"decimal" --- Decimal fixed point and floating point arithmetic
***************************************************************

**Source code:** Lib/decimal.py

======================================================================

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

* Decimal "is based on a floating-point model which was designed
  with people in mind, and necessarily has a paramount guiding
  principle -- computers must provide an arithmetic that works in the
  same way as the arithmetic that people learn at school." -- excerpt
  from the decimal arithmetic specification.

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

* The exactness carries over into arithmetic.  In decimal floating
  point, "0.1 + 0.1 + 0.1 - 0.3" is exactly equal to zero.  In binary
  floating point, the result is "5.5511151231257827e-017".  While near
  to zero, the differences prevent reliable equality testing and
  differences can accumulate. For this reason, decimal is preferred in
  accounting applications which have strict equality invariants.

* The decimal module incorporates a notion of significant places so
  that "1.30 + 1.20" is "2.50".  The trailing zero is kept to indicate
  significance. This is the customary presentation for monetary
  applications. For multiplication, the "schoolbook" approach uses all
  the figures in the multiplicands.  For instance, "1.3 * 1.2" gives
  "1.56" while "1.30 * 1.20" gives "1.5600".

* Unlike hardware based binary floating point, the decimal module
  has a user alterable precision (defaulting to 28 places) which can
  be as large as needed for a given problem:

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

* Both binary and decimal floating point are implemented in terms of
  published standards.  While the built-in float type exposes only a
  modest portion of its capabilities, the decimal module exposes all
  required parts of the standard. When needed, the programmer has full
  control over rounding and signal handling. This includes an option
  to enforce exact arithmetic by using exceptions to block any inexact
  operations.

* The decimal module was designed to support "without prejudice,
  both exact unrounded decimal arithmetic (sometimes called fixed-
  point arithmetic) and rounded floating-point arithmetic."  --
  excerpt from the decimal arithmetic specification.

The module design is centered around three concepts:  the decimal
number, the context for arithmetic, and signals.

A decimal number is immutable.  It has a sign, coefficient digits, and
an exponent.  To preserve significance, the coefficient digits do not
truncate trailing zeros.  Decimals also include special values such as
"Infinity", "-Infinity", and "NaN".  The standard also differentiates
"-0" from "+0".

The context for arithmetic is an environment specifying precision,
rounding rules, limits on exponents, flags indicating the results of
operations, and trap enablers which determine whether signals are
treated as exceptions.  Rounding options include "ROUND_CEILING",
"ROUND_DOWN", "ROUND_FLOOR", "ROUND_HALF_DOWN", "ROUND_HALF_EVEN",
"ROUND_HALF_UP", "ROUND_UP", and "ROUND_05UP".

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

For each signal there is a flag and a trap enabler.  When a signal is
encountered, its flag is set to one, then, if the trap enabler is set
to one, an exception is raised.  Flags are sticky, so the user needs
to reset them before monitoring a calculation.

也參考:

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


Quick-start Tutorial
====================

The usual start to using decimals is importing the module, viewing the
current context with "getcontext()" and, if necessary, setting new
values for precision, rounding, or enabled traps:

   >>> 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

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

   >>> getcontext().prec = 28
   >>> Decimal(10)
   Decimal('10')
   >>> Decimal('3.14')
   Decimal('3.14')
   >>> Decimal(3.14)
   Decimal('3.140000000000000124344978758017532527446746826171875')
   >>> Decimal((0, (3, 1, 4), -2))
   Decimal('3.14')
   >>> Decimal(str(2.0 ** 0.5))
   Decimal('1.4142135623730951')
   >>> Decimal(2) ** Decimal('0.5')
   Decimal('1.414213562373095048801688724')
   >>> Decimal('NaN')
   Decimal('NaN')
   >>> Decimal('-Infinity')
   Decimal('-Infinity')

If the "FloatOperation" signal is trapped, accidental mixing of
decimals and floats in constructors or ordering comparisons raises an
exception:

   >>> 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 版新加入.

The significance of a new Decimal is determined solely by the number
of digits input.  Context precision and rounding only come into play
during arithmetic operations.

   >>> 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')

If the internal limits of the C version are exceeded, constructing a
decimal raises "InvalidOperation":

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

3.3 版更變.

Decimals interact well with much of the rest of Python.  Here is a
small decimal floating point flying circus:

   >>> 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')

And some mathematical functions are also available to 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')

The "quantize()" method rounds a number to a fixed exponent.  This
method is useful for monetary applications that often round results to
a fixed number of places:

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

As shown above, the "getcontext()" function accesses the current
context and allows the settings to be changed.  This approach meets
the needs of most applications.

For more advanced work, it may be useful to create alternate contexts
using the Context() constructor.  To make an alternate active, use the
"setcontext()" function.

In accordance with the standard, the "decimal" module provides two
ready to use standard contexts, "BasicContext" and "ExtendedContext".
The former is especially useful for debugging because many of the
traps are enabled:

   >>> 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

Contexts also have signal flags for monitoring exceptional conditions
encountered during computations.  The flags remain set until
explicitly cleared, so it is best to clear the flags before each set
of monitored computations by using the "clear_flags()" method.

   >>> 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=[])

The *flags* entry shows that the rational approximation to "Pi" was
rounded (digits beyond the context precision were thrown away) and
that the result is inexact (some of the discarded digits were non-
zero).

Individual traps are set using the dictionary in the "traps" field of
a 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

Most programs adjust the current context only once, at the beginning
of the program.  And, in many applications, data is converted to
"Decimal" with a single cast inside a loop.  With context set and
decimals created, the bulk of the program manipulates the data no
differently than with other Python numeric types.


Decimal objects
===============

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

   Construct a new "Decimal" object based from *value*.

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

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

   Other Unicode decimal digits are also permitted where "digit"
   appears above.  These include decimal digits from various other
   alphabets (for example, Arabic-Indic and Devanāgarī digits) along
   with the fullwidth digits "'\uff10'" through "'\uff19'".

   If *value* is a "tuple", it should have three components, a sign
   ("0" for positive or "1" for negative), a "tuple" of digits, and an
   integer exponent. For example, "Decimal((0, (1, 4, 1, 4), -3))"
   returns "Decimal('1.414')".

   If *value* is a "float", the binary floating point value is
   losslessly converted to its exact decimal equivalent.  This
   conversion can often require 53 or more digits of precision.  For
   example, "Decimal(float('1.1'))" converts to
   "Decimal('1.100000000000000088817841970012523233890533447265625')".

   The *context* precision does not affect how many digits are stored.
   That is determined exclusively by the number of digits in *value*.
   For example, "Decimal('3.00000')" records all five zeros even if
   the context precision is only three.

   The purpose of the *context* argument is determining what to do if
   *value* is a malformed string.  If the context traps
   "InvalidOperation", an exception is raised; otherwise, the
   constructor returns a new Decimal with the value of "NaN".

   Once constructed, "Decimal" objects are immutable.

   3.2 版更變: The argument to the constructor is now permitted to be
   a "float" instance.

   3.3 版更變: "float" arguments raise an exception if the
   "FloatOperation" trap is set. By default the trap is off.

   3.6 版更變: Underscores are allowed for grouping, as with integral
   and floating-point literals in code.

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

   There are some small differences between arithmetic on Decimal
   objects and arithmetic on integers and floats.  When the remainder
   operator "%" is applied to Decimal objects, the sign of the result
   is the sign of the *dividend* rather than the sign of the divisor:

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

   The integer division operator "//" behaves analogously, returning
   the integer part of the true quotient (truncating towards zero)
   rather than its floor, so as to preserve the usual identity "x ==
   (x // y) * y + x % y":

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

   The "%" and "//" operators implement the "remainder" and "divide-
   integer" operations (respectively) as described in the
   specification.

   Decimal objects cannot generally be combined with floats or
   instances of "fractions.Fraction" in arithmetic operations: an
   attempt to add a "Decimal" to a "float", for example, will raise a
   "TypeError".  However, it is possible to use Python's comparison
   operators to compare a "Decimal" instance "x" with another number
   "y".  This avoids confusing results when doing equality comparisons
   between numbers of different types.

   3.2 版更變: Mixed-type comparisons between "Decimal" instances and
   other numeric types are now fully supported.

   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)

      The conversion is exact.  Raise OverflowError on infinities and
      ValueError on NaNs.

   3.6 版新加入.

   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*.

      備註: 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')

      3.1 版新加入.

   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.


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".


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 版新加入.

   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 版新加入.

   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.

      3.3 版更變: 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.


Constants
=========

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".


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.


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)


Floating Point Notes
====================


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')


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')


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()
    . . .


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


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')

Q. Is the CPython implementation fast for large numbers?

A. Yes.  In the CPython and PyPy3 implementations, the C/CFFI versions
of the decimal module integrate the high speed libmpdec library for
arbitrary precision correctly-rounded decimal floating point
arithmetic. "libmpdec" uses Karatsuba multiplication for medium-sized
numbers and the Number Theoretic Transform for very large numbers.
However, to realize this performance gain, the context needs to be set
for unrounded calculations.

>>> c = getcontext()
>>> c.prec = MAX_PREC
>>> c.Emax = MAX_EMAX
>>> c.Emin = MIN_EMIN

3.3 版新加入.
