from typing import Union, Dict, Tuple, Any, Sequence, Optional from numbers import Number from types import CodeType import warnings import builtins import math import sympy import numpy try: import scipy.special as _special_functions except ImportError: _special_functions = {fname: numpy.vectorize(fobject) for fname, fobject in math.__dict__.items() if not fname.startswith('_') and fname not in numpy.__dict__} warnings.warn('scipy is not installed. This reduces the set of available functions to those present in numpy + ' 'manually vectorized functions in math.') __all__ = ["sympify", "substitute_with_eval", "to_numpy", "get_variables", "get_free_symbols", "recursive_substitution", "evaluate_lambdified", "get_most_simple_representation"] Sympifyable = Union[str, Number, sympy.Expr, numpy.str_] class IndexedBasedFinder(dict): """Acts as a symbol lookup and determines which symbols in an expression a subscripted.""" def __init__(self): super().__init__() self.symbols = set() self.indexed_base = set() self.indices = set() class SubscriptionChecker(sympy.Symbol): """A symbol stand-in which detects whether the symbol is subscripted.""" def __getitem__(s, k): self.indexed_base.add(str(s)) self.indices.add(k) if isinstance(k, SubscriptionChecker): k = sympy.Symbol(str(k)) return sympy.IndexedBase(str(s))[k] self.SubscriptionChecker = SubscriptionChecker def unimplementded(*args, **kwargs): raise NotImplementedError("Not a full dict") for m in vars(dict).keys(): if not m.startswith('_'): setattr(self, m, unimplementded) def __getitem__(self, k) -> sympy.Expr: """Return an instance of the internal SubscriptionChecker class for each symbol to determine which symbols are indexed/subscripted. __getitem__ is (apparently) called by symbol for each token and gets either symbol names or type names such as 'Integer', 'Float', etc. We have to take care of returning correct types for symbols (-> SubscriptionChecker) and the base types (-> Integer, Float, etc). """ if hasattr(sympy, k): # if k is a sympy base type identifier, return the base type return getattr(sympy, k) # otherwise track the symbol name and return a SubscriptionChecker instance self.symbols.add(k) return self.SubscriptionChecker(k) def __setitem__(self, key, value): raise NotImplementedError("Not a full dict") def __delitem__(self, key): raise NotImplementedError("Not a full dict") def __contains__(self, k) -> bool: return True class Broadcast(sympy.Function): """Broadcast x to the specified shape using numpy.broadcast_to Examples: >>> bc = Broadcast('a', (3,)) >>> assert bc.subs({'a': 2}) == sympy.Array([2, 2, 2]) >>> assert bc.subs({'a': (1, 2, 3)}) == sympy.Array([1, 2, 3]) """ @classmethod def eval(cls, x, shape) -> Optional[sympy.Array]: if hasattr(shape, 'free_symbols') and shape.free_symbols: # cannot do anything return None if hasattr(x, '__len__') or not x.free_symbols: return sympy.Array(numpy.broadcast_to(x, shape)) class Len(sympy.Function): nargs = 1 @classmethod def eval(cls, arg) -> Optional[sympy.Integer]: if hasattr(arg, '__len__'): return sympy.Integer(len(arg)) is_Integer = True Len.__name__ = 'len' sympify_namespace = {'len': Len, 'Len': Len, 'Broadcast': Broadcast} def numpy_compatible_mul(*args) -> Union[sympy.Mul, sympy.Array]: if any(isinstance(a, sympy.NDimArray) for a in args): result = 1 for a in args: result = result * (numpy.array(a.tolist()) if isinstance(a, sympy.NDimArray) else a) return sympy.Array(result) else: return sympy.Mul(*args) def numpy_compatible_ceiling(input_value: Any) -> Any: if isinstance(input_value, numpy.ndarray): return numpy.ceil(input_value).astype(numpy.int64) else: return sympy.ceiling(input_value) def to_numpy(sympy_array: sympy.NDimArray) -> numpy.ndarray: if isinstance(sympy_array, sympy.DenseNDimArray): if len(sympy_array.shape) == 2: return numpy.asarray(sympy_array.tomatrix()) elif len(sympy_array.shape) == 1: return numpy.asarray(sympy_array) return numpy.array(sympy_array.tolist()) def get_subscripted_symbols(expression: str) -> set: # track all symbols that are subscipted in here indexed_base_finder = IndexedBasedFinder() sympy.sympify(expression, locals=indexed_base_finder) return indexed_base_finder.indexed_base def sympify(expr: Union[str, Number, sympy.Expr, numpy.str_], **kwargs) -> sympy.Expr: if isinstance(expr, numpy.str_): # putting numpy.str_ in sympy.sympify behaves unexpected in version 1.1.1 # It seems to ignore the locals argument expr = str(expr) try: return sympy.sympify(expr, **kwargs, locals=sympify_namespace) except TypeError as err: if True:#err.args[0] == "'Symbol' object is not subscriptable": indexed_base = get_subscripted_symbols(expr) return sympy.sympify(expr, **kwargs, locals={**{k: sympy.IndexedBase(k) for k in indexed_base}, **sympify_namespace}) else: raise def get_most_simple_representation(expression: sympy.Expr) -> Union[str, int, float]: if expression.free_symbols: return str(expression) elif expression.is_Integer: return int(expression) elif expression.is_Float: return float(expression) else: return str(expression) def get_free_symbols(expression: sympy.Expr) -> Sequence[sympy.Symbol]: return tuple(symbol for symbol in expression.free_symbols if not isinstance(symbol, sympy.Indexed)) def get_variables(expression: sympy.Expr) -> Sequence[str]: return tuple(map(str, get_free_symbols(expression))) def substitute_with_eval(expression: sympy.Expr, substitutions: Dict[str, Union[sympy.Expr, numpy.ndarray, str]]) -> sympy.Expr: """Substitutes only sympy.Symbols. Workaround for numpy like array behaviour. ~Factor 3 slower compared to subs""" substitutions = {k: v if isinstance(v, sympy.Expr) else sympify(v) for k, v in substitutions.items()} for symbol in get_free_symbols(expression): symbol_name = str(symbol) if symbol_name not in substitutions: substitutions[symbol_name] = symbol string_representation = sympy.srepr(expression) return eval(string_representation, sympy.__dict__, {'Symbol': substitutions.__getitem__, 'Mul': numpy_compatible_mul}) def _recursive_substitution(expression: sympy.Expr, substitutions: Dict[sympy.Symbol, sympy.Expr]) -> sympy.Expr: if not expression.free_symbols: return expression elif expression.func is sympy.Symbol: return substitutions.get(expression, expression) elif expression.func is sympy.Mul: func = numpy_compatible_mul else: func = expression.func substitutions = {s: substitutions.get(s, s) for s in get_free_symbols(expression)} return func(*(_recursive_substitution(arg, substitutions) for arg in expression.args)) def recursive_substitution(expression: sympy.Expr, substitutions: Dict[str, Union[sympy.Expr, numpy.ndarray, str]]) -> sympy.Expr: substitutions = {sympy.Symbol(k): sympify(v) for k, v in substitutions.items()} for s in get_free_symbols(expression): substitutions.setdefault(s, s) return _recursive_substitution(expression, substitutions) _base_environment = {'builtins': builtins, '__builtins__': builtins} _math_environment = {**_base_environment, **math.__dict__} _numpy_environment = {**_base_environment, **numpy.__dict__} _sympy_environment = {**_base_environment, **sympy.__dict__} _lambdify_modules = [{'ceiling': numpy_compatible_ceiling, 'Broadcast': numpy.broadcast_to}, 'numpy', _special_functions] def evaluate_compiled(expression: sympy.Expr, parameters: Dict[str, Union[numpy.ndarray, Number]], compiled: CodeType=None, mode=None) -> Tuple[any, CodeType]: if compiled is None: compiled = compile(sympy.printing.lambdarepr.lambdarepr(expression), '<string>', 'eval') if mode == 'numeric' or mode is None: result = eval(compiled, parameters.copy(), _numpy_environment) elif mode == 'exact': result = eval(compiled, parameters.copy(), _sympy_environment) else: raise ValueError("Unknown mode: '{}'".format(mode)) return result, compiled def evaluate_lambdified(expression: Union[sympy.Expr, numpy.ndarray], variables: Sequence[str], parameters: Dict[str, Union[numpy.ndarray, Number]], lambdified) -> Tuple[Any, Any]: lambdified = lambdified or sympy.lambdify(variables, expression, _lambdify_modules) return lambdified(**parameters), lambdified def almost_equal(lhs: sympy.Expr, rhs: sympy.Expr, epsilon: float=1e-15) -> Optional[bool]: """Returns True (or False) if the two expressions are almost equal (or not). Returns None if this cannot be determined.""" relation = sympy.simplify(sympy.Abs(lhs - rhs) <= epsilon) if relation is sympy.true: return True elif relation is sympy.false: return False else: return None