How To Stop Floating Point Arithmetic Errors in Python
We expect precision, consistency, and accuracy when we code. After all, it’s a computer doing the work. But your arithmetic may have been off the entire time and you didn’t even know.
If you’ve experienced floating point arithmetic errors, then you know what we’re talking about. If you’re…
Числа с фиксированной точностью. Класс Decimal
Числа с фиксированной точностью – это числа типа Decimal , которые при вычислениях используют фиксированное количество знаков после запятой. Тип Decimal – это специально разработанный класс (начиная с версии Python 2.4).
Понятие «фиксированная точность» означает, что с помощью таких чисел можно сохранять значение которое будет всегда иметь определенное количество знаков после запятой.
Например, нужно сохранять числа строго с числом 6 знаков после запятой.
Класс Decimal реализован в модуле decimal . Чтобы использовать возможности класса Decimal нужно выполнить команду
2. Как с помощью класса Decimal задать нужную фиксированную точность? Примеры
Чтобы создать объект класса Decimal используется конструктор этого класса. Конструктор получает строку с числом, в котором указывается заданная точность, например
3. Пример необходимости применения класса Decimal в программах на Python
В примере продемонстрирована необходимость написания программ с использованием класса Decimal для случаев, когда точность вычисления крайне важна.
Результат работы программы
Сначала создается объект (переменная) с именем a , в которую записывается сумма
Затем значение этой переменной выводится на экран. Как видно из результата, результат вычисления переменной a содержит погрешность. Это связано с тем, что память, которая выделяется для чисел вещественного типа, ограничена. Другими словами, количество бит в представлении вещественных чисел есть недостаточным.
На следующем шаге создается объект с именем b , в который записывается сумма с использованием класса Decimal
После вывода значения b на экран, видно что значение переменной b представлено точно без погрешности.
4. Пример использования класса Decimal и функции str()
В предыдущем примере конструктор класса Decimal получал строку с числом
в котором определялась точность (1 знак после запятой) и значение числа 0.2.
Возможна ситуация, когда нужно передать непосредственно число а не строку. В этом случае удобно использовать функцию str() , как показано ниже
Функция str() получает число и переводит его в строку
5. Использование фиксированной точности для чисел с разной точностью представлений. Пример
Возможна ситуация, когда в выражении, содержащем класс Decimal , имеются числа с разной точностью представлений. В этом случае точность результата автоматически устанавливается равной точности числа с наибольшей точностью представления.
Например. При сложении трех чисел
автоматически устанавливается точность 3 знака после запятой, поскольку конструктор
определяет число 0.001 с наибольшей точностью представления.
6. Создание объектов класса Decimal из вещественных чисел. Пример
Для случаев, когда есть в наличии вещественное число, можно создать объект класса Decimal . В этом случае используется метод from_float() класса Decimal .
Результат выполнения вышеприведенного кода
Как видно из результата, не всегда удается получить фиксированную точность при использовании метода from_float() .
7. Глобальная настройка точности. Пример
Бывают случаи, когда точность в программе нужно задать для всех операций текущего потока управления. Это может быть, например, представление денежных сумм с учетом копеек (2 знака после запятой).
Пример.
Результат выполнения программы
В вышеприведенном примере глобальная точность для класса Decimal задается с помощью функции getcontext() , которая возвращает объект контекста в этом модуле. Точность задается в текущем потоке управления.
decimal — Decimal fixed point and floating point arithmetic¶
The decimal module provides support for fast correctly rounded decimal floating point arithmetic. It offers several advantages over the float datatype:
Decimal “is based on a 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 and 2.2 do not have exact representations in binary floating point. End users typically would not expect 1.1 + 2.2 to display as 3.3000000000000003 as it does with binary floating point.
The exactness carries over into arithmetic. In decimal floating point, 0.1 + 0.1 + 0.1 — 0.3 is exactly equal to zero. In binary floating point, the result is 5.5511151231257827e017 . 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 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:
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.
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:
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 :
If the FloatOperation signal is trapped, accidental mixing of decimals and floats in constructors or ordering comparisons raises an exception:
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.
If the internal limits of the C version are exceeded, constructing a decimal raises 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:
And some mathematical functions are also available to Decimal:
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:
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:
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.
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 attribute of a context:
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¶
Construct a new Decimal object based from value.
value can be an integer, string, tuple, float , or another Decimal object. If no value is given, returns Decimal(‘0’) . If value is a string, it should conform to the decimal numeric string syntax after leading and trailing whitespace characters, as well as underscores throughout, are removed:
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 or 1 for negative), a tuple of digits, and an integer exponent. For example, Decimal((0, (1, 4, 1, 4), 3)) returns Decimal(‘1.414’) .
If value is a float , the binary floating point value is losslessly converted to its exact decimal equivalent. This conversion can often require 53 or more digits of precision. For example, Decimal(float(‘1.1’)) converts to Decimal(‘1.100000000000000088817841970012523233890533447265625’) .
The context precision does not affect how many digits are stored. That is determined exclusively by the number of digits in value. For example, Decimal(‘3.00000’) records all five zeros even if the context precision is only three.
The purpose of the context argument is determining what to do if value is a malformed string. If the context traps InvalidOperation , an exception is raised; otherwise, the constructor returns a new Decimal with the value of NaN .
Once constructed, Decimal objects are immutable.
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 the FloatOperation 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 and int . All of the usual math operations and special methods apply. Likewise, decimal objects can be copied, pickled, printed, used as dictionary keys, used as set elements, compared, sorted, and coerced to another type (such as float or int ).
There are some small differences between arithmetic on Decimal objects and arithmetic on integers and floats. When the remainder operator % is applied to Decimal objects, the sign of the result is the sign of the dividend rather than the sign of the divisor:
The integer division operator // behaves analogously, returning the integer part of the true quotient (truncating towards zero) rather than its floor, so as to preserve the usual identity x == (x // y) * y + x % y :
The % and // operators implement the remainder and divideinteger operations (respectively) as described in the specification.
Decimal objects cannot generally be combined with floats or instances of fractions.Fraction in arithmetic operations: an attempt to add a Decimal to a float , for example, will raise a TypeError . However, it is possible to use Python’s comparison operators to compare a Decimal instance x with another number y . This avoids confusing results when doing equality comparisons between numbers of different types.
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:
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.
Return a pair (n, d) of integers that represent the given Decimal instance as a fraction, in lowest terms and with a positive denominator:
The conversion is exact. Raise OverflowError on infinities and ValueError on NaNs.
New in version 3.6.
Return a named tuple representation of the number: DecimalTuple(sign, digits, exponent) .
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:
This operation is identical to the compare() method, except that all NaNs signal. That is, if neither operand is a signaling NaN then any quiet NaN operand is treated as though it were a signaling NaN.
compare_total ( other , context = None ) ¶
Compare two operands using their abstract representation rather than their numerical value. Similar to the compare() method, but the result gives a total ordering on Decimal instances. Two Decimal instances with the same numeric value but different representations compare unequal in this ordering:
Quiet and signaling NaNs are also included in the total ordering. The result of this function is Decimal(‘0’) if both operands have the same representation, Decimal(‘1’) if the first operand is lower in the total order than the second, and Decimal(‘1’) if the first operand is higher in the total order than the second operand. See the specification for details of the total order.
This operation is unaffected by context and is quiet: no flags are changed and no rounding is performed. As an exception, the C version may raise InvalidOperation if the second operand cannot be converted exactly.
compare_total_mag ( other , context = None ) ¶
Compare two operands using their abstract representation rather than their value as in compare_total() , but ignoring the sign of each operand. x.compare_total_mag(y) is equivalent to x.copy_abs().compare_total(y.copy_abs()) .
This operation is unaffected by context and is quiet: no flags are changed and no rounding is performed. As an exception, the C version may raise InvalidOperation if the second operand cannot be converted exactly.
Just returns self, this method is only to comply with the Decimal Specification.
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.
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:
This operation is unaffected by context and is quiet: no flags are changed and no rounding is performed. As an exception, the C version may raise InvalidOperation if the second operand cannot be converted exactly.
exp ( context = None ) ¶
Return the value of the (natural) exponential function e**x at the given number. The result is correctly rounded using the ROUND_HALF_EVEN rounding mode.
Alternative constructor that only accepts instances of float or int .
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 .
From Python 3.2 onwards, a Decimal instance can also be constructed directly from a float .
New in version 3.1.
Fused multiplyadd. Return self*other+third with no rounding of the intermediate product self*other.
Return True if the argument is canonical and False otherwise. Currently, a Decimal instance is always canonical, so this operation always returns True .
Return True if the argument is a finite number, and False if the argument is an infinity or a NaN.
Return True if the argument is either positive or negative infinity and False otherwise.
Return True if the argument is a (quiet or signaling) NaN and False otherwise.
is_normal ( context = None ) ¶
Return True if the argument is a normal finite number. Return False if the argument is zero, subnormal, infinite or a NaN.
Return True if the argument is a quiet NaN, and False otherwise.
Return True if the argument has a negative sign and False otherwise. Note that zeros and NaNs can both carry signs.
Return True if the argument is a signaling NaN and False otherwise.
is_subnormal ( context = None ) ¶
Return True if the argument is subnormal, and False otherwise.
Return True if the argument is a (positive or negative) zero and False otherwise.
ln ( context = None ) ¶
Return the natural (base e) logarithm of the operand. The result is correctly rounded using the ROUND_HALF_EVEN rounding mode.
log10 ( context = None ) ¶
Return the base ten logarithm of the operand. The result is correctly rounded using the ROUND_HALF_EVEN rounding mode.
logb ( context = None ) ¶
For a nonzero number, return the adjusted exponent of its operand as a Decimal instance. If the operand is a zero then Decimal(‘Infinity’) is returned and the DivisionByZero flag is raised. If the operand is an infinity then Decimal(‘Infinity’) is returned.
logical_and ( other , context = None ) ¶
logical_and() is a logical operation which takes two logical operands (see Logical operands ). The result is the digitwise and 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 digitwise or of the two operands.
logical_xor ( other , context = None ) ¶
logical_xor() is a logical operation which takes two logical operands (see Logical operands ). The result is the 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 that NaN values are either signaled or ignored (depending on the context and whether they are signaling or quiet).
max_mag ( other , context = None ) ¶
Similar to the max() method, but the comparison is done using the absolute values of the operands.
min ( other , context = None ) ¶
Like min(self, other) except that the context rounding rule is applied before returning and that NaN values are either signaled or ignored (depending on the context and whether they are signaling or quiet).
min_mag ( other , context = None ) ¶
Similar to the min() method, but the comparison is done using the absolute values of the operands.
next_minus ( context = None ) ¶
Return the largest number representable in the given context (or in the current thread’s context if no context is given) that is smaller than the given operand.
next_plus ( context = None ) ¶
Return the smallest number representable in the given context (or in the current thread’s context if no context is given) that is larger than the given operand.
next_toward ( other , context = None ) ¶
If the two operands are unequal, return the number closest to the first operand in the direction of the second operand. If both operands are numerically equal, return a copy of the first operand with the sign set to be the same as the sign of the second operand.
normalize ( context = None ) ¶
Normalize the number by stripping the rightmost trailing zeros and converting any result equal to Decimal(‘0’) to Decimal(‘0e0’) . Used for producing canonical values for attributes of an equivalence class. For example, Decimal(‘32.100’) and Decimal(‘0.321000e+2’) both normalize to the equivalent value Decimal(‘32.1’) .
number_class ( context = None ) ¶
Return a string describing the class of the operand. The returned value is one of the following ten strings.
"Infinity" , indicating that the operand is negative infinity.
"Normal" , indicating that the operand is a negative normal number.
"Subnormal" , indicating that the operand is negative and subnormal.
"Zero" , indicating that the operand is a negative zero.
"+Zero" , indicating that the operand is a positive zero.
"+Subnormal" , indicating that the operand is positive and subnormal.
"+Normal" , indicating that the operand is a positive normal number.
"+Infinity" , indicating that the operand is positive infinity.
"NaN" , indicating that the operand is a quiet NaN (Not a Number).
"sNaN" , indicating that the operand is a signaling NaN.
Return a value equal to the first operand after rounding and having the exponent of the second operand.
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 given context argument; if neither argument is given the rounding mode of the current thread’s context is used.
An error is returned whenever the resulting exponent is greater than Emax or less than Etiny() .
Return Decimal(10) , the radix (base) in which the Decimal class does all its arithmetic. Included for compatibility with the specification.
remainder_near ( other , context = None ) ¶
Return the remainder from dividing self by other. This differs from self % other in that the sign of the remainder is chosen so as to minimize its absolute value. More precisely, the return value is self — n * other where n is the integer nearest to the exact value of self / other , and if two integers are equally near then the even one is chosen.
If the result is zero then its sign will be the sign of self.
Return the result of rotating the digits of the first operand by an amount specified by the second operand. The second operand must be an integer in the range precision through precision. The absolute value of the second operand gives the number of places to rotate. If the second operand is positive then rotation is to the left; otherwise rotation is to the right. The coefficient of the first operand is padded on the left with zeros to length precision if necessary. The sign and exponent of the first operand are unchanged.
same_quantum ( other , context = None ) ¶
Test whether self and other have the same exponent or whether both are NaN .
This operation is unaffected by context and is quiet: no flags are changed and no rounding is performed. As an exception, the C version may raise InvalidOperation if the second operand cannot be converted exactly.
scaleb ( other , context = None ) ¶
Return the first operand with exponent adjusted by the second. Equivalently, return the first operand multiplied by 10**other . The second operand must be an integer.
shift ( other , context = None ) ¶
Return the result of shifting the digits of the first operand by an amount specified by the second operand. The second operand must be an integer in the range precision through precision. The absolute value of the second operand gives the number of places to shift. If the second operand is positive then the shift is to the left; otherwise the shift is to the right. Digits shifted into the coefficient are zeros. The sign and exponent of the first operand are unchanged.
sqrt ( context = None ) ¶
Return the square root of the argument to full precision.
to_eng_string ( context = None ) ¶
Convert to a string, using engineering notation if an exponent is needed.
Engineering notation has an exponent which is a multiple of 3. This can leave up to 3 digits to the left of the decimal place and may require the addition of either one or two trailing zeros.
For example, this converts Decimal(‘123E+1’) to Decimal(‘1.23E+3’) .
to_integral ( rounding = None , context = None ) ¶
Identical to the to_integral_value() method. The to_integral name has been kept for compatibility with older versions.
to_integral_exact ( rounding = None , context = None ) ¶
Round to the nearest integer, signaling Inexact or Rounded as appropriate if rounding occurs. The rounding mode is determined by the rounding parameter if given, else by the given context . If neither parameter is given then the rounding mode of the current context is used.
to_integral_value ( rounding = None , context = None ) ¶
Round to the nearest integer without signaling Inexact or Rounded . If given, applies rounding; otherwise, uses the rounding method in either the supplied context or the current context.
Logical operands¶
The logical_and() , logical_invert() , logical_or() , and logical_xor() methods expect their arguments to be logical operands. A logical operand is a Decimal instance whose exponent and sign are both zero, and whose digits are all either 0 or 1 .
Context objects¶
Contexts are environments for arithmetic operations. They govern precision, set rules for rounding, determine which signals are treated as exceptions, and limit the range for exponents.
Each thread has its own current context which is accessed or changed using the getcontext() and setcontext() functions:
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 , \*\*kwargs ) ¶
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. The kwargs argument is used to set the attributes of the new context.
For example, the following code sets the current decimal precision to 42 places, performs a calculation, and then automatically restores the previous context:
Using keyword arguments, the code would be the following:
Raises TypeError if kwargs supplies an attribute that Context doesn’t support. Raises either TypeError or ValueError if kwargs supplies an invalid value for an attribute.
Changed in version 3.11: localcontext() now supports setting context attributes through the use of keyword arguments.
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) except Inexact , Rounded , and Subnormal .
Because many of the traps are enabled, this context is useful for debugging.
class decimal. ExtendedContext ¶
This is a standard context defined by the General Decimal Arithmetic Specification. Precision is set to nine. Rounding is set to ROUND_HALF_EVEN . All flags are cleared. No traps are enabled (so that exceptions are not raised during computations).
Because the traps are disabled, this context is useful for applications that prefer to have result value of NaN or Infinity instead of raising exceptions. This allows an application to complete a run in the presence of conditions that would otherwise halt the program.
class decimal. DefaultContext ¶
This context is used by the Context constructor as a prototype for new contexts. Changing a field (such a precision) has the effect of changing the default for new contexts created by the Context constructor.
This context is most useful in 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 Context.prec = 28 , Context.rounding = ROUND_HALF_EVEN , and enabled traps for Overflow , InvalidOperation , and DivisionByZero .
In addition to the three supplied contexts, new contexts can be created with the Context constructor.
class decimal. Context ( prec = None , rounding = None , Emin = None , Emax = None , capitals = None , clamp = None , flags = None , traps = None ) ¶
Creates a new context. If a field is not specified or is None , the default values are copied from the DefaultContext . If the flags field is not specified or is None , all flags are cleared.
prec is an integer in the range [ 1 , MAX_PREC ] that sets the precision for arithmetic operations in the context.
The rounding option is one of the constants listed in the section Rounding Modes.
The traps and flags fields list any signals to be set. Generally, new contexts should only set traps and leave the flags clear.
The Emin and Emax fields are integers specifying the outer limits allowable for exponents. Emin must be in the range [ MIN_EMIN , 0 ], Emax in the range [ 0 , MAX_EMAX ].
The capitals field is either 0 or 1 (the default). If set to 1 , exponents are printed with a capital E ; otherwise, a lowercase e is used: Decimal(‘6.02e+23’) .
The clamp field is either 0 (the default) or 1 . If set to 1 , the exponent e of a Decimal instance representable in this context is strictly limited to the range Emin — prec + 1 <= e <= Emax — prec + 1 . If clamp is 0 then a weaker condition holds: the adjusted exponent of the Decimal instance is at most Emax . When clamp is 1 , a large normal number will, where possible, have its exponent reduced and a corresponding number of zeros added to its coefficient, in order to fit the exponent constraints; this preserves the value of the number but loses information about significant trailing zeros. For example:
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 the Decimal methods described above (with the exception of the adjusted() and as_tuple() methods) there is a corresponding Context method. For example, for a Context instance C and Decimal instance x , C.exp(x) is equivalent to x.exp(context=C) . Each Context method accepts a Python integer (an instance of int ) anywhere that a Decimal instance is accepted.
Resets all of the flags to 0 .
Resets all of the traps to 0 .
New in version 3.3.
Return a duplicate of the context.
Return a copy of the Decimal instance 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:
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.
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.
New in version 3.1.
Returns a value equal to Emin — prec + 1 which is the minimum exponent value for subnormal results. When underflow occurs, the exponent is set to Etiny .
Returns a value equal to Emax — prec + 1 .
The usual approach to working with decimals is to create Decimal instances and then apply arithmetic operations which take place within the current context for the active thread. An alternative approach is to use context methods for calculating within a specific context. The methods are similar to those for the Decimal class and are only briefly recounted here.
Returns the absolute value of x.
Return the sum of x and y.
Returns the same Decimal object x.
Compares x and y numerically.
Compares the values of the two operands numerically.
Compares two operands using their abstract representation.
Compares two operands using their abstract representation, ignoring sign.
Returns a copy of x with the sign set to 0.
Returns a copy of x with the sign inverted.
Copies the sign from y to x.
Return x divided by y, truncated to an integer.
Divides two numbers and returns the integer part of the result.
Returns x multiplied by y, plus z.
Returns True if x is canonical; otherwise returns False .
Returns True if x is finite; otherwise returns False .
Returns True if x is infinite; otherwise returns False .
Returns True if x is a qNaN or sNaN; otherwise returns False .
Returns True if x is a normal number; otherwise returns False .
Returns True if x is a quiet NaN; otherwise returns False .
Returns True if x is negative; otherwise returns False .
Returns True if x is a signaling NaN; otherwise returns False .
Returns True if x is subnormal; otherwise returns False .
Returns True if x is a zero; otherwise returns False .
Returns the natural (base e) logarithm of x.
Returns the base 10 logarithm of x.
Returns the exponent of the magnitude of the operand’s MSD.
Applies the logical operation and between each operand’s digits.
Invert all the digits in x.
Applies the logical operation or between each operand’s digits.
Applies the logical operation xor between each operand’s digits.
Compares two values numerically and returns the maximum.
Compares the values numerically with their sign ignored.
Compares two values numerically and returns the minimum.
Compares the values numerically with their sign ignored.
Minus corresponds to the unary prefix minus operator in Python.
Return the product of x and y.
Returns the largest representable number smaller than x.
Returns the smallest representable number larger than x.
Returns the number closest to x, in direction towards y.
Reduces x to its simplest form.
Returns an indication of the class of 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.
Return x to the power of y , reduced modulo modulo if given.
With two arguments, compute x**y . If x is negative then y must be integral. The result will be inexact unless y is integral and the result is finite and can be expressed exactly in ‘precision’ digits. The rounding mode of the context is used. Results are always correctly rounded in the Python version.
Decimal(0) ** Decimal(0) results in InvalidOperation , and if InvalidOperation is not trapped, then results in Decimal(‘NaN’) .
Changed in version 3.3: The C module computes power() in terms of the correctly rounded exp() and ln() functions. The result is welldefined but only “almost always correctly rounded”.
With three arguments, compute (x**y) % modulo . For the three argument form, the following restrictions on the arguments hold:

all three arguments must be integral

y must be nonnegative

at least one of x or y must be nonzero

modulo must be nonzero and have at most ‘precision’ digits
The value resulting from Context.power(x, y, modulo) is equal to the value that would be obtained by computing (x**y) % modulo with unbounded precision, but is computed more efficiently. The exponent of the result is zero, regardless of the exponents of x , y and modulo . The result is always exact.
Returns a value equal to x (rounded), having the exponent of y.
Just returns 10, as this is Decimal, 🙂
Returns the remainder from integer division.
The sign of the result, if nonzero, is the same as that of the original dividend.
Returns x — y * n , where n is the integer nearest the exact value of x / y (if the result is 0 then its sign will be the sign of x).
Returns a rotated copy of x, y times.
Returns True if the two operands have the same exponent.
Returns the first operand after adding the second value its exp.
Returns a shifted copy of x, y times.
Square root of a nonnegative number to context precision.
Return the difference between x and y.
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.
Rounds to an integer.
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.
The value is True . Deprecated, because Python now always has threads.
Deprecated since version 3.9.
The default value is True . If Python is configured using the —withoutdecimalcontextvar option , the C version uses a threadlocal rather than a coroutinelocal context and the value is False . This is slightly faster in some nested context scenarios.
New in version 3.9: backported to 3.7 and 3.8.
Rounding modes¶
Round towards Infinity .
Round towards zero.
Round towards Infinity .
Round to nearest with ties going towards zero.
Round to nearest with ties going to nearest even integer.
Round to nearest with ties going away from zero.
Round away from zero.
Round away from zero if last digit after rounding towards zero would have been 0 or 5; otherwise round towards zero.
Signals¶
Signals represent conditions that arise during computation. Each corresponds to one context flag and one context trap enabler.
The context flag is set whenever the condition is encountered. After the computation, flags may be checked for informational purposes (for instance, to determine whether a computation was exact). After checking the flags, be sure to clear all flags before starting the next computation.
If the context’s trap enabler is set for the signal, then the condition causes a Python exception to be raised. For example, if the DivisionByZero trap is set, then a DivisionByZero exception is raised upon encountering the condition.
class decimal. Clamped ¶
Altered an exponent to fit representation constraints.
Typically, clamping occurs when an exponent falls outside the context’s Emin and Emax limits. If possible, the exponent is reduced to fit by adding zeros to the coefficient.
class decimal. DecimalException ¶
Base class for other signals and a subclass of ArithmeticError .
class decimal. DivisionByZero ¶
Signals the division of a 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 or Infinity with the sign determined by the inputs to the calculation.
class decimal. Inexact ¶
Indicates that rounding occurred and the result is not exact.
Signals when 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:
Indicates the exponent is larger than Context.Emax after rounding has occurred. If not trapped, the result depends on the rounding mode, either pulling inward to the largest representable finite number or rounding outward to Infinity . In either case, Inexact and Rounded are also signaled.
class decimal. Rounded ¶
Rounding occurred though possibly no information was lost.
Signaled whenever rounding discards digits; even if those digits are zero (such as rounding 5.00 to 5.0 ). If not trapped, returns the result unchanged. This signal is used to detect loss of significant digits.
class decimal. Subnormal ¶
Exponent was lower than Emin prior to rounding.
Occurs when an operation result is subnormal (the exponent is too small). If not trapped, returns the result unchanged.
class decimal. Underflow ¶
Numerical underflow with result rounded to zero.
Occurs when a subnormal result is pushed to zero by rounding. Inexact and Subnormal are also signaled.
class decimal. FloatOperation ¶
Enable stricter semantics for mixing floats and Decimals.
If the signal is not trapped (default), mixing floats and Decimals is permitted in the Decimal constructor, create_decimal() and all comparison operators. Both conversion and comparisons are exact. Any occurrence of a mixed operation is silently recorded by setting FloatOperation in the context flags. Explicit conversions with from_float() or create_decimal_from_float() do not set the flag.
Otherwise (the signal is trapped), only equality comparisons and explicit conversions are silent. All other mixed operations raise FloatOperation .
The following table summarizes the hierarchy of signals:
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:
The decimal module makes it possible to restore the identities by expanding the precision sufficiently to avoid loss of significance:
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 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:
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:
Recipes¶
Here are a few recipes that serve as utility functions and that demonstrate ways to work with the Decimal class:
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:
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:
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:
In developing fixedpoint applications, it is convenient to define functions to handle the quantize() step:
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:
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:
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:
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:
The solution is either to increase precision or to force rounding of inputs using the unary plus operation:
Alternatively, inputs can be rounded upon creation using the Context.create_decimal() method:
Q. Is the CPython implementation fast for large numbers?
A. Yes. In the CPython and PyPy3 implementations, the C/CFFI versions of the decimal module integrate the high speed libmpdec library for arbitrary precision correctly rounded decimal floating point arithmetic 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:
For inexact results, MAX_PREC is far too large on 64bit platforms and the available memory will be insufficient:
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:
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.
New in version 3.3.
Changed in version 3.9: This approach now works for all exact results except for noninteger powers.
Число с фиксированной точностью
Надо вывести число a типа float на экран c точностью не менее n знаков после запятой.
Python 3, без использования библиотек.
Или (на более ранних версиях):
Вот какой вариант решения получается у меня:
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