Expression constants (#831)

This comes from #828. Here a new module is added to contain the
expression constants of the form `DirectedInfinity[...]`.

mathics/builtin/arithmetic.py

@@ -23,6 +23,8 @@
 from mathics.builtin.inference import evaluate_predicate, get_assumptions_list
 from mathics.builtin.scoping import dynamic_scoping
 from mathics.core.atoms import (
+    MATHICS3_COMPLEX_I,
+    MATHICS3_COMPLEX_I_NEG,
     Complex,
     Integer,
     Integer0,
@@ -43,9 +45,17 @@
 )
 from mathics.core.convert.expression import to_expression
 from mathics.core.convert.sympy import SympyExpression, from_sympy, sympy_symbol_prefix
-from mathics.core.element import BaseElement, ElementsProperties
+from mathics.core.element import BaseElement
 from mathics.core.evaluation import Evaluation
 from mathics.core.expression import Expression
+from mathics.core.expression_predefined import (
+    MATHICS3_COMPLEX_INFINITY,
+    MATHICS3_I_INFINITY,
+    MATHICS3_I_NEG_INFINITY,
+    MATHICS3_INFINITY,
+    MATHICS3_NEG_INFINITY,
+    PredefinedExpression,
+)
 from mathics.core.list import ListExpression
 from mathics.core.number import dps, min_prec
 from mathics.core.symbols import (
@@ -73,7 +83,13 @@
 # This tells documentation how to sort this module
 sort_order = "mathics.builtin.mathematical-functions"
 
-ExpressionComplexInfinity = Expression(SymbolDirectedInfinity)
+
+map_direction_infinity = {
+    Integer1: MATHICS3_INFINITY,
+    IntegerM1: MATHICS3_NEG_INFINITY,
+    MATHICS3_COMPLEX_I: MATHICS3_I_INFINITY,
+    MATHICS3_COMPLEX_I_NEG: MATHICS3_I_NEG_INFINITY,
+}
 
 
 class _MPMathFunction(SympyFunction):
@@ -647,12 +663,18 @@ class DirectedInfinity(SympyFunction):
         "DirectedInfinity[z_?NumericQ]": "HoldForm[z Infinity]",
     }
 
+    def eval_complex_infinity(self, evaluation: Evaluation):
+        """DirectedInfinity[]"""
+        return MATHICS3_COMPLEX_INFINITY
+
     def eval_directed_infinity(self, direction, evaluation: Evaluation):
         """DirectedInfinity[direction_]"""
-        if direction in (Integer1, IntegerM1):
-            return None
+        result = map_direction_infinity.get(direction, None)
+        if result:
+            return result
+
         if direction.is_zero:
-            return ExpressionComplexInfinity
+            return MATHICS3_COMPLEX_INFINITY
 
         normalized_direction = eval_Sign(direction)
         # TODO: improve eval_Sign, to avoid the need of the
@@ -677,10 +699,9 @@ def eval_directed_infinity(self, direction, evaluation: Evaluation):
 
         if normalized_direction is None:
             return None
-        return Expression(
+        return PredefinedExpression(
             SymbolDirectedInfinity,
             normalized_direction.evaluate(evaluation),
-            elements_properties=ElementsProperties(True, False, False),
         )
 
     def to_sympy(self, expr, **kwargs):

mathics/builtin/atomic/strings.py

@@ -19,12 +19,12 @@
 from mathics.core.convert.python import from_bool
 from mathics.core.evaluation import Evaluation
 from mathics.core.expression import Expression
+from mathics.core.expression_predefined import MATHICS3_INFINITY
 from mathics.core.list import ListExpression
 from mathics.core.parser import MathicsFileLineFeeder, parse
 from mathics.core.symbols import Symbol, SymbolTrue
 from mathics.core.systemsymbols import (
     SymbolBlank,
-    SymbolDirectedInfinity,
     SymbolFailed,
     SymbolInputForm,
     SymbolOutputForm,
@@ -800,7 +800,7 @@ def convert_rule(r):
         # convert n
         if n is None:
             py_n = 0
-        elif n == Expression(SymbolDirectedInfinity, Integer1):
+        elif n.sameQ(MATHICS3_INFINITY):
             py_n = 0
         else:
             py_n = n.get_int_value()

mathics/builtin/binary/io.py

@@ -15,14 +15,16 @@
 from mathics.core.convert.expression import to_expression, to_mathics_list
 from mathics.core.convert.mpmath import from_mpmath
 from mathics.core.expression import Expression
+from mathics.core.expression_predefined import (
+    MATHICS3_I_INFINITY,
+    MATHICS3_I_NEG_INFINITY,
+    MATHICS3_INFINITY,
+    MATHICS3_NEG_INFINITY,
+)
 from mathics.core.read import SymbolEndOfFile
 from mathics.core.streams import stream_manager
 from mathics.core.symbols import Symbol
-from mathics.core.systemsymbols import (
-    SymbolComplex,
-    SymbolDirectedInfinity,
-    SymbolIndeterminate,
-)
+from mathics.core.systemsymbols import SymbolIndeterminate
 from mathics.eval.nevaluator import eval_N
 
 SymbolBinaryWrite = Symbol("BinaryWrite")
@@ -38,27 +40,21 @@ def _IEEE_real(real):
         if math.isnan(real):
             return SymbolIndeterminate
         elif math.isinf(real):
-            return Expression(SymbolDirectedInfinity, Integer((-1) ** (real < 0)))
+            return MATHICS3_NEG_INFINITY if real < 0 else MATHICS3_INFINITY
         else:
             return Real(real)
 
     @staticmethod
     def _IEEE_cmplx(real, imag):
         if math.isnan(real) or math.isnan(imag):
             return SymbolIndeterminate
-        elif math.isinf(real) or math.isinf(imag):
-            if math.isinf(real) and math.isinf(imag):
+        if math.isinf(real):
+            if math.isinf(imag):
                 return SymbolIndeterminate
-            return Expression(
-                SymbolDirectedInfinity,
-                to_expression(
-                    SymbolComplex,
-                    (-1) ** (real < 0) if math.isinf(real) else 0,
-                    (-1) ** (imag < 0) if math.isinf(imag) else 0,
-                ),
-            )
-        else:
-            return Complex(MachineReal(real), MachineReal(imag))
+            return MATHICS3_NEG_INFINITY if real < 0 else MATHICS3_INFINITY
+        if math.isinf(imag):
+            return MATHICS3_I_NEG_INFINITY if imag < 0 else MATHICS3_I_INFINITY
+        return Complex(MachineReal(real), MachineReal(imag))
 
     @classmethod
     def get_readers(cls):
@@ -172,7 +168,7 @@ def _Real128_reader(s):
             return Real(sympy.Float(0, 4965))
         elif expbits == 0x7FFF:
             if fracbits == 0:
-                return Expression(SymbolDirectedInfinity, Integer((-1) ** signbit))
+                return MATHICS3_NEG_INFINITY if signbit else MATHICS3_INFINITY
             else:
                 return SymbolIndeterminate
 

mathics/builtin/functional/functional_iteration.py

@@ -10,8 +10,8 @@
 from mathics.core.convert.python import from_python
 from mathics.core.evaluation import Evaluation
 from mathics.core.expression import Expression
+from mathics.core.expression_predefined import MATHICS3_INFINITY
 from mathics.core.symbols import Symbol, SymbolTrue
-from mathics.core.systemsymbols import SymbolDirectedInfinity
 
 # This tells documentation how to sort this module
 sort_order = "mathics.builtin.iteratively-applying-functions"
@@ -51,7 +51,7 @@ class FixedPoint(Builtin):
 
     def eval(self, f, expr, n, evaluation: Evaluation, options: dict):
         "FixedPoint[f_, expr_, n_:DirectedInfinity[1], OptionsPattern[FixedPoint]]"
-        if n == Expression(SymbolDirectedInfinity, Integer1):
+        if n.sameQ(MATHICS3_INFINITY):
             count = None
         else:
             count = n.get_int_value()
@@ -131,7 +131,7 @@ class FixedPointList(Builtin):
     def eval(self, f, expr, n, evaluation: Evaluation):
         "FixedPointList[f_, expr_, n_:DirectedInfinity[1]]"
 
-        if n == Expression(SymbolDirectedInfinity, Integer1):
+        if n.sameQ(MATHICS3_INFINITY):
             count = None
         else:
             count = n.get_int_value()

mathics/builtin/list/rearrange.py

@@ -11,18 +11,14 @@
 from typing import Callable
 
 from mathics.builtin.base import Builtin, MessageException
-from mathics.core.atoms import Integer, Integer0, Integer1
+from mathics.core.atoms import Integer, Integer0
 from mathics.core.attributes import A_FLAT, A_ONE_IDENTITY, A_PROTECTED
 from mathics.core.evaluation import Evaluation
 from mathics.core.expression import Expression, structure
+from mathics.core.expression_predefined import MATHICS3_INFINITY
 from mathics.core.list import ListExpression
 from mathics.core.symbols import Atom, Symbol, SymbolTrue
-from mathics.core.systemsymbols import (
-    SymbolDirectedInfinity,
-    SymbolMap,
-    SymbolReverse,
-    SymbolSplit,
-)
+from mathics.core.systemsymbols import SymbolMap, SymbolReverse, SymbolSplit
 from mathics.eval.parts import walk_levels
 
 
@@ -813,7 +809,7 @@ def insert_element(elements):
     def eval(self, expr, n, h, evaluation):
         "Flatten[expr_, n_, h_]"
 
-        if n == Expression(SymbolDirectedInfinity, Integer1):
+        if n.sameQ(MATHICS3_INFINITY):
             n = -1  # a negative number indicates an unbounded level
         else:
             n_int = n.get_int_value()

mathics/builtin/numbers/algebra.py

@@ -29,6 +29,10 @@
 from mathics.core.element import BaseElement
 from mathics.core.evaluation import Evaluation
 from mathics.core.expression import Expression
+from mathics.core.expression_predefined import (
+    MATHICS3_COMPLEX_INFINITY,
+    MATHICS3_NEG_INFINITY,
+)
 from mathics.core.list import ListExpression
 from mathics.core.rules import Pattern
 from mathics.core.symbols import (
@@ -46,12 +50,10 @@
     SymbolAlternatives,
     SymbolAssumptions,
     SymbolAutomatic,
-    SymbolComplexInfinity,
     SymbolCos,
     SymbolCosh,
     SymbolCot,
     SymbolCoth,
-    SymbolDirectedInfinity,
     SymbolEqual,
     SymbolIndeterminate,
     SymbolLess,
@@ -1377,7 +1379,7 @@ def eval_novar(self, expr, evaluation):
     def eval(self, expr, form, h, evaluation):
         "Exponent[expr_, form_, h_]"
         if expr == Integer0:
-            return Expression(SymbolDirectedInfinity, Integer(-1))
+            return MATHICS3_NEG_INFINITY
 
         if not form.has_form("List", None):
             # TODO: add ElementProperties in Expression interface refactor branch:
@@ -1647,7 +1649,7 @@ def eval_power_of_zero(self, b, evaluation):
         if self.eval(Expression(SymbolLess, Integer0, b), evaluation) is SymbolTrue:
             return Integer0
         if self.eval(Expression(SymbolLess, b, Integer0), evaluation) is SymbolTrue:
-            return Symbol(SymbolComplexInfinity)
+            return MATHICS3_COMPLEX_INFINITY
         if self.eval(Expression(SymbolEqual, b, Integer0), evaluation) is SymbolTrue:
             return Symbol(SymbolIndeterminate)
         return Expression(SymbolPower, Integer0, b)

mathics/builtin/string/operations.py

@@ -25,11 +25,11 @@
 from mathics.core.convert.python import from_python
 from mathics.core.evaluation import Evaluation
 from mathics.core.expression import BoxError, Expression, string_list
+from mathics.core.expression_predefined import MATHICS3_INFINITY
 from mathics.core.list import ListExpression
 from mathics.core.symbols import SymbolFalse, SymbolFullForm, SymbolList, SymbolTrue
 from mathics.core.systemsymbols import (
     SymbolAll,
-    SymbolDirectedInfinity,
     SymbolOutputForm,
     SymbolStringInsert,
     SymbolStringJoin,
@@ -521,7 +521,7 @@ def eval(self, string, patt, evaluation: Evaluation, options: dict):
         return self.eval_n(
             string,
             patt,
-            Expression(SymbolDirectedInfinity, Integer1),
+            MATHICS3_INFINITY,
             evaluation,
             options,
         )

mathics/builtin/testing_expressions/equality_inequality.py

@@ -19,11 +19,15 @@
 )
 from mathics.core.convert.expression import to_expression, to_numeric_args
 from mathics.core.expression import Expression
+from mathics.core.expression_predefined import (
+    MATHICS3_COMPLEX_INFINITY,
+    MATHICS3_INFINITY,
+    MATHICS3_NEG_INFINITY,
+)
 from mathics.core.number import dps
 from mathics.core.symbols import Atom, Symbol, SymbolFalse, SymbolList, SymbolTrue
 from mathics.core.systemsymbols import (
     SymbolAnd,
-    SymbolComplexInfinity,
     SymbolDirectedInfinity,
     SymbolExactNumberQ,
     SymbolInequality,
@@ -110,6 +114,8 @@ def get_pairs(args):
                 yield (args[i], args[j])
 
     def expr_equal(self, lhs, rhs, max_extra_prec=None) -> Optional[bool]:
+        if rhs is lhs:
+            return True
         if isinstance(rhs, Expression):
             lhs, rhs = rhs, lhs
         if not isinstance(lhs, Expression):
@@ -129,34 +135,23 @@ def expr_equal(self, lhs, rhs, max_extra_prec=None) -> Optional[bool]:
         return True
 
     def infty_equal(self, lhs, rhs, max_extra_prec=None) -> Optional[bool]:
-        if rhs.get_head().sameQ(SymbolDirectedInfinity):
-            lhs, rhs = rhs, lhs
-        if not lhs.get_head().sameQ(SymbolDirectedInfinity):
+        if (
+            lhs.get_head() is not SymbolDirectedInfinity
+            or rhs.get_head() is not SymbolDirectedInfinity
+        ):
             return None
-        if rhs.sameQ(SymbolInfinity) or rhs.sameQ(SymbolComplexInfinity):
-            if len(lhs.elements) == 0:
-                return True
-            else:
-                return self.equal2(
-                    to_expression(SymbolSign, lhs.elements[0]), Integer1, max_extra_prec
-                )
-        if rhs.is_numeric():
-            return False
-        elif isinstance(rhs, Atom):
+        lhs_elements, rhs_elements = lhs.elements, rhs.elements
+
+        if len(lhs_elements) != len(rhs_elements):
             return None
-        if rhs.get_head().sameQ(lhs.get_head()):
-            dir1 = dir2 = Integer1
-            if len(lhs.elements) == 1:
-                dir1 = lhs.elements[0]
-            if len(rhs.elements) == 1:
-                dir2 = rhs.elements[0]
-            if self.equal2(dir1, dir2, max_extra_prec):
-                return True
-            # Now, compare the signs:
-            dir1_sign = Expression(SymbolSign, dir1)
-            dir2_sign = Expression(SymbolSign, dir2)
-            return self.equal2(dir1_sign, dir2_sign, max_extra_prec)
-        return
+        # Both are complex infinity?
+        if len(lhs_elements) == 0:
+            return True
+        if len(lhs_elements) == 1:
+            # Check directions: Notice that they are already normalized...
+            return self.equal2(lhs_elements[0], rhs_elements[0], max_extra_prec)
+        # DirectedInfinity with more than two elements cannot be compared here...
+        return None
 
     def sympy_equal(self, lhs, rhs, max_extra_prec=None) -> Optional[bool]:
         try:
@@ -200,12 +195,12 @@ def equal2(self, lhs: Any, rhs: Any, max_extra_prec=None) -> Optional[bool]:
         """
         Two-argument Equal[]
         """
+        if lhs is rhs or lhs.sameQ(rhs):
+            return True
         if hasattr(lhs, "equal2"):
             result = lhs.equal2(rhs)
             if result is not None:
                 return result
-        elif lhs.sameQ(rhs):
-            return True
         # TODO: Check $Assumptions
         # Still we didn't have a result. Try with the following
         # tests
@@ -294,7 +289,7 @@ def eval(self, items, evaluation):
                     results.append(element)
 
         if not results:
-            return Expression(SymbolDirectedInfinity, Integer(-self.sense))
+            return MATHICS3_INFINITY if self.sense < 0 else MATHICS3_NEG_INFINITY
         if len(results) == 1:
             return results.pop()
         if len(results) < len(items):

mathics/core/atoms.py

@@ -942,6 +942,9 @@ def is_zero(self) -> bool:
 
 
 RationalOneHalf = Rational(1, 2)
+RationalMinusOneHalf = Rational(-1, 2)
+MATHICS3_COMPLEX_I = Complex(Integer0, Integer1)
+MATHICS3_COMPLEX_I_NEG = Complex(Integer0, IntegerM1)
 
 
 class String(Atom, BoxElementMixin):