# 9.4. `decimal` — Arithmétique décimale en virgule fixe et flottante

Code source : Lib/decimal.py

Le module `decimal` fournit une arithmétique en virgule flottante rapide et produisant des arrondis mathématiquement corrects. Il possède plusieurs avantages en comparaison au type `float` :

- 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.

- Les nombres décimaux peuvent être représentés exactement en base décimale flottante. En revanche, des nombres tels que `1.1` ou `1.2` n’ont pas de représentation exacte en base binaire flottante. L’utilisateur final ne s’attend typiquement pas à obtenir `3.3000000000000003` lorsqu’il saisit `1.1 + 2.2`, ce qui se passe en arithmétique binaire à virgule flottante.

- 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`.

Les signaux sont des groupes de conditions exceptionnelles qui surviennent durant le calcul. Selon les besoins de l’application, les signaux peuvent être ignorés, considérés comme de l’information, ou bien traités comme des exceptions. Les signaux dans le module `decimal` sont : `Clamped`, `InvalidOperation`, `DivisionByZero`, `Inexact`, `Rounded`, `Subnormal`, `Overflow`, `Underflow` et `FloatOperation`.

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

- La spécification d’IBM sur l’arithmétique décimale : 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=-999999, Emax=999999,
            capitals=1, clamp=0, flags=[], traps=[Overflow, DivisionByZero,
            InvalidOperation])

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

Les instances de `Decimal` peuvent être construites avec des `int`, des `str`, des `floats` ou des `tuples`. La construction depuis un entier ou un `float` effectue la conversion exacte de cet entier ou de ce `float`. Les nombres décimaux incluent des valeurs spéciales telles que `NaN` qui signifie en anglais « Not a number », en français « pas un nombre », des `Infinity` positifs ou négatifs et `-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('1.4142135623730951')
    >>> Decimal(2) ** Decimal('0.5')
    Decimal('1.414213562373095048801688724')
    >>> Decimal('NaN')
    Decimal('NaN')
    >>> Decimal('-Infinity')
    Decimal('-Infinity')

Si un signal `FloatOperation` est détecté, un mélange accidentel d’objets `Decimal` et de `float` dans les constructeurs ou des opérations de comparaisons, une exception est levée :

    >>> c = getcontext()
    >>> c.traps[FloatOperation] = True
    >>> Decimal(3.14)
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
    decimal.FloatOperation: [<class 'decimal.FloatOperation'>]
    >>> Decimal('3.5') < 3.7
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
    decimal.FloatOperation: [<class 'decimal.FloatOperation'>]
    >>> Decimal('3.5') == 3.5
    True

Nouveau dans la version 3.3.

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')

Si les limites internes de la version en C sont dépassées, la construction d’un objet décimal lève l’exception `InvalidOperation` :

    >>> Decimal("1e9999999999999999999")
    Traceback (most recent call last):
      File "<stdin>", line 1, in <module>
    decimal.InvalidOperation: [<class 'decimal.InvalidOperation'>]

Modifié dans la version 3.3.

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

    >>> data = list(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)
    Decimal('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=-999999, Emax=999999,
            capitals=1, clamp=0, 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=-999999, Emax=999999,
            capitals=1, clamp=0, flags=[Inexact, Rounded], 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="0", context=None)

  Construire un nouvel objet `Decimal` à partir de value.

  value peut être un entier, une chaîne de caractères, un tuple, `float`, ou une autre instance de `Decimal`. Si value n’est pas fourni, le constructeur renvoie `Decimal('0')`. Si value est une chaîne de caractère, elle doit correspondre à la syntaxe décimale en dehors des espaces de début et de fin, ou des tirets bas, qui sont enlevés :

      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

  Les chiffres codés en Unicode sont aussi autorisés, là ou `digit` apparaît. Cela inclut des chiffres décimaux venant d’autres alphabets (par exemple les chiffres indo-arabes ou Devanagari) ainsi que les chiffres de pleine largeur `'\uff10'` jusqu’à `'\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 3.2: L’argument du constructeur peut désormais être un objet `float`.

  Modifié dans la version 3.3: Un argument `float` lève une exception si l’option `FloatOperation` est activé. Par défaut l’option ne l’est pas.

  Modifié dans la version 3.6: Les tirets bas sont autorisés pour regrouper, tout comme pour l’arithmétique en virgule fixe et flottante.

  Les objets `Decimal` partagent beaucoup de propriétés avec les autres types numériques natifs tels que `float` et `int`. Toutes les opérations mathématiques et méthodes sont conservées. De même les objets `Decimal` peuvent être copiés, sérialisés via le module `pickle`, affichés, utilisés comme clé de dictionnaire, éléments d’ensembles, comparés, classés, et convertis vers un autre type (tel que `float` ou `int`).

  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.

  Les objets `Decimal` ne peuvent généralement pas être combinés avec des `float` ou des objets `fractions.Fraction` lors d’opérations arithmétiques : tout addition entre un `Decimal` avec un `float`, par exemple, lève une exception `TypeError`. Cependant, il est possible d’utiliser les opérateurs de comparaison entre instances de `Decimal` avec les autres types numériques. Cela évite d’avoir des résultats absurdes lors des tests d’égalité entre différents types.

  Modifié dans la version 3.2: Les comparaisons inter-types entre `Decimal` et les autres types numériques sont désormais intégralement gérés.

  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_integer_ratio`()

    Return a pair `(n, d)` of integers that represent the given `Decimal` instance as a fraction, in lowest terms and with a positive denominator:

        >>> Decimal('-3.14').as_integer_ratio()
        (-157, 50)

    La conversion est exacte. Lève une `OverflowError` sur l’infini et `ValueError` sur les `Nan`’s.

  Nouveau dans la version 3.6.

  `as_tuple`()

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

  `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.

  `compare`(other, context=None)

    Compare the values of two Decimal instances. `compare()` returns a Decimal instance, 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=None)

    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.

  `compare_total`(other, context=None)

    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.

    This operation is unaffected by context and is quiet: no flags are changed and no rounding is performed. As an exception, the C version may raise InvalidOperation if the second operand cannot be converted exactly.

  `compare_total_mag`(other, context=None)

    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())`.

    This operation is unaffected by context and is quiet: no flags are changed and no rounding is performed. As an exception, the C version may raise InvalidOperation if the second operand cannot be converted exactly.

  `conjugate`()

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

  `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.

  `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.

  `copy_sign`(other, context=None)

    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 context and is quiet: no flags are changed and no rounding is performed. As an exception, the C version may raise InvalidOperation if the second operand cannot be converted exactly.

  `exp`(context=None)

    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')

  `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 3.2 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 3.1.

  `fma`(other, third, context=None)

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

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

  `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`.

  `is_finite`()

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

  `is_infinite`()

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

  `is_nan`()

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

  `is_normal`(context=None)

    Return `True` if the argument is a normal finite number. Return `False` if the argument is zero, subnormal, infinite or a NaN.

  `is_qnan`()

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

  `is_signed`()

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

  `is_snan`()

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

  `is_subnormal`(context=None)

    Return `True` if the argument is subnormal, and `False` otherwise.

  `is_zero`()

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

  `ln`(context=None)

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

  `log10`(context=None)

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

  `logb`(context=None)

    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.

  `logical_and`(other, context=None)

    `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.

  `logical_invert`(context=None)

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

  `logical_or`(other, context=None)

    `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.

  `logical_xor`(other, context=None)

    `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.

  `max`(other, context=None)

    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=None)

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

  `min`(other, context=None)

    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=None)

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

  `next_minus`(context=None)

    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.

  `next_plus`(context=None)

    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.

  `next_toward`(other, context=None)

    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.

  `normalize`(context=None)

    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=None)

    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.

  `quantize`(exp, rounding=None, context=None)

    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.

    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.

  `remainder_near`(other, context=None)

    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=None)

    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.

  `same_quantum`(other, context=None)

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

    This operation is unaffected by context and is quiet: no flags are changed and no rounding is performed. As an exception, the C version may raise InvalidOperation if the second operand cannot be converted exactly.

  `scaleb`(other, context=None)

    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.

  `shift`(other, context=None)

    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.

  `sqrt`(context=None)

    Return the square root of the argument to full precision.

  `to_eng_string`(context=None)

    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=None, context=None)

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

  `to_integral_exact`(rounding=None, context=None)

    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.

  `to_integral_value`(rounding=None, context=None)

    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.

### 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.

You can also use the `with` statement and the `localcontext()` function to temporarily change the active context.

`decimal.``localcontext`(ctx=None)

  Return a context manager that will set the current context for the active thread to a copy of ctx 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.

  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

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 `prec`=`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, Emin=None, Emax=None, capitals=None, clamp=None, flags=None, traps=None)

  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.

  prec is an integer in the range [`1`, `MAX_PREC`] that sets the precision for arithmetic operations in the context.

  The rounding option is one of the constants listed in the section Rounding Modes.

  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. Emin must be in the range [`MIN_EMIN`, `0`], Emax in the range [`0`, `MAX_EMAX`].

  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')`.

  The clamp field is either `0` (the default) or `1`. If set to `1`, the exponent `e` of a `Decimal` instance representable in this context is strictly limited to the range `Emin - prec + 1 <= e <= Emax - prec + 1`. If clamp is `0` then a weaker condition holds: the adjusted exponent of the `Decimal` instance is at most `Emax`. When clamp is `1`, a large normal number will, where possible, have its exponent reduced and a corresponding number of zeros added to its coefficient, in order to fit the exponent constraints; this preserves the value of the number but loses information about significant trailing zeros. For example:

      >>> Context(prec=6, Emax=999, clamp=1).create_decimal('1.23e999')
      Decimal('1.23000E+999')

  A clamp value of `1` allows compatibility with the fixed-width decimal interchange formats specified in IEEE 754.

  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`) anywhere that a Decimal instance is accepted.

  `clear_flags`()

    Resets all of the flags to `0`.

  `clear_traps`()

    Resets all of the traps to `0`.

    Nouveau dans la version 3.3.

  `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 or underscores are 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):
            ...
        decimal.Inexact: None

    Nouveau dans la version 3.1.

  `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=None)

    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 rounding mode of the context is used. Results are always correctly-rounded in the Python version.

    Modifié dans la version 3.3: The C module computes `power()` in terms of the correctly-rounded `exp()` and `ln()` functions. The result is well-defined but only « almost always correctly-rounded ».

    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.

  `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. Constantes

The constants in this section are only relevant for the C module. They are also included in the pure Python version for compatibility.

| 32-bit | 64-bit
--- | --- | ---
`decimal.``MAX_PREC` | `425000000` | `999999999999999999`
`decimal.``MAX_EMAX` | `425000000` | `999999999999999999`
`decimal.``MIN_EMIN` | `-425000000` | `-999999999999999999`
`decimal.``MIN_ETINY` | `-849999999` | `-1999999999999999997`

`decimal.``HAVE_THREADS`

  The default value is `True`. If Python is compiled without threads, the C version automatically disables the expensive thread local context machinery. In this case, the value is `False`.

## 9.4.5. Rounding modes

`decimal.``ROUND_CEILING`

  Round towards `Infinity`.

`decimal.``ROUND_DOWN`

  Round towards zero.

`decimal.``ROUND_FLOOR`

  Round towards `-Infinity`.

`decimal.``ROUND_HALF_DOWN`

  Round to nearest with ties going towards zero.

`decimal.``ROUND_HALF_EVEN`

  Round to nearest with ties going to nearest even integer.

`decimal.``ROUND_HALF_UP`

  Round to nearest with ties going away from zero.

`decimal.``ROUND_UP`

  Round away from zero.

`decimal.``ROUND_05UP`

  Round away from zero if last digit after rounding towards zero would have been 0 or 5; otherwise round towards zero.

## 9.4.6. 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
      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.

class `decimal.``FloatOperation`

  Enable stricter semantics for mixing floats and Decimals.

  If the signal is not trapped (default), mixing floats and Decimals is permitted in the `Decimal` constructor, `create_decimal()` and all comparison operators. Both conversion and comparisons are exact. Any occurrence of a mixed operation is silently recorded by setting `FloatOperation` in the context flags. Explicit conversions with `from_float()` or `create_decimal_from_float()` do not set the flag.

  Otherwise (the signal is trapped), only equality comparisons and explicit conversions are silent. All other mixed operations raise `FloatOperation`.

The following table summarizes the hierarchy of signals:

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

## 9.4.7. Floating Point Notes

### 9.4.7.1. 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')

### 9.4.7.2. 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-1000026')

## 9.4.8. 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()
     . . .

## 9.4.9. 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 = list(map(str, digits))
        build, next = result.append, digits.pop
        if sign:
            build(trailneg)
        for i in range(places):
            build(next() if digits else '0')
        if places:
            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.

        The Taylor series approximation works best for a small value of x.
        For larger values, first compute x = x % (2 * pi).

        >>> 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.

        The Taylor series approximation works best for a small value of x.
        For larger values, first compute x = x % (2 * pi).

        >>> 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

## 9.4.10. 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 zeroes, losing significance, but keeping the value unchanged:

    >>> def remove_exponent(d):
    ...     return d.quantize(Decimal(1)) if d == d.to_integral() else d.normalize()

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

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')