2 :mod:`decimal` --- Decimal fixed point and floating point arithmetic
3 ====================================================================
6 :synopsis: Implementation of the General Decimal Arithmetic Specification.
9 .. moduleauthor:: Eric Price <eprice at tjhsst.edu>
10 .. moduleauthor:: Facundo Batista <facundo at taniquetil.com.ar>
11 .. moduleauthor:: Raymond Hettinger <python at rcn.com>
12 .. moduleauthor:: Aahz <aahz at pobox.com>
13 .. moduleauthor:: Tim Peters <tim.one at comcast.net>
16 .. sectionauthor:: Raymond D. Hettinger <python at rcn.com>
21 The :mod:`decimal` module provides support for decimal floating point
22 arithmetic. It offers several advantages over the :class:`float` datatype:
24 * Decimal "is based on a floating-point model which was designed with people
25 in mind, and necessarily has a paramount guiding principle -- computers must
26 provide an arithmetic that works in the same way as the arithmetic that
27 people learn at school." -- excerpt from the decimal arithmetic specification.
29 * Decimal numbers can be represented exactly. In contrast, numbers like
30 :const:`1.1` do not have an exact representation in binary floating point. End
31 users typically would not expect :const:`1.1` to display as
32 :const:`1.1000000000000001` as it does with binary floating point.
34 * The exactness carries over into arithmetic. In decimal floating point, ``0.1
35 + 0.1 + 0.1 - 0.3`` is exactly equal to zero. In binary floating point, the result
36 is :const:`5.5511151231257827e-017`. While near to zero, the differences
37 prevent reliable equality testing and differences can accumulate. For this
38 reason, decimal is preferred in accounting applications which have strict
41 * The decimal module incorporates a notion of significant places so that ``1.30
42 + 1.20`` is :const:`2.50`. The trailing zero is kept to indicate significance.
43 This is the customary presentation for monetary applications. For
44 multiplication, the "schoolbook" approach uses all the figures in the
45 multiplicands. For instance, ``1.3 * 1.2`` gives :const:`1.56` while ``1.30 *
46 1.20`` gives :const:`1.5600`.
48 * Unlike hardware based binary floating point, the decimal module has a user
49 alterable precision (defaulting to 28 places) which can be as large as needed for
52 >>> getcontext().prec = 6
53 >>> Decimal(1) / Decimal(7)
55 >>> getcontext().prec = 28
56 >>> Decimal(1) / Decimal(7)
57 Decimal('0.1428571428571428571428571429')
59 * Both binary and decimal floating point are implemented in terms of published
60 standards. While the built-in float type exposes only a modest portion of its
61 capabilities, the decimal module exposes all required parts of the standard.
62 When needed, the programmer has full control over rounding and signal handling.
63 This includes an option to enforce exact arithmetic by using exceptions
64 to block any inexact operations.
66 * The decimal module was designed to support "without prejudice, both exact
67 unrounded decimal arithmetic (sometimes called fixed-point arithmetic)
68 and rounded floating-point arithmetic." -- excerpt from the decimal
69 arithmetic specification.
71 The module design is centered around three concepts: the decimal number, the
72 context for arithmetic, and signals.
74 A decimal number is immutable. It has a sign, coefficient digits, and an
75 exponent. To preserve significance, the coefficient digits do not truncate
76 trailing zeros. Decimals also include special values such as
77 :const:`Infinity`, :const:`-Infinity`, and :const:`NaN`. The standard also
78 differentiates :const:`-0` from :const:`+0`.
80 The context for arithmetic is an environment specifying precision, rounding
81 rules, limits on exponents, flags indicating the results of operations, and trap
82 enablers which determine whether signals are treated as exceptions. Rounding
83 options include :const:`ROUND_CEILING`, :const:`ROUND_DOWN`,
84 :const:`ROUND_FLOOR`, :const:`ROUND_HALF_DOWN`, :const:`ROUND_HALF_EVEN`,
85 :const:`ROUND_HALF_UP`, :const:`ROUND_UP`, and :const:`ROUND_05UP`.
87 Signals are groups of exceptional conditions arising during the course of
88 computation. Depending on the needs of the application, signals may be ignored,
89 considered as informational, or treated as exceptions. The signals in the
90 decimal module are: :const:`Clamped`, :const:`InvalidOperation`,
91 :const:`DivisionByZero`, :const:`Inexact`, :const:`Rounded`, :const:`Subnormal`,
92 :const:`Overflow`, and :const:`Underflow`.
94 For each signal there is a flag and a trap enabler. When a signal is
95 encountered, its flag is incremented from zero and, then, if the trap enabler is
96 set to one, an exception is raised. Flags are sticky, so the user needs to
97 reset them before monitoring a calculation.
102 * IBM's General Decimal Arithmetic Specification, `The General Decimal Arithmetic
103 Specification <http://www2.hursley.ibm.com/decimal/decarith.html>`_.
105 * IEEE standard 854-1987, `Unofficial IEEE 854 Text
106 <http://754r.ucbtest.org/standards/854.pdf>`_.
108 .. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
111 .. _decimal-tutorial:
116 The usual start to using decimals is importing the module, viewing the current
117 context with :func:`getcontext` and, if necessary, setting new values for
118 precision, rounding, or enabled traps::
120 >>> from decimal import *
122 Context(prec=28, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
123 capitals=1, flags=[], traps=[Overflow, InvalidOperation,
126 >>> getcontext().prec = 7 # Set a new precision
128 Decimal instances can be constructed from integers, strings, or tuples. To
129 create a Decimal from a :class:`float`, first convert it to a string. This
130 serves as an explicit reminder of the details of the conversion (including
131 representation error). Decimal numbers include special values such as
132 :const:`NaN` which stands for "Not a number", positive and negative
133 :const:`Infinity`, and :const:`-0`. ::
139 >>> Decimal((0, (3, 1, 4), -2))
141 >>> Decimal(str(2.0 ** 0.5))
142 Decimal('1.41421356237')
143 >>> Decimal(2) ** Decimal('0.5')
144 Decimal('1.414213562373095048801688724')
147 >>> Decimal('-Infinity')
150 The significance of a new Decimal is determined solely by the number of digits
151 input. Context precision and rounding only come into play during arithmetic
154 >>> getcontext().prec = 6
157 >>> Decimal('3.1415926535')
158 Decimal('3.1415926535')
159 >>> Decimal('3.1415926535') + Decimal('2.7182818285')
161 >>> getcontext().rounding = ROUND_UP
162 >>> Decimal('3.1415926535') + Decimal('2.7182818285')
165 Decimals interact well with much of the rest of Python. Here is a small decimal
166 floating point flying circus::
168 >>> data = map(Decimal, '1.34 1.87 3.45 2.35 1.00 0.03 9.25'.split())
174 [Decimal('0.03'), Decimal('1.00'), Decimal('1.34'), Decimal('1.87'),
175 Decimal('2.35'), Decimal('3.45'), Decimal('9.25')]
183 >>> round(a, 1) # round() first converts to binary floating point
194 And some mathematical functions are also available to Decimal::
196 >>> Decimal(2).sqrt()
197 Decimal('1.414213562373095048801688724')
199 Decimal('2.718281828459045235360287471')
200 >>> Decimal('10').ln()
201 Decimal('2.302585092994045684017991455')
202 >>> Decimal('10').log10()
205 The :meth:`quantize` method rounds a number to a fixed exponent. This method is
206 useful for monetary applications that often round results to a fixed number of
209 >>> Decimal('7.325').quantize(Decimal('.01'), rounding=ROUND_DOWN)
211 >>> Decimal('7.325').quantize(Decimal('1.'), rounding=ROUND_UP)
214 As shown above, the :func:`getcontext` function accesses the current context and
215 allows the settings to be changed. This approach meets the needs of most
218 For more advanced work, it may be useful to create alternate contexts using the
219 Context() constructor. To make an alternate active, use the :func:`setcontext`
222 In accordance with the standard, the :mod:`Decimal` module provides two ready to
223 use standard contexts, :const:`BasicContext` and :const:`ExtendedContext`. The
224 former is especially useful for debugging because many of the traps are
227 >>> myothercontext = Context(prec=60, rounding=ROUND_HALF_DOWN)
228 >>> setcontext(myothercontext)
229 >>> Decimal(1) / Decimal(7)
230 Decimal('0.142857142857142857142857142857142857142857142857142857142857')
233 Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
234 capitals=1, flags=[], traps=[])
235 >>> setcontext(ExtendedContext)
236 >>> Decimal(1) / Decimal(7)
237 Decimal('0.142857143')
238 >>> Decimal(42) / Decimal(0)
241 >>> setcontext(BasicContext)
242 >>> Decimal(42) / Decimal(0)
243 Traceback (most recent call last):
244 File "<pyshell#143>", line 1, in -toplevel-
245 Decimal(42) / Decimal(0)
246 DivisionByZero: x / 0
248 Contexts also have signal flags for monitoring exceptional conditions
249 encountered during computations. The flags remain set until explicitly cleared,
250 so it is best to clear the flags before each set of monitored computations by
251 using the :meth:`clear_flags` method. ::
253 >>> setcontext(ExtendedContext)
254 >>> getcontext().clear_flags()
255 >>> Decimal(355) / Decimal(113)
256 Decimal('3.14159292')
258 Context(prec=9, rounding=ROUND_HALF_EVEN, Emin=-999999999, Emax=999999999,
259 capitals=1, flags=[Inexact, Rounded], traps=[])
261 The *flags* entry shows that the rational approximation to :const:`Pi` was
262 rounded (digits beyond the context precision were thrown away) and that the
263 result is inexact (some of the discarded digits were non-zero).
265 Individual traps are set using the dictionary in the :attr:`traps` field of a
268 >>> Decimal(1) / Decimal(0)
270 >>> getcontext().traps[DivisionByZero] = 1
271 >>> Decimal(1) / Decimal(0)
272 Traceback (most recent call last):
273 File "<pyshell#112>", line 1, in -toplevel-
274 Decimal(1) / Decimal(0)
275 DivisionByZero: x / 0
277 Most programs adjust the current context only once, at the beginning of the
278 program. And, in many applications, data is converted to :class:`Decimal` with
279 a single cast inside a loop. With context set and decimals created, the bulk of
280 the program manipulates the data no differently than with other Python numeric
283 .. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
292 .. class:: Decimal([value [, context]])
294 Construct a new :class:`Decimal` object based from *value*.
296 *value* can be an integer, string, tuple, or another :class:`Decimal`
297 object. If no *value* is given, returns ``Decimal('0')``. If *value* is a
298 string, it should conform to the decimal numeric string syntax after leading
299 and trailing whitespace characters are removed::
302 digit ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
303 indicator ::= 'e' | 'E'
304 digits ::= digit [digit]...
305 decimal-part ::= digits '.' [digits] | ['.'] digits
306 exponent-part ::= indicator [sign] digits
307 infinity ::= 'Infinity' | 'Inf'
308 nan ::= 'NaN' [digits] | 'sNaN' [digits]
309 numeric-value ::= decimal-part [exponent-part] | infinity
310 numeric-string ::= [sign] numeric-value | [sign] nan
312 If *value* is a :class:`tuple`, it should have three components, a sign
313 (:const:`0` for positive or :const:`1` for negative), a :class:`tuple` of
314 digits, and an integer exponent. For example, ``Decimal((0, (1, 4, 1, 4), -3))``
315 returns ``Decimal('1.414')``.
317 The *context* precision does not affect how many digits are stored. That is
318 determined exclusively by the number of digits in *value*. For example,
319 ``Decimal('3.00000')`` records all five zeros even if the context precision is
322 The purpose of the *context* argument is determining what to do if *value* is a
323 malformed string. If the context traps :const:`InvalidOperation`, an exception
324 is raised; otherwise, the constructor returns a new Decimal with the value of
327 Once constructed, :class:`Decimal` objects are immutable.
329 .. versionchanged:: 2.6
330 leading and trailing whitespace characters are permitted when
331 creating a Decimal instance from a string.
333 Decimal floating point objects share many properties with the other built-in
334 numeric types such as :class:`float` and :class:`int`. All of the usual math
335 operations and special methods apply. Likewise, decimal objects can be copied,
336 pickled, printed, used as dictionary keys, used as set elements, compared,
337 sorted, and coerced to another type (such as :class:`float` or :class:`long`).
339 In addition to the standard numeric properties, decimal floating point objects
340 also have a number of specialized methods:
343 .. method:: Decimal.adjusted()
345 Return the adjusted exponent after shifting out the coefficient's rightmost
346 digits until only the lead digit remains: ``Decimal('321e+5').adjusted()``
347 returns seven. Used for determining the position of the most significant digit
348 with respect to the decimal point.
351 .. method:: Decimal.as_tuple()
353 Return a :term:`named tuple` representation of the number:
354 ``DecimalTuple(sign, digits, exponent)``.
356 .. versionchanged:: 2.6
360 .. method:: Decimal.canonical()
362 Return the canonical encoding of the argument. Currently, the
363 encoding of a :class:`Decimal` instance is always canonical, so
364 this operation returns its argument unchanged.
366 .. versionadded:: 2.6
368 .. method:: Decimal.compare(other[, context])
370 Compare the values of two Decimal instances. This operation
371 behaves in the same way as the usual comparison method
372 :meth:`__cmp__`, except that :meth:`compare` returns a Decimal
373 instance rather than an integer, and if either operand is a NaN
374 then the result is a NaN::
376 a or b is a NaN ==> Decimal('NaN')
377 a < b ==> Decimal('-1')
378 a == b ==> Decimal('0')
379 a > b ==> Decimal('1')
381 .. method:: Decimal.compare_signal(other[, context])
383 This operation is identical to the :meth:`compare` method, except
384 that all NaNs signal. That is, if neither operand is a signaling
385 NaN then any quiet NaN operand is treated as though it were a
388 .. versionadded:: 2.6
390 .. method:: Decimal.compare_total(other)
392 Compare two operands using their abstract representation rather
393 than their numerical value. Similar to the :meth:`compare` method,
394 but the result gives a total ordering on :class:`Decimal`
395 instances. Two :class:`Decimal` instances with the same numeric
396 value but different representations compare unequal in this
399 >>> Decimal('12.0').compare_total(Decimal('12'))
402 Quiet and signaling NaNs are also included in the total ordering.
403 The result of this function is ``Decimal('0')`` if both operands
404 have the same representation, ``Decimal('-1')`` if the first
405 operand is lower in the total order than the second, and
406 ``Decimal('1')`` if the first operand is higher in the total order
407 than the second operand. See the specification for details of the
410 .. versionadded:: 2.6
412 .. method:: Decimal.compare_total_mag(other)
414 Compare two operands using their abstract representation rather
415 than their value as in :meth:`compare_total`, but ignoring the sign
416 of each operand. ``x.compare_total_mag(y)`` is equivalent to
417 ``x.copy_abs().compare_total(y.copy_abs())``.
419 .. versionadded:: 2.6
421 .. method:: Decimal.copy_abs()
423 Return the absolute value of the argument. This operation is
424 unaffected by the context and is quiet: no flags are changed and no
425 rounding is performed.
427 .. versionadded:: 2.6
429 .. method:: Decimal.copy_negate()
431 Return the negation of the argument. This operation is unaffected
432 by the context and is quiet: no flags are changed and no rounding
435 .. versionadded:: 2.6
437 .. method:: Decimal.copy_sign(other)
439 Return a copy of the first operand with the sign set to be the
440 same as the sign of the second operand. For example::
442 >>> Decimal('2.3').copy_sign(Decimal('-1.5'))
445 This operation is unaffected by the context and is quiet: no flags
446 are changed and no rounding is performed.
448 .. versionadded:: 2.6
450 .. method:: Decimal.exp([context])
452 Return the value of the (natural) exponential function ``e**x`` at the
453 given number. The result is correctly rounded using the
454 :const:`ROUND_HALF_EVEN` rounding mode.
457 Decimal('2.718281828459045235360287471')
458 >>> Decimal(321).exp()
459 Decimal('2.561702493119680037517373933E+139')
461 .. versionadded:: 2.6
463 .. method:: Decimal.fma(other, third[, context])
465 Fused multiply-add. Return self*other+third with no rounding of
466 the intermediate product self*other.
468 >>> Decimal(2).fma(3, 5)
471 .. versionadded:: 2.6
473 .. method:: Decimal.is_canonical()
475 Return :const:`True` if the argument is canonical and
476 :const:`False` otherwise. Currently, a :class:`Decimal` instance
477 is always canonical, so this operation always returns
480 .. versionadded:: 2.6
482 .. method:: is_finite()
484 Return :const:`True` if the argument is a finite number, and
485 :const:`False` if the argument is an infinity or a NaN.
487 .. versionadded:: 2.6
489 .. method:: is_infinite()
491 Return :const:`True` if the argument is either positive or
492 negative infinity and :const:`False` otherwise.
494 .. versionadded:: 2.6
498 Return :const:`True` if the argument is a (quiet or signaling)
499 NaN and :const:`False` otherwise.
501 .. versionadded:: 2.6
503 .. method:: is_normal()
505 Return :const:`True` if the argument is a *normal* finite number.
506 Return :const:`False` if the argument is zero, subnormal, infinite
509 .. versionadded:: 2.6
511 .. method:: is_qnan()
513 Return :const:`True` if the argument is a quiet NaN, and
514 :const:`False` otherwise.
516 .. versionadded:: 2.6
518 .. method:: is_signed()
520 Return :const:`True` if the argument has a negative sign and
521 :const:`False` otherwise. Note that zeros and NaNs can both carry
524 .. versionadded:: 2.6
526 .. method:: is_snan()
528 Return :const:`True` if the argument is a signaling NaN and
529 :const:`False` otherwise.
531 .. versionadded:: 2.6
533 .. method:: is_subnormal()
535 Return :const:`True` if the argument is subnormal, and
536 :const:`False` otherwise.
538 .. versionadded:: 2.6
540 .. method:: is_zero()
542 Return :const:`True` if the argument is a (positive or negative)
543 zero and :const:`False` otherwise.
545 .. versionadded:: 2.6
547 .. method:: Decimal.ln([context])
549 Return the natural (base e) logarithm of the operand. The result
550 is correctly rounded using the :const:`ROUND_HALF_EVEN` rounding
553 .. versionadded:: 2.6
555 .. method:: Decimal.log10([context])
557 Return the base ten logarithm of the operand. The result is
558 correctly rounded using the :const:`ROUND_HALF_EVEN` rounding mode.
560 .. versionadded:: 2.6
562 .. method:: Decimal.logb([context])
564 For a nonzero number, return the adjusted exponent of its operand
565 as a :class:`Decimal` instance. If the operand is a zero then
566 ``Decimal('-Infinity')`` is returned and the
567 :const:`DivisionByZero` flag is raised. If the operand is an
568 infinity then ``Decimal('Infinity')`` is returned.
570 .. versionadded:: 2.6
572 .. method:: Decimal.logical_and(other[, context])
574 :meth:`logical_and` is a logical operation which takes two
575 *logical operands* (see :ref:`logical_operands_label`). The result
576 is the digit-wise ``and`` of the two operands.
578 .. versionadded:: 2.6
580 .. method:: Decimal.logical_invert(other[, context])
582 :meth:`logical_invert` is a logical operation. The argument must
583 be a *logical operand* (see :ref:`logical_operands_label`). The
584 result is the digit-wise inversion of the operand.
586 .. versionadded:: 2.6
588 .. method:: Decimal.logical_or(other[, context])
590 :meth:`logical_or` is a logical operation which takes two *logical
591 operands* (see :ref:`logical_operands_label`). The result is the
592 digit-wise ``or`` of the two operands.
594 .. versionadded:: 2.6
596 .. method:: Decimal.logical_xor(other[, context])
598 :meth:`logical_xor` is a logical operation which takes two
599 *logical operands* (see :ref:`logical_operands_label`). The result
600 is the digit-wise exclusive or of the two operands.
602 .. versionadded:: 2.6
604 .. method:: Decimal.max(other[, context])
606 Like ``max(self, other)`` except that the context rounding rule is applied
607 before returning and that :const:`NaN` values are either signaled or ignored
608 (depending on the context and whether they are signaling or quiet).
610 .. method:: Decimal.max_mag(other[, context])
612 Similar to the :meth:`max` method, but the comparison is done using
613 the absolute values of the operands.
615 .. versionadded:: 2.6
617 .. method:: Decimal.min(other[, context])
619 Like ``min(self, other)`` except that the context rounding rule is applied
620 before returning and that :const:`NaN` values are either signaled or ignored
621 (depending on the context and whether they are signaling or quiet).
623 .. method:: Decimal.min_mag(other[, context])
625 Similar to the :meth:`min` method, but the comparison is done using
626 the absolute values of the operands.
628 .. versionadded:: 2.6
630 .. method:: Decimal.next_minus([context])
632 Return the largest number representable in the given context (or
633 in the current thread's context if no context is given) that is smaller
634 than the given operand.
636 .. versionadded:: 2.6
638 .. method:: Decimal.next_plus([context])
640 Return the smallest number representable in the given context (or
641 in the current thread's context if no context is given) that is
642 larger than the given operand.
644 .. versionadded:: 2.6
646 .. method:: Decimal.next_toward(other[, context])
648 If the two operands are unequal, return the number closest to the
649 first operand in the direction of the second operand. If both
650 operands are numerically equal, return a copy of the first operand
651 with the sign set to be the same as the sign of the second operand.
653 .. versionadded:: 2.6
655 .. method:: Decimal.normalize([context])
657 Normalize the number by stripping the rightmost trailing zeros and converting
658 any result equal to :const:`Decimal('0')` to :const:`Decimal('0e0')`. Used for
659 producing canonical values for members of an equivalence class. For example,
660 ``Decimal('32.100')`` and ``Decimal('0.321000e+2')`` both normalize to the
661 equivalent value ``Decimal('32.1')``.
663 .. method:: Decimal.number_class([context])
665 Return a string describing the *class* of the operand. The
666 returned value is one of the following ten strings.
668 * ``"-Infinity"``, indicating that the operand is negative infinity.
669 * ``"-Normal"``, indicating that the operand is a negative normal number.
670 * ``"-Subnormal"``, indicating that the operand is negative and subnormal.
671 * ``"-Zero"``, indicating that the operand is a negative zero.
672 * ``"+Zero"``, indicating that the operand is a positive zero.
673 * ``"+Subnormal"``, indicating that the operand is positive and subnormal.
674 * ``"+Normal"``, indicating that the operand is a positive normal number.
675 * ``"+Infinity"``, indicating that the operand is positive infinity.
676 * ``"NaN"``, indicating that the operand is a quiet NaN (Not a Number).
677 * ``"sNaN"``, indicating that the operand is a signaling NaN.
679 .. versionadded:: 2.6
681 .. method:: Decimal.quantize(exp[, rounding[, context[, watchexp]]])
683 Return a value equal to the first operand after rounding and
684 having the exponent of the second operand.
686 >>> Decimal('1.41421356').quantize(Decimal('1.000'))
689 Unlike other operations, if the length of the coefficient after the
690 quantize operation would be greater than precision, then an
691 :const:`InvalidOperation` is signaled. This guarantees that, unless
692 there is an error condition, the quantized exponent is always equal
693 to that of the right-hand operand.
695 Also unlike other operations, quantize never signals Underflow,
696 even if the result is subnormal and inexact.
698 If the exponent of the second operand is larger than that of the
699 first then rounding may be necessary. In this case, the rounding
700 mode is determined by the ``rounding`` argument if given, else by
701 the given ``context`` argument; if neither argument is given the
702 rounding mode of the current thread's context is used.
704 If *watchexp* is set (default), then an error is returned whenever the
705 resulting exponent is greater than :attr:`Emax` or less than :attr:`Etiny`.
707 .. method:: Decimal.radix()
709 Return ``Decimal(10)``, the radix (base) in which the
710 :class:`Decimal` class does all its arithmetic. Included for
711 compatibility with the specification.
713 .. versionadded:: 2.6
715 .. method:: Decimal.remainder_near(other[, context])
717 Compute the modulo as either a positive or negative value depending on which is
718 closest to zero. For instance, ``Decimal(10).remainder_near(6)`` returns
719 ``Decimal('-2')`` which is closer to zero than ``Decimal('4')``.
721 If both are equally close, the one chosen will have the same sign as *self*.
723 .. method:: Decimal.rotate(other[, context])
725 Return the result of rotating the digits of the first operand by
726 an amount specified by the second operand. The second operand
727 must be an integer in the range -precision through precision. The
728 absolute value of the second operand gives the number of places to
729 rotate. If the second operand is positive then rotation is to the
730 left; otherwise rotation is to the right. The coefficient of the
731 first operand is padded on the left with zeros to length precision
732 if necessary. The sign and exponent of the first operand are
735 .. versionadded:: 2.6
737 .. method:: Decimal.same_quantum(other[, context])
739 Test whether self and other have the same exponent or whether both are
742 .. method:: Decimal.scaleb(other[, context])
744 Return the first operand with exponent adjusted by the second.
745 Equivalently, return the first operand multiplied by ``10**other``.
746 The second operand must be an integer.
748 .. versionadded:: 2.6
750 .. method:: Decimal.shift(other[, context])
752 Return the result of shifting the digits of the first operand by
753 an amount specified by the second operand. The second operand must
754 be an integer in the range -precision through precision. The
755 absolute value of the second operand gives the number of places to
756 shift. If the second operand is positive then the shift is to the
757 left; otherwise the shift is to the right. Digits shifted into the
758 coefficient are zeros. The sign and exponent of the first operand
761 .. versionadded:: 2.6
763 .. method:: Decimal.sqrt([context])
765 Return the square root of the argument to full precision.
768 .. method:: Decimal.to_eng_string([context])
770 Convert to an engineering-type string.
772 Engineering notation has an exponent which is a multiple of 3, so there are up
773 to 3 digits left of the decimal place. For example, converts
774 ``Decimal('123E+1')`` to ``Decimal('1.23E+3')``
776 .. method:: Decimal.to_integral([rounding[, context]])
778 Identical to the :meth:`to_integral_value` method. The ``to_integral``
779 name has been kept for compatibility with older versions.
781 .. method:: Decimal.to_integral_exact([rounding[, context]])
783 Round to the nearest integer, signaling
784 :const:`Inexact` or :const:`Rounded` as appropriate if rounding
785 occurs. The rounding mode is determined by the ``rounding``
786 parameter if given, else by the given ``context``. If neither
787 parameter is given then the rounding mode of the current context is
790 .. versionadded:: 2.6
792 .. method:: Decimal.to_integral_value([rounding[, context]])
794 Round to the nearest integer without signaling :const:`Inexact` or
795 :const:`Rounded`. If given, applies *rounding*; otherwise, uses the rounding
796 method in either the supplied *context* or the current context.
798 .. versionchanged:: 2.6
799 renamed from ``to_integral`` to ``to_integral_value``. The old name
800 remains valid for compatibility.
802 .. method:: Decimal.trim()
804 Return the decimal with *insignificant* trailing zeros removed.
805 Here, a trailing zero is considered insignificant either if it
806 follows the decimal point, or if the exponent of the argument (that
807 is, the last element of the :meth:`as_tuple` representation) is
810 .. versionadded:: 2.6
812 .. _logical_operands_label:
817 The :meth:`logical_and`, :meth:`logical_invert`, :meth:`logical_or`,
818 and :meth:`logical_xor` methods expect their arguments to be *logical
819 operands*. A *logical operand* is a :class:`Decimal` instance whose
820 exponent and sign are both zero, and whose digits are all either
821 :const:`0` or :const:`1`.
823 .. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
831 Contexts are environments for arithmetic operations. They govern precision, set
832 rules for rounding, determine which signals are treated as exceptions, and limit
833 the range for exponents.
835 Each thread has its own current context which is accessed or changed using the
836 :func:`getcontext` and :func:`setcontext` functions:
839 .. function:: getcontext()
841 Return the current context for the active thread.
844 .. function:: setcontext(c)
846 Set the current context for the active thread to *c*.
848 Beginning with Python 2.5, you can also use the :keyword:`with` statement and
849 the :func:`localcontext` function to temporarily change the active context.
852 .. function:: localcontext([c])
854 Return a context manager that will set the current context for the active thread
855 to a copy of *c* on entry to the with-statement and restore the previous context
856 when exiting the with-statement. If no context is specified, a copy of the
857 current context is used.
859 .. versionadded:: 2.5
861 For example, the following code sets the current decimal precision to 42 places,
862 performs a calculation, and then automatically restores the previous context::
864 from decimal import localcontext
866 with localcontext() as ctx:
867 ctx.prec = 42 # Perform a high precision calculation
868 s = calculate_something()
869 s = +s # Round the final result back to the default precision
871 New contexts can also be created using the :class:`Context` constructor
872 described below. In addition, the module provides three pre-made contexts:
875 .. class:: BasicContext
877 This is a standard context defined by the General Decimal Arithmetic
878 Specification. Precision is set to nine. Rounding is set to
879 :const:`ROUND_HALF_UP`. All flags are cleared. All traps are enabled (treated
880 as exceptions) except :const:`Inexact`, :const:`Rounded`, and
883 Because many of the traps are enabled, this context is useful for debugging.
886 .. class:: ExtendedContext
888 This is a standard context defined by the General Decimal Arithmetic
889 Specification. Precision is set to nine. Rounding is set to
890 :const:`ROUND_HALF_EVEN`. All flags are cleared. No traps are enabled (so that
891 exceptions are not raised during computations).
893 Because the traps are disabled, this context is useful for applications that
894 prefer to have result value of :const:`NaN` or :const:`Infinity` instead of
895 raising exceptions. This allows an application to complete a run in the
896 presence of conditions that would otherwise halt the program.
899 .. class:: DefaultContext
901 This context is used by the :class:`Context` constructor as a prototype for new
902 contexts. Changing a field (such a precision) has the effect of changing the
903 default for new contexts creating by the :class:`Context` constructor.
905 This context is most useful in multi-threaded environments. Changing one of the
906 fields before threads are started has the effect of setting system-wide
907 defaults. Changing the fields after threads have started is not recommended as
908 it would require thread synchronization to prevent race conditions.
910 In single threaded environments, it is preferable to not use this context at
911 all. Instead, simply create contexts explicitly as described below.
913 The default values are precision=28, rounding=ROUND_HALF_EVEN, and enabled traps
914 for Overflow, InvalidOperation, and DivisionByZero.
916 In addition to the three supplied contexts, new contexts can be created with the
917 :class:`Context` constructor.
920 .. class:: Context(prec=None, rounding=None, traps=None, flags=None, Emin=None, Emax=None, capitals=1)
922 Creates a new context. If a field is not specified or is :const:`None`, the
923 default values are copied from the :const:`DefaultContext`. If the *flags*
924 field is not specified or is :const:`None`, all flags are cleared.
926 The *prec* field is a positive integer that sets the precision for arithmetic
927 operations in the context.
929 The *rounding* option is one of:
931 * :const:`ROUND_CEILING` (towards :const:`Infinity`),
932 * :const:`ROUND_DOWN` (towards zero),
933 * :const:`ROUND_FLOOR` (towards :const:`-Infinity`),
934 * :const:`ROUND_HALF_DOWN` (to nearest with ties going towards zero),
935 * :const:`ROUND_HALF_EVEN` (to nearest with ties going to nearest even integer),
936 * :const:`ROUND_HALF_UP` (to nearest with ties going away from zero), or
937 * :const:`ROUND_UP` (away from zero).
938 * :const:`ROUND_05UP` (away from zero if last digit after rounding towards zero
939 would have been 0 or 5; otherwise towards zero)
941 The *traps* and *flags* fields list any signals to be set. Generally, new
942 contexts should only set traps and leave the flags clear.
944 The *Emin* and *Emax* fields are integers specifying the outer limits allowable
947 The *capitals* field is either :const:`0` or :const:`1` (the default). If set to
948 :const:`1`, exponents are printed with a capital :const:`E`; otherwise, a
949 lowercase :const:`e` is used: :const:`Decimal('6.02e+23')`.
951 .. versionchanged:: 2.6
952 The :const:`ROUND_05UP` rounding mode was added.
954 The :class:`Context` class defines several general purpose methods as
955 well as a large number of methods for doing arithmetic directly in a
956 given context. In addition, for each of the :class:`Decimal` methods
957 described above (with the exception of the :meth:`adjusted` and
958 :meth:`as_tuple` methods) there is a corresponding :class:`Context`
959 method. For example, ``C.exp(x)`` is equivalent to
960 ``x.exp(context=C)``.
962 .. method:: Context.clear_flags()
964 Resets all of the flags to :const:`0`.
967 .. method:: Context.copy()
969 Return a duplicate of the context.
971 .. method:: Context.copy_decimal(num)
973 Return a copy of the Decimal instance num.
975 .. method:: Context.create_decimal(num)
977 Creates a new Decimal instance from *num* but using *self* as context. Unlike
978 the :class:`Decimal` constructor, the context precision, rounding method, flags,
979 and traps are applied to the conversion.
981 This is useful because constants are often given to a greater precision than is
982 needed by the application. Another benefit is that rounding immediately
983 eliminates unintended effects from digits beyond the current precision. In the
984 following example, using unrounded inputs means that adding zero to a sum can
987 >>> getcontext().prec = 3
988 >>> Decimal('3.4445') + Decimal('1.0023')
990 >>> Decimal('3.4445') + Decimal(0) + Decimal('1.0023')
993 This method implements the to-number operation of the IBM
994 specification. If the argument is a string, no leading or trailing
995 whitespace is permitted.
997 .. method:: Context.Etiny()
999 Returns a value equal to ``Emin - prec + 1`` which is the minimum exponent value
1000 for subnormal results. When underflow occurs, the exponent is set to
1004 .. method:: Context.Etop()
1006 Returns a value equal to ``Emax - prec + 1``.
1008 The usual approach to working with decimals is to create :class:`Decimal`
1009 instances and then apply arithmetic operations which take place within the
1010 current context for the active thread. An alternative approach is to use context
1011 methods for calculating within a specific context. The methods are similar to
1012 those for the :class:`Decimal` class and are only briefly recounted here.
1015 .. method:: Context.abs(x)
1017 Returns the absolute value of *x*.
1020 .. method:: Context.add(x, y)
1022 Return the sum of *x* and *y*.
1025 .. method:: Context.divide(x, y)
1027 Return *x* divided by *y*.
1030 .. method:: Context.divide_int(x, y)
1032 Return *x* divided by *y*, truncated to an integer.
1035 .. method:: Context.divmod(x, y)
1037 Divides two numbers and returns the integer part of the result.
1040 .. method:: Context.minus(x)
1042 Minus corresponds to the unary prefix minus operator in Python.
1045 .. method:: Context.multiply(x, y)
1047 Return the product of *x* and *y*.
1050 .. method:: Context.plus(x)
1052 Plus corresponds to the unary prefix plus operator in Python. This operation
1053 applies the context precision and rounding, so it is *not* an identity
1057 .. method:: Context.power(x, y[, modulo])
1059 Return ``x`` to the power of ``y``, reduced modulo ``modulo`` if
1062 With two arguments, compute ``x**y``. If ``x`` is negative then
1063 ``y`` must be integral. The result will be inexact unless ``y`` is
1064 integral and the result is finite and can be expressed exactly in
1065 'precision' digits. The result should always be correctly rounded,
1066 using the rounding mode of the current thread's context.
1068 With three arguments, compute ``(x**y) % modulo``. For the three
1069 argument form, the following restrictions on the arguments hold:
1071 - all three arguments must be integral
1072 - ``y`` must be nonnegative
1073 - at least one of ``x`` or ``y`` must be nonzero
1074 - ``modulo`` must be nonzero and have at most 'precision' digits
1076 The result of ``Context.power(x, y, modulo)`` is identical to
1077 the result that would be obtained by computing ``(x**y) %
1078 modulo`` with unbounded precision, but is computed more
1079 efficiently. It is always exact.
1081 .. versionchanged:: 2.6
1082 ``y`` may now be nonintegral in ``x**y``.
1083 Stricter requirements for the three-argument version.
1086 .. method:: Context.remainder(x, y)
1088 Returns the remainder from integer division.
1090 The sign of the result, if non-zero, is the same as that of the original
1093 .. method:: Context.subtract(x, y)
1095 Return the difference between *x* and *y*.
1097 .. method:: Context.to_sci_string(x)
1099 Converts a number to a string using scientific notation.
1101 .. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1104 .. _decimal-signals:
1109 Signals represent conditions that arise during computation. Each corresponds to
1110 one context flag and one context trap enabler.
1112 The context flag is incremented whenever the condition is encountered. After the
1113 computation, flags may be checked for informational purposes (for instance, to
1114 determine whether a computation was exact). After checking the flags, be sure to
1115 clear all flags before starting the next computation.
1117 If the context's trap enabler is set for the signal, then the condition causes a
1118 Python exception to be raised. For example, if the :class:`DivisionByZero` trap
1119 is set, then a :exc:`DivisionByZero` exception is raised upon encountering the
1125 Altered an exponent to fit representation constraints.
1127 Typically, clamping occurs when an exponent falls outside the context's
1128 :attr:`Emin` and :attr:`Emax` limits. If possible, the exponent is reduced to
1129 fit by adding zeros to the coefficient.
1132 .. class:: DecimalException
1134 Base class for other signals and a subclass of :exc:`ArithmeticError`.
1137 .. class:: DivisionByZero
1139 Signals the division of a non-infinite number by zero.
1141 Can occur with division, modulo division, or when raising a number to a negative
1142 power. If this signal is not trapped, returns :const:`Infinity` or
1143 :const:`-Infinity` with the sign determined by the inputs to the calculation.
1148 Indicates that rounding occurred and the result is not exact.
1150 Signals when non-zero digits were discarded during rounding. The rounded result
1151 is returned. The signal flag or trap is used to detect when results are
1155 .. class:: InvalidOperation
1157 An invalid operation was performed.
1159 Indicates that an operation was requested that does not make sense. If not
1160 trapped, returns :const:`NaN`. Possible causes include::
1167 x._rescale( non-integer )
1178 Indicates the exponent is larger than :attr:`Emax` after rounding has occurred.
1179 If not trapped, the result depends on the rounding mode, either pulling inward
1180 to the largest representable finite number or rounding outward to
1181 :const:`Infinity`. In either case, :class:`Inexact` and :class:`Rounded` are
1187 Rounding occurred though possibly no information was lost.
1189 Signaled whenever rounding discards digits; even if those digits are zero (such
1190 as rounding :const:`5.00` to :const:`5.0`). If not trapped, returns the result
1191 unchanged. This signal is used to detect loss of significant digits.
1194 .. class:: Subnormal
1196 Exponent was lower than :attr:`Emin` prior to rounding.
1198 Occurs when an operation result is subnormal (the exponent is too small). If not
1199 trapped, returns the result unchanged.
1202 .. class:: Underflow
1204 Numerical underflow with result rounded to zero.
1206 Occurs when a subnormal result is pushed to zero by rounding. :class:`Inexact`
1207 and :class:`Subnormal` are also signaled.
1209 The following table summarizes the hierarchy of signals::
1211 exceptions.ArithmeticError(exceptions.StandardError)
1214 DivisionByZero(DecimalException, exceptions.ZeroDivisionError)
1216 Overflow(Inexact, Rounded)
1217 Underflow(Inexact, Rounded, Subnormal)
1222 .. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1227 Floating Point Notes
1228 --------------------
1231 Mitigating round-off error with increased precision
1232 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
1234 The use of decimal floating point eliminates decimal representation error
1235 (making it possible to represent :const:`0.1` exactly); however, some operations
1236 can still incur round-off error when non-zero digits exceed the fixed precision.
1238 The effects of round-off error can be amplified by the addition or subtraction
1239 of nearly offsetting quantities resulting in loss of significance. Knuth
1240 provides two instructive examples where rounded floating point arithmetic with
1241 insufficient precision causes the breakdown of the associative and distributive
1242 properties of addition::
1244 # Examples from Seminumerical Algorithms, Section 4.2.2.
1245 >>> from decimal import Decimal, getcontext
1246 >>> getcontext().prec = 8
1248 >>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
1250 Decimal('9.5111111')
1254 >>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
1258 Decimal('0.0060000')
1260 The :mod:`decimal` module makes it possible to restore the identities by
1261 expanding the precision sufficiently to avoid loss of significance::
1263 >>> getcontext().prec = 20
1264 >>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
1266 Decimal('9.51111111')
1268 Decimal('9.51111111')
1270 >>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
1272 Decimal('0.0060000')
1274 Decimal('0.0060000')
1280 The number system for the :mod:`decimal` module provides special values
1281 including :const:`NaN`, :const:`sNaN`, :const:`-Infinity`, :const:`Infinity`,
1282 and two zeros, :const:`+0` and :const:`-0`.
1284 Infinities can be constructed directly with: ``Decimal('Infinity')``. Also,
1285 they can arise from dividing by zero when the :exc:`DivisionByZero` signal is
1286 not trapped. Likewise, when the :exc:`Overflow` signal is not trapped, infinity
1287 can result from rounding beyond the limits of the largest representable number.
1289 The infinities are signed (affine) and can be used in arithmetic operations
1290 where they get treated as very large, indeterminate numbers. For instance,
1291 adding a constant to infinity gives another infinite result.
1293 Some operations are indeterminate and return :const:`NaN`, or if the
1294 :exc:`InvalidOperation` signal is trapped, raise an exception. For example,
1295 ``0/0`` returns :const:`NaN` which means "not a number". This variety of
1296 :const:`NaN` is quiet and, once created, will flow through other computations
1297 always resulting in another :const:`NaN`. This behavior can be useful for a
1298 series of computations that occasionally have missing inputs --- it allows the
1299 calculation to proceed while flagging specific results as invalid.
1301 A variant is :const:`sNaN` which signals rather than remaining quiet after every
1302 operation. This is a useful return value when an invalid result needs to
1303 interrupt a calculation for special handling.
1305 The behavior of Python's comparison operators can be a little surprising where a
1306 :const:`NaN` is involved. A test for equality where one of the operands is a
1307 quiet or signaling :const:`NaN` always returns :const:`False` (even when doing
1308 ``Decimal('NaN')==Decimal('NaN')``), while a test for inequality always returns
1309 :const:`True`. An attempt to compare two Decimals using any of the ``<``,
1310 ``<=``, ``>`` or ``>=`` operators will raise the :exc:`InvalidOperation` signal
1311 if either operand is a :const:`NaN`, and return :const:`False` if this signal is
1312 not trapped. Note that the General Decimal Arithmetic specification does not
1313 specify the behavior of direct comparisons; these rules for comparisons
1314 involving a :const:`NaN` were taken from the IEEE 854 standard (see Table 3 in
1315 section 5.7). To ensure strict standards-compliance, use the :meth:`compare`
1316 and :meth:`compare-signal` methods instead.
1318 The signed zeros can result from calculations that underflow. They keep the sign
1319 that would have resulted if the calculation had been carried out to greater
1320 precision. Since their magnitude is zero, both positive and negative zeros are
1321 treated as equal and their sign is informational.
1323 In addition to the two signed zeros which are distinct yet equal, there are
1324 various representations of zero with differing precisions yet equivalent in
1325 value. This takes a bit of getting used to. For an eye accustomed to
1326 normalized floating point representations, it is not immediately obvious that
1327 the following calculation returns a value equal to zero::
1329 >>> 1 / Decimal('Infinity')
1330 Decimal('0E-1000000026')
1332 .. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1335 .. _decimal-threads:
1337 Working with threads
1338 --------------------
1340 The :func:`getcontext` function accesses a different :class:`Context` object for
1341 each thread. Having separate thread contexts means that threads may make
1342 changes (such as ``getcontext.prec=10``) without interfering with other threads.
1344 Likewise, the :func:`setcontext` function automatically assigns its target to
1347 If :func:`setcontext` has not been called before :func:`getcontext`, then
1348 :func:`getcontext` will automatically create a new context for use in the
1351 The new context is copied from a prototype context called *DefaultContext*. To
1352 control the defaults so that each thread will use the same values throughout the
1353 application, directly modify the *DefaultContext* object. This should be done
1354 *before* any threads are started so that there won't be a race condition between
1355 threads calling :func:`getcontext`. For example::
1357 # Set applicationwide defaults for all threads about to be launched
1358 DefaultContext.prec = 12
1359 DefaultContext.rounding = ROUND_DOWN
1360 DefaultContext.traps = ExtendedContext.traps.copy()
1361 DefaultContext.traps[InvalidOperation] = 1
1362 setcontext(DefaultContext)
1364 # Afterwards, the threads can be started
1370 .. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1373 .. _decimal-recipes:
1378 Here are a few recipes that serve as utility functions and that demonstrate ways
1379 to work with the :class:`Decimal` class::
1381 def moneyfmt(value, places=2, curr='', sep=',', dp='.',
1382 pos='', neg='-', trailneg=''):
1383 """Convert Decimal to a money formatted string.
1385 places: required number of places after the decimal point
1386 curr: optional currency symbol before the sign (may be blank)
1387 sep: optional grouping separator (comma, period, space, or blank)
1388 dp: decimal point indicator (comma or period)
1389 only specify as blank when places is zero
1390 pos: optional sign for positive numbers: '+', space or blank
1391 neg: optional sign for negative numbers: '-', '(', space or blank
1392 trailneg:optional trailing minus indicator: '-', ')', space or blank
1394 >>> d = Decimal('-1234567.8901')
1395 >>> moneyfmt(d, curr='$')
1397 >>> moneyfmt(d, places=0, sep='.', dp='', neg='', trailneg='-')
1399 >>> moneyfmt(d, curr='$', neg='(', trailneg=')')
1401 >>> moneyfmt(Decimal(123456789), sep=' ')
1403 >>> moneyfmt(Decimal('-0.02'), neg='<', trailneg='>')
1407 q = Decimal(10) ** -places # 2 places --> '0.01'
1408 sign, digits, exp = value.quantize(q).as_tuple()
1410 digits = map(str, digits)
1411 build, next = result.append, digits.pop
1414 for i in range(places):
1415 build(next() if digits else '0')
1421 if i == 3 and digits:
1425 build(neg if sign else pos)
1426 return ''.join(reversed(result))
1429 """Compute Pi to the current precision.
1432 3.141592653589793238462643383
1435 getcontext().prec += 2 # extra digits for intermediate steps
1436 three = Decimal(3) # substitute "three=3.0" for regular floats
1437 lasts, t, s, n, na, d, da = 0, three, 3, 1, 0, 0, 24
1444 getcontext().prec -= 2
1445 return +s # unary plus applies the new precision
1448 """Return e raised to the power of x. Result type matches input type.
1450 >>> print exp(Decimal(1))
1451 2.718281828459045235360287471
1452 >>> print exp(Decimal(2))
1453 7.389056098930650227230427461
1460 getcontext().prec += 2
1461 i, lasts, s, fact, num = 0, 0, 1, 1, 1
1468 getcontext().prec -= 2
1472 """Return the cosine of x as measured in radians.
1474 >>> print cos(Decimal('0.5'))
1475 0.8775825618903727161162815826
1478 >>> print cos(0.5+0j)
1482 getcontext().prec += 2
1483 i, lasts, s, fact, num, sign = 0, 0, 1, 1, 1, 1
1490 s += num / fact * sign
1491 getcontext().prec -= 2
1495 """Return the sine of x as measured in radians.
1497 >>> print sin(Decimal('0.5'))
1498 0.4794255386042030002732879352
1501 >>> print sin(0.5+0j)
1505 getcontext().prec += 2
1506 i, lasts, s, fact, num, sign = 1, 0, x, 1, x, 1
1513 s += num / fact * sign
1514 getcontext().prec -= 2
1518 .. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1526 Q. It is cumbersome to type ``decimal.Decimal('1234.5')``. Is there a way to
1527 minimize typing when using the interactive interpreter?
1529 A. Some users abbreviate the constructor to just a single letter::
1531 >>> D = decimal.Decimal
1532 >>> D('1.23') + D('3.45')
1535 Q. In a fixed-point application with two decimal places, some inputs have many
1536 places and need to be rounded. Others are not supposed to have excess digits
1537 and need to be validated. What methods should be used?
1539 A. The :meth:`quantize` method rounds to a fixed number of decimal places. If
1540 the :const:`Inexact` trap is set, it is also useful for validation::
1542 >>> TWOPLACES = Decimal(10) ** -2 # same as Decimal('0.01')
1544 >>> # Round to two places
1545 >>> Decimal('3.214').quantize(TWOPLACES)
1548 >>> # Validate that a number does not exceed two places
1549 >>> Decimal('3.21').quantize(TWOPLACES, context=Context(traps=[Inexact]))
1552 >>> Decimal('3.214').quantize(TWOPLACES, context=Context(traps=[Inexact]))
1553 Traceback (most recent call last):
1555 Inexact: Changed in rounding
1557 Q. Once I have valid two place inputs, how do I maintain that invariant
1558 throughout an application?
1560 A. Some operations like addition, subtraction, and multiplication by an integer
1561 will automatically preserve fixed point. Others operations, like division and
1562 non-integer multiplication, will change the number of decimal places and need to
1563 be followed-up with a :meth:`quantize` step::
1565 >>> a = Decimal('102.72') # Initial fixed-point values
1566 >>> b = Decimal('3.17')
1567 >>> a + b # Addition preserves fixed-point
1571 >>> a * 42 # So does integer multiplication
1573 >>> (a * b).quantize(TWOPLACES) # Must quantize non-integer multiplication
1575 >>> (b / a).quantize(TWOPLACES) # And quantize division
1578 In developing fixed-point applications, it is convenient to define functions
1579 to handle the :meth:`quantize` step::
1581 >>> def mul(x, y, fp=TWOPLACES):
1582 ... return (x * y).quantize(fp)
1583 >>> def div(x, y, fp=TWOPLACES):
1584 ... return (x / y).quantize(fp)
1586 >>> mul(a, b) # Automatically preserve fixed-point
1591 Q. There are many ways to express the same value. The numbers :const:`200`,
1592 :const:`200.000`, :const:`2E2`, and :const:`.02E+4` all have the same value at
1593 various precisions. Is there a way to transform them to a single recognizable
1596 A. The :meth:`normalize` method maps all equivalent values to a single
1599 >>> values = map(Decimal, '200 200.000 2E2 .02E+4'.split())
1600 >>> [v.normalize() for v in values]
1601 [Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2')]
1603 Q. Some decimal values always print with exponential notation. Is there a way
1604 to get a non-exponential representation?
1606 A. For some values, exponential notation is the only way to express the number
1607 of significant places in the coefficient. For example, expressing
1608 :const:`5.0E+3` as :const:`5000` keeps the value constant but cannot show the
1609 original's two-place significance.
1611 If an application does not care about tracking significance, it is easy to
1612 remove the exponent and trailing zeroes, losing significance, but keeping the
1615 >>> def remove_exponent(d):
1616 ... return d.quantize(Decimal(1)) if d == d.to_integral() else d.normalize()
1618 >>> remove_exponent(Decimal('5E+3'))
1621 Q. Is there a way to convert a regular float to a :class:`Decimal`?
1623 A. Yes, all binary floating point numbers can be exactly expressed as a
1624 Decimal. An exact conversion may take more precision than intuition would
1625 suggest, so we trap :const:`Inexact` to signal a need for more precision::
1627 def float_to_decimal(f):
1628 "Convert a floating point number to a Decimal with no loss of information"
1629 n, d = f.as_integer_ratio()
1630 with localcontext() as ctx:
1631 ctx.traps[Inexact] = True
1634 return Decimal(n) / Decimal(d)
1638 >>> float_to_decimal(math.pi)
1639 Decimal('3.141592653589793115997963468544185161590576171875')
1641 Q. Why isn't the :func:`float_to_decimal` routine included in the module?
1643 A. There is some question about whether it is advisable to mix binary and
1644 decimal floating point. Also, its use requires some care to avoid the
1645 representation issues associated with binary floating point::
1647 >>> float_to_decimal(1.1)
1648 Decimal('1.100000000000000088817841970012523233890533447265625')
1650 Q. Within a complex calculation, how can I make sure that I haven't gotten a
1651 spurious result because of insufficient precision or rounding anomalies.
1653 A. The decimal module makes it easy to test results. A best practice is to
1654 re-run calculations using greater precision and with various rounding modes.
1655 Widely differing results indicate insufficient precision, rounding mode issues,
1656 ill-conditioned inputs, or a numerically unstable algorithm.
1658 Q. I noticed that context precision is applied to the results of operations but
1659 not to the inputs. Is there anything to watch out for when mixing values of
1660 different precisions?
1662 A. Yes. The principle is that all values are considered to be exact and so is
1663 the arithmetic on those values. Only the results are rounded. The advantage
1664 for inputs is that "what you type is what you get". A disadvantage is that the
1665 results can look odd if you forget that the inputs haven't been rounded::
1667 >>> getcontext().prec = 3
1668 >>> Decimal('3.104') + D('2.104')
1670 >>> Decimal('3.104') + D('0.000') + D('2.104')
1673 The solution is either to increase precision or to force rounding of inputs
1674 using the unary plus operation::
1676 >>> getcontext().prec = 3
1677 >>> +Decimal('1.23456789') # unary plus triggers rounding
1680 Alternatively, inputs can be rounded upon creation using the
1681 :meth:`Context.create_decimal` method::
1683 >>> Context(prec=5, rounding=ROUND_DOWN).create_decimal('1.2345678')