sympyextensions.py 22.5 KB
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import itertools
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import warnings
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import operator
from functools import reduce, partial
from collections import defaultdict, Counter
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import sympy as sp
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from sympy.functions import Abs
from typing import Optional, Union, List, TypeVar, Iterable, Sequence, Callable, Dict, Tuple
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from pystencils.data_types import get_type_of_expression, get_base_type, cast_func
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from pystencils.assignment import Assignment
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T = TypeVar('T')

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def prod(seq: Iterable[T]) -> T:
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    """Takes a sequence and returns the product of all elements"""
    return reduce(operator.mul, seq, 1)


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def remove_small_floats(expr, threshold):
    """Removes all sp.Float objects whose absolute value is smaller than threshold

    >>> expr = sp.sympify("x + 1e-15 * y")
    >>> remove_small_floats(expr, 1e-14)
    x
    """
    if isinstance(expr, sp.Float) and sp.Abs(expr) < threshold:
        return 0
    else:
        new_args = [remove_small_floats(c, threshold) for c in expr.args]
        return expr.func(*new_args) if new_args else expr


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def is_integer_sequence(sequence: Iterable) -> bool:
    """Checks if all elements of the passed sequence can be cast to integers"""
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    try:
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        for i in sequence:
            int(i)
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        return True
    except TypeError:
        return False


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def scalar_product(a: Iterable[T], b: Iterable[T]) -> T:
    """Scalar product between two sequences."""
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    return sum(a_i * b_i for a_i, b_i in zip(a, b))


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def kronecker_delta(*args):
    """Kronecker delta for variable number of arguments, 1 if all args are equal, otherwise 0"""
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    for a in args:
        if a != args[0]:
            return 0
    return 1


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def tanh_step_function_approximation(x, step_location, kind='right', steepness=0.0001):
    """Approximation of step function by a tanh function

    >>> tanh_step_function_approximation(1.2, step_location=1.0, kind='right')
    1.00000000000000
    >>> tanh_step_function_approximation(0.9, step_location=1.0, kind='right')
    0
    >>> tanh_step_function_approximation(1.1, step_location=1.0, kind='left')
    0
    >>> tanh_step_function_approximation(0.9, step_location=1.0, kind='left')
    1.00000000000000
    >>> tanh_step_function_approximation(0.5, step_location=(0, 1), kind='middle')
    1
    """
    if kind == 'left':
        return (1 - sp.tanh((x - step_location) / steepness)) / 2
    elif kind == 'right':
        return (1 + sp.tanh((x - step_location) / steepness)) / 2
    elif kind == 'middle':
        x1, x2 = step_location
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        return 1 - (tanh_step_function_approximation(x, x1, 'left', steepness)
                    + tanh_step_function_approximation(x, x2, 'right', steepness))
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def multidimensional_sum(i, dim):
    """Multidimensional summation
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    Example:
        >>> list(multidimensional_sum(2, dim=3))
        [(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
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    """
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    prod_args = [range(dim)] * i
    return itertools.product(*prod_args)


def normalize_product(product: sp.Expr) -> List[sp.Expr]:
    """Expects a sympy expression that can be interpreted as a product and returns a list of all factors.

    Removes sp.Pow nodes that have integer exponent by representing them as single factors in list.

    Returns:
        * for a Mul node list of factors ('args')
        * for a Pow node with positive integer exponent a list of factors
        * for other node types [product] is returned
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    """
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    def handle_pow(power):
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        if power.exp.is_integer and power.exp.is_number and power.exp > 0:
            return [power.base] * power.exp
        else:
            return [power]

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    if isinstance(product, sp.Pow):
        return handle_pow(product)
    elif isinstance(product, sp.Mul):
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        result = []
        for a in product.args:
            if a.func == sp.Pow:
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                result += handle_pow(a)
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            else:
                result.append(a)
        return result
    else:
        return [product]


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def symmetric_product(*args, with_diagonal: bool = True) -> Iterable:
    """Similar to itertools.product but yields only values where the index is ascending i.e. values below/up to diagonal

    Examples:
        >>> list(symmetric_product([1, 2, 3], ['a', 'b', 'c']))
        [(1, 'a'), (1, 'b'), (1, 'c'), (2, 'b'), (2, 'c'), (3, 'c')]
        >>> list(symmetric_product([1, 2, 3], ['a', 'b', 'c'], with_diagonal=False))
        [(1, 'b'), (1, 'c'), (2, 'c')]
    """
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    ranges = [range(len(a)) for a in args]
    for idx in itertools.product(*ranges):
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        valid_index = True
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        for t in range(1, len(idx)):
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            if (with_diagonal and idx[t - 1] > idx[t]) or (not with_diagonal and idx[t - 1] >= idx[t]):
                valid_index = False
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                break
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        if valid_index:
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            yield tuple(a[i] for a, i in zip(args, idx))


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def fast_subs(expression: T, substitutions: Dict,
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              skip: Optional[Callable[[sp.Expr], bool]] = None) -> T:
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    """Similar to sympy subs function.
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    Args:
        expression: expression where parts should be substituted
        substitutions: dict defining substitutions by mapping from old to new terms
        skip: function that marks expressions to be skipped (if True is returned) - that means that in these skipped
              expressions no substitutions are done

    This version is much faster for big substitution dictionaries than sympy version
    """
    if type(expression) is sp.Matrix:
        return expression.copy().applyfunc(partial(fast_subs, substitutions=substitutions))

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    def visit(expr):
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        if skip and skip(expr):
            return expr
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        if hasattr(expr, "fast_subs"):
            return expr.fast_subs(substitutions)
        if expr in substitutions:
            return substitutions[expr]
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        if not hasattr(expr, 'args'):
            return expr
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        param_list = [visit(a) for a in expr.args]
        return expr if not param_list else expr.func(*param_list)
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    if len(substitutions) == 0:
        return expression
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    else:
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        return visit(expression)

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def is_constant(expr):
    """Simple version of checking if a sympy expression is constant.
    Works also for piecewise defined functions - sympy's is_constant() has a problem there, see:
    https://github.com/sympy/sympy/issues/16662
    """
    return len(expr.free_symbols) == 0

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def subs_additive(expr: sp.Expr, replacement: sp.Expr, subexpression: sp.Expr,
                  required_match_replacement: Optional[Union[int, float]] = 0.5,
                  required_match_original: Optional[Union[int, float]] = None) -> sp.Expr:
    """Transformation for replacing a given subexpression inside a sum.

    Examples:
        The next example demonstrates the advantage of replace_additive compared to sympy.subs:
        >>> x, y, z, k = sp.symbols("x y z k")
        >>> subs_additive(3*x + 3*y, replacement=k, subexpression=x + y)
        3*k

        Terms that don't match completely can be substituted at the cost of additional terms.
        This trade-off is managed using the required_match parameters.
        >>> subs_additive(3*x + 3*y + z, replacement=k, subexpression=x+y+z, required_match_original=1.0)
        3*x + 3*y + z
        >>> subs_additive(3*x + 3*y + z, replacement=k, subexpression=x+y+z, required_match_original=0.5)
        3*k - 2*z
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        >>> subs_additive(3*x + 3*y + z, replacement=k, subexpression=x+y+z, required_match_original=2)
        3*k - 2*z
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    Args:
        expr: input expression
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        replacement: expression that is inserted for subexpression (if found)
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        subexpression: expression to replace
        required_match_replacement:
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             * if float: the percentage of terms of the subexpression that has to be matched in order to replace
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             * if integer: the total number of terms that has to be matched in order to replace
             * None: is equal to integer 1
             * if both match parameters are given, both restrictions have to be fulfilled (i.e. logical AND)
        required_match_original:
             * if float: the percentage of terms of the original addition expression that has to be matched
             * if integer: the total number of terms that has to be matched in order to replace
             * None: is equal to integer 1

    Returns:
        new expression with replacement
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    """
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    def normalize_match_parameter(match_parameter, expression_length):
        if match_parameter is None:
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            return 1
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        elif isinstance(match_parameter, float):
            assert 0 <= match_parameter <= 1
            res = int(match_parameter * expression_length)
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            return max(res, 1)
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        elif isinstance(match_parameter, int):
            assert match_parameter > 0
            return match_parameter
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        raise ValueError("Invalid parameter")

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    normalized_replacement_match = normalize_match_parameter(required_match_replacement, len(subexpression.args))
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    def visit(current_expr):
        if current_expr.is_Add:
            expr_max_length = max(len(current_expr.args), len(subexpression.args))
            normalized_current_expr_match = normalize_match_parameter(required_match_original, expr_max_length)
            expr_coefficients = current_expr.as_coefficients_dict()
            subexpression_coefficient_dict = subexpression.as_coefficients_dict()
            intersection = set(subexpression_coefficient_dict.keys()).intersection(set(expr_coefficients))
            if len(intersection) >= max(normalized_replacement_match, normalized_current_expr_match):
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                # find common factor
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                factors = defaultdict(int)
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                skips = 0
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                for common_symbol in subexpression_coefficient_dict.keys():
                    if common_symbol not in expr_coefficients:
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                        skips += 1
                        continue
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                    factor = expr_coefficients[common_symbol] / subexpression_coefficient_dict[common_symbol]
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                    factors[sp.simplify(factor)] += 1

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                common_factor = max(factors.items(), key=operator.itemgetter(1))[0]
                if factors[common_factor] >= max(normalized_current_expr_match, normalized_replacement_match):
                    return current_expr - common_factor * subexpression + common_factor * replacement
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        # if no subexpression was found
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        param_list = [visit(a) for a in current_expr.args]
        if not param_list:
            return current_expr
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        else:
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            if current_expr.func == sp.Mul and sp.numbers.Zero() in param_list:
                return sp.numbers.Zero()
            else:
                return current_expr.func(*param_list, evaluate=False)
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    return visit(expr)


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def replace_second_order_products(expr: sp.Expr, search_symbols: Iterable[sp.Symbol],
                                  positive: Optional[bool] = None,
                                  replace_mixed: Optional[List[Assignment]] = None) -> sp.Expr:
    """Replaces second order mixed terms like x*y by 2*( (x+y)**2 - x**2 - y**2 ).

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    This makes the term longer - simplify usually is undoing these - however this
    transformation can be done to find more common sub-expressions
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    Args:
        expr: input expression
        search_symbols: symbols that are searched for
                         for example, given [x,y,z] terms like x*y, x*z, z*y are replaced
        positive: there are two ways to do this substitution, either with term
                 (x+y)**2 or (x-y)**2 . if positive=True the first version is done,
                 if positive=False the second version is done, if positive=None the
                 sign is determined by the sign of the mixed term that is replaced
        replace_mixed: if a list is passed here, the expr x+y or x-y is replaced by a special new symbol
                       and the replacement equation is added to the list
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    """
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    mixed_symbols_replaced = set([e.lhs for e in replace_mixed]) if replace_mixed is not None else set()
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    if expr.is_Mul:
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        distinct_search_symbols = set()
        nr_of_search_terms = 0
        other_factors = 1
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        for t in expr.args:
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            if t in search_symbols:
                nr_of_search_terms += 1
                distinct_search_symbols.add(t)
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            else:
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                other_factors *= t
        if len(distinct_search_symbols) == 2 and nr_of_search_terms == 2:
            u, v = sorted(list(distinct_search_symbols), key=lambda symbol: symbol.name)
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            if positive is None:
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                other_factors_without_symbols = other_factors
                for s in other_factors.atoms(sp.Symbol):
                    other_factors_without_symbols = other_factors_without_symbols.subs(s, 1)
                positive = other_factors_without_symbols.is_positive
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                assert positive is not None
            sign = 1 if positive else -1
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            if replace_mixed is not None:
                new_symbol_str = 'P' if positive else 'M'
                mixed_symbol_name = u.name + new_symbol_str + v.name
                mixed_symbol = sp.Symbol(mixed_symbol_name.replace("_", ""))
                if mixed_symbol not in mixed_symbols_replaced:
                    mixed_symbols_replaced.add(mixed_symbol)
                    replace_mixed.append(Assignment(mixed_symbol, u + sign * v))
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            else:
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                mixed_symbol = u + sign * v
            return sp.Rational(1, 2) * sign * other_factors * (mixed_symbol ** 2 - u ** 2 - v ** 2)
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    param_list = [replace_second_order_products(a, search_symbols, positive, replace_mixed) for a in expr.args]
    result = expr.func(*param_list, evaluate=False) if param_list else expr
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    return result


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def remove_higher_order_terms(expr: sp.Expr, symbols: Sequence[sp.Symbol], order: int = 3) -> sp.Expr:
    """Removes all terms that contain more than 'order' factors of given 'symbols'
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    Example:
        >>> x, y = sp.symbols("x y")
        >>> term = x**2 * y + y**2 * x + y**3 + x + y ** 2
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        >>> remove_higher_order_terms(term, order=2, symbols=[x, y])
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        x + y**2
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    """
    from sympy.core.power import Pow
    from sympy.core.add import Add, Mul

    result = 0
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    expr = expr.expand()
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    def velocity_factors_in_product(product):
        factor_count = 0
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        if type(product) is Mul:
            for factor in product.args:
                if type(factor) == Pow:
                    if factor.args[0] in symbols:
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                        factor_count += factor.args[1]
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                if factor in symbols:
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                    factor_count += 1
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        elif type(product) is Pow:
            if product.args[0] in symbols:
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                factor_count += product.args[1]
        return factor_count
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    if type(expr) == Mul or type(expr) == Pow:
        if velocity_factors_in_product(expr) <= order:
            return expr
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        else:
            return sp.Rational(0, 1)

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    if type(expr) != Add:
        return expr
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    for sum_term in expr.args:
        if velocity_factors_in_product(sum_term) <= order:
            result += sum_term
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    return result


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def complete_the_square(expr: sp.Expr, symbol_to_complete: sp.Symbol,
                        new_variable: sp.Symbol) -> Tuple[sp.Expr, Optional[Tuple[sp.Symbol, sp.Expr]]]:
    """Transforms second order polynomial into only squared part.
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    Examples:
        >>> a, b, c, s, n = sp.symbols("a b c s n")
        >>> expr = a * s**2 + b * s + c
        >>> completed_expr, substitution = complete_the_square(expr, symbol_to_complete=s, new_variable=n)
        >>> completed_expr
        a*n**2 + c - b**2/(4*a)
        >>> substitution
        (n, s + b/(2*a))
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    Returns:
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        (replaced_expr, tuple to pass to subs, such that old expr comes out again)
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    """
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    p = sp.Poly(expr, symbol_to_complete)
    coefficients = p.all_coeffs()
    if len(coefficients) != 3:
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        return expr, None
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    a, b, _ = coefficients
    expr = expr.subs(symbol_to_complete, new_variable - b / (2 * a))
    return sp.simplify(expr), (new_variable, symbol_to_complete + b / (2 * a))
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def complete_the_squares_in_exp(expr: sp.Expr, symbols_to_complete: Sequence[sp.Symbol]):
    """Completes squares in arguments of exponential which makes them simpler to integrate.

    Very useful for integrating Maxwell-Boltzmann equilibria and its moment generating function
    """
    dummies = [sp.Dummy() for _ in symbols_to_complete]
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    def visit(term):
        if term.func == sp.exp:
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            exp_arg = term.args[0]
            for symbol_to_complete, dummy in zip(symbols_to_complete, dummies):
                exp_arg, substitution = complete_the_square(exp_arg, symbol_to_complete, dummy)
            return sp.exp(sp.expand(exp_arg))
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        else:
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            param_list = [visit(a) for a in term.args]
            if not param_list:
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                return term
            else:
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                return term.func(*param_list)
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    result = visit(expr)
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    for s, d in zip(symbols_to_complete, dummies):
        result = result.subs(d, s)
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    return result


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def extract_most_common_factor(term):
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    """Processes a sum of fractions: determines the most common factor and splits term in common factor and rest"""
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    coefficient_dict = term.as_coefficients_dict()
    counter = Counter([Abs(v) for v in coefficient_dict.values()])
    common_factor, occurrences = max(counter.items(), key=operator.itemgetter(1))
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    if occurrences == 1 and (1 in counter):
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        common_factor = 1
    return common_factor, term / common_factor
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def count_operations(term: Union[sp.Expr, List[sp.Expr]],
                     only_type: Optional[str] = 'real') -> Dict[str, int]:
    """Counts the number of additions, multiplications and division.
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    Args:
        term: a sympy expression (term, assignment) or sequence of sympy objects
        only_type: 'real' or 'int' to count only operations on these types, or None for all
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    Returns:
        dict with 'adds', 'muls' and 'divs' keys
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    """
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    from pystencils.fast_approximation import fast_sqrt, fast_inv_sqrt, fast_division

    result = {'adds': 0, 'muls': 0, 'divs': 0, 'sqrts': 0,
              'fast_sqrts': 0, 'fast_inv_sqrts': 0, 'fast_div': 0}
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    if isinstance(term, Sequence):
        for element in term:
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            r = count_operations(element, only_type)
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            for operation_name in result.keys():
                result[operation_name] += r[operation_name]
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        return result
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    elif isinstance(term, Assignment):
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        term = term.rhs

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    if hasattr(term, 'evalf'):
        term = term.evalf()
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    def check_type(e):
        if only_type is None:
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            return True
        try:
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            base_type = get_base_type(get_type_of_expression(e))
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        except ValueError:
            return False
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        if only_type == 'int' and (base_type.is_int() or base_type.is_uint()):
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            return True
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        if only_type == 'real' and (base_type.is_float()):
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            return True
        else:
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            return base_type == only_type
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    def visit(t):
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        visit_children = True
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        if t.func is sp.Add:
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            if check_type(t):
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                result['adds'] += len(t.args) - 1
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        elif t.func in [sp.Or, sp.And]:
            pass
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        elif t.func is sp.Mul:
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            if check_type(t):
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                result['muls'] += len(t.args) - 1
                for a in t.args:
                    if a == 1 or a == -1:
                        result['muls'] -= 1
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        elif isinstance(t, sp.Float) or isinstance(t, sp.Rational):
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            pass
        elif isinstance(t, sp.Symbol):
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            visit_children = False
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        elif isinstance(t, sp.tensor.Indexed):
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            visit_children = False
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        elif t.is_integer:
            pass
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        elif isinstance(t, cast_func):
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            visit_children = False
            visit(t.args[0])
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        elif t.func is fast_sqrt:
            result['fast_sqrts'] += 1
        elif t.func is fast_inv_sqrt:
            result['fast_inv_sqrts'] += 1
        elif t.func is fast_division:
            result['fast_div'] += 1
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        elif t.func is sp.Pow:
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            if check_type(t.args[0]):
                visit_children = False
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                if t.exp.is_integer and t.exp.is_number:
                    if t.exp >= 0:
                        result['muls'] += int(t.exp) - 1
                    else:
                        result['muls'] -= 1
                        result['divs'] += 1
                        result['muls'] += (-int(t.exp)) - 1
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                elif sp.nsimplify(t.exp) == sp.Rational(1, 2):
                    result['sqrts'] += 1
                else:
                    warnings.warn("Cannot handle exponent", t.exp, " of sp.Pow node")
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            else:
                warnings.warn("Counting operations: only integer exponents are supported in Pow, "
                              "counting will be inaccurate")
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        elif t.func is sp.Piecewise:
            for child_term, condition in t.args:
                visit(child_term)
            visit_children = False
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        elif isinstance(t, sp.Rel):
            pass
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        else:
            warnings.warn("Unknown sympy node of type " + str(t.func) + " counting will be inaccurate")

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        if visit_children:
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            for a in t.args:
                visit(a)

    visit(term)
    return result
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def count_operations_in_ast(ast) -> Dict[str, int]:
    """Counts number of operations in an abstract syntax tree, see also :func:`count_operations`"""
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    from pystencils.astnodes import SympyAssignment
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    result = defaultdict(int)
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    def visit(node):
        if isinstance(node, SympyAssignment):
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            r = count_operations(node.rhs)
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            for k, v in r.items():
                result[k] += v
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        else:
            for arg in node.args:
                visit(arg)
    visit(ast)
    return result


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def common_denominator(expr: sp.Expr) -> sp.Expr:
    """Finds least common multiple of all denominators occurring in an expression"""
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    denominators = [r.q for r in expr.atoms(sp.Rational)]
    return sp.lcm(denominators)
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def get_symmetric_part(expr: sp.Expr, symbols: Iterable[sp.Symbol]) -> sp.Expr:
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    """
    Returns the symmetric part of a sympy expressions.

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    Args:
        expr: sympy expression, labeled here as :math:`f`
        symbols: sequence of symbols which are considered as degrees of freedom, labeled here as :math:`x_0, x_1,...`

    Returns:
        :math:`\frac{1}{2} [ f(x_0, x_1, ..) + f(-x_0, -x_1) ]`
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    """
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    substitution_dict = {e: -e for e in symbols}
    return sp.Rational(1, 2) * (expr + expr.subs(substitution_dict))
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def sort_assignments_topologically(assignments: Sequence[Assignment]) -> List[Assignment]:
    """Sorts assignments in topological order, such that symbols used on rhs occur first on a lhs"""
    res = sp.cse_main.reps_toposort([[e.lhs, e.rhs] for e in assignments])
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    return [Assignment(a, b) for a, b in res]
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class SymbolCreator:
    def __getattribute__(self, name):
        return sp.Symbol(name)