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)
[docs]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
[docs]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
[docs]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)
[docs]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))
[docs]def get_variables(expression: sympy.Expr) -> Sequence[str]:
return tuple(map(str, get_free_symbols(expression)))
[docs]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))
[docs]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
[docs]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