Improve infinity and Precision (#767)

This would be the simplest part of #766, including tests and changes in
`Infinity` and `Precision`. Some of the new tests are commented / marked
as xfail because require the other changes.

CHANGES.rst

@@ -91,7 +91,10 @@ Bugs
 #. Some scikit image routines line ``EdgeDetect`` were getting omitted due to overly stringent PYPI requirements
 
 #. Units and Quantities were sometimes failing. Also they were omitted from documentation.
+#. Better handling of ``Infinite`` quantities.
+#. Fix ``Precision`` compatibility with WMA.
 
+
 PyPI Package requirements
 +++++++++++++++++++++++++
 

mathics/builtin/arithfns/basic.py

@@ -979,7 +979,7 @@ def eval(self, items, evaluation):
                 )
             elif item.get_head().sameQ(SymbolDirectedInfinity):
                 infinity_factor = True
-                if len(item.elements) > 1:
+                if len(item.elements) > 0:
                     direction = item.elements[0]
                     if isinstance(direction, Number):
                         numbers.append(direction)

mathics/builtin/arithmetic.py

@@ -49,6 +49,7 @@
 from mathics.core.convert.expression import to_expression
 from mathics.core.convert.mpmath import from_mpmath
 from mathics.core.convert.sympy import SympyExpression, from_sympy, sympy_symbol_prefix
+from mathics.core.element import ElementsProperties
 from mathics.core.expression import Expression
 from mathics.core.list import ListExpression
 from mathics.core.number import SpecialValueError, dps, min_prec
@@ -658,9 +659,9 @@ class DirectedInfinity(SympyFunction):
         "DirectedInfinity[Indeterminate]": "Indeterminate",
         "DirectedInfinity[args___] ^ -1": "0",
         "0 * DirectedInfinity[args___]": "Message[Infinity::indet, Unevaluated[0 DirectedInfinity[args]]]; Indeterminate",
-        "DirectedInfinity[a_?NumericQ] /; N[Abs[a]] != 1": "DirectedInfinity[a / Abs[a]]",
-        "DirectedInfinity[a_] * DirectedInfinity[b_]": "DirectedInfinity[a*b]",
-        "DirectedInfinity[] * DirectedInfinity[args___]": "DirectedInfinity[]",
+        # "DirectedInfinity[a_?NumericQ] /; N[Abs[a]] != 1": "DirectedInfinity[a / Abs[a]]",
+        # "DirectedInfinity[a_] * DirectedInfinity[b_]": "DirectedInfinity[a*b]",
+        # "DirectedInfinity[] * DirectedInfinity[args___]": "DirectedInfinity[]",
         # Rules already implemented in Times.eval
         #        "z_?NumberQ * DirectedInfinity[]": "DirectedInfinity[]",
         #        "z_?NumberQ * DirectedInfinity[a_]": "DirectedInfinity[z * a]",
@@ -685,9 +686,7 @@ class DirectedInfinity(SympyFunction):
             "  Unevaluated[DirectedInfinity[0.]]];"
             "Indeterminate"
         ),
-        "DirectedInfinity[ComplexInfinity]": "ComplexInfinity",
-        "DirectedInfinity[Infinity]": "Infinity",
-        "DirectedInfinity[-Infinity]": "-Infinity",
+        "DirectedInfinity[DirectedInfinity[x___]]": "DirectedInfinity[x]",
     }
 
     formats = {
@@ -698,6 +697,19 @@ class DirectedInfinity(SympyFunction):
         "DirectedInfinity[z_?NumericQ]": "HoldForm[z Infinity]",
     }
 
+    def eval(self, z, evaluation):
+        """DirectedInfinity[z_]"""
+        if z in (Integer1, IntegerM1):
+            return None
+        if isinstance(z, Number) or isinstance(eval_N(z, evaluation), Number):
+            direction = (z / Expression(SymbolAbs, z)).evaluate(evaluation)
+            return Expression(
+                SymbolDirectedInfinity,
+                direction,
+                elements_properties=ElementsProperties(True, True, True),
+            )
+        return None
+
     def to_sympy(self, expr, **kwargs):
         if len(expr.elements) == 1:
             dir = expr.elements[0].get_int_value()

mathics/builtin/numbers/hyperbolic.py

@@ -58,9 +58,9 @@ class ArcCosh(_MPMathFunction):
 
     rules = {
         "ArcCosh[Undefined]": "Undefined",
-        "ArcCosh[I Infinity]": "Infinity",
-        "ArcCosh[-I Infinity]": "Infinity",
-        "ArcCosh[ComplexInfinity]": "Infinity",
+        "ArcCosh[DirectedInfinity[I]]": "Infinity",
+        "ArcCosh[DirectedInfinity[-I]]": "Infinity",
+        "ArcCosh[DirectedInfinity[]]": "Infinity",
         "Derivative[1][ArcCosh]": "1/(Sqrt[#-1]*Sqrt[#+1])&",
     }
     summary_text = "inverse hyperbolic cosine function"
@@ -324,15 +324,8 @@ class Gudermannian(Builtin):
         "Gudermannian[Undefined]": "Undefined",
         "Gudermannian[0]": "0",
         "Gudermannian[2*Pi*I]": "0",
-        # FIXME: handling DirectedInfinity[-I] leads to problems
-        # "Gudermannian[6/4 Pi I]": "DirectedInfinity[-I]",
-        # Handled already
-        # "Gudermannian[Infinity]": "Pi/2",
-        # FIXME: Pi/2 from Sympy seems to take precedence
-        "Gudermannian[-Infinity]": "-Pi/2",
-        # Below, we don't use instead of ComplexInfinity that gets
-        # substituted out for DirectedInfinity[] before we match on
-        # Gudermannian[...]
+        "Gudermannian[3 I / 2 Pi]": "DirectedInfinity[-I]",
+        "Gudermannian[DirectedInfinity[-1]]": "-Pi/2",
         "Gudermannian[DirectedInfinity[]]": "Indeterminate",
         "Gudermannian[z_]": "2 ArcTan[Tanh[z / 2]]",
         # Commented out because := might not work properly

mathics/core/convert/mpmath.py

@@ -11,7 +11,12 @@
 @lru_cache(maxsize=1024)
 def from_mpmath(value, prec=None, acc=None):
     "Converts mpf or mpc to Number."
+    if mpmath.isnan(value):
+        return SymbolIndeterminate
     if isinstance(value, mpmath.mpf):
+        if mpmath.isinf(value):
+            direction = Integer1 if value > 0 else IntegerM1
+            return Expression(SymbolDirectedInfinity, direction)
         # if accuracy is given, override
         # prec:
         if acc is not None:
@@ -28,6 +33,8 @@ def from_mpmath(value, prec=None, acc=None):
         # HACK: use str here to prevent loss of precision
         return PrecisionReal(sympy.Float(str(value), prec))
     elif isinstance(value, mpmath.mpc):
+        if mpmath.isinf(value):
+            return SymbolComplexInfinity
         if value.imag == 0.0:
             return from_mpmath(value.real, prec, acc)
         real = from_mpmath(value.real, prec, acc)

mathics/core/convert/sympy.py

@@ -252,10 +252,14 @@ def from_sympy(expr):
         elif expr is sympy.false:
             return SymbolFalse
 
-    elif expr.is_number and all([x.is_Number for x in expr.as_real_imag()]):
+    if expr.is_number and all([x.is_Number for x in expr.as_real_imag()]):
         # Hack to convert <Integer> * I to Complex[0, <Integer>]
-        return Complex(*[from_sympy(arg) for arg in expr.as_real_imag()])
-    elif expr.is_Add:
+        try:
+            return Complex(*[from_sympy(arg) for arg in expr.as_real_imag()])
+        except ValueError:
+            # The exception happens if one of the components is infinity
+            pass
+    if expr.is_Add:
         return to_expression(
             SymbolPlus, *sorted([from_sympy(arg) for arg in expr.args])
         )

mathics/core/expression.py

@@ -862,7 +862,13 @@ def get_sort_key(self, pattern_sort=False) -> tuple:
                         exps[name] = exps.get(name, 0) + 1
             elif self.has_form("Power", 2):
                 var = self._elements[0].get_name()
-                exp = self._elements[1].round_to_float()
+                # TODO: Check if this is the expected behaviour.
+                # round_to_float is an attribute of Expression,
+                # but not for Atoms.
+                try:
+                    exp = self._elements[1].round_to_float()
+                except AttributeError:
+                    exp = None
                 if var and exp is not None:
                     exps[var] = exps.get(var, 0) + exp
             if exps:

test/builtin/arithmetic/test_abs.py

@@ -4,10 +4,45 @@
 """
 from test.helper import check_evaluation
 
+import pytest
 
-def test_abs():
-    for str_expr, str_expected in [
-        ("Abs[a - b]", "Abs[a - b]"),
-        ("Abs[Sqrt[3]]", "Sqrt[3]"),
-    ]:
-        check_evaluation(str_expr, str_expected)
+
+@pytest.mark.parametrize(
+    ("str_expr", "str_expected", "msg"),
+    [
+        ("Abs[a - b]", "Abs[a - b]", None),
+        ("Abs[Sqrt[3]]", "Sqrt[3]", None),
+        ("Abs[Sqrt[3]/5]", "Sqrt[3]/5", None),
+        ("Abs[-2/3]", "2/3", None),
+        ("Abs[2+3 I]", "Sqrt[13]", None),
+        ("Abs[2.+3 I]", "3.60555", None),
+        # TODO: Implement rules for these cases.
+        # ("Abs[4^(2 Pi)]", "4^(2 Pi)", None),
+    ],
+)
+def test_abs(str_expr, str_expected, msg):
+    check_evaluation(str_expr, str_expected, failure_message=msg)
+
+
+@pytest.mark.parametrize(
+    ("str_expr", "str_expected", "msg"),
+    [
+        ("Sign[a - b]", "Sign[a - b]", None),
+        ("Sign[Sqrt[3]]", "1", None),
+        ("Sign[0]", "0", None),
+        ("Sign[0.]", "0", None),
+        ("Sign[(1 + I)]", "(1/2 +  I/2)Sqrt[2]", None),
+        ("Sign[(1. + I)]", "(0.707107 + 0.707107 I)", None),
+        ("Sign[(1 + I)/Sqrt[2]]", "(1 + I)/Sqrt[2]", None),
+        ("Sign[(1 + I)/Sqrt[2.]]", "(0.707107 + 0.707107 I)", None),
+        ("Sign[-2/3]", "-1", None),
+        ("Sign[2+3 I]", "(2 + 3 I)/(13^(1/2))", None),
+        ("Sign[2.+3 I]", "0.5547 + 0.83205 I", None),
+        ("Sign[4^(2 Pi)]", "1", None),
+        # FixME: add rules to handle this kind of case
+        # ("Sign[I^(2 Pi)]", "I^(2 Pi)", None),
+        # ("Sign[4^(2 Pi I)]", "1", None),
+    ],
+)
+def test_sign(str_expr, str_expected, msg):
+    check_evaluation(str_expr, str_expected, failure_message=msg)