More compatibility improvements in Accuracy and Precision. Adding tes…

…ts. (#769)

This PR is another fix coming from issues found when I was working on
#766. In particular, the way in which Mathics handles Precision and
Accuracy for real numbers with 0. nominal value. Also, a bug that
prevents parsing Real numbers with fixed accuracy was fixed.

CHANGES.rst

@@ -38,6 +38,8 @@ New Builtins
 #. ``Kurtosis``
 #. ``ListLogPlot``
 #. ``LogPlot``
+#. ``$MaxMachineNumber``
+#. ``$MinMachineNumber``
 #. ``NumberLinePlot``
 #. ``PauliMatrix``
 #. ``Remove``
@@ -92,9 +94,10 @@ Bugs
 
 #. Units and Quantities were sometimes failing. Also they were omitted from documentation.
 #. Better handling of ``Infinite`` quantities.
-#. Fix ``Precision`` compatibility with WMA.
+#. Improved ``Precision`` and ``Accuracy``compatibility with WMA. In particular, ``Precision[0.]`` and ``Accuracy[0.]``
+#. Accuracy in numbers using the notation ``` n.nnn``acc ```  now is properly handled.
+
 
-
 PyPI Package requirements
 +++++++++++++++++++++++++
 
@@ -115,6 +118,8 @@ Enhancements
 #. ``Grid`` compatibility with WMA was improved.  Now it supports non-uniform list of lists and lists with general elements.
 #. Support for BigEndian Big TIFF
 
+
+
 5.0.2
 -----
 

SYMBOLS_MANIFEST.txt

@@ -40,7 +40,9 @@ System`$Machine
 System`$MachineEpsilon
 System`$MachineName
 System`$MachinePrecision
+System`$MaxMachineNumber
 System`$MaxPrecision
+System`$MinMachineNumber
 System`$MinPrecision
 System`$ModuleNumber
 System`$OperatingSystem
@@ -245,8 +247,8 @@ System`Continue
 System`ContinuedFraction
 System`Convert`B64Dump`B64Decode
 System`Convert`B64Dump`B64Encode
-System`ConvertersDump`$extensionMappings
-System`ConvertersDump`$formatMappings
+System`ConvertersDump`$ExtensionMappings
+System`ConvertersDump`$FormatMappings
 System`CoprimeQ
 System`CopyDirectory
 System`CopyFile
@@ -613,7 +615,6 @@ System`MachinePrecision
 System`Magenta
 System`MakeBoxes
 System`ManhattanDistance
-System`Manipulate
 System`MantissaExponent
 System`Map
 System`MapAt
@@ -687,10 +688,10 @@ System`Null
 System`NullSpace
 System`Number
 System`NumberForm
+System`NumberLinePlot
 System`NumberQ
 System`NumberString
 System`Numerator
-System`NumberLinePlot
 System`NumericFunction
 System`NumericQ
 System`O
@@ -739,7 +740,6 @@ System`Pi
 System`Pick
 System`PieChart
 System`Piecewise
-System`PillowImageFilter
 System`Pink
 System`PixelValue
 System`PixelValuePositions
@@ -781,7 +781,6 @@ System`Print
 System`PrintTrace
 System`Private`$ContextPathStack
 System`Private`$ContextStack
-System`Private`ManipulateParameter
 System`Product
 System`ProductLog
 System`Projection
@@ -830,6 +829,7 @@ System`Reap
 System`Record
 System`Rectangle
 System`RectangleBox
+System`Rectangular
 System`Red
 System`RegularExpression
 System`RegularPolygon

mathics/builtin/atomic/numbers.py

@@ -24,32 +24,35 @@
 
 from mathics.builtin.base import Builtin, Predefined, Test
 from mathics.core.atoms import (
-    Complex,
     Integer,
     Integer0,
     Integer10,
     MachineReal,
-    MachineReal0,
     Number,
     Rational,
-    Real,
 )
 from mathics.core.attributes import A_LISTABLE, A_PROTECTED
 from mathics.core.convert.python import from_python
 from mathics.core.expression import Expression
 from mathics.core.list import ListExpression
-from mathics.core.number import dps, machine_epsilon, machine_precision
+from mathics.core.number import (
+    MACHINE_PRECISION_VALUE,
+    machine_epsilon,
+    machine_precision,
+)
 from mathics.core.symbols import Symbol, SymbolDivide
 from mathics.core.systemsymbols import (
     SymbolIndeterminate,
     SymbolInfinity,
     SymbolLog,
+    SymbolMachinePrecision,
     SymbolN,
     SymbolPrecision,
     SymbolRealDigits,
     SymbolRound,
 )
 from mathics.eval.nevaluator import eval_N
+from mathics.eval.numbers import eval_Accuracy, eval_Precision
 
 SymbolIntegerDigits = Symbol("IntegerDigits")
 SymbolIntegerExponent = Symbol("IntegerExponent")
@@ -157,7 +160,8 @@ class Accuracy(Builtin):
       <dd>examines the number of significant digits of $expr$ after the \
       decimal point in the number x.
     </dl>
-    <i>This is rather a proof-of-concept than a full implementation.</i>
+    <i>Notice that the result could be slighly different than the obtained\
+    in WMA, due to differencs in the internal representation of the real numbers.</i>
 
     Accuracy of a real number is estimated from its value and its precision:
 
@@ -183,11 +187,17 @@ class Accuracy(Builtin):
      = Infinity
 
     >> Accuracy[F[1.3, Pi, A]]
-     = 14.8861
+     = ...
 
     'Accuracy' for the value 0 is a fixed-precision Real number:
      >> 0``2
       = 0.00
+     >> Accuracy[0.``2]
+      = 2.
+
+    For 0.`, the accuracy satisfies:
+    >> Accuracy[0.`] == $MachinePrecision - Log[10, $MinMachineNumber]
+     = True
 
     In compound expressions, the 'Accuracy' is fixed by the number with
     the lowest 'Accuracy':
@@ -203,33 +213,10 @@ class Accuracy(Builtin):
 
     def eval(self, z, evaluation):
         "Accuracy[z_]"
-        if isinstance(z, Real):
-            if z.is_zero:
-                return MachineReal(dps(z.get_precision()))
-            z_f = z.to_python()
-            log10_z = mpmath.log((-z_f if z_f < 0 else z_f), 10)
-            return MachineReal(dps(z.get_precision()) - log10_z)
-
-        if isinstance(z, Complex):
-            acc_real = self.eval(z.real, evaluation)
-            acc_imag = self.eval(z.imag, evaluation)
-            if acc_real is SymbolInfinity:
-                return acc_imag
-            if acc_imag is SymbolInfinity:
-                return acc_real
-            return Real(min(acc_real.to_python(), acc_imag.to_python()))
-
-        if isinstance(z, Expression):
-            result = None
-            for element in z.elements:
-                candidate = self.eval(element, evaluation)
-                if isinstance(candidate, Real):
-                    candidate_f = candidate.to_python()
-                    if result is None or candidate_f < result:
-                        result = candidate_f
-            if result is not None:
-                return Real(result)
-        return SymbolInfinity
+        acc = eval_Accuracy(z)
+        if acc is None:
+            return SymbolInfinity
+        return MachineReal(acc)
 
 
 class ExactNumberQ(Test):
@@ -927,8 +914,8 @@ class Precision(Builtin):
       <dt>'Precision[$expr$]'
       <dd>examines the number of significant digits of $expr$.
     </dl>
-
-    <i>This is rather a proof-of-concept than a full implementation.</i>
+    <i>Notice that the result could be slighly different than the obtained\
+    in WMA, due to differencs in the internal representation of the real numbers.</i>
 
     The precision of an exact number, e.g. an Integer, is 'Infinity':
 
@@ -954,43 +941,39 @@ class Precision(Builtin):
     >> Precision[{{1, 1.`},{1.`5, 1.`10}}]
      = 5.
 
+    For non-zero Real values, it holds in general:
+
+    'Accuracy'[$z$] == 'Precision'[$z$] + 'Log'[$z$]
+
+    >> (Accuracy[z] == Precision[z] + Log[z])/.z-> 37.`
+     = True
+
+    The case of `0.` values is special. Following WMA, in a Machine Real\
+    representation, the precision is set to 'MachinePrecision':
+    >> Precision[0.]
+     = MachinePrecision
+
+    On the other hand, for a Precision Real with fixed accuracy,\
+    the precision is evaluated to 0.:
+    >> Precision[0.``3]
+     = 0.
+
 
     See also <url>
     :'Accuracy':
     /doc/reference-of-built-in-symbols/atomic-elements-of-expressions/representation-of-numbers/accuracy/</url>.
     """
 
-    rules = {
-        "Precision[z_?MachineNumberQ]": "MachinePrecision",
-    }
-
     summary_text = "find the precision of a number"
 
     def eval(self, z, evaluation):
         "Precision[z_]"
-        if isinstance(z, Real):
-            if z.is_zero:
-                return MachineReal0
-            return MachineReal(dps(z.get_precision()))
-
-        if isinstance(z, Complex):
-            prec_real = self.eval(z.real, evaluation)
-            prec_imag = self.eval(z.imag, evaluation)
-            if prec_real is SymbolInfinity:
-                return prec_imag
-            if prec_imag is SymbolInfinity:
-                return prec_real
-
-            return Real(min(prec_real.to_python(), prec_imag.to_python()))
-
-        if isinstance(z, Expression):
-            result = None
-            for element in z.elements:
-                candidate = self.eval(element, evaluation)
-                if isinstance(candidate, Real):
-                    candidate_f = candidate.to_python()
-                    if result is None or candidate_f < result:
-                        result = candidate_f
-            if result is not None:
-                return Real(result)
-        return SymbolInfinity
+        if isinstance(z, MachineReal):
+            return SymbolMachinePrecision
+
+        prec = eval_Precision(z)
+        if prec is None:
+            return SymbolInfinity
+        if prec == MACHINE_PRECISION_VALUE:
+            return SymbolMachinePrecision
+        return MachineReal(prec)

mathics/builtin/numbers/constants.py

@@ -19,7 +19,14 @@
 from mathics.builtin.base import Builtin, Predefined, SympyObject
 from mathics.core.atoms import MachineReal, PrecisionReal
 from mathics.core.attributes import A_CONSTANT, A_PROTECTED, A_READ_PROTECTED
-from mathics.core.number import PrecisionValueError, get_precision, machine_precision
+from mathics.core.evaluation import Evaluation
+from mathics.core.number import (
+    MAX_MACHINE_NUMBER,
+    MIN_MACHINE_NUMBER,
+    PrecisionValueError,
+    get_precision,
+    machine_precision,
+)
 from mathics.core.symbols import Atom, Symbol, strip_context
 from mathics.core.systemsymbols import SymbolIndeterminate
 
@@ -575,6 +582,59 @@ class Overflow(Builtin):
     summary_text = "overflow in numeric evaluation"
 
 
+class MaxMachineNumber(Predefined):
+    """
+    Largest normalizable machine number (<url>
+    :WMA:
+    https://reference.wolfram.com/language/ref/$MaxMachineNumber.html
+    </url>)
+
+    <dl>
+      <dt>'$MaxMachineNumber'
+      <dd>Represents the largest positive number that can be represented \
+          as a normalized machine number in the system.
+    </dl>
+
+    The product of '$MaxMachineNumber' and  '$MinMachineNumber' is a constant:
+    >> $MaxMachineNumber * $MinMachineNumber
+     = 4.
+
+    """
+
+    name = "$MaxMachineNumber"
+    summary_text = "largest normalized positive machine number"
+
+    def evaluate(self, evaluation: Evaluation) -> MachineReal:
+        return MachineReal(MAX_MACHINE_NUMBER)
+
+
+class MinMachineNumber(Predefined):
+    """
+    Smallest normalizable machine number (<url>
+    :WMA:
+    https://reference.wolfram.com/language/ref/$MinMachineNumber.html
+    </url>)
+
+    <dl>
+      <dt>'$MinMachineNumber'
+      <dd>Represents the smallest positive number that can be represented \
+          as a normalized machine number in the system.
+    </dl>
+
+    'MachinePrecision' minus the 'Log' base 10 of this number is the\
+    'Accuracy' of 0`:
+    >> MachinePrecision -Log[10., $MinMachineNumber]==Accuracy[0`]
+     = True
+
+    """
+
+    name = "$MinMachineNumber"
+    summary_text = "smallest normalized positive machine number"
+
+    def evaluate(self, evaluation: Evaluation) -> MachineReal:
+        return MachineReal(MIN_MACHINE_NUMBER)
+
+
 class Pi(_MPMathConstant, _SympyConstant):
     """
     <url>

mathics/core/expression.py

@@ -54,6 +54,7 @@
     SymbolDirectedInfinity,
     SymbolFunction,
     SymbolMinus,
+    SymbolOverflow,
     SymbolPattern,
     SymbolPower,
     SymbolSequence,
@@ -1266,7 +1267,11 @@ def rules():
                     yield rule
 
         for rule in rules():
-            result = rule.apply(new, evaluation, fully=False)
+            try:
+                result = rule.apply(new, evaluation, fully=False)
+            except OverflowError:
+                evaluation.message("General", "ovfl")
+                return Expression(SymbolOverflow), False
             if result is not None:
                 if not isinstance(result, EvalMixin):
                     return result, False