9.4. "decimal" — Arithmétique décimale en virgule fixe et flottante
*******************************************************************

Nouveau dans la version 2.4.

The "decimal" module provides support for decimal floating point
arithmetic.  It offers several advantages over the "float" datatype:

* Le module "decimal" « est basé sur un modèle en virgule flottante
  conçu pour les humains, qui suit ce principe directeur :
  l’ordinateur doit fournir un modèle de calcul qui fonctionne de la
  même manière que le calcul qu’on apprend à l’école » – extrait
  (traduit) de la spécification de l’arithmétique décimale.

* Decimal numbers can be represented exactly.  In contrast, numbers
  like "1.1" and "2.2" do not have exact representations in binary
  floating point.  End users typically would not expect "1.1 + 2.2" to
  display as "3.3000000000000003" as it does with binary floating
  point.

* Ces inexactitudes ont des conséquences en arithmétique. En base
  décimale à virgule flottante, "0.1 + 0.1 + 0.1 - 0.3" est exactement
  égal à zéro. En virgule flottante binaire, l’ordinateur l’évalue à
  "5.5511151231257827e-017". Bien que très proche de zéro, cette
  différence induit des erreurs lors des tests d’égalité, erreurs qui
  peuvent s’accumuler. Pour ces raisons "decimal" est le module
  utilisé pour des applications comptables ayant des contraintes
  strictes de fiabilité.

* Le module "decimal" incorpore la notion de chiffres significatifs,
  tels que "1.30 + 1.20" est égal à "2.50". Le dernier zéro n’est
  conservé que pour respecter le nombre de chiffres significatifs.
  C’est également l’affichage préféré pour représenter des sommes
  d’argent. Pour la multiplication, l’approche « scolaire » utilise
  tout les chiffres présents dans les facteurs. Par exemple, "1.3 *
  1.2" donnerait "1.56" tandis que "1.30 * 1.20" donnerait "1.5600".

* Contrairement à l’arithmétique en virgule flottante binaire, le
  module "decimal" possède un paramètre de précision ajustable (par
  défaut à 28 chiffres significatifs) qui peut être aussi élevée que
  nécessaire pour un problème donné :

  >>> from decimal import *
  >>> getcontext().prec = 6
  >>> Decimal(1) / Decimal(7)
  Decimal('0.142857')
  >>> getcontext().prec = 28
  >>> Decimal(1) / Decimal(7)
  Decimal('0.1428571428571428571428571429')

* L’arithmétique binaire et décimale en virgule flottante sont
  implémentées selon des standards publiés. Alors que le type "float"
  n’expose qu’une faible portion de ses capacités, le module "decimal"
  expose tous les composants nécessaires du standard. Lorsque
  nécessaire, le développeur a un contrôle total de la gestion de
  signal et de l’arrondi. Cela inclut la possibilité de forcer une
  arithmétique exacte en utilisant des exceptions pour bloquer toute
  opération inexacte.

* Le module "decimal" a été conçu pour gérer « sans préjugé, à la
  fois une arithmétique décimale non-arrondie (aussi appelée
  arithmétique en virgule fixe) et à la fois une arithmétique en
  virgule flottante. » (extrait traduit de la spécification de
  l’arithmétique décimale).

Le module est conçu autour de trois concepts : le nombre décimal, le
contexte arithmétique et les signaux.

Un "Decimal" est immuable. Il a un signe, un coefficient, et un
exposant. Pour préserver le nombre de chiffres significatifs, les
zéros en fin de chaîne ne sont pas tronqués. Les décimaux incluent
aussi des valeurs spéciales telles que "Infinity", "-Infinity", et
"NaN". Le standard fait également la différence entre "-0" et "+0".

Le contexte de l’arithmétique est un environnement qui permet de
configurer une précision, une règle pour l’arrondi, des limites sur
l’exposant, des options indiquant le résultat des opérations et si les
signaux (remontés lors d’opérations illégales) sont traités comme des
exceptions Python. Les options d’arrondi incluent "ROUND_CEILING",
"ROUND_DOWN", "ROUND_FLOOR", "ROUND_HALF_DOWN", "ROUND_HALF_EVEN",
"ROUND_HALF_UP", "ROUND_UP", et "ROUND_05UP".

Signals are groups of exceptional conditions arising during the course
of computation.  Depending on the needs of the application, signals
may be ignored, considered as informational, or treated as exceptions.
The signals in the decimal module are: "Clamped", "InvalidOperation",
"DivisionByZero", "Inexact", "Rounded", "Subnormal", "Overflow", and
"Underflow".

Chaque signal est configurable indépendamment. Quand une opération
illégale survient, le signal est mis à "1", puis s’il est configuré
pour, une exception est levée. La mise à "1" est persistante,
l’utilisateur doit donc les remettre à zéro avant de commencer un
calcul qu’il souhaite surveiller.

Voir aussi:

  * IBM’s General Decimal Arithmetic Specification, The General
    Decimal Arithmetic Specification.


9.4.1. Introduction pratique
============================

Commençons par importer le module, regarder le contexte actuel avec
"getcontext()", et si nécessaire configurer la précision, l’arrondi,
et la gestion des signaux :

   >>> from decimal import *
   >>> getcontext()
   Context(prec=28, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
           capitals=1, flags=[], traps=[Overflow, DivisionByZero,
           InvalidOperation])

   >>> getcontext().prec = 7       # Set a new precision

Decimal instances can be constructed from integers, strings, floats,
or tuples. Construction from an integer or a float performs an exact
conversion of the value of that integer or float.  Decimal numbers
include special values such as "NaN" which stands for « Not a number
», positive and negative "Infinity", and "-0".

>>> getcontext().prec = 28
>>> Decimal(10)
Decimal('10')
>>> Decimal('3.14')
Decimal('3.14')
>>> Decimal(3.14)
Decimal('3.140000000000000124344978758017532527446746826171875')
>>> Decimal((0, (3, 1, 4), -2))
Decimal('3.14')
>>> Decimal(str(2.0 ** 0.5))
Decimal('1.41421356237')
>>> Decimal(2) ** Decimal('0.5')
Decimal('1.414213562373095048801688724')
>>> Decimal('NaN')
Decimal('NaN')
>>> Decimal('-Infinity')
Decimal('-Infinity')

Le nombre de chiffres significatifs d’un nouvel objet "Decimal" est
déterminé entièrement par le nombre de chiffres saisis. La précision
et les règles d’arrondis n’interviennent que lors d’opérations
arithmétiques.

   >>> getcontext().prec = 6
   >>> Decimal('3.0')
   Decimal('3.0')
   >>> Decimal('3.1415926535')
   Decimal('3.1415926535')
   >>> Decimal('3.1415926535') + Decimal('2.7182818285')
   Decimal('5.85987')
   >>> getcontext().rounding = ROUND_UP
   >>> Decimal('3.1415926535') + Decimal('2.7182818285')
   Decimal('5.85988')

Les objets "Decimal" interagissent très bien avec le reste de Python.
Voici quelques exemple d’opérations avec des décimaux :

   >>> data = map(Decimal, '1.34 1.87 3.45 2.35 1.00 0.03 9.25'.split())
   >>> max(data)
   Decimal('9.25')
   >>> min(data)
   Decimal('0.03')
   >>> sorted(data)
   [Decimal('0.03'), Decimal('1.00'), Decimal('1.34'), Decimal('1.87'),
    Decimal('2.35'), Decimal('3.45'), Decimal('9.25')]
   >>> sum(data)
   Decimal('19.29')
   >>> a,b,c = data[:3]
   >>> str(a)
   '1.34'
   >>> float(a)
   1.34
   >>> round(a, 1)     # round() first converts to binary floating point
   1.3
   >>> int(a)
   1
   >>> a * 5
   Decimal('6.70')
   >>> a * b
   Decimal('2.5058')
   >>> c % a
   Decimal('0.77')

Et certaines fonctions mathématiques sont également disponibles sur
des instances de "Decimal" :

>>> getcontext().prec = 28
>>> Decimal(2).sqrt()
Decimal('1.414213562373095048801688724')
>>> Decimal(1).exp()
Decimal('2.718281828459045235360287471')
>>> Decimal('10').ln()
Decimal('2.302585092994045684017991455')
>>> Decimal('10').log10()
Decimal('1')

La méthode "quantize()" arrondit un nombre à un exposant fixe. Cette
méthode est utile pour des applications monétaires qui arrondissent
souvent un résultat à un nombre de chiffres significatifs exact :

>>> Decimal('7.325').quantize(Decimal('.01'), rounding=ROUND_DOWN)
Decimal('7.32')
>>> Decimal('7.325').quantize(Decimal('1.'), rounding=ROUND_UP)
Decimal('8')

Comme montré plus haut, la fonction "getcontext()" accède au contexte
actuel et permet de modifier les paramètres. Cette approche répond aux
besoins de la plupart des applications.

Pour un travail plus avancé, il peut être utile de créer des contextes
alternatifs en utilisant le constructeur de "Context". Pour activer
cet objet "Context", utilisez la fonction "setcontext()".

En accord avec le standard, le module "decimal" fournit des objets
Context standards, "BasicContext" et "ExtendedContext". Le premier est
particulièrement utile pour le débogage car beaucoup des pièges sont
activés dans cet objet.

   >>> myothercontext = Context(prec=60, rounding=ROUND_HALF_DOWN)
   >>> setcontext(myothercontext)
   >>> Decimal(1) / Decimal(7)
   Decimal('0.142857142857142857142857142857142857142857142857142857142857')

   >>> ExtendedContext
   Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
           capitals=1, flags=[], traps=[])
   >>> setcontext(ExtendedContext)
   >>> Decimal(1) / Decimal(7)
   Decimal('0.142857143')
   >>> Decimal(42) / Decimal(0)
   Decimal('Infinity')

   >>> setcontext(BasicContext)
   >>> Decimal(42) / Decimal(0)
   Traceback (most recent call last):
     File "<pyshell#143>", line 1, in -toplevel-
       Decimal(42) / Decimal(0)
   DivisionByZero: x / 0

Les objets "Context" ont aussi des options pour détecter des
opérations illégales lors des calculs. Ces options restent activées
jusqu’à ce qu’elles soit remises à zéro de manière explicite. Il
convient donc de remettre à zéro ces options avant chaque inspection
de chaque calcul, avec la méthode "clear_flags()".

   >>> setcontext(ExtendedContext)
   >>> getcontext().clear_flags()
   >>> Decimal(355) / Decimal(113)
   Decimal('3.14159292')
   >>> getcontext()
   Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
           capitals=1, flags=[Rounded, Inexact], traps=[])

Les options montrent que l’approximation de "Pi" par une fraction a
été arrondie (les chiffres au delà de la précision spécifiée par
l’objet Context ont été tronqués) et que le résultat est différent
(certains des chiffres tronqués étaient différents de zéro).

L’activation des pièges se fait en utilisant un dictionnaire dans
l’attribut "traps" de l’objet Context :

   >>> setcontext(ExtendedContext)
   >>> Decimal(1) / Decimal(0)
   Decimal('Infinity')
   >>> getcontext().traps[DivisionByZero] = 1
   >>> Decimal(1) / Decimal(0)
   Traceback (most recent call last):
     File "<pyshell#112>", line 1, in -toplevel-
       Decimal(1) / Decimal(0)
   DivisionByZero: x / 0

La plupart des applications n’ajustent l’objet "Context" qu’une seule
fois, au démarrage. Et, dans beaucoup d’applications, les données sont
convertie une fois pour toutes en "Decimal". Une fois le "Context"
initialisé, et les objets "Decimal" créés, l’essentiel du programme
manipule la donnée de la même manière qu’avec les autres types
numériques Python.


9.4.2. Les objets Decimal
=========================

class decimal.Decimal([value[, context]])

   Construire un nouvel objet "Decimal" à partir de *value*.

   *value* can be an integer, string, tuple, "float", or another
   "Decimal" object. If no *value* is given, returns "Decimal('0')".
   If *value* is a string, it should conform to the decimal numeric
   string syntax after leading and trailing whitespace characters are
   removed:

      sign           ::=  '+' | '-'
      digit          ::=  '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
      indicator      ::=  'e' | 'E'
      digits         ::=  digit [digit]...
      decimal-part   ::=  digits '.' [digits] | ['.'] digits
      exponent-part  ::=  indicator [sign] digits
      infinity       ::=  'Infinity' | 'Inf'
      nan            ::=  'NaN' [digits] | 'sNaN' [digits]
      numeric-value  ::=  decimal-part [exponent-part] | infinity
      numeric-string ::=  [sign] numeric-value | [sign] nan

   If *value* is a unicode string then other Unicode decimal digits
   are also permitted where "digit" appears above.  These include
   decimal digits from various other alphabets (for example, Arabic-
   Indic and Devanāgarī digits) along with the fullwidth digits
   "u'\uff10'" through "u'\uff19'".

   Si *value* est un "tuple", il doit avoir 3 éléments, le signe ("0"
   pour positif ou "1" pour négatif), un "tuple" de chiffres, et un
   entier représentant l’exposant. Par exemple, "Decimal((0, (1, 4, 1,
   4), -3))" construit l’objet "Decimal('1.414')".

   Si *value* est un "float", la valeur en binaire flottant est
   convertie exactement à son équivalent décimal. Cette conversion
   peut parfois nécessiter 53 chiffres significatifs ou plus. Par
   exemple, "Decimal(float('1.1'))" devient
   "Decimal('1.100000000000000088817841970012523233890533447265625')".

   La précision spécifiée dans Context n’affecte pas le nombre de
   chiffres stockés. Cette valeur est déterminée exclusivement par le
   nombre de chiffres dans *value*. Par exemple, "Decimal('3.00000')"
   enregistre les 5 zéros même si la précision du contexte est de 3.

   L’objectif de l’argument *context* est de déterminer ce que Python
   doit faire si *value* est une chaîne avec un mauvais format. Si
   l’option "InvalidOperation" est activée, une exception est levée,
   sinon le constructeur renvoie un objet "Decimal" avec la valeur
   "NaN".

   Une fois construit, les objets "Decimal"  sont immuables.

   Modifié dans la version 2.6: leading and trailing whitespace
   characters are permitted when creating a Decimal instance from a
   string.

   Modifié dans la version 2.7: L’argument du constructeur peut
   désormais être un objet "float".

   Decimal floating point objects share many properties with the other
   built-in numeric types such as "float" and "int".  All of the usual
   math operations and special methods apply.  Likewise, decimal
   objects can be copied, pickled, printed, used as dictionary keys,
   used as set elements, compared, sorted, and coerced to another type
   (such as "float" or "long").

   Il existe quelques différences mineures entre l’arithmétique entre
   les objets décimaux et l’arithmétique avec les entiers et les
   "float". Quand l’opérateur modulo "%" est appliqué sur des objets
   décimaux, le signe du résultat est le signe du *dividend* plutôt
   que le signe du diviseur:

      >>> (-7) % 4
      1
      >>> Decimal(-7) % Decimal(4)
      Decimal('-3')

   L’opérateur division entière, "//" se comporte de la même manière,
   retournant la partie entière du quotient, plutôt que son arrondi,
   de manière à préserver l’identité d’Euclide "x == (x // y) * y + x
   % y":

      >>> -7 // 4
      -2
      >>> Decimal(-7) // Decimal(4)
      Decimal('-1')

   Les opérateurs "//" et "%" implémentent la division entière et le
   reste (ou modulo), respectivement, tel que décrit dans la
   spécification.

   Decimal objects cannot generally be combined with floats in
   arithmetic operations: an attempt to add a "Decimal" to a "float",
   for example, will raise a "TypeError". There’s one exception to
   this rule: it’s possible to use Python’s comparison operators to
   compare a "float" instance "x" with a "Decimal" instance "y".
   Without this exception, comparisons between "Decimal" and "float"
   instances would follow the general rules for comparing objects of
   different types described in the Expressions section of the
   reference manual, leading to confusing results.

   Modifié dans la version 2.7: A comparison between a "float"
   instance "x" and a "Decimal" instance "y" now returns a result
   based on the values of "x" and "y".  In earlier versions "x < y"
   returned the same (arbitrary) result for any "Decimal" instance "x"
   and any "float" instance "y".

   In addition to the standard numeric properties, decimal floating
   point objects also have a number of specialized methods:

   adjusted()

      Return the adjusted exponent after shifting out the
      coefficient’s rightmost digits until only the lead digit
      remains: "Decimal('321e+5').adjusted()" returns seven.  Used for
      determining the position of the most significant digit with
      respect to the decimal point.

   as_tuple()

      Return a *named tuple* representation of the number:
      "DecimalTuple(sign, digits, exponent)".

      Modifié dans la version 2.6: Use a named tuple.

   canonical()

      Return the canonical encoding of the argument.  Currently, the
      encoding of a "Decimal" instance is always canonical, so this
      operation returns its argument unchanged.

      Nouveau dans la version 2.6.

   compare(other[, context])

      Compare the values of two Decimal instances.  This operation
      behaves in the same way as the usual comparison method
      "__cmp__()", except that "compare()" returns a Decimal instance
      rather than an integer, and if either operand is a NaN then the
      result is a NaN:

         a or b is a NaN ==> Decimal('NaN')
         a < b           ==> Decimal('-1')
         a == b          ==> Decimal('0')
         a > b           ==> Decimal('1')

   compare_signal(other[, context])

      This operation is identical to the "compare()" method, except
      that all NaNs signal.  That is, if neither operand is a
      signaling NaN then any quiet NaN operand is treated as though it
      were a signaling NaN.

      Nouveau dans la version 2.6.

   compare_total(other)

      Compare two operands using their abstract representation rather
      than their numerical value.  Similar to the "compare()" method,
      but the result gives a total ordering on "Decimal" instances.
      Two "Decimal" instances with the same numeric value but
      different representations compare unequal in this ordering:

      >>> Decimal('12.0').compare_total(Decimal('12'))
      Decimal('-1')

      Quiet and signaling NaNs are also included in the total
      ordering.  The result of this function is "Decimal('0')" if both
      operands have the same representation, "Decimal('-1')" if the
      first operand is lower in the total order than the second, and
      "Decimal('1')" if the first operand is higher in the total order
      than the second operand.  See the specification for details of
      the total order.

      Nouveau dans la version 2.6.

   compare_total_mag(other)

      Compare two operands using their abstract representation rather
      than their value as in "compare_total()", but ignoring the sign
      of each operand. "x.compare_total_mag(y)" is equivalent to
      "x.copy_abs().compare_total(y.copy_abs())".

      Nouveau dans la version 2.6.

   conjugate()

      Just returns self, this method is only to comply with the
      Decimal Specification.

      Nouveau dans la version 2.6.

   copy_abs()

      Return the absolute value of the argument.  This operation is
      unaffected by the context and is quiet: no flags are changed and
      no rounding is performed.

      Nouveau dans la version 2.6.

   copy_negate()

      Return the negation of the argument.  This operation is
      unaffected by the context and is quiet: no flags are changed and
      no rounding is performed.

      Nouveau dans la version 2.6.

   copy_sign(other)

      Return a copy of the first operand with the sign set to be the
      same as the sign of the second operand.  For example:

      >>> Decimal('2.3').copy_sign(Decimal('-1.5'))
      Decimal('-2.3')

      This operation is unaffected by the context and is quiet: no
      flags are changed and no rounding is performed.

      Nouveau dans la version 2.6.

   exp([context])

      Return the value of the (natural) exponential function "e**x" at
      the given number.  The result is correctly rounded using the
      "ROUND_HALF_EVEN" rounding mode.

      >>> Decimal(1).exp()
      Decimal('2.718281828459045235360287471')
      >>> Decimal(321).exp()
      Decimal('2.561702493119680037517373933E+139')

      Nouveau dans la version 2.6.

   from_float(f)

      Classmethod that converts a float to a decimal number, exactly.

      Note *Decimal.from_float(0.1)* is not the same as
      *Decimal(“0.1”)*. Since 0.1 is not exactly representable in
      binary floating point, the value is stored as the nearest
      representable value which is *0x1.999999999999ap-4*.  That
      equivalent value in decimal is
      *0.1000000000000000055511151231257827021181583404541015625*.

      Note: From Python 2.7 onwards, a "Decimal" instance can also
        be constructed directly from a "float".

         >>> Decimal.from_float(0.1)
         Decimal('0.1000000000000000055511151231257827021181583404541015625')
         >>> Decimal.from_float(float('nan'))
         Decimal('NaN')
         >>> Decimal.from_float(float('inf'))
         Decimal('Infinity')
         >>> Decimal.from_float(float('-inf'))
         Decimal('-Infinity')

      Nouveau dans la version 2.7.

   fma(other, third[, context])

      Fused multiply-add.  Return self*other+third with no rounding of
      the intermediate product self*other.

      >>> Decimal(2).fma(3, 5)
      Decimal('11')

      Nouveau dans la version 2.6.

   is_canonical()

      Return "True" if the argument is canonical and "False"
      otherwise.  Currently, a "Decimal" instance is always canonical,
      so this operation always returns "True".

      Nouveau dans la version 2.6.

   is_finite()

      Return "True" if the argument is a finite number, and "False" if
      the argument is an infinity or a NaN.

      Nouveau dans la version 2.6.

   is_infinite()

      Return "True" if the argument is either positive or negative
      infinity and "False" otherwise.

      Nouveau dans la version 2.6.

   is_nan()

      Return "True" if the argument is a (quiet or signaling) NaN and
      "False" otherwise.

      Nouveau dans la version 2.6.

   is_normal()

      Return "True" if the argument is a *normal* finite non-zero
      number with an adjusted exponent greater than or equal to
      *Emin*. Return "False" if the argument is zero, subnormal,
      infinite or a NaN.  Note, the term *normal* is used here in a
      different sense with the "normalize()" method which is used to
      create canonical values.

      Nouveau dans la version 2.6.

   is_qnan()

      Return "True" if the argument is a quiet NaN, and "False"
      otherwise.

      Nouveau dans la version 2.6.

   is_signed()

      Return "True" if the argument has a negative sign and "False"
      otherwise.  Note that zeros and NaNs can both carry signs.

      Nouveau dans la version 2.6.

   is_snan()

      Return "True" if the argument is a signaling NaN and "False"
      otherwise.

      Nouveau dans la version 2.6.

   is_subnormal()

      Return "True" if the argument is subnormal, and "False"
      otherwise. A number is subnormal is if it is nonzero, finite,
      and has an adjusted exponent less than *Emin*.

      Nouveau dans la version 2.6.

   is_zero()

      Return "True" if the argument is a (positive or negative) zero
      and "False" otherwise.

      Nouveau dans la version 2.6.

   ln([context])

      Return the natural (base e) logarithm of the operand.  The
      result is correctly rounded using the "ROUND_HALF_EVEN" rounding
      mode.

      Nouveau dans la version 2.6.

   log10([context])

      Return the base ten logarithm of the operand.  The result is
      correctly rounded using the "ROUND_HALF_EVEN" rounding mode.

      Nouveau dans la version 2.6.

   logb([context])

      For a nonzero number, return the adjusted exponent of its
      operand as a "Decimal" instance.  If the operand is a zero then
      "Decimal('-Infinity')" is returned and the "DivisionByZero" flag
      is raised.  If the operand is an infinity then
      "Decimal('Infinity')" is returned.

      Nouveau dans la version 2.6.

   logical_and(other[, context])

      "logical_and()" is a logical operation which takes two *logical
      operands* (see Logical operands).  The result is the digit-wise
      "and" of the two operands.

      Nouveau dans la version 2.6.

   logical_invert([context])

      "logical_invert()" is a logical operation.  The result is the
      digit-wise inversion of the operand.

      Nouveau dans la version 2.6.

   logical_or(other[, context])

      "logical_or()" is a logical operation which takes two *logical
      operands* (see Logical operands).  The result is the digit-wise
      "or" of the two operands.

      Nouveau dans la version 2.6.

   logical_xor(other[, context])

      "logical_xor()" is a logical operation which takes two *logical
      operands* (see Logical operands).  The result is the digit-wise
      exclusive or of the two operands.

      Nouveau dans la version 2.6.

   max(other[, context])

      Like "max(self, other)" except that the context rounding rule is
      applied before returning and that "NaN" values are either
      signaled or ignored (depending on the context and whether they
      are signaling or quiet).

   max_mag(other[, context])

      Similar to the "max()" method, but the comparison is done using
      the absolute values of the operands.

      Nouveau dans la version 2.6.

   min(other[, context])

      Like "min(self, other)" except that the context rounding rule is
      applied before returning and that "NaN" values are either
      signaled or ignored (depending on the context and whether they
      are signaling or quiet).

   min_mag(other[, context])

      Similar to the "min()" method, but the comparison is done using
      the absolute values of the operands.

      Nouveau dans la version 2.6.

   next_minus([context])

      Return the largest number representable in the given context (or
      in the current thread’s context if no context is given) that is
      smaller than the given operand.

      Nouveau dans la version 2.6.

   next_plus([context])

      Return the smallest number representable in the given context
      (or in the current thread’s context if no context is given) that
      is larger than the given operand.

      Nouveau dans la version 2.6.

   next_toward(other[, context])

      If the two operands are unequal, return the number closest to
      the first operand in the direction of the second operand.  If
      both operands are numerically equal, return a copy of the first
      operand with the sign set to be the same as the sign of the
      second operand.

      Nouveau dans la version 2.6.

   normalize([context])

      Normalize the number by stripping the rightmost trailing zeros
      and converting any result equal to "Decimal('0')" to
      "Decimal('0e0')". Used for producing canonical values for
      attributes of an equivalence class. For example,
      "Decimal('32.100')" and "Decimal('0.321000e+2')" both normalize
      to the equivalent value "Decimal('32.1')".

   number_class([context])

      Return a string describing the *class* of the operand.  The
      returned value is one of the following ten strings.

      * ""-Infinity"", indicating that the operand is negative
        infinity.

      * ""-Normal"", indicating that the operand is a negative
        normal number.

      * ""-Subnormal"", indicating that the operand is negative and
        subnormal.

      * ""-Zero"", indicating that the operand is a negative zero.

      * ""+Zero"", indicating that the operand is a positive zero.

      * ""+Subnormal"", indicating that the operand is positive and
        subnormal.

      * ""+Normal"", indicating that the operand is a positive
        normal number.

      * ""+Infinity"", indicating that the operand is positive
        infinity.

      * ""NaN"", indicating that the operand is a quiet NaN (Not a
        Number).

      * ""sNaN"", indicating that the operand is a signaling NaN.

      Nouveau dans la version 2.6.

   quantize(exp[, rounding[, context[, watchexp]]])

      Return a value equal to the first operand after rounding and
      having the exponent of the second operand.

      >>> Decimal('1.41421356').quantize(Decimal('1.000'))
      Decimal('1.414')

      Unlike other operations, if the length of the coefficient after
      the quantize operation would be greater than precision, then an
      "InvalidOperation" is signaled. This guarantees that, unless
      there is an error condition, the quantized exponent is always
      equal to that of the right-hand operand.

      Also unlike other operations, quantize never signals Underflow,
      even if the result is subnormal and inexact.

      If the exponent of the second operand is larger than that of the
      first then rounding may be necessary.  In this case, the
      rounding mode is determined by the "rounding" argument if given,
      else by the given "context" argument; if neither argument is
      given the rounding mode of the current thread’s context is used.

      If *watchexp* is set (default), then an error is returned
      whenever the resulting exponent is greater than "Emax" or less
      than "Etiny".

   radix()

      Return "Decimal(10)", the radix (base) in which the "Decimal"
      class does all its arithmetic.  Included for compatibility with
      the specification.

      Nouveau dans la version 2.6.

   remainder_near(other[, context])

      Return the remainder from dividing *self* by *other*.  This
      differs from "self % other" in that the sign of the remainder is
      chosen so as to minimize its absolute value.  More precisely,
      the return value is "self - n * other" where "n" is the integer
      nearest to the exact value of "self / other", and if two
      integers are equally near then the even one is chosen.

      If the result is zero then its sign will be the sign of *self*.

      >>> Decimal(18).remainder_near(Decimal(10))
      Decimal('-2')
      >>> Decimal(25).remainder_near(Decimal(10))
      Decimal('5')
      >>> Decimal(35).remainder_near(Decimal(10))
      Decimal('-5')

   rotate(other[, context])

      Return the result of rotating the digits of the first operand by
      an amount specified by the second operand.  The second operand
      must be an integer in the range -precision through precision.
      The absolute value of the second operand gives the number of
      places to rotate.  If the second operand is positive then
      rotation is to the left; otherwise rotation is to the right. The
      coefficient of the first operand is padded on the left with
      zeros to length precision if necessary.  The sign and exponent
      of the first operand are unchanged.

      Nouveau dans la version 2.6.

   same_quantum(other[, context])

      Test whether self and other have the same exponent or whether
      both are "NaN".

   scaleb(other[, context])

      Return the first operand with exponent adjusted by the second.
      Equivalently, return the first operand multiplied by
      "10**other".  The second operand must be an integer.

      Nouveau dans la version 2.6.

   shift(other[, context])

      Return the result of shifting the digits of the first operand by
      an amount specified by the second operand.  The second operand
      must be an integer in the range -precision through precision.
      The absolute value of the second operand gives the number of
      places to shift.  If the second operand is positive then the
      shift is to the left; otherwise the shift is to the right.
      Digits shifted into the coefficient are zeros.  The sign and
      exponent of the first operand are unchanged.

      Nouveau dans la version 2.6.

   sqrt([context])

      Return the square root of the argument to full precision.

   to_eng_string([context])

      Convert to a string, using engineering notation if an exponent
      is needed.

      Engineering notation has an exponent which is a multiple of 3.
      This can leave up to 3 digits to the left of the decimal place
      and may require the addition of either one or two trailing
      zeros.

      For example, this converts "Decimal('123E+1')" to
      "Decimal('1.23E+3')".

   to_integral([rounding[, context]])

      Identical to the "to_integral_value()" method.  The
      "to_integral" name has been kept for compatibility with older
      versions.

   to_integral_exact([rounding[, context]])

      Round to the nearest integer, signaling "Inexact" or "Rounded"
      as appropriate if rounding occurs.  The rounding mode is
      determined by the "rounding" parameter if given, else by the
      given "context".  If neither parameter is given then the
      rounding mode of the current context is used.

      Nouveau dans la version 2.6.

   to_integral_value([rounding[, context]])

      Round to the nearest integer without signaling "Inexact" or
      "Rounded".  If given, applies *rounding*; otherwise, uses the
      rounding method in either the supplied *context* or the current
      context.

      Modifié dans la version 2.6: renamed from "to_integral" to
      "to_integral_value".  The old name remains valid for
      compatibility.


9.4.2.1. Logical operands
-------------------------

The "logical_and()", "logical_invert()", "logical_or()", and
"logical_xor()" methods expect their arguments to be *logical
operands*.  A *logical operand* is a "Decimal" instance whose exponent
and sign are both zero, and whose digits are all either "0" or "1".


9.4.3. Context objects
======================

Contexts are environments for arithmetic operations.  They govern
precision, set rules for rounding, determine which signals are treated
as exceptions, and limit the range for exponents.

Each thread has its own current context which is accessed or changed
using the "getcontext()" and "setcontext()" functions:

decimal.getcontext()

   Return the current context for the active thread.

decimal.setcontext(c)

   Set the current context for the active thread to *c*.

Beginning with Python 2.5, you can also use the "with" statement and
the "localcontext()" function to temporarily change the active
context.

decimal.localcontext([c])

   Return a context manager that will set the current context for the
   active thread to a copy of *c* on entry to the with-statement and
   restore the previous context when exiting the with-statement. If no
   context is specified, a copy of the current context is used.

   Nouveau dans la version 2.5.

   For example, the following code sets the current decimal precision
   to 42 places, performs a calculation, and then automatically
   restores the previous context:

      from decimal import localcontext

      with localcontext() as ctx:
          ctx.prec = 42   # Perform a high precision calculation
          s = calculate_something()
      s = +s  # Round the final result back to the default precision

      with localcontext(BasicContext):      # temporarily use the BasicContext
          print Decimal(1) / Decimal(7)
          print Decimal(355) / Decimal(113)

New contexts can also be created using the "Context" constructor
described below. In addition, the module provides three pre-made
contexts:

class decimal.BasicContext

   This is a standard context defined by the General Decimal
   Arithmetic Specification.  Precision is set to nine.  Rounding is
   set to "ROUND_HALF_UP".  All flags are cleared.  All traps are
   enabled (treated as exceptions) except "Inexact", "Rounded", and
   "Subnormal".

   Because many of the traps are enabled, this context is useful for
   debugging.

class decimal.ExtendedContext

   This is a standard context defined by the General Decimal
   Arithmetic Specification.  Precision is set to nine.  Rounding is
   set to "ROUND_HALF_EVEN".  All flags are cleared.  No traps are
   enabled (so that exceptions are not raised during computations).

   Because the traps are disabled, this context is useful for
   applications that prefer to have result value of "NaN" or
   "Infinity" instead of raising exceptions.  This allows an
   application to complete a run in the presence of conditions that
   would otherwise halt the program.

class decimal.DefaultContext

   This context is used by the "Context" constructor as a prototype
   for new contexts.  Changing a field (such a precision) has the
   effect of changing the default for new contexts created by the
   "Context" constructor.

   This context is most useful in multi-threaded environments.
   Changing one of the fields before threads are started has the
   effect of setting system-wide defaults.  Changing the fields after
   threads have started is not recommended as it would require thread
   synchronization to prevent race conditions.

   In single threaded environments, it is preferable to not use this
   context at all.  Instead, simply create contexts explicitly as
   described below.

   The default values are precision=28, rounding=ROUND_HALF_EVEN, and
   enabled traps for Overflow, InvalidOperation, and DivisionByZero.

In addition to the three supplied contexts, new contexts can be
created with the "Context" constructor.

class decimal.Context(prec=None, rounding=None, traps=None, flags=None, Emin=None, Emax=None, capitals=1)

   Creates a new context.  If a field is not specified or is "None",
   the default values are copied from the "DefaultContext".  If the
   *flags* field is not specified or is "None", all flags are cleared.

   The *prec* field is a positive integer that sets the precision for
   arithmetic operations in the context.

   The *rounding* option is one of:

   * "ROUND_CEILING" (towards "Infinity"),

   * "ROUND_DOWN" (towards zero),

   * "ROUND_FLOOR" (towards "-Infinity"),

   * "ROUND_HALF_DOWN" (to nearest with ties going towards zero),

   * "ROUND_HALF_EVEN" (to nearest with ties going to nearest even
     integer),

   * "ROUND_HALF_UP" (to nearest with ties going away from zero), or

   * "ROUND_UP" (away from zero).

   * "ROUND_05UP" (away from zero if last digit after rounding
     towards zero would have been 0 or 5; otherwise towards zero)

   The *traps* and *flags* fields list any signals to be set.
   Generally, new contexts should only set traps and leave the flags
   clear.

   The *Emin* and *Emax* fields are integers specifying the outer
   limits allowable for exponents.

   The *capitals* field is either "0" or "1" (the default). If set to
   "1", exponents are printed with a capital "E"; otherwise, a
   lowercase "e" is used: "Decimal('6.02e+23')".

   Modifié dans la version 2.6: The "ROUND_05UP" rounding mode was
   added.

   The "Context" class defines several general purpose methods as well
   as a large number of methods for doing arithmetic directly in a
   given context. In addition, for each of the "Decimal" methods
   described above (with the exception of the "adjusted()" and
   "as_tuple()" methods) there is a corresponding "Context" method.
   For example, for a "Context" instance "C" and "Decimal" instance
   "x", "C.exp(x)" is equivalent to "x.exp(context=C)".  Each
   "Context" method accepts a Python integer (an instance of "int" or
   "long") anywhere that a Decimal instance is accepted.

   clear_flags()

      Resets all of the flags to "0".

   copy()

      Return a duplicate of the context.

   copy_decimal(num)

      Return a copy of the Decimal instance num.

   create_decimal(num)

      Creates a new Decimal instance from *num* but using *self* as
      context. Unlike the "Decimal" constructor, the context
      precision, rounding method, flags, and traps are applied to the
      conversion.

      This is useful because constants are often given to a greater
      precision than is needed by the application.  Another benefit is
      that rounding immediately eliminates unintended effects from
      digits beyond the current precision. In the following example,
      using unrounded inputs means that adding zero to a sum can
      change the result:

         >>> getcontext().prec = 3
         >>> Decimal('3.4445') + Decimal('1.0023')
         Decimal('4.45')
         >>> Decimal('3.4445') + Decimal(0) + Decimal('1.0023')
         Decimal('4.44')

      This method implements the to-number operation of the IBM
      specification. If the argument is a string, no leading or
      trailing whitespace is permitted.

   create_decimal_from_float(f)

      Creates a new Decimal instance from a float *f* but rounding
      using *self* as the context.  Unlike the "Decimal.from_float()"
      class method, the context precision, rounding method, flags, and
      traps are applied to the conversion.

         >>> context = Context(prec=5, rounding=ROUND_DOWN)
         >>> context.create_decimal_from_float(math.pi)
         Decimal('3.1415')
         >>> context = Context(prec=5, traps=[Inexact])
         >>> context.create_decimal_from_float(math.pi)
         Traceback (most recent call last):
             ...
         Inexact: None

      Nouveau dans la version 2.7.

   Etiny()

      Returns a value equal to "Emin - prec + 1" which is the minimum
      exponent value for subnormal results.  When underflow occurs,
      the exponent is set to "Etiny".

   Etop()

      Returns a value equal to "Emax - prec + 1".

   The usual approach to working with decimals is to create "Decimal"
   instances and then apply arithmetic operations which take place
   within the current context for the active thread.  An alternative
   approach is to use context methods for calculating within a
   specific context.  The methods are similar to those for the
   "Decimal" class and are only briefly recounted here.

   abs(x)

      Renvoie la valeur absolue de *x*.

   add(x, y)

      Return the sum of *x* and *y*.

   canonical(x)

      Returns the same Decimal object *x*.

   compare(x, y)

      Compares *x* and *y* numerically.

   compare_signal(x, y)

      Compares the values of the two operands numerically.

   compare_total(x, y)

      Compares two operands using their abstract representation.

   compare_total_mag(x, y)

      Compares two operands using their abstract representation,
      ignoring sign.

   copy_abs(x)

      Returns a copy of *x* with the sign set to 0.

   copy_negate(x)

      Returns a copy of *x* with the sign inverted.

   copy_sign(x, y)

      Copies the sign from *y* to *x*.

   divide(x, y)

      Return *x* divided by *y*.

   divide_int(x, y)

      Return *x* divided by *y*, truncated to an integer.

   divmod(x, y)

      Divides two numbers and returns the integer part of the result.

   exp(x)

      Returns *e ** x*.

   fma(x, y, z)

      Returns *x* multiplied by *y*, plus *z*.

   is_canonical(x)

      Returns "True" if *x* is canonical; otherwise returns "False".

   is_finite(x)

      Returns "True" if *x* is finite; otherwise returns "False".

   is_infinite(x)

      Returns "True" if *x* is infinite; otherwise returns "False".

   is_nan(x)

      Returns "True" if *x* is a qNaN or sNaN; otherwise returns
      "False".

   is_normal(x)

      Returns "True" if *x* is a normal number; otherwise returns
      "False".

   is_qnan(x)

      Returns "True" if *x* is a quiet NaN; otherwise returns "False".

   is_signed(x)

      Returns "True" if *x* is negative; otherwise returns "False".

   is_snan(x)

      Returns "True" if *x* is a signaling NaN; otherwise returns
      "False".

   is_subnormal(x)

      Returns "True" if *x* is subnormal; otherwise returns "False".

   is_zero(x)

      Returns "True" if *x* is a zero; otherwise returns "False".

   ln(x)

      Returns the natural (base e) logarithm of *x*.

   log10(x)

      Returns the base 10 logarithm of *x*.

   logb(x)

      Returns the exponent of the magnitude of the operand’s MSD.

   logical_and(x, y)

      Applies the logical operation *and* between each operand’s
      digits.

   logical_invert(x)

      Invert all the digits in *x*.

   logical_or(x, y)

      Applies the logical operation *or* between each operand’s
      digits.

   logical_xor(x, y)

      Applies the logical operation *xor* between each operand’s
      digits.

   max(x, y)

      Compares two values numerically and returns the maximum.

   max_mag(x, y)

      Compares the values numerically with their sign ignored.

   min(x, y)

      Compares two values numerically and returns the minimum.

   min_mag(x, y)

      Compares the values numerically with their sign ignored.

   minus(x)

      Minus corresponds to the unary prefix minus operator in Python.

   multiply(x, y)

      Return the product of *x* and *y*.

   next_minus(x)

      Returns the largest representable number smaller than *x*.

   next_plus(x)

      Returns the smallest representable number larger than *x*.

   next_toward(x, y)

      Returns the number closest to *x*, in direction towards *y*.

   normalize(x)

      Reduces *x* to its simplest form.

   number_class(x)

      Returns an indication of the class of *x*.

   plus(x)

      Plus corresponds to the unary prefix plus operator in Python.
      This operation applies the context precision and rounding, so it
      is *not* an identity operation.

   power(x, y[, modulo])

      Return "x" to the power of "y", reduced modulo "modulo" if
      given.

      With two arguments, compute "x**y".  If "x" is negative then "y"
      must be integral.  The result will be inexact unless "y" is
      integral and the result is finite and can be expressed exactly
      in “precision” digits. The result should always be correctly
      rounded, using the rounding mode of the current thread’s
      context.

      With three arguments, compute "(x**y) % modulo".  For the three
      argument form, the following restrictions on the arguments hold:

         * all three arguments must be integral

         * "y" must be nonnegative

         * at least one of "x" or "y" must be nonzero

         * "modulo" must be nonzero and have at most “precision”
           digits

      The value resulting from "Context.power(x, y, modulo)" is equal
      to the value that would be obtained by computing "(x**y) %
      modulo" with unbounded precision, but is computed more
      efficiently.  The exponent of the result is zero, regardless of
      the exponents of "x", "y" and "modulo".  The result is always
      exact.

      Modifié dans la version 2.6: "y" may now be nonintegral in
      "x**y". Stricter requirements for the three-argument version.

   quantize(x, y)

      Returns a value equal to *x* (rounded), having the exponent of
      *y*.

   radix()

      Just returns 10, as this is Decimal, :)

   remainder(x, y)

      Returns the remainder from integer division.

      The sign of the result, if non-zero, is the same as that of the
      original dividend.

   remainder_near(x, y)

      Returns "x - y * n", where *n* is the integer nearest the exact
      value of "x / y" (if the result is 0 then its sign will be the
      sign of *x*).

   rotate(x, y)

      Returns a rotated copy of *x*, *y* times.

   same_quantum(x, y)

      Returns "True" if the two operands have the same exponent.

   scaleb(x, y)

      Returns the first operand after adding the second value its exp.

   shift(x, y)

      Returns a shifted copy of *x*, *y* times.

   sqrt(x)

      Square root of a non-negative number to context precision.

   subtract(x, y)

      Return the difference between *x* and *y*.

   to_eng_string(x)

      Convert to a string, using engineering notation if an exponent
      is needed.

      Engineering notation has an exponent which is a multiple of 3.
      This can leave up to 3 digits to the left of the decimal place
      and may require the addition of either one or two trailing
      zeros.

   to_integral_exact(x)

      Rounds to an integer.

   to_sci_string(x)

      Converts a number to a string using scientific notation.


9.4.4. Signals
==============

Signals represent conditions that arise during computation. Each
corresponds to one context flag and one context trap enabler.

The context flag is set whenever the condition is encountered. After
the computation, flags may be checked for informational purposes (for
instance, to determine whether a computation was exact). After
checking the flags, be sure to clear all flags before starting the
next computation.

If the context’s trap enabler is set for the signal, then the
condition causes a Python exception to be raised.  For example, if the
"DivisionByZero" trap is set, then a "DivisionByZero" exception is
raised upon encountering the condition.

class decimal.Clamped

   Altered an exponent to fit representation constraints.

   Typically, clamping occurs when an exponent falls outside the
   context’s "Emin" and "Emax" limits.  If possible, the exponent is
   reduced to fit by adding zeros to the coefficient.

class decimal.DecimalException

   Base class for other signals and a subclass of "ArithmeticError".

class decimal.DivisionByZero

   Signals the division of a non-infinite number by zero.

   Can occur with division, modulo division, or when raising a number
   to a negative power.  If this signal is not trapped, returns
   "Infinity" or "-Infinity" with the sign determined by the inputs to
   the calculation.

class decimal.Inexact

   Indicates that rounding occurred and the result is not exact.

   Signals when non-zero digits were discarded during rounding. The
   rounded result is returned.  The signal flag or trap is used to
   detect when results are inexact.

class decimal.InvalidOperation

   An invalid operation was performed.

   Indicates that an operation was requested that does not make sense.
   If not trapped, returns "NaN".  Possible causes include:

      Infinity - Infinity
      0 * Infinity
      Infinity / Infinity
      x % 0
      Infinity % x
      x._rescale( non-integer )
      sqrt(-x) and x > 0
      0 ** 0
      x ** (non-integer)
      x ** Infinity

class decimal.Overflow

   Numerical overflow.

   Indicates the exponent is larger than "Emax" after rounding has
   occurred.  If not trapped, the result depends on the rounding mode,
   either pulling inward to the largest representable finite number or
   rounding outward to "Infinity".  In either case, "Inexact" and
   "Rounded" are also signaled.

class decimal.Rounded

   Rounding occurred though possibly no information was lost.

   Signaled whenever rounding discards digits; even if those digits
   are zero (such as rounding "5.00" to "5.0").  If not trapped,
   returns the result unchanged.  This signal is used to detect loss
   of significant digits.

class decimal.Subnormal

   Exponent was lower than "Emin" prior to rounding.

   Occurs when an operation result is subnormal (the exponent is too
   small). If not trapped, returns the result unchanged.

class decimal.Underflow

   Numerical underflow with result rounded to zero.

   Occurs when a subnormal result is pushed to zero by rounding.
   "Inexact" and "Subnormal" are also signaled.

The following table summarizes the hierarchy of signals:

   exceptions.ArithmeticError(exceptions.StandardError)
       DecimalException
           Clamped
           DivisionByZero(DecimalException, exceptions.ZeroDivisionError)
           Inexact
               Overflow(Inexact, Rounded)
               Underflow(Inexact, Rounded, Subnormal)
           InvalidOperation
           Rounded
           Subnormal


9.4.5. Floating Point Notes
===========================


9.4.5.1. Mitigating round-off error with increased precision
------------------------------------------------------------

The use of decimal floating point eliminates decimal representation
error (making it possible to represent "0.1" exactly); however, some
operations can still incur round-off error when non-zero digits exceed
the fixed precision.

The effects of round-off error can be amplified by the addition or
subtraction of nearly offsetting quantities resulting in loss of
significance.  Knuth provides two instructive examples where rounded
floating point arithmetic with insufficient precision causes the
breakdown of the associative and distributive properties of addition:

   # Examples from Seminumerical Algorithms, Section 4.2.2.
   >>> from decimal import Decimal, getcontext
   >>> getcontext().prec = 8

   >>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
   >>> (u + v) + w
   Decimal('9.5111111')
   >>> u + (v + w)
   Decimal('10')

   >>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
   >>> (u*v) + (u*w)
   Decimal('0.01')
   >>> u * (v+w)
   Decimal('0.0060000')

The "decimal" module makes it possible to restore the identities by
expanding the precision sufficiently to avoid loss of significance:

   >>> getcontext().prec = 20
   >>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
   >>> (u + v) + w
   Decimal('9.51111111')
   >>> u + (v + w)
   Decimal('9.51111111')
   >>>
   >>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
   >>> (u*v) + (u*w)
   Decimal('0.0060000')
   >>> u * (v+w)
   Decimal('0.0060000')


9.4.5.2. Special values
-----------------------

The number system for the "decimal" module provides special values
including "NaN", "sNaN", "-Infinity", "Infinity", and two zeros, "+0"
and "-0".

Infinities can be constructed directly with:  "Decimal('Infinity')".
Also, they can arise from dividing by zero when the "DivisionByZero"
signal is not trapped.  Likewise, when the "Overflow" signal is not
trapped, infinity can result from rounding beyond the limits of the
largest representable number.

The infinities are signed (affine) and can be used in arithmetic
operations where they get treated as very large, indeterminate
numbers.  For instance, adding a constant to infinity gives another
infinite result.

Some operations are indeterminate and return "NaN", or if the
"InvalidOperation" signal is trapped, raise an exception.  For
example, "0/0" returns "NaN" which means « not a number ».  This
variety of "NaN" is quiet and, once created, will flow through other
computations always resulting in another "NaN".  This behavior can be
useful for a series of computations that occasionally have missing
inputs — it allows the calculation to proceed while flagging specific
results as invalid.

A variant is "sNaN" which signals rather than remaining quiet after
every operation.  This is a useful return value when an invalid result
needs to interrupt a calculation for special handling.

The behavior of Python’s comparison operators can be a little
surprising where a "NaN" is involved.  A test for equality where one
of the operands is a quiet or signaling "NaN" always returns "False"
(even when doing "Decimal('NaN')==Decimal('NaN')"), while a test for
inequality always returns "True".  An attempt to compare two Decimals
using any of the "<", "<=", ">" or ">=" operators will raise the
"InvalidOperation" signal if either operand is a "NaN", and return
"False" if this signal is not trapped.  Note that the General Decimal
Arithmetic specification does not specify the behavior of direct
comparisons; these rules for comparisons involving a "NaN" were taken
from the IEEE 854 standard (see Table 3 in section 5.7).  To ensure
strict standards-compliance, use the "compare()" and "compare-
signal()" methods instead.

The signed zeros can result from calculations that underflow. They
keep the sign that would have resulted if the calculation had been
carried out to greater precision.  Since their magnitude is zero, both
positive and negative zeros are treated as equal and their sign is
informational.

In addition to the two signed zeros which are distinct yet equal,
there are various representations of zero with differing precisions
yet equivalent in value.  This takes a bit of getting used to.  For an
eye accustomed to normalized floating point representations, it is not
immediately obvious that the following calculation returns a value
equal to zero:

>>> 1 / Decimal('Infinity')
Decimal('0E-1000000026')


9.4.6. Working with threads
===========================

The "getcontext()" function accesses a different "Context" object for
each thread.  Having separate thread contexts means that threads may
make changes (such as "getcontext.prec=10") without interfering with
other threads.

Likewise, the "setcontext()" function automatically assigns its target
to the current thread.

If "setcontext()" has not been called before "getcontext()", then
"getcontext()" will automatically create a new context for use in the
current thread.

The new context is copied from a prototype context called
*DefaultContext*. To control the defaults so that each thread will use
the same values throughout the application, directly modify the
*DefaultContext* object. This should be done *before* any threads are
started so that there won’t be a race condition between threads
calling "getcontext()". For example:

   # Set applicationwide defaults for all threads about to be launched
   DefaultContext.prec = 12
   DefaultContext.rounding = ROUND_DOWN
   DefaultContext.traps = ExtendedContext.traps.copy()
   DefaultContext.traps[InvalidOperation] = 1
   setcontext(DefaultContext)

   # Afterwards, the threads can be started
   t1.start()
   t2.start()
   t3.start()
    . . .


9.4.7. Cas pratiques
====================

Here are a few recipes that serve as utility functions and that
demonstrate ways to work with the "Decimal" class:

   def moneyfmt(value, places=2, curr='', sep=',', dp='.',
                pos='', neg='-', trailneg=''):
       """Convert Decimal to a money formatted string.

       places:  required number of places after the decimal point
       curr:    optional currency symbol before the sign (may be blank)
       sep:     optional grouping separator (comma, period, space, or blank)
       dp:      decimal point indicator (comma or period)
                only specify as blank when places is zero
       pos:     optional sign for positive numbers: '+', space or blank
       neg:     optional sign for negative numbers: '-', '(', space or blank
       trailneg:optional trailing minus indicator:  '-', ')', space or blank

       >>> d = Decimal('-1234567.8901')
       >>> moneyfmt(d, curr='$')
       '-$1,234,567.89'
       >>> moneyfmt(d, places=0, sep='.', dp='', neg='', trailneg='-')
       '1.234.568-'
       >>> moneyfmt(d, curr='$', neg='(', trailneg=')')
       '($1,234,567.89)'
       >>> moneyfmt(Decimal(123456789), sep=' ')
       '123 456 789.00'
       >>> moneyfmt(Decimal('-0.02'), neg='<', trailneg='>')
       '<0.02>'

       """
       q = Decimal(10) ** -places      # 2 places --> '0.01'
       sign, digits, exp = value.quantize(q).as_tuple()
       result = []
       digits = map(str, digits)
       build, next = result.append, digits.pop
       if sign:
           build(trailneg)
       for i in range(places):
           build(next() if digits else '0')
       build(dp)
       if not digits:
           build('0')
       i = 0
       while digits:
           build(next())
           i += 1
           if i == 3 and digits:
               i = 0
               build(sep)
       build(curr)
       build(neg if sign else pos)
       return ''.join(reversed(result))

   def pi():
       """Compute Pi to the current precision.

       >>> print pi()
       3.141592653589793238462643383

       """
       getcontext().prec += 2  # extra digits for intermediate steps
       three = Decimal(3)      # substitute "three=3.0" for regular floats
       lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24
       while s != lasts:
           lasts = s
           n, na = n+na, na+8
           d, da = d+da, da+32
           t = (t * n) / d
           s += t
       getcontext().prec -= 2
       return +s               # unary plus applies the new precision

   def exp(x):
       """Return e raised to the power of x.  Result type matches input type.

       >>> print exp(Decimal(1))
       2.718281828459045235360287471
       >>> print exp(Decimal(2))
       7.389056098930650227230427461
       >>> print exp(2.0)
       7.38905609893
       >>> print exp(2+0j)
       (7.38905609893+0j)

       """
       getcontext().prec += 2
       i, lasts, s, fact, num = 0, 0, 1, 1, 1
       while s != lasts:
           lasts = s
           i += 1
           fact *= i
           num *= x
           s += num / fact
       getcontext().prec -= 2
       return +s

   def cos(x):
       """Return the cosine of x as measured in radians.

       >>> print cos(Decimal('0.5'))
       0.8775825618903727161162815826
       >>> print cos(0.5)
       0.87758256189
       >>> print cos(0.5+0j)
       (0.87758256189+0j)

       """
       getcontext().prec += 2
       i, lasts, s, fact, num, sign = 0, 0, 1, 1, 1, 1
       while s != lasts:
           lasts = s
           i += 2
           fact *= i * (i-1)
           num *= x * x
           sign *= -1
           s += num / fact * sign
       getcontext().prec -= 2
       return +s

   def sin(x):
       """Return the sine of x as measured in radians.

       >>> print sin(Decimal('0.5'))
       0.4794255386042030002732879352
       >>> print sin(0.5)
       0.479425538604
       >>> print sin(0.5+0j)
       (0.479425538604+0j)

       """
       getcontext().prec += 2
       i, lasts, s, fact, num, sign = 1, 0, x, 1, x, 1
       while s != lasts:
           lasts = s
           i += 2
           fact *= i * (i-1)
           num *= x * x
           sign *= -1
           s += num / fact * sign
       getcontext().prec -= 2
       return +s


9.4.8. FAQ *decimal*
====================

Q. C’est fastidieux de taper "decimal.Decimal('1234.5')".  Y a-t-il un
moyen de réduire la frappe quand on utilise l’interpréteur interactif
?

R. Certains utilisateurs abrègent le constructeur en une seule lettre
:

>>> D = decimal.Decimal
>>> D('1.23') + D('3.45')
Decimal('4.68')

Q. In a fixed-point application with two decimal places, some inputs
have many places and need to be rounded.  Others are not supposed to
have excess digits and need to be validated.  What methods should be
used?

A. The "quantize()" method rounds to a fixed number of decimal places.
If the "Inexact" trap is set, it is also useful for validation:

>>> TWOPLACES = Decimal(10) ** -2       # same as Decimal('0.01')

>>> # Round to two places
>>> Decimal('3.214').quantize(TWOPLACES)
Decimal('3.21')

>>> # Validate that a number does not exceed two places
>>> Decimal('3.21').quantize(TWOPLACES, context=Context(traps=[Inexact]))
Decimal('3.21')

>>> Decimal('3.214').quantize(TWOPLACES, context=Context(traps=[Inexact]))
Traceback (most recent call last):
   ...
Inexact: None

Q. Once I have valid two place inputs, how do I maintain that
invariant throughout an application?

A. Some operations like addition, subtraction, and multiplication by
an integer will automatically preserve fixed point.  Others
operations, like division and non-integer multiplication, will change
the number of decimal places and need to be followed-up with a
"quantize()" step:

>>> a = Decimal('102.72')           # Initial fixed-point values
>>> b = Decimal('3.17')
>>> a + b                           # Addition preserves fixed-point
Decimal('105.89')
>>> a - b
Decimal('99.55')
>>> a * 42                          # So does integer multiplication
Decimal('4314.24')
>>> (a * b).quantize(TWOPLACES)     # Must quantize non-integer multiplication
Decimal('325.62')
>>> (b / a).quantize(TWOPLACES)     # And quantize division
Decimal('0.03')

In developing fixed-point applications, it is convenient to define
functions to handle the "quantize()" step:

>>> def mul(x, y, fp=TWOPLACES):
...     return (x * y).quantize(fp)
>>> def div(x, y, fp=TWOPLACES):
...     return (x / y).quantize(fp)

>>> mul(a, b)                       # Automatically preserve fixed-point
Decimal('325.62')
>>> div(b, a)
Decimal('0.03')

Q. There are many ways to express the same value.  The numbers "200",
"200.000", "2E2", and "02E+4" all have the same value at various
precisions. Is there a way to transform them to a single recognizable
canonical value?

A. The "normalize()" method maps all equivalent values to a single
representative:

>>> values = map(Decimal, '200 200.000 2E2 .02E+4'.split())
>>> [v.normalize() for v in values]
[Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2')]

Q. Some decimal values always print with exponential notation.  Is
there a way to get a non-exponential representation?

A. For some values, exponential notation is the only way to express
the number of significant places in the coefficient.  For example,
expressing "5.0E+3" as "5000" keeps the value constant but cannot show
the original’s two-place significance.

If an application does not care about tracking significance, it is
easy to remove the exponent and trailing zeros, losing significance,
but keeping the value unchanged:

   def remove_exponent(d):
       '''Remove exponent and trailing zeros.

       >>> remove_exponent(Decimal('5E+3'))
       Decimal('5000')

       '''
       return d.quantize(Decimal(1)) if d == d.to_integral() else d.normalize()

Q. Is there a way to convert a regular float to a "Decimal"?

A. Yes, any binary floating point number can be exactly expressed as a
Decimal though an exact conversion may take more precision than
intuition would suggest:

   >>> Decimal(math.pi)
   Decimal('3.141592653589793115997963468544185161590576171875')

Q. Within a complex calculation, how can I make sure that I haven’t
gotten a spurious result because of insufficient precision or rounding
anomalies.

A. The decimal module makes it easy to test results.  A best practice
is to re-run calculations using greater precision and with various
rounding modes. Widely differing results indicate insufficient
precision, rounding mode issues, ill-conditioned inputs, or a
numerically unstable algorithm.

Q. I noticed that context precision is applied to the results of
operations but not to the inputs.  Is there anything to watch out for
when mixing values of different precisions?

A. Yes.  The principle is that all values are considered to be exact
and so is the arithmetic on those values.  Only the results are
rounded.  The advantage for inputs is that « what you type is what you
get ».  A disadvantage is that the results can look odd if you forget
that the inputs haven’t been rounded:

   >>> getcontext().prec = 3
   >>> Decimal('3.104') + Decimal('2.104')
   Decimal('5.21')
   >>> Decimal('3.104') + Decimal('0.000') + Decimal('2.104')
   Decimal('5.20')

The solution is either to increase precision or to force rounding of
inputs using the unary plus operation:

   >>> getcontext().prec = 3
   >>> +Decimal('1.23456789')      # unary plus triggers rounding
   Decimal('1.23')

Alternatively, inputs can be rounded upon creation using the
"Context.create_decimal()" method:

>>> Context(prec=5, rounding=ROUND_DOWN).create_decimal('1.2345678')
Decimal('1.2345')
