New in version 2.4.
The decimal module provides support for decimal floating point arithmetic. It offers several advantages over the float datatype:
Decimal “is based on a floating-point model which was designed with people in mind, and necessarily has a paramount guiding principle – computers must provide an arithmetic that works in the same way as the arithmetic that people learn at school.” – excerpt from the decimal arithmetic specification.
Decimal numbers can be represented exactly. In contrast, numbers like 1.1 do not have an exact representation in binary floating point. End users typically would not expect 1.1 to display as 1.1000000000000001 as it does with binary floating point.
The exactness carries over into arithmetic. In decimal floating point, 0.1 + 0.1 + 0.1 - 0.3 is exactly equal to zero. In binary floating point, the result is 5.5511151231257827e-017. While near to zero, the differences prevent reliable equality testing and differences can accumulate. For this reason, decimal is preferred in accounting applications which have strict equality invariants.
The decimal module incorporates a notion of significant places so that 1.30 + 1.20 is 2.50. The trailing zero is kept to indicate significance. This is the customary presentation for monetary applications. For multiplication, the “schoolbook” approach uses all the figures in the multiplicands. For instance, 1.3 * 1.2 gives 1.56 while 1.30 * 1.20 gives 1.5600.
Unlike hardware based binary floating point, the decimal module has a user alterable precision (defaulting to 28 places) which can be as large as needed for a given problem:
>>> getcontext().prec = 6
>>> Decimal(1) / Decimal(7)
Decimal('0.142857')
>>> getcontext().prec = 28
>>> Decimal(1) / Decimal(7)
Decimal('0.1428571428571428571428571429')
Both binary and decimal floating point are implemented in terms of published standards. While the built-in float type exposes only a modest portion of its capabilities, the decimal module exposes all required parts of the standard. When needed, the programmer has full control over rounding and signal handling. This includes an option to enforce exact arithmetic by using exceptions to block any inexact operations.
The decimal module was designed to support “without prejudice, both exact unrounded decimal arithmetic (sometimes called fixed-point arithmetic) and rounded floating-point arithmetic.” – excerpt from the decimal arithmetic specification.
The module design is centered around three concepts: the decimal number, the context for arithmetic, and signals.
A decimal number is immutable. It has a sign, coefficient digits, and an exponent. To preserve significance, the coefficient digits do not truncate trailing zeros. Decimals also include special values such as Infinity, -Infinity, and NaN. The standard also differentiates -0 from +0.
The context for arithmetic is an environment specifying precision, rounding rules, limits on exponents, flags indicating the results of operations, and trap enablers which determine whether signals are treated as exceptions. Rounding options include ROUND_CEILING, ROUND_DOWN, ROUND_FLOOR, ROUND_HALF_DOWN, ROUND_HALF_EVEN, ROUND_HALF_UP, ROUND_UP, and ROUND_05UP.
Signals are groups of exceptional conditions arising during the course of computation. Depending on the needs of the application, signals may be ignored, considered as informational, or treated as exceptions. The signals in the decimal module are: Clamped, InvalidOperation, DivisionByZero, Inexact, Rounded, Subnormal, Overflow, and Underflow.
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
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=-999999999, Emax=999999999,
capitals=1, flags=[], traps=[Overflow, DivisionByZero,
InvalidOperation])
>>> getcontext().prec = 7 # Set a new precision
Decimal instances can be constructed from integers, strings, or tuples. To create a Decimal from a float, first convert it to a string. This serves as an explicit reminder of the details of the conversion (including representation error). 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((0, (3, 1, 4), -2))
Decimal('3.14')
>>> Decimal(str(2.0 ** 0.5))
Decimal('1.41421356237')
>>> Decimal(2) ** Decimal('0.5')
Decimal('1.414213562373095048801688724')
>>> Decimal('NaN')
Decimal('NaN')
>>> Decimal('-Infinity')
Decimal('-Infinity')
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')
Decimals interact well with much of the rest of Python. Here is a small decimal floating point flying circus:
>>> data = 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)
1.3400000000000001
>>> round(a, 1) # round() first converts to binary floating point
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=-999999999, Emax=999999999,
capitals=1, 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=-999999999, Emax=999999999,
capitals=1, flags=[Rounded, Inexact], traps=[])
The flags entry shows that the rational approximation to Pi was rounded (digits beyond the context precision were thrown away) and that the result is inexact (some of the discarded digits were non-zero).
Individual traps are set using the dictionary in the traps field of a context:
>>> setcontext(ExtendedContext)
>>> Decimal(1) / Decimal(0)
Decimal('Infinity')
>>> getcontext().traps[DivisionByZero] = 1
>>> Decimal(1) / Decimal(0)
Traceback (most recent call last):
File "<pyshell#112>", line 1, in -toplevel-
Decimal(1) / Decimal(0)
DivisionByZero: x / 0
Most programs adjust the current context only once, at the beginning of the program. And, in many applications, data is converted to Decimal with a single cast inside a loop. With context set and decimals created, the bulk of the program manipulates the data no differently than with other Python numeric types.
Construct a new Decimal object based from value.
value can be an integer, string, tuple, or another Decimal object. If no value is given, returns Decimal('0'). If value is a string, it should conform to the decimal numeric string syntax after leading and trailing whitespace characters are removed:
sign ::= '+' | '-'
digit ::= '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
indicator ::= 'e' | 'E'
digits ::= digit [digit]...
decimal-part ::= digits '.' [digits] | ['.'] digits
exponent-part ::= indicator [sign] digits
infinity ::= 'Infinity' | 'Inf'
nan ::= 'NaN' [digits] | 'sNaN' [digits]
numeric-value ::= decimal-part [exponent-part] | infinity
numeric-string ::= [sign] numeric-value | [sign] nan
If value is a unicode string then other Unicode decimal digits are also permitted where digit appears above. These include decimal digits from various other alphabets (for example, Arabic-Indic and Devanāgarī digits) along with the fullwidth digits u'\uff10' through u'\uff19'.
If value is a tuple, it should have three components, a sign (0 for positive or 1 for negative), a tuple of digits, and an integer exponent. For example, Decimal((0, (1, 4, 1, 4), -3)) returns Decimal('1.414').
The context precision does not affect how many digits are stored. That is determined exclusively by the number of digits in value. For example, Decimal('3.00000') records all five zeros even if the context precision is only three.
The purpose of the context argument is determining what to do if value is a malformed string. If the context traps InvalidOperation, an exception is raised; otherwise, the constructor returns a new Decimal with the value of NaN.
Once constructed, Decimal objects are immutable.
Changed in version 2.6: leading and trailing whitespace characters are permitted when creating a Decimal instance from a string.
Decimal floating point objects share many properties with the other built-in numeric types such as float and int. All of the usual math operations and special methods apply. Likewise, decimal objects can be copied, pickled, printed, used as dictionary keys, used as set elements, compared, sorted, and coerced to another type (such as float or long).
In addition to the standard numeric properties, decimal floating point objects also have a number of specialized methods:
Return a named tuple representation of the number: DecimalTuple(sign, digits, exponent).
Changed in version 2.6: Use a named tuple.
Return the canonical encoding of the argument. Currently, the encoding of a Decimal instance is always canonical, so this operation returns its argument unchanged.
New in version 2.6.
Compare the values of two Decimal instances. This operation behaves in the same way as the usual comparison method __cmp__(), except that compare() returns a Decimal instance rather than an integer, 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')
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.
New in version 2.6.
Compare two operands using their abstract representation rather than their numerical value. Similar to the compare() method, but the result gives a total ordering on Decimal instances. Two Decimal instances with the same numeric value but different representations compare unequal in this ordering:
>>> Decimal('12.0').compare_total(Decimal('12'))
Decimal('-1')
Quiet and signaling NaNs are also included in the total ordering. The result of this function is Decimal('0') if both operands have the same representation, Decimal('-1') if the first operand is lower in the total order than the second, and Decimal('1') if the first operand is higher in the total order than the second operand. See the specification for details of the total order.
New in version 2.6.
Compare two operands using their abstract representation rather than their value as in compare_total(), but ignoring the sign of each operand. x.compare_total_mag(y) is equivalent to x.copy_abs().compare_total(y.copy_abs()).
New in version 2.6.
Just returns self, this method is only to comply with the Decimal Specification.
New in version 2.6.
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.
New in version 2.6.
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.
New in version 2.6.
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 the context and is quiet: no flags are changed and no rounding is performed.
New in version 2.6.
Return the value of the (natural) exponential function e**x at the given number. The result is correctly rounded using the ROUND_HALF_EVEN rounding mode.
>>> Decimal(1).exp()
Decimal('2.718281828459045235360287471')
>>> Decimal(321).exp()
Decimal('2.561702493119680037517373933E+139')
New in version 2.6.
Fused multiply-add. Return self*other+third with no rounding of the intermediate product self*other.
>>> Decimal(2).fma(3, 5)
Decimal('11')
New in version 2.6.
Return True if the argument is canonical and False otherwise. Currently, a Decimal instance is always canonical, so this operation always returns True.
New in version 2.6.
Return True if the argument is a finite number, and False if the argument is an infinity or a NaN.
New in version 2.6.
Return True if the argument is either positive or negative infinity and False otherwise.
New in version 2.6.
Return True if the argument is a (quiet or signaling) NaN and False otherwise.
New in version 2.6.
Return True if the argument is a normal finite number. Return False if the argument is zero, subnormal, infinite or a NaN.
New in version 2.6.
Return True if the argument has a negative sign and False otherwise. Note that zeros and NaNs can both carry signs.
New in version 2.6.
Return True if the argument is a (positive or negative) zero and False otherwise.
New in version 2.6.
Return the natural (base e) logarithm of the operand. The result is correctly rounded using the ROUND_HALF_EVEN rounding mode.
New in version 2.6.
Return the base ten logarithm of the operand. The result is correctly rounded using the ROUND_HALF_EVEN rounding mode.
New in version 2.6.
For a nonzero number, return the adjusted exponent of its operand as a Decimal instance. If the operand is a zero then Decimal('-Infinity') is returned and the DivisionByZero flag is raised. If the operand is an infinity then Decimal('Infinity') is returned.
New in version 2.6.
logical_and() is a logical operation which takes two logical operands (see Logical operands). The result is the digit-wise and of the two operands.
New in version 2.6.
logical_invert() is a logical operation. The argument must be a logical operand (see Logical operands). The result is the digit-wise inversion of the operand.
New in version 2.6.
logical_or() is a logical operation which takes two logical operands (see Logical operands). The result is the digit-wise or of the two operands.
New in version 2.6.
logical_xor() is a logical operation which takes two logical operands (see Logical operands). The result is the digit-wise exclusive or of the two operands.
New in version 2.6.
Similar to the max() method, but the comparison is done using the absolute values of the operands.
New in version 2.6.
Similar to the min() method, but the comparison is done using the absolute values of the operands.
New in version 2.6.
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.
New in version 2.6.
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.
New in version 2.6.
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.
New in version 2.6.
Return a string describing the class of the operand. The returned value is one of the following ten strings.
New in version 2.6.
Return a value equal to the first operand after rounding and having the exponent of the second operand.
>>> Decimal('1.41421356').quantize(Decimal('1.000'))
Decimal('1.414')
Unlike other operations, if the length of the coefficient after the quantize operation would be greater than precision, then an InvalidOperation is signaled. This guarantees that, unless there is an error condition, the quantized exponent is always equal to that of the right-hand operand.
Also unlike other operations, quantize never signals Underflow, even if the result is subnormal and inexact.
If the exponent of the second operand is larger than that of the first then rounding may be necessary. In this case, the rounding mode is determined by the rounding argument if given, else by the given context argument; if neither argument is given the rounding mode of the current thread’s context is used.
If watchexp is set (default), then an error is returned whenever the resulting exponent is greater than Emax or less than Etiny.
Return Decimal(10), the radix (base) in which the Decimal class does all its arithmetic. Included for compatibility with the specification.
New in version 2.6.
Compute the modulo as either a positive or negative value depending on which is closest to zero. For instance, Decimal(10).remainder_near(6) returns Decimal('-2') which is closer to zero than Decimal('4').
If both are equally close, the one chosen will have the same sign as self.
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.
New in version 2.6.
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.
New in version 2.6.
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.
New in version 2.6.
Convert to an engineering-type string.
Engineering notation has an exponent which is a multiple of 3, so there are up to 3 digits left of the decimal place. For example, converts Decimal('123E+1') to Decimal('1.23E+3')
Round to the nearest integer, signaling Inexact or Rounded as appropriate if rounding occurs. The rounding mode is determined by the rounding parameter if given, else by the given context. If neither parameter is given then the rounding mode of the current context is used.
New in version 2.6.
Round to the nearest integer without signaling Inexact or Rounded. If given, applies rounding; otherwise, uses the rounding method in either the supplied context or the current context.
Changed in version 2.6: renamed from to_integral to to_integral_value. The old name remains valid for compatibility.
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:
Beginning with Python 2.5, you can also use the with statement and the localcontext() function to temporarily change the active context.
Return a context manager that will set the current context for the active thread to a copy of c on entry to the with-statement and restore the previous context when exiting the with-statement. If no context is specified, a copy of the current context is used.
New in version 2.5.
For example, the following code sets the current decimal precision to 42 places, performs a calculation, and then automatically restores the previous context:
from decimal import localcontext
with localcontext() as ctx:
ctx.prec = 42 # Perform a high precision calculation
s = calculate_something()
s = +s # Round the final result back to the default precision
New contexts can also be created using the Context constructor described below. In addition, the module provides three pre-made contexts:
This is a standard context defined by the General Decimal Arithmetic Specification. Precision is set to nine. Rounding is set to ROUND_HALF_UP. All flags are cleared. All traps are enabled (treated as exceptions) except Inexact, Rounded, and Subnormal.
Because many of the traps are enabled, this context is useful for debugging.
This is a standard context defined by the General Decimal Arithmetic Specification. Precision is set to nine. Rounding is set to ROUND_HALF_EVEN. All flags are cleared. No traps are enabled (so that exceptions are not raised during computations).
Because the traps are disabled, this context is useful for applications that prefer to have result value of NaN or Infinity instead of raising exceptions. This allows an application to complete a run in the presence of conditions that would otherwise halt the program.
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 creating by the Context constructor.
This context is most useful in multi-threaded environments. Changing one of the fields before threads are started has the effect of setting system-wide defaults. Changing the fields after threads have started is not recommended as it would require thread synchronization to prevent race conditions.
In single threaded environments, it is preferable to not use this context at all. Instead, simply create contexts explicitly as described below.
The default values are precision=28, rounding=ROUND_HALF_EVEN, and enabled traps for Overflow, InvalidOperation, and DivisionByZero.
In addition to the three supplied contexts, new contexts can be created with the Context constructor.
Creates a new context. If a field is not specified or is None, the default values are copied from the DefaultContext. If the flags field is not specified or is None, all flags are cleared.
The prec field is a positive integer that sets the precision for arithmetic operations in the context.
The rounding option is one of:
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.
The capitals field is either 0 or 1 (the default). If set to 1, exponents are printed with a capital E; otherwise, a lowercase e is used: Decimal('6.02e+23').
Changed in version 2.6: The ROUND_05UP rounding mode was added.
The Context class defines several general purpose methods as well as a large number of methods for doing arithmetic directly in a given context. In addition, for each of the Decimal methods described above (with the exception of the adjusted() and as_tuple() methods) there is a corresponding Context method. For example, C.exp(x) is equivalent to x.exp(context=C).
Creates a new Decimal instance from num but using self as context. Unlike the Decimal constructor, the context precision, rounding method, flags, and traps are applied to the conversion.
This is useful because constants are often given to a greater precision than is needed by the application. Another benefit is that rounding immediately eliminates unintended effects from digits beyond the current precision. In the following example, using unrounded inputs means that adding zero to a sum can change the result:
>>> getcontext().prec = 3
>>> Decimal('3.4445') + Decimal('1.0023')
Decimal('4.45')
>>> Decimal('3.4445') + Decimal(0) + Decimal('1.0023')
Decimal('4.44')
This method implements the to-number operation of the IBM specification. If the argument is a string, no leading or trailing whitespace is permitted.
The usual approach to working with decimals is to create Decimal instances and then apply arithmetic operations which take place within the current context for the active thread. An alternative approach is to use context methods for calculating within a specific context. The methods are similar to those for the Decimal class and are only briefly recounted here.
Return x to the power of y, reduced modulo modulo if given.
With two arguments, compute x**y. If x is negative then y must be integral. The result will be inexact unless y is integral and the result is finite and can be expressed exactly in ‘precision’ digits. The result should always be correctly rounded, using the rounding mode of the current thread’s context.
With three arguments, compute (x**y) % modulo. For the three argument form, the following restrictions on the arguments hold:
- all three arguments must be integral
- y must be nonnegative
- at least one of x or y must be nonzero
- modulo must be nonzero and have at most ‘precision’ digits
The result of Context.power(x, y, modulo) is identical to the result that would be obtained by computing (x**y) % modulo with unbounded precision, but is computed more efficiently. It is always exact.
Changed in version 2.6: y may now be nonintegral in x**y. Stricter requirements for the three-argument version.
Returns the remainder from integer division.
The sign of the result, if non-zero, is the same as that of the original dividend.
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.
Altered an exponent to fit representation constraints.
Typically, clamping occurs when an exponent falls outside the context’s Emin and Emax limits. If possible, the exponent is reduced to fit by adding zeros to the coefficient.
Signals the division of a non-infinite number by zero.
Can occur with division, modulo division, or when raising a number to a negative power. If this signal is not trapped, returns Infinity or -Infinity with the sign determined by the inputs to the calculation.
Indicates that rounding occurred and the result is not exact.
Signals when non-zero digits were discarded during rounding. The rounded result is returned. The signal flag or trap is used to detect when results are inexact.
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
x._rescale( non-integer )
sqrt(-x) and x > 0
0 ** 0
x ** (non-integer)
x ** Infinity
Numerical overflow.
Indicates the exponent is larger than Emax after rounding has occurred. If not trapped, the result depends on the rounding mode, either pulling inward to the largest representable finite number or rounding outward to Infinity. In either case, Inexact and Rounded are also signaled.
Rounding occurred though possibly no information was lost.
Signaled whenever rounding discards digits; even if those digits are zero (such as rounding 5.00 to 5.0). If not trapped, returns the result unchanged. This signal is used to detect loss of significant digits.
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.
Numerical underflow with result rounded to zero.
Occurs when a subnormal result is pushed to zero by rounding. Inexact and Subnormal are also signaled.
The following table summarizes the hierarchy of signals:
exceptions.ArithmeticError(exceptions.StandardError)
DecimalException
Clamped
DivisionByZero(DecimalException, exceptions.ZeroDivisionError)
Inexact
Overflow(Inexact, Rounded)
Underflow(Inexact, Rounded, Subnormal)
InvalidOperation
Rounded
Subnormal
The use of decimal floating point eliminates decimal representation error (making it possible to represent 0.1 exactly); however, some operations can still incur round-off error when non-zero digits exceed the fixed precision.
The effects of round-off error can be amplified by the addition or subtraction of nearly offsetting quantities resulting in loss of significance. Knuth provides two instructive examples where rounded floating point arithmetic with insufficient precision causes the breakdown of the associative and distributive properties of addition:
# Examples from Seminumerical Algorithms, Section 4.2.2.
>>> from decimal import Decimal, getcontext
>>> getcontext().prec = 8
>>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
>>> (u + v) + w
Decimal('9.5111111')
>>> u + (v + w)
Decimal('10')
>>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
>>> (u*v) + (u*w)
Decimal('0.01')
>>> u * (v+w)
Decimal('0.0060000')
The decimal module makes it possible to restore the identities by expanding the precision sufficiently to avoid loss of significance:
>>> getcontext().prec = 20
>>> u, v, w = Decimal(11111113), Decimal(-11111111), Decimal('7.51111111')
>>> (u + v) + w
Decimal('9.51111111')
>>> u + (v + w)
Decimal('9.51111111')
>>>
>>> u, v, w = Decimal(20000), Decimal(-6), Decimal('6.0000003')
>>> (u*v) + (u*w)
Decimal('0.0060000')
>>> u * (v+w)
Decimal('0.0060000')
The number system for the decimal module provides special values including NaN, sNaN, -Infinity, Infinity, and two zeros, +0 and -0.
Infinities can be constructed directly with: Decimal('Infinity'). Also, they can arise from dividing by zero when the DivisionByZero signal is not trapped. Likewise, when the Overflow signal is not trapped, infinity can result from rounding beyond the limits of the largest representable number.
The infinities are signed (affine) and can be used in arithmetic operations where they get treated as very large, indeterminate numbers. For instance, adding a constant to infinity gives another infinite result.
Some operations are indeterminate and return NaN, or if the InvalidOperation signal is trapped, raise an exception. For example, 0/0 returns NaN which means “not a number”. This variety of NaN is quiet and, once created, will flow through other computations always resulting in another NaN. This behavior can be useful for a series of computations that occasionally have missing inputs — it allows the calculation to proceed while flagging specific results as invalid.
A variant is sNaN which signals rather than remaining quiet after every operation. This is a useful return value when an invalid result needs to interrupt a calculation for special handling.
The behavior of Python’s comparison operators can be a little surprising where a NaN is involved. A test for equality where one of the operands is a quiet or signaling NaN always returns False (even when doing Decimal('NaN')==Decimal('NaN')), while a test for inequality always returns True. An attempt to compare two Decimals using any of the <, <=, > or >= operators will raise the InvalidOperation signal if either operand is a NaN, and return False if this signal is not trapped. Note that the General Decimal Arithmetic specification does not specify the behavior of direct comparisons; these rules for comparisons involving a NaN were taken from the IEEE 854 standard (see Table 3 in section 5.7). To ensure strict standards-compliance, use the compare() and compare-signal() methods instead.
The signed zeros can result from calculations that underflow. They keep the sign that would have resulted if the calculation had been carried out to greater precision. Since their magnitude is zero, both positive and negative zeros are treated as equal and their sign is informational.
In addition to the two signed zeros which are distinct yet equal, there are various representations of zero with differing precisions yet equivalent in value. This takes a bit of getting used to. For an eye accustomed to normalized floating point representations, it is not immediately obvious that the following calculation returns a value equal to zero:
>>> 1 / Decimal('Infinity')
Decimal('0E-1000000026')
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()
. . .
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 = map(str, digits)
build, next = result.append, digits.pop
if sign:
build(trailneg)
for i in range(places):
build(next() if digits else '0')
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.
>>> print cos(Decimal('0.5'))
0.8775825618903727161162815826
>>> print cos(0.5)
0.87758256189
>>> print cos(0.5+0j)
(0.87758256189+0j)
"""
getcontext().prec += 2
i, lasts, s, fact, num, sign = 0, 0, 1, 1, 1, 1
while s != lasts:
lasts = s
i += 2
fact *= i * (i-1)
num *= x * x
sign *= -1
s += num / fact * sign
getcontext().prec -= 2
return +s
def sin(x):
"""Return the sine of x as measured in radians.
>>> print sin(Decimal('0.5'))
0.4794255386042030002732879352
>>> print sin(0.5)
0.479425538604
>>> print sin(0.5+0j)
(0.479425538604+0j)
"""
getcontext().prec += 2
i, lasts, s, fact, num, sign = 1, 0, x, 1, x, 1
while s != lasts:
lasts = s
i += 2
fact *= i * (i-1)
num *= x * x
sign *= -1
s += num / fact * sign
getcontext().prec -= 2
return +s
Q. It is cumbersome to type decimal.Decimal('1234.5'). Is there a way to minimize typing when using the interactive interpreter?
A. Some users abbreviate the constructor to just a single letter:
>>> D = decimal.Decimal
>>> D('1.23') + D('3.45')
Decimal('4.68')
Q. In a fixed-point application with two decimal places, some inputs have many places and need to be rounded. Others are not supposed to have excess digits and need to be validated. What methods should be used?
A. The quantize() method rounds to a fixed number of decimal places. If the Inexact trap is set, it is also useful for validation:
>>> TWOPLACES = Decimal(10) ** -2 # same as Decimal('0.01')>>> # Round to two places >>> Decimal('3.214').quantize(TWOPLACES) Decimal('3.21')>>> # Validate that a number does not exceed two places >>> Decimal('3.21').quantize(TWOPLACES, context=Context(traps=[Inexact])) Decimal('3.21')>>> Decimal('3.214').quantize(TWOPLACES, context=Context(traps=[Inexact])) Traceback (most recent call last): ... Inexact: None
Q. Once I have valid two place inputs, how do I maintain that invariant throughout an application?
A. Some operations like addition, subtraction, and multiplication by an integer will automatically preserve fixed point. Others operations, like division and non-integer multiplication, will change the number of decimal places and need to be followed-up with a quantize() step:
>>> a = Decimal('102.72') # Initial fixed-point values
>>> b = Decimal('3.17')
>>> a + b # Addition preserves fixed-point
Decimal('105.89')
>>> a - b
Decimal('99.55')
>>> a * 42 # So does integer multiplication
Decimal('4314.24')
>>> (a * b).quantize(TWOPLACES) # Must quantize non-integer multiplication
Decimal('325.62')
>>> (b / a).quantize(TWOPLACES) # And quantize division
Decimal('0.03')
In developing fixed-point applications, it is convenient to define functions to handle the quantize() step:
>>> def mul(x, y, fp=TWOPLACES): ... return (x * y).quantize(fp) >>> def div(x, y, fp=TWOPLACES): ... return (x / y).quantize(fp)>>> mul(a, b) # Automatically preserve fixed-point Decimal('325.62') >>> div(b, a) Decimal('0.03')
Q. There are many ways to express the same value. The numbers 200, 200.000, 2E2, and 02E+4 all have the same value at various precisions. Is there a way to transform them to a single recognizable canonical value?
A. The normalize() method maps all equivalent values to a single representative:
>>> values = map(Decimal, '200 200.000 2E2 .02E+4'.split())
>>> [v.normalize() for v in values]
[Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2'), Decimal('2E+2')]
Q. Some decimal values always print with exponential notation. Is there a way to get a non-exponential representation?
A. For some values, exponential notation is the only way to express the number of significant places in the coefficient. For example, expressing 5.0E+3 as 5000 keeps the value constant but cannot show the original’s two-place significance.
If an application does not care about tracking significance, it is easy to remove the exponent and trailing zeroes, losing significance, but keeping the value unchanged:
>>> def remove_exponent(d): ... return d.quantize(Decimal(1)) if d == d.to_integral() else d.normalize()>>> remove_exponent(Decimal('5E+3')) Decimal('5000')
Q. Is there a way to convert a regular float to a Decimal?
A. Yes, all binary floating point numbers can be exactly expressed as a Decimal. An exact conversion may take more precision than intuition would suggest, so we trap Inexact to signal a need for more precision:
def float_to_decimal(f):
"Convert a floating point number to a Decimal with no loss of information"
n, d = f.as_integer_ratio()
numerator, denominator = Decimal(n), Decimal(d)
ctx = Context(prec=60)
result = ctx.divide(numerator, denominator)
while ctx.flags[Inexact]:
ctx.flags[Inexact] = False
ctx.prec *= 2
result = ctx.divide(numerator, denominator)
return result
>>> float_to_decimal(math.pi)
Decimal('3.141592653589793115997963468544185161590576171875')
Q. Why isn’t the float_to_decimal() routine included in the module?
A. There is some question about whether it is advisable to mix binary and decimal floating point. Also, its use requires some care to avoid the representation issues associated with binary floating point:
>>> float_to_decimal(1.1)
Decimal('1.100000000000000088817841970012523233890533447265625')
Q. Within a complex calculation, how can I make sure that I haven’t gotten a spurious result because of insufficient precision or rounding anomalies.
A. The decimal module makes it easy to test results. A best practice is to re-run calculations using greater precision and with various rounding modes. Widely differing results indicate insufficient precision, rounding mode issues, ill-conditioned inputs, or a numerically unstable algorithm.
Q. I noticed that context precision is applied to the results of operations but not to the inputs. Is there anything to watch out for when mixing values of different precisions?
A. Yes. The principle is that all values are considered to be exact and so is the arithmetic on those values. Only the results are rounded. The advantage for inputs is that “what you type is what you get”. A disadvantage is that the results can look odd if you forget that the inputs haven’t been rounded:
>>> getcontext().prec = 3
>>> Decimal('3.104') + Decimal('2.104')
Decimal('5.21')
>>> Decimal('3.104') + Decimal('0.000') + Decimal('2.104')
Decimal('5.20')
The solution is either to increase precision or to force rounding of inputs using the unary plus operation:
>>> getcontext().prec = 3
>>> +Decimal('1.23456789') # unary plus triggers rounding
Decimal('1.23')
Alternatively, inputs can be rounded upon creation using the Context.create_decimal() method:
>>> Context(prec=5, rounding=ROUND_DOWN).create_decimal('1.2345678')
Decimal('1.2345')