# 9.4. `decimal` — 十进制定点和浮点运算

源码： Lib/decimal.py

`decimal` 模块为快速正确舍入的十进制浮点运算提供支持。 它提供了 `float` 数据类型以外的几个优点：

- Decimal “基于一个浮点模型，它是为人们设计的，并且必然具有最重要的指导原则 —— 计算机必须提供与人们在学校学习的算法相同的算法。” —— 摘自十进制算术规范。

- 十进制数字可以准确表示。 相比之下，数字如 `1.1` 和 `2.2` 在二进制浮点中没有精确的表示。 最终用户通常不希望``1.1 + 2.2``显示为 `3.3000000000000003` ，就像二进制浮点一样。

- 精确性延续到算术中。 在十进制浮点数中，`0.1 + 0.1 + 0.1 - 0.3` 恰好等于零。 在二进制浮点数中，结果为 `5.5511151231257827e-017` 。 虽然接近于零，但差异妨碍了可靠的相等性检验，并且差异可能会累积。 因此，在具有严格相等不变量的会计应用程序中， decimal 是首选。

- 十进制模块包含一个重要位置的概念，因此 `1.30 + 1.20` 是 `2.50` 。 保留尾随零以表示重要性。 这是货币申请的惯常陈述。 对于乘法，“教科书”方法使用被乘数中的所有数字。 例如， `1.3 * 1.2` 给出 `1.56` 而 `1.30 * 1.20` 给出 `1.5600` 。

- 与基于硬件的二进制浮点不同，十进制模块具有用户可更改的精度（默认为28个位置），可以与给定问题所需的一样大：

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

- 二进制和十进制浮点都是根据已发布的标准实现的。 虽然内置浮点类型只公开其功能的一小部分，但十进制模块公开了标准的所有必需部分。 在需要时，程序员可以完全控制舍入和信号处理。 这包括通过使用异常来阻止任何不精确操作来强制执行精确算术的选项。

- 十进制模块旨在支持“无偏见，精确的非连续十进制算术（有时称为定点算术）和舍入浮点算术”。 —— 摘自十进制算术规范。

模块设计以三个概念为中心：十进制数，算术上下文和信号。

十进制数是不可变的。 它有一个符号，系数数字和一个指数。 为了保持重要性，系数数字不会截断尾随零。十进制数也包括特殊值，例如 `Infinity` ，`-Infinity` ，和 `NaN` 。 该标准还区分 `-0` 和 `+0` 。

算术的上下文是指定精度、舍入规则、指数限制、指示操作结果的标志以及确定符号是否被视为异常的陷阱启用器的环境。 舍入选项包括 `ROUND_CEILING` 、 `ROUND_DOWN` 、 `ROUND_FLOOR` 、 `ROUND_HALF_DOWN`, `ROUND_HALF_EVEN` 、 `ROUND_HALF_UP` 、 `ROUND_UP` 以及 `ROUND_05UP`.

信号是在计算过程中出现的异常条件组。 根据应用程序的需要，信号可能会被忽略，被视为信息，或被视为异常。 十进制模块中的信号有：`Clamped` 、 `InvalidOperation` 、 `DivisionByZero` 、 `Inexact` 、 `Rounded` 、 `Subnormal` 、 `Overflow` 、 `Underflow` 以及 `FloatOperation` 。

对于每个信号，都有一个标志和一个陷阱启动器。 遇到信号时，其标志设置为 1 ，然后，如果陷阱启用器设置为 1 ，则引发异常。 标志是粘性的，因此用户需要在监控计算之前重置它们。

参见

- IBM的通用十进制算术规范， The General Decimal Arithmetic Specification.

## 9.4.1. 快速入门教程

通常使用小数的开始是导入模块，使用 `getcontext()` 查看当前上下文，并在必要时为精度、舍入或启用的陷阱设置新值:

    >>> 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

可以从整数、字符串、浮点数或元组构造十进制实例。 从整数或浮点构造将执行该整数或浮点值的精确转换。 十进制数包括特殊值，例如 `NaN` 代表“非数字”，正的和负的 `Infinity`，和 `-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')

如果 `FloatOperation` 信号被捕获，构造函数中的小数和浮点数的意外混合或排序比较会引发异常

    >>> 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

3.3 新版功能.

新 Decimal 的重要性仅由输入的位数决定。 上下文精度和舍入仅在算术运算期间发挥作用。

    >>> 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')

如果超出了C版本的内部限制，则构造一个十进制将引发 `InvalidOperation`

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

在 3.3 版更改.

小数与 Python 的其余部分很好地交互。 这是一个小的十进制浮点飞行杂技团：

    >>> 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')

Decimal 也可以使用一些数学函数：

    >>> 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')

`quantize()` 方法将数字四舍五入为固定指数。 此方法对于将结果舍入到固定的位置的货币应用程序非常有用：

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

如上所示，`getcontext()` 函数访问当前上下文并允许更改设置。 这种方法满足大多数应用程序的需求。

对于更高级的工作，使用 Context() 构造函数创建备用上下文可能很有用。 要使用备用活动，请使用 `setcontext()` 函数。

根据标准，`decimal` 模块提供了两个现成的标准上下文 `BasicContext` 和 `ExtendedContext` 。 前者对调试特别有用，因为许多陷阱都已启用：

    >>> 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

上下文还具有用于监视计算期间遇到的异常情况的信号标志。 标志保持设置直到明确清除，因此最好通过使用 `clear_flags()` 方法清除每组受监控计算之前的标志。:

    >>> 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=[])

flags 条目显示对 `Pi` 的有理逼近被舍入（超出上下文精度的数字被抛弃）并且结果是不精确的（一些丢弃的数字不为零）。

使用上下文的 `traps` 字段中的字典设置单个陷阱：

    >>> 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

大多数程序仅在程序开始时调整当前上下文一次。 并且，在许多应用程序中，数据在循环内单个强制转换为 `Decimal` 。 通过创建上下文集和小数，程序的大部分操作数据与其他 Python 数字类型没有区别。

## 9.4.2. Decimal 对象

class `decimal.``Decimal`(value="0", context=None)

  根据 value 构造一个新的 `Decimal` 对象。

  value 可以是整数，字符串，元组，`float` ，或另一个 `Decimal` 对象。 如果没有给出 value，则返回 `Decimal('0')`。 如果 value 是一个字符串，它应该在前导和尾随空格字符以及下划线被删除之后符合十进制数字字符串语法:

      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

  当上面出现 `digit` 时也允许其他十进制数码。 其中包括来自各种其他语言系统的十进制数码（例如阿拉伯-印地语和天城文的数码）以及全宽数码 `'\uff10'` 到 `'\uff19'`。

  如果 value 是一个 `tuple` ，它应该有三个组件，一个符号（ `0` 表示正数或 `1` 表示负数），一个数字的 `tuple` 和整数指数。 例如， `Decimal((0, (1, 4, 1, 4), -3))` 返回 `Decimal('1.414')`。

  如果 value 是 `float` ，则二进制浮点值无损地转换为其精确的十进制等效值。 此转换通常需要53位或更多位数的精度。 例如， `Decimal(float('1.1'))` 转换为``Decimal(‘1.100000000000000088817841970012523233890533447265625’)``。

  context 精度不会影响存储的位数。 这完全由 value 中的位数决定。 例如，`Decimal('3.00000')` 记录所有五个零，即使上下文精度只有三。

  context 参数的目的是确定 value 是格式错误的字符串时该怎么做。 如果上下文陷阱 `InvalidOperation`，则引发异常；否则，构造函数返回一个新的 Decimal，其值为 `NaN`。

  构造完成后， `Decimal` 对象是不可变的。

  在 3.2 版更改: 现在允许构造函数的参数为 `float` 实例。

  在 3.3 版更改: `float` 参数在设置 `FloatOperation` 陷阱时引发异常。 默认情况下，陷阱已关闭。

  在 3.6 版更改: 允许下划线进行分组，就像代码中的整数和浮点文字一样。

  十进制浮点对象与其他内置数值类型共享许多属性，例如 `float` 和 `int` 。 所有常用的数学运算和特殊方法都适用。 同样，十进制对象可以复制、pickle、打印、用作字典键、用作集合元素、比较、排序和强制转换为另一种类型（例如 `float` 或 `int` ）。

  算术对十进制对象和算术对整数和浮点数有一些小的差别。 当余数运算符 `%` 应用于Decimal对象时，结果的符号是 被除数 的符号，而不是除数的符号:

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

  整数除法运算符 `//` 的行为类似，返回真商的整数部分（截断为零）而不是它的向下取整，以便保留通常的标识 `x == (x // y) * y + x % y`:

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

  `%` 和 `//` 运算符实现了 `remainder` 和 `divide-integer` 操作（分别），如规范中所述。

  十进制对象通常不能与浮点数或 `fractions.Fraction` 实例在算术运算中结合使用：例如,尝试将 `Decimal` 加到 `float` ，将引发 `TypeError`。 但是，可以使用 Python 的比较运算符来比较 `Decimal` 实例 `x` 和另一个数字 `y` 。 这样可以避免在对不同类型的数字进行相等比较时混淆结果。

  在 3.2 版更改: 现在完全支持 `Decimal` 实例和其他数字类型之间的混合类型比较。

  除了标准的数字属性，十进制浮点对象还有许多专门的方法：

  `adjusted`()

    在移出系数最右边的数字之后返回调整后的指数，直到只剩下前导数字：`Decimal('321e+5').adjusted()` 返回 7 。 用于确定最高有效位相对于小数点的位置。

  `as_integer_ratio`()

    返回一对 `(n, d)` 整数，表示给定的 `Decimal` 实例作为分数、最简形式项并带有正分母:

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

    转换是精确的。 在 Infinity 上引发 OverflowError ，在 NaN 上引起 ValueError 。

  3.6 新版功能.

  `as_tuple`()

    返回一个 named tuple 表示的数字： `DecimalTuple(sign, digits, exponent)`。

  `canonical`()

    返回参数的规范编码。 目前，一个 `Decimal` 实例的编码始终是规范的，因此该操作返回其参数不变。

  `compare`(other, context=None)

    比较两个 Decimal 实例的值。 `compare()` 返回一个 Decimal 实例，如果任一操作数是 NaN ，那么结果是 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)

    除了所有 NaN 信号之外，此操作与 `compare()` 方法相同。 也就是说，如果两个操作数都不是信令NaN，那么任何静默的 NaN 操作数都被视为信令NaN。

  `compare_total`(other, context=None)

    使用它们的抽象表示而不是它们的数值来比较两个操作数。 类似于 `compare()` 方法，但结果给出了一个总排序 `Decimal` 实例。 两个 `Decimal` 实例具有相同的数值但不同的表示形式在此排序中比较不相等：

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

    静默和发出信号的 NaN 也包括在总排序中。 这个函数的结果是 `Decimal('0')` 如果两个操作数具有相同的表示，或是 `Decimal('-1')` 如果第一个操作数的总顺序低于第二个操作数，或是 `Decimal('1')` 如果第一个操作数在总顺序中高于第二个操作数。 有关总排序的详细信息，请参阅规范。

    此操作不受上下文影响且静默：不更改任何标志且不执行舍入。 作为例外，如果无法准确转换第二个操作数，则C版本可能会引发InvalidOperation。

  `compare_total_mag`(other, context=None)

    比较两个操作数使用它们的抽象表示而不是它们的值，如 `compare_total()`，但忽略每个操作数的符号。 `x.compare_total_mag(y)` 相当于 `x.copy_abs().compare_total(y.copy_abs())`。

    此操作不受上下文影响且静默：不更改任何标志且不执行舍入。 作为例外，如果无法准确转换第二个操作数，则C版本可能会引发InvalidOperation。

  `conjugate`()

    只返回self，这种方法只符合 Decimal 规范。

  `copy_abs`()

    返回参数的绝对值。 此操作不受上下文影响并且是静默的：没有更改标志且不执行舍入。

  `copy_negate`()

    回到参数的否定。 此操作不受上下文影响并且是静默的：没有标志更改且不执行舍入。

  `copy_sign`(other, context=None)

    返回第一个操作数的副本，其符号设置为与第二个操作数的符号相同。 例如：

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

    此操作不受上下文影响且静默：不更改任何标志且不执行舍入。 作为例外，如果无法准确转换第二个操作数，则C版本可能会引发InvalidOperation。

  `exp`(context=None)

    返回给定数字的（自然）指数函数``e**x``的值。结果使用 `ROUND_HALF_EVEN` 舍入模式正确舍入。

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

  `from_float`(f)

    将浮点数转换为十进制数的类方法。

    注意， Decimal.from_float(0.1) 与 Decimal(‘0.1’) 不同。 由于 0.1 在二进制浮点中不能精确表示，因此该值存储为最接近的可表示值，即 0x1.999999999999ap-4 。 十进制的等效值是`0.1000000000000000055511151231257827021181583404541015625`。

    注解

    从 Python 3.2 开始，`Decimal` 实例也可以直接从 `float` 构造。

        >>> 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')

    3.1 新版功能.

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

    混合乘法加法。 返回 self*other+third ，中间乘积 self*other 没有四舍五入。

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

  `is_canonical`()

    如果参数是规范的，则为返回 `True`，否则为 `False` 。 目前，`Decimal` 实例总是规范的，所以这个操作总是返回 `True` 。

  `is_finite`()

    如果参数是一个有限的数，则返回为 `True` ；如果参数为无穷大或 NaN ，则返回为 `False`。

  `is_infinite`()

    如果参数为正负无穷大，则返回为 `True` ，否则为 `False` 。

  `is_nan`()

    如果参数为 NaN （无论是否静默），则返回为 `True` ，否则为 `False` 。

  `is_normal`(context=None)

    如果参数是一个有限正规数，返回 `True`，如果参数是0、次正规数、无穷大或是NaN，返回 `False`。

  `is_qnan`()

    如果参数为静默 NaN，返回 `True`，否则返回 `False`。

  `is_signed`()

    如果参数带有负号，则返回为 `True`，否则返回 `False`。注意，0 和 NaN 都可带有符号。

  `is_snan`()

    如果参数为显式 NaN，则返回 `True`，否则返回 `False`。

  `is_subnormal`(context=None)

    如果参数为次正规数，则返回 `True`，否则返回 `False`。

  `is_zero`()

    如果参数是0（正负皆可），则返回 `True`，否则返回 `False`。

  `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`.

    此操作不受上下文影响且静默：不更改任何标志且不执行舍入。 作为例外，如果无法准确转换第二个操作数，则C版本可能会引发InvalidOperation。

  `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`.

    3.3 新版功能.

  `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

    3.1 新版功能.

  `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)

    Returns the absolute value of 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.

    在 3.3 版更改: 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. 常数

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. Recipes

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. Decimal FAQ

Q. It is cumbersome to type `decimal.Decimal('1234.5')`. Is there a way to minimize typing when using the interactive interpreter?

A. Some users abbreviate the constructor to just a single letter:

    >>> 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')