', 'exp', 'log',
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'sqrt', 'floor', 'conjugate',
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]
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# Functions with different names in numpy
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numpy_functions_different = {
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"acos": "arccos",
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"acosh": "arccosh",
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"arg": "angle",
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"asin": "arcsin",
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"asinh": "arcsinh",
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"atan": "arctan",
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"atan2": "arctan2",
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"atanh": "arctanh",
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"ceiling": "ceil",
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"im": "imag",
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"ln": "log",
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"Max": "amax",
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"Min": "amin",
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"re": "real",
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"Abs": "abs",
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}
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# Strings that should be translated
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numpy_not_functions = {
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'pi': 'np.pi',
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'oo': 'np.inf',
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'E': 'np.e',
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}
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###
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# python math
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###
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# Functions that are the same in math
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math_functions_same = [
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'sin', 'cos', 'tan', 'asin', 'acos', 'atan', 'atan2',
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'sinh', 'cosh', 'tanh', 'asinh', 'acosh', 'atanh',
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'exp', 'log', 'erf', 'sqrt', 'floor', 'factorial', 'gamma',
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]
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# Functions with different names in math
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math_functions_different = {
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'ceiling': 'ceil',
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'ln': 'log',
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'loggamma': 'lgamma'
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}
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# Strings that should be translated
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math_not_functions = {
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'pi': 'math.pi',
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'E': 'math.e',
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}
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###
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# python cmath
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###
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# Functions that are the same in cmath
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cmath_functions_same = [
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'sin', 'cos', 'tan', 'asin', 'acos', 'atan',
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'sinh', 'cosh', 'tanh', 'asinh', 'acosh', 'atanh',
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'exp', 'log', 'sqrt',
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]
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# Functions with different names in cmath
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cmath_functions_different = {
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'ln': 'log',
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'arg': 'phase',
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}
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# Strings that should be translated
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cmath_not_functions = {
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'pi': 'cmath.pi',
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'E': 'cmath.e',
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}
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###
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# intervalmath
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###
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interval_not_functions = {
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'pi': 'math.pi',
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'E': 'math.e'
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}
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interval_functions_same = [
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'sin', 'cos', 'exp', 'tan', 'atan', 'log',
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'sqrt', 'cosh', 'sinh', 'tanh', 'floor',
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'acos', 'asin', 'acosh', 'asinh', 'atanh',
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'Abs', 'And', 'Or'
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]
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interval_functions_different = {
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'Min': 'imin',
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'Max': 'imax',
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'ceiling': 'ceil',
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}
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###
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# mpmath, etc
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###
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#TODO
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###
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# Create the final ordered tuples of dictionaries
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###
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# For strings
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def get_dict_str(self):
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dict_str = dict(self.builtin_not_functions)
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if self.use_np:
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dict_str.update(self.numpy_not_functions)
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if self.use_python_math:
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dict_str.update(self.math_not_functions)
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if self.use_python_cmath:
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dict_str.update(self.cmath_not_functions)
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if self.use_interval:
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dict_str.update(self.interval_not_functions)
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return dict_str
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# For functions
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def get_dict_fun(self):
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dict_fun = dict(self.builtin_functions_different)
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if self.use_np:
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for s in self.numpy_functions_same:
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dict_fun[s] = 'np.' + s
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for k, v in self.numpy_functions_different.items():
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dict_fun[k] = 'np.' + v
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if self.use_python_math:
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for s in self.math_functions_same:
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dict_fun[s] = 'math.' + s
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for k, v in self.math_functions_different.items():
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dict_fun[k] = 'math.' + v
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if self.use_python_cmath:
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for s in self.cmath_functions_same:
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dict_fun[s] = 'cmath.' + s
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for k, v in self.cmath_functions_different.items():
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dict_fun[k] = 'cmath.' + v
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if self.use_interval:
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for s in self.interval_functions_same:
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dict_fun[s] = 'imath.' + s
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for k, v in self.interval_functions_different.items():
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dict_fun[k] = 'imath.' + v
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return dict_fun
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##############################################################################
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# The translator functions, tree parsers, etc.
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##############################################################################
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def str2tree(self, exprstr):
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"""Converts an expression string to a tree.
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Explanation
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===========
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Functions are represented by ('func_name(', tree_of_arguments).
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Other expressions are (head_string, mid_tree, tail_str).
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Expressions that do not contain functions are directly returned.
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Examples
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========
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>>> from sympy.abc import x, y, z
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>>> from sympy import Integral, sin
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>>> from sympy.plotting.experimental_lambdify import Lambdifier
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>>> str2tree = Lambdifier([x], x).str2tree
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>>> str2tree(str(Integral(x, (x, 1, y))))
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('', ('Integral(', 'x, (x, 1, y)'), ')')
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>>> str2tree(str(x+y))
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'x + y'
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>>> str2tree(str(x+y*sin(z)+1))
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('x + y*', ('sin(', 'z'), ') + 1')
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>>> str2tree('sin(y*(y + 1.1) + (sin(y)))')
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('', ('sin(', ('y*(y + 1.1) + (', ('sin(', 'y'), '))')), ')')
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"""
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#matches the first 'function_name('
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first_par = re.search(r'(\w+\()', exprstr)
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if first_par is None:
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return exprstr
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else:
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start = first_par.start()
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end = first_par.end()
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head = exprstr[:start]
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func = exprstr[start:end]
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tail = exprstr[end:]
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count = 0
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for i, c in enumerate(tail):
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if c == '(':
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count += 1
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elif c == ')':
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count -= 1
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if count == -1:
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break
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func_tail = self.str2tree(tail[:i])
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tail = self.str2tree(tail[i:])
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return (head, (func, func_tail), tail)
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@classmethod
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def tree2str(cls, tree):
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"""Converts a tree to string without translations.
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Examples
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========
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>>> from sympy.abc import x, y, z
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>>> from sympy import sin
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>>> from sympy.plotting.experimental_lambdify import Lambdifier
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>>> str2tree = Lambdifier([x], x).str2tree
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>>> tree2str = Lambdifier([x], x).tree2str
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|
|
>>> tree2str(str2tree(str(x+y*sin(z)+1)))
|
|
'x + y*sin(z) + 1'
|
|
"""
|
|
if isinstance(tree, str):
|
|
return tree
|
|
else:
|
|
return ''.join(map(cls.tree2str, tree))
|
|
|
|
def tree2str_translate(self, tree):
|
|
"""Converts a tree to string with translations.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
Function names are translated by translate_func.
|
|
Other strings are translated by translate_str.
|
|
"""
|
|
if isinstance(tree, str):
|
|
return self.translate_str(tree)
|
|
elif isinstance(tree, tuple) and len(tree) == 2:
|
|
return self.translate_func(tree[0][:-1], tree[1])
|
|
else:
|
|
return ''.join([self.tree2str_translate(t) for t in tree])
|
|
|
|
def translate_str(self, estr):
|
|
"""Translate substrings of estr using in order the dictionaries in
|
|
dict_tuple_str."""
|
|
for pattern, repl in self.dict_str.items():
|
|
estr = re.sub(pattern, repl, estr)
|
|
return estr
|
|
|
|
def translate_func(self, func_name, argtree):
|
|
"""Translate function names and the tree of arguments.
|
|
|
|
Explanation
|
|
===========
|
|
|
|
If the function name is not in the dictionaries of dict_tuple_fun then the
|
|
function is surrounded by a float((...).evalf()).
|
|
|
|
The use of float is necessary as np.<function>(sympy.Float(..)) raises an
|
|
error."""
|
|
if func_name in self.dict_fun:
|
|
new_name = self.dict_fun[func_name]
|
|
argstr = self.tree2str_translate(argtree)
|
|
return new_name + '(' + argstr
|
|
elif func_name in ['Eq', 'Ne']:
|
|
op = {'Eq': '==', 'Ne': '!='}
|
|
return "(lambda x, y: x {} y)({}".format(op[func_name], self.tree2str_translate(argtree))
|
|
else:
|
|
template = '(%s(%s)).evalf(' if self.use_evalf else '%s(%s'
|
|
if self.float_wrap_evalf:
|
|
template = 'float(%s)' % template
|
|
elif self.complex_wrap_evalf:
|
|
template = 'complex(%s)' % template
|
|
|
|
# Wrapping should only happen on the outermost expression, which
|
|
# is the only thing we know will be a number.
|
|
float_wrap_evalf = self.float_wrap_evalf
|
|
complex_wrap_evalf = self.complex_wrap_evalf
|
|
self.float_wrap_evalf = False
|
|
self.complex_wrap_evalf = False
|
|
ret = template % (func_name, self.tree2str_translate(argtree))
|
|
self.float_wrap_evalf = float_wrap_evalf
|
|
self.complex_wrap_evalf = complex_wrap_evalf
|
|
return ret
|
|
|
|
##############################################################################
|
|
# The namespace constructors
|
|
##############################################################################
|
|
|
|
@classmethod
|
|
def sympy_expression_namespace(cls, expr):
|
|
"""Traverses the (func, args) tree of an expression and creates a sympy
|
|
namespace. All other modules are imported only as a module name. That way
|
|
the namespace is not polluted and rests quite small. It probably causes much
|
|
more variable lookups and so it takes more time, but there are no tests on
|
|
that for the moment."""
|
|
if expr is None:
|
|
return {}
|
|
else:
|
|
funcname = str(expr.func)
|
|
# XXX Workaround
|
|
# Here we add an ugly workaround because str(func(x))
|
|
# is not always the same as str(func). Eg
|
|
# >>> str(Integral(x))
|
|
# "Integral(x)"
|
|
# >>> str(Integral)
|
|
# "<class 'sympy.integrals.integrals.Integral'>"
|
|
# >>> str(sqrt(x))
|
|
# "sqrt(x)"
|
|
# >>> str(sqrt)
|
|
# "<function sqrt at 0x3d92de8>"
|
|
# >>> str(sin(x))
|
|
# "sin(x)"
|
|
# >>> str(sin)
|
|
# "sin"
|
|
# Either one of those can be used but not all at the same time.
|
|
# The code considers the sin example as the right one.
|
|
regexlist = [
|
|
r'<class \'sympy[\w.]*?.([\w]*)\'>$',
|
|
# the example Integral
|
|
r'<function ([\w]*) at 0x[\w]*>$', # the example sqrt
|
|
]
|
|
for r in regexlist:
|
|
m = re.match(r, funcname)
|
|
if m is not None:
|
|
funcname = m.groups()[0]
|
|
# End of the workaround
|
|
# XXX debug: print funcname
|
|
args_dict = {}
|
|
for a in expr.args:
|
|
if (isinstance(a, Symbol) or
|
|
isinstance(a, NumberSymbol) or
|
|
a in [I, zoo, oo]):
|
|
continue
|
|
else:
|
|
args_dict.update(cls.sympy_expression_namespace(a))
|
|
args_dict.update({funcname: expr.func})
|
|
return args_dict
|
|
|
|
@staticmethod
|
|
def sympy_atoms_namespace(expr):
|
|
"""For no real reason this function is separated from
|
|
sympy_expression_namespace. It can be moved to it."""
|
|
atoms = expr.atoms(Symbol, NumberSymbol, I, zoo, oo)
|
|
d = {}
|
|
for a in atoms:
|
|
# XXX debug: print 'atom:' + str(a)
|
|
d[str(a)] = a
|
|
return d
|