decimal
— Decimal fixed point and floating point arithmetic#
Source code: Lib/decimal.py
The decimal
module provides support for fast correctlyrounded
decimal floating point arithmetic. It offers several advantages over the
float
datatype:
Decimal “is based on a floatingpoint 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
and2.2
do not have exact representations in binary floating point. End users typically would not expect1.1 + 2.2
to display as3.3000000000000003
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 is5.5511151231257827e017
. 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
is2.50
. 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
gives1.56
while1.30 * 1.20
gives1.5600
.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 builtin 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 fixedpoint arithmetic) and rounded floatingpoint 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.
See also
IBM’s General Decimal Arithmetic Specification, The General Decimal Arithmetic Specification.
Quickstart 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
New in version 3.3.
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'>]
Changed in version 3.3.
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 nonzero).
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 anotherDecimal
object. If no value is given, returnsDecimal('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]... decimalpart ::= digits '.' [digits]  ['.'] digits exponentpart ::= indicator [sign] digits infinity ::= 'Infinity'  'Inf' nan ::= 'NaN' [digits]  'sNaN' [digits] numericvalue ::= decimalpart [exponentpart]  infinity numericstring ::= [sign] numericvalue  [sign] nan
Other Unicode decimal digits are also permitted where
digit
appears above. These include decimal digits from various other alphabets (for example, ArabicIndic 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 or1
for negative), atuple
of digits, and an integer exponent. For example,Decimal((0, (1, 4, 1, 4), 3))
returnsDecimal('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 toDecimal('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 ofNaN
.Once constructed,
Decimal
objects are immutable.Changed in version 3.2: The argument to the constructor is now permitted to be a
float
instance.Changed in version 3.3:
float
arguments raise an exception if theFloatOperation
trap is set. By default the trap is off.Changed in version 3.6: Underscores are allowed for grouping, as with integral and floatingpoint literals in code.
Decimal floating point objects share many properties with the other builtin numeric types such as
float
andint
. 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 asfloat
orint
).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 identityx == (x // y) * y + x % y
:>>> 7 // 4 2 >>> Decimal(7) // Decimal(4) Decimal('1')
The
%
and//
operators implement theremainder
anddivideinteger
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 aDecimal
to afloat
, for example, will raise aTypeError
. However, it is possible to use Python’s comparison operators to compare aDecimal
instancex
with another numbery
. This avoids confusing results when doing equality comparisons between numbers of different types.Changed in version 3.2: Mixedtype 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 givenDecimal
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.
New in version 3.6.
 as_tuple()#
Return a named tuple representation of the number:
DecimalTuple(sign, digits, exponent)
.
 canonical()#
Return the canonical encoding of the argument. Currently, the encoding of a
Decimal
instance is always canonical, so this operation returns its argument unchanged.
 compare(other, context=None)#
Compare the values of two Decimal instances.
compare()
returns a Decimal instance, and if either operand is a NaN then the result is a NaN:a or b is a NaN ==> Decimal('NaN') a < b ==> Decimal('1') a == b ==> Decimal('0') a > b ==> Decimal('1')
 compare_signal(other, context=None)#
This operation is identical to the
compare()
method, except that all NaNs signal. That is, if neither operand is a signaling NaN then any quiet NaN operand is treated as though it were a signaling NaN.
 compare_total(other, context=None)#
Compare two operands using their abstract representation rather than their numerical value. Similar to the
compare()
method, but the result gives a total ordering onDecimal
instances. TwoDecimal
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, andDecimal('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 tox.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 theROUND_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 0x1.999999999999ap4. That equivalent value in decimal is 0.1000000000000000055511151231257827021181583404541015625.
>>> 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')
New in version 3.1.
 fma(other, third, context=None)#
Fused multiplyadd. 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 andFalse
otherwise. Currently, aDecimal
instance is always canonical, so this operation always returnsTrue
.
 is_finite()#
Return
True
if the argument is a finite number, andFalse
if the argument is an infinity or a NaN.
 is_infinite()#
Return
True
if the argument is either positive or negative infinity andFalse
otherwise.
 is_normal(context=None)#
Return
True
if the argument is a normal finite number. ReturnFalse
if the argument is zero, subnormal, infinite or a NaN.
 is_signed()#
Return
True
if the argument has a negative sign andFalse
otherwise. Note that zeros and NaNs can both carry signs.
 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 thenDecimal('Infinity')
is returned and theDivisionByZero
flag is raised. If the operand is an infinity thenDecimal('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 digitwiseand
of the two operands.
 logical_invert(context=None)#
logical_invert()
is a logical operation. The result is the digitwise 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 digitwiseor
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 digitwise 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 thatNaN
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 thatNaN
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')
toDecimal('0e0')
. Used for producing canonical values for attributes of an equivalence class. For example,Decimal('32.100')
andDecimal('0.321000e+2')
both normalize to the equivalent valueDecimal('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 righthand 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 givencontext
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 thanEtiny
.
 radix()#
Return
Decimal(10)
, the radix (base) in which theDecimal
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 isself  n * other
wheren
is the integer nearest to the exact value ofself / 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')
toDecimal('1.23E+3')
.
 to_integral(rounding=None, context=None)#
Identical to the
to_integral_value()
method. Theto_integral
name has been kept for compatibility with older versions.
 to_integral_exact(rounding=None, context=None)#
Round to the nearest integer, signaling
Inexact
orRounded
as appropriate if rounding occurs. The rounding mode is determined by therounding
parameter if given, else by the givencontext
. If neither parameter is given then the rounding mode of the current context is used.
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 withstatement and restore the previous context when exiting the withstatement. 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 premade 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) exceptInexact
,Rounded
, andSubnormal
.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
orInfinity
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 theContext
constructor.This context is most useful in multithreaded environments. Changing one of the fields before threads are started has the effect of setting systemwide 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 forOverflow
,InvalidOperation
, andDivisionByZero
.
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 theDefaultContext
. If the flags field is not specified or isNone
, 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
or1
(the default). If set to1
, exponents are printed with a capitalE
; otherwise, a lowercasee
is used:Decimal('6.02e+23')
.The clamp field is either
0
(the default) or1
. If set to1
, the exponente
of aDecimal
instance representable in this context is strictly limited to the rangeEmin  prec + 1 <= e <= Emax  prec + 1
. If clamp is0
then a weaker condition holds: the adjusted exponent of theDecimal
instance is at mostEmax
. When clamp is1
, 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 fixedwidth 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 theDecimal
methods described above (with the exception of theadjusted()
andas_tuple()
methods) there is a correspondingContext
method. For example, for aContext
instanceC
andDecimal
instancex
,C.exp(x)
is equivalent tox.exp(context=C)
. EachContext
method accepts a Python integer (an instance ofint
) 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
.New in version 3.3.
 copy()#
Return a duplicate of the context.
 copy_decimal(num)#
Return a copy of the Decimal instance num.
 create_decimal(num)#
Creates a new Decimal instance from num but using self as context. Unlike the
Decimal
constructor, the context precision, rounding method, flags, and traps are applied to the conversion.This is useful because constants are often given to a greater precision than is needed by the application. Another benefit is that rounding immediately eliminates unintended effects from digits beyond the current precision. In the following example, using unrounded inputs means that adding zero to a sum can change the result:
>>> getcontext().prec = 3 >>> Decimal('3.4445') + Decimal('1.0023') Decimal('4.45') >>> Decimal('3.4445') + Decimal(0) + Decimal('1.0023') Decimal('4.44')
This method implements the tonumber 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
New in version 3.1.
 Etiny()#
Returns a value equal to
Emin  prec + 1
which is the minimum exponent value for subnormal results. When underflow occurs, the exponent is set toEtiny
.
 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 theDecimal
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 returnsFalse
.
 is_finite(x)#
Returns
True
if x is finite; otherwise returnsFalse
.
 is_infinite(x)#
Returns
True
if x is infinite; otherwise returnsFalse
.
 is_nan(x)#
Returns
True
if x is a qNaN or sNaN; otherwise returnsFalse
.
 is_normal(x)#
Returns
True
if x is a normal number; otherwise returnsFalse
.
 is_qnan(x)#
Returns
True
if x is a quiet NaN; otherwise returnsFalse
.
 is_signed(x)#
Returns
True
if x is negative; otherwise returnsFalse
.
 is_snan(x)#
Returns
True
if x is a signaling NaN; otherwise returnsFalse
.
 is_subnormal(x)#
Returns
True
if x is subnormal; otherwise returnsFalse
.
 is_zero(x)#
Returns
True
if x is a zero; otherwise returnsFalse
.
 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 ofy
, reduced modulomodulo
if given.With two arguments, compute
x**y
. Ifx
is negative theny
must be integral. The result will be inexact unlessy
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 correctlyrounded in the Python version.Decimal(0) ** Decimal(0)
results inInvalidOperation
, and ifInvalidOperation
is not trapped, then results inDecimal('NaN')
.Changed in version 3.3: The C module computes
power()
in terms of the correctlyroundedexp()
andln()
functions. The result is welldefined but only “almost always correctlyrounded”.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 nonnegativeat least one of
x
ory
must be nonzeromodulo
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 ofx
,y
andmodulo
. 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 nonzero, 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 ofx / 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 nonnegative 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.
32bit 
64bit 














 decimal.HAVE_THREADS#
The value is
True
. Deprecated, because Python now always has threads.
Deprecated since version 3.9.
 decimal.HAVE_CONTEXTVAR#
The default value is
True
. If Python isconfigured using the withoutdecimalcontextvar option
, the C version uses a threadlocal rather than a coroutinelocal context and the value isFalse
. This is slightly faster in some nested context scenarios.
New in version 3.9: backported to 3.7 and 3.8.
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
andEmax
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 noninfinite 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
orInfinity
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 nonzero 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 ** (noninteger) 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 toInfinity
. In either case,Inexact
andRounded
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
to5.0
). 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
andSubnormal
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 settingFloatOperation
in the context flags. Explicit conversions withfrom_float()
orcreate_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 roundoff 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 roundoff error when nonzero digits exceed the fixed precision.
The effects of roundoff 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 standardscompliance, use the compare()
and comparesignal()
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('0E1000026')
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 * (i1)
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 * (i1)
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 fixedpoint 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
noninteger multiplication, will change the number of decimal places and need to
be followedup with a quantize()
step:
>>> a = Decimal('102.72') # Initial fixedpoint values
>>> b = Decimal('3.17')
>>> a + b # Addition preserves fixedpoint
Decimal('105.89')
>>> a  b
Decimal('99.55')
>>> a * 42 # So does integer multiplication
Decimal('4314.24')
>>> (a * b).quantize(TWOPLACES) # Must quantize noninteger multiplication
Decimal('325.62')
>>> (b / a).quantize(TWOPLACES) # And quantize division
Decimal('0.03')
In developing fixedpoint 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 fixedpoint
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 nonexponential 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 twoplace 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 rerun calculations using greater precision and with various rounding modes. Widely differing results indicate insufficient precision, rounding mode issues, illconditioned 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 correctlyrounded decimal floating point arithmetic 1.
libmpdec
uses Karatsuba multiplication
for mediumsized numbers and the Number Theoretic Transform
for very large numbers.
The context must be adapted for exact arbitrary precision arithmetic. Emin
and Emax
should always be set to the maximum values, clamp
should always be 0 (the default). Setting prec
requires some care.
The easiest approach for trying out bignum arithmetic is to use the maximum
value for prec
as well 2:
>>> setcontext(Context(prec=MAX_PREC, Emax=MAX_EMAX, Emin=MIN_EMIN))
>>> x = Decimal(2) ** 256
>>> x / 128
Decimal('904625697166532776746648320380374280103671755200316906558262375061821325312')
For inexact results, MAX_PREC
is far too large on 64bit platforms and
the available memory will be insufficient:
>>> Decimal(1) / 3
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
MemoryError
On systems with overallocation (e.g. Linux), a more sophisticated approach is to
adjust prec
to the amount of available RAM. Suppose that you have 8GB of
RAM and expect 10 simultaneous operands using a maximum of 500MB each:
>>> import sys
>>>
>>> # Maximum number of digits for a single operand using 500MB in 8byte words
>>> # with 19 digits per word (4byte and 9 digits for the 32bit build):
>>> maxdigits = 19 * ((500 * 1024**2) // 8)
>>>
>>> # Check that this works:
>>> c = Context(prec=maxdigits, Emax=MAX_EMAX, Emin=MIN_EMIN)
>>> c.traps[Inexact] = True
>>> setcontext(c)
>>>
>>> # Fill the available precision with nines:
>>> x = Decimal(0).logical_invert() * 9
>>> sys.getsizeof(x)
524288112
>>> x + 2
Traceback (most recent call last):
File "<stdin>", line 1, in <module>
decimal.Inexact: [<class 'decimal.Inexact'>]
In general (and especially on systems without overallocation), it is recommended
to estimate even tighter bounds and set the Inexact
trap if all calculations
are expected to be exact.