InputForm of a low-precision number shows the digits it keeps, as in WMA

mathics/format/box/numberform.py

@@ -168,8 +168,24 @@ def int_to_string_shorter_repr(
     return value_str
 
 
+def _stored_digits(extended: sympy.Float) -> tuple[str, int]:
+    """
+    The shortest decimal string that reads back as ``extended``, and its
+    number of significant digits. As in WMA, InputForm shows these for a
+    number of low precision: 3.1415926535897932385`3.
+    """
+    bits = extended._prec
+    with mpmath.workprec(bits):
+        value = extended._to_mpmath(bits)
+        for n in range(1, dps(bits) + 4):
+            s = mpmath.nstr(value, n)
+            if sympy.Float(s, precision=bits)._mpf_ == extended._mpf_:
+                break
+    return s, n
+
+
 def real_to_tuple_info(
-    real: Real, digits: Optional[int]
+    real: Real, digits: Optional[int], stored: bool = False
 ) -> tuple[str, int, bool, int, int]:
     """
     Convert ``real`` to a tuple representing that value. The tuple consists of:
@@ -180,7 +196,9 @@ def real_to_tuple_info(
     * Updated value of digits, according to the number precision.
     * the decimal precision.
 
-    If ``digits`` is None, we use the default precision.
+    If ``digits`` is None, we use the default precision. If ``stored`` is
+    True, a PrecisionReal that keeps more bits than its precision shows
+    all the digits it keeps.
     """
     binary_precision = real.precision
     if isinstance(real, PrecisionReal):
@@ -202,7 +220,10 @@ def real_to_tuple_info(
         is_nonnegative = True
         return s, exponent, is_nonnegative, digits, precision
 
-    if digits is None:
+    if stored and isinstance(real, PrecisionReal) and real._extended is not None:
+        s, digits = _stored_digits(real._extended)
+        value = real._extended
+    elif digits is None:
         if real.is_machine_precision:
             value = real.value
             s = repr(value)
@@ -462,7 +483,11 @@ def numberform_to_boxes(
         if digits_after_decimal_point is None:
             is_int = True
     elif isinstance(value, Real):
-        s, exp, is_nonnegative, digits, precision = real_to_tuple_info(value, digits)
+        # As in WMA, InputForm and FullForm show all the digits kept.
+        stored = digits is None and form in ("System`InputForm", "System`FullForm")
+        s, exp, is_nonnegative, digits, precision = real_to_tuple_info(
+            value, digits, stored
+        )
     else:
         raise ValueError("Expected Real or Integer.")
 

test/builtin/atomic/test_numbers.py

@@ -1070,6 +1070,52 @@ def test_low_precision_storage(str_expr, str_expected, msg):
     check_evaluation(str_expr, str_expected, failure_message=msg)
 
 
+@pytest.mark.parametrize(
+    ("str_expr", "str_expected", "msg"),
+    [
+        (
+            "ToString[3.1415926535897932385`3, InputForm]",
+            '"3.1415926535897932385`3."',
+            "InputForm shows the digits that are kept, not the precision",
+        ),
+        (
+            "ToString[FullForm[3.1415926535897932385`3]]",
+            '"3.1415926535897932385`3."',
+            None,
+        ),
+        (
+            "ToString[-3.1415926535897932385`3, InputForm]",
+            '"-3.1415926535897932385`3."',
+            None,
+        ),
+        ("ToString[3.1415926535897932385`3]", '"3.14"', "OutputForm: the precision"),
+        (
+            "StringTake[ToString[N[Pi, 20] + 0``3, InputForm], 22]",
+            '"3.1415926535897932385`"',
+            None,
+        ),
+        (
+            "Precision[1.5`20 - ToExpression[ToString[1.4`2, InputForm]]]",
+            "0.853872",
+            "reading InputForm back gives the same kept value",
+        ),
+        (
+            "ToExpression[ToString[#, InputForm]] === # &[N[1/3, 3]]",
+            "True",
+            None,
+        ),
+        (
+            "Precision[1.5`20 - ToExpression[ToString[N[1/3, 3], InputForm]]]",
+            "Precision[1.5`20 - N[1/3, 3]]",
+            None,
+        ),
+    ],
+)
+def test_low_precision_input_form(str_expr, str_expected, msg):
+    """Compared with WMA 15"""
+    check_evaluation(str_expr, str_expected, failure_message=msg)
+
+
 @pytest.mark.parametrize(
     ("str_expr", "str_expected", "msg"),
     [

test/format/format_tests.yaml

@@ -171,17 +171,17 @@
 -4.326563712`2:
   msg: A negative Real number
   latex:
-    System`InputForm: '-4.33'
+    System`InputForm: '-4.326563712'
     System`OutputForm: '\text{-4.3}'
     System`StandardForm: '-4.33'
     System`TraditionalForm: '-4.33'
   mathml:
-    System`InputForm: <mn>-4.33</mn>
+    System`InputForm: <mn>-4.326563712</mn>
     System`OutputForm: <mtext>-4.3</mtext>
     System`StandardForm: "<mrow>\n <mn>-</mn>\n <mn>4.33</mn>\n</mrow>"
     System`TraditionalForm: "<mrow>\n <mn>-</mn>\n <mn>4.33</mn>\n</mrow>"
   text:
-    System`InputForm: -4.33`2.
+    System`InputForm: -4.326563712`2.
     System`OutputForm: '-4.3'
     System`StandardForm: -4.33`2.
     System`TraditionalForm: -4.33`2.