Decimal numbers with fixed-point arithmetic¶
The package decimalfp provides a Decimal
number type which can
represent decimal numbers of (nearly) arbitrary magnitude and very large
precision, i.e. with a very large number of fractional digits.
Usage¶
decimalfp.Decimal
instances are created by giving a value (default:
0) and a precision (i.e the number of fractional digits, default: None).
>>> Decimal()
Decimal(0)
If precision is given, it must be of type int and >= 0.
>>> Decimal(5, 3)
Decimal(5, 3)
>>> Decimal(5555, -3)
ValueError: Precision must be >= 0.
If value is given, it must either be a string, an instance of numbers.Integral, number.Rational (for example fractions.Fraction), decimal.Decimal, a finite instance of numbers.Real (for example float) or be convertable to a float or an int.
The value is always adjusted to the given precision or the precision is calculated from the given value, if no precision is given.
>>> Decimal(b'12.345')
Decimal('12.345')
>>> Decimal(u'12.345')
Decimal('12.345')
>>> Decimal(u'12.345', 5)
Decimal('12.345', 5)
>>> Decimal(15, 4)
Decimal(15, 4)
>>> Decimal(10 ** 23)
Decimal(100000000000000000000000)
>>> Decimal(10 ** 48)
Decimal(1000000000000000000000000000000000000000000000000)
>>> Decimal(2 ** 48)
Decimal(281474976710656)
>>> Decimal(2 ** 98)
Decimal(316912650057057350374175801344)
>>> Decimal(Fraction(7, 56))
Decimal('0.125')
>>> Decimal(Fraction(8106479329266893, 4503599627370496), 7)
Decimal('1.8', 7)
>>> Decimal(1.8, 7)
Decimal('1.8', 7)
>>> Decimal(decimal.Decimal('-19.26', 5), 3)
Decimal('-19.26', 3)
When the given precision is lower than the precision of the given value, the result is rounded, according to the current default rounding mode (which itself defaults to ROUND_HALF_EVEN).
>>> Decimal(u'12.345', 2)
Decimal('12.34')
>>> Decimal(u'12.3456', 3)
Decimal('12.346')
>>> Decimal(0.2, 3)
Decimal('0.2', 3)
>>> Decimal(0.2, 17)
Decimal('0.20000000000000001')
>>> Decimal(0.2, 55)
Decimal('0.200000000000000011102230246251565404236316680908203125', 55)
When no precision is given and the given value is a float or a
numbers.Rational (but no Decimal
), the Decimal
constructor
tries to convert value exactly. But this is done only up a fixed limit of
fractional digits (imposed by the implementation, currently 65535). If value
can not be represented as a Decimal
within this limit, an exception
is raised.
>>> Decimal(Fraction(1, 7))
ValueError: Can't convert Fraction(1, 7) exactly to Decimal.
Decimal
does not deal with infinity, division by 0 always raises a
ZeroDivisionError. Likewise, infinite instances of type float or
decimal.Decimal can not be converted to Decimal
instances. The same
is true for the ‘not a number’ instances of these types.
Computations¶
When importing decimalfp, its Decimal
type is registered in Pythons
numerical stack as number.Rational. It supports all operations defined for
that base class and its instances can be mixed in computations with instances
of all numeric types mentioned above.
All numerical operations give an exact result, i.e. they are not automatically
constraint to the precision of the operands or to a number of significant
digits (like the floating-point Decimal type from the standard module
decimal). When the result can not exactly be represented by a
Decimal
instance within the limit of fractional digits, an instance
of fractions.Fraction is returned.
Addition and subtraction¶
Adding or subtracting Decimal
instances results in a Decimal
instance with a precision equal to the maximum of the precisions of the
operands.
>>> Decimal('7.3') + Decimal('8.275')
Decimal('15.575')
>>> Decimal('-7.3', 4) + Decimal('8.275')
Decimal('0.975', 4)
In operations with other numerical types the precision of the result is at
least equal to the precision of the involved Decimal
instance, but
may be greater, if neccessary. If the needed precision exceeds the limit of
fractional digits, an instance of fractions.Fraction is returned.
>>> 0.25 + Decimal(3)
Decimal('3.25')
>>> 0.25 - Decimal(-3, 5)
Decimal('3.25', 5)
>>> 0.725 + Decimal('3')
Decimal('3.72499999999999997779553950749686919152736663818359375')
>>> Decimal('3') + Fraction(1, 7)
Fraction(22, 7)
Multiplication and division¶
Multiplying Decimal
instances results in a Decimal
instance
with precision equal to the sum of the precisions of the operands.
>>> Decimal('5.000') * Decimal('2.5')
Decimal('12.5', 4)
Dividing Decimal
instances results in a Decimal
instance
with precision at least equal to max(0, numerator.precision -
denominator.precision), but may be greater, if needed.
>>> Decimal('5.2000') / Decimal('2.5')
Decimal('2.08', 3)
>>> Decimal('5.2003') / Decimal('2.5')
Decimal('2.08012')
In operations with other numerical types the precision of the result is at
least equal to the precision of the involved Decimal
instance, but
may be greater, if neccessary. If the needed precision exceeds the limit of
fractional digits, an instance of fractions.Fraction is returned.
>>> 3 * Decimal('7.5')
Decimal('22.5')
>>> Decimal(5) * 0.25
Decimal('1.25')
>>> Decimal('3') * Fraction(1, 7)
Fraction(3, 7)
Rounding¶
Decimal
supports rounding via the built-in function round.
Note
In Python 3.x the function round uses the rounding mode ROUND_HALF_EVEN and returns an int when called with one argument, otherwise the same type as the number to be rounded.
>>> round(Decimal('12.345'))
12
>>> round(Decimal('12.345'), 2)
Decimal('12.34')
>>> round(Decimal('1234.5'), -2)
Decimal(1200)
In addition, via the method adjusted()
a Decimal
with a
different precision can be derived, supporting all rounding modes defined by
the standard library module decimal.
The rounding modes defined in decimal are wrapped into the Enum
ROUNDING
.
>>> d = Decimal('12.345')
>>> d.adjusted(2) # default rounding mode is ROUND_HALF_EVEN !
Decimal('12.34')
>>> d.adjusted(2, ROUNDING.ROUND_HALF_UP)
Decimal('12.35')
>>> d.adjusted(1, ROUNDING.ROUND_UP)
Decimal('12.4')
For the details of the different rounding modes see the documentation of the standard library module decimal.
round and adjusted only allow to round to a quantum that’s a power to 10.
The method quantize()
can be used to round to any quantum and it does
also support all rounding modes mentioned above.
>>> d = Decimal('12.345')
>>># equivalent to round(d, 2) or d.adjusted(2)
>>># (default rounding mode ROUNDING.ROUND_HALF_EVEN):
>>> d.quantize(Decimal('0.01'))
Decimal('12.34')
>>> d.quantize(Decimal('0.05'))
Decimal('12.35')
>>> d.quantize('0.6')
Decimal('12.6')
>>> d.quantize(4)
Decimal('12')
Class Decimal¶
DecValueT = Union[SupportsInt, SupportsFloat, SupportsAsIntegerRatio, str]
- class Decimal¶
Decimal number with a given number of fractional digits.
- Parameters
value – numerical value (default: None)
precision – number of fractional digits (default: None)
If value is given, it must either be a string, an instance of int, number.Rational (for example fractions.Fraction), decimal.Decimal, a finite instance of numbers.Real (for example float) or be convertable to a float or an int.
If a string is given as value, it must be a string in one of two formats:
[+|-]<int>[.<frac>][<e|E>[+|-]<exp>] or
[+|-].<frac>[<e|E>[+|-]<exp>].
If given value is None, Decimal(0) is returned.
- Returns
Decimal
instance derived from value accordingto precision
The value is always adjusted to the given precision or the precision is calculated from the given value, if no precision is given.
- Raises
TypeError – precision is given, but not of type int.
TypeError – value is not an instance of the types listed above and not convertable to float or int.
ValueError – precision is given, but not >= 0.
ValueError – precision is given, but not <= MAX_DEC_PRECISION.
ValueError – value can not be converted to a Decimal (with a number of fractional digits <= MAX_DEC_PRECISION).
Decimal
instances are immutable.- __add__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational ¶
self + other
- __bytes__() bytes ¶
bytes(self)
- __ceil__() int ¶
math.ceil(self)
- __eq__(other: Any) bool ¶
self == other
- __floor__() int ¶
math.floor(self)
- __format__(fmt_spec: str) str ¶
Return self converted to a string according to fmt_spec.
- Parameters
fmt_spec – a standard format specifier for a number
- Returns
str – self converted to a string according to fmt_spec
- __ge__(other: Any) bool ¶
self >= other
- __gt__(other: Any) bool ¶
self > other
- __hash__() int ¶
hash(self)
- __init__()¶
- __int__() int ¶
math.trunc(self)
- __le__(other: Any) bool ¶
self <= other
- __lt__(other: Any) bool ¶
self < other
- __mul__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational ¶
self * other
- static __new__(cls, value: DecValueT = None, precision: Optional[Integer] = None) Decimal ¶
Create and return new Decimal instance.
- __pow__(other: SupportsIntOrFloat, mod: Optional[Any] = None) numbers.Complex ¶
self ** other
If other is an integer (or a Rational with denominator = 1), the result will be a Decimal or a Fraction. Otherwise, the result will be a float or a complex.
mod must always be None (otherwise a TypeError is raised).
- __radd__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational ¶
other + self
- __repr__() str ¶
repr(self)
- __rmul__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational ¶
other * self
- __round__(ndigits: Optional[int] = None) Union[int, Decimal] ¶
round(self [, n_digits])
Round self to a given precision in decimal digits (default 0). n_digits may be negative.
This method is called by the built-in round function. It returns an int when called with one argument, otherwise a
Decimal
.
- __rsub__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational ¶
other - self
- __rtruediv__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational ¶
other / self
- __str__() str ¶
str(self)
- __sub__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational ¶
self - other
- __truediv__(other: decimalfp.SupportsAsIntegerRatio) numbers.Rational ¶
self / other
- __trunc__() int ¶
math.trunc(self)
- adjusted(precision: Optional[int] = None, rounding: Optional[ROUNDING] = None) Decimal ¶
Return adjusted copy of self.
- Parameters
precision – number of fractional digits (default: None)
rounding – rounding mode (default: None)
- Returns
Decimal
instance derived from self, adjustedto the given precision, using the given rounding mode
If no precision is given, the result is adjusted to the minimum precision preserving x == x.adjusted().
If no rounding mode is given, the current default rounding mode is used.
If the given precision is less than the precision of self, the result is rounded and thus information may be lost.
- as_fraction() fractions.Fraction ¶
Return an instance of Fraction equal to self.
Returns the Fraction with the smallest positive denominator, whose ratio is equal to self.
- as_integer_ratio() Tuple[int, int] ¶
Return a pair of integers whose ratio is equal to self.
Returns the pair of numerator and denominator with the smallest positive denominator, whose ratio is equal to self.
- as_tuple() Tuple[int, int, int] ¶
Return a tuple (sign, coeff, exp) equivalent to self.
self == sign * coeff * 10 ** exp.
sign in (-1, 0, 1), for self < 0, = 0, > 0. coeff = 0 only if self = 0.
- classmethod from_decimal(d: Union[Decimal, int, decimal.Decimal]) Decimal ¶
Convert a finite decimal number to a
Decimal
.
- classmethod from_float(f: Union[float, int]) Decimal ¶
Convert a finite float (or int) to a
Decimal
.- Parameters
f – number to be converted to a Decimal
- Returns
Decimal
instance derived from f- Raises
TypeError – f is neither a float nor an int.
ValueError – f can not be converted to a
Decimal
with a precision <= MAX_DEC_PRECISION.
Beware that Decimal.from_float(0.3) != Decimal(‘0.3’).
- classmethod from_real(r: numbers.Real, exact: bool = True) Decimal ¶
Convert a finite Real number to a
Decimal
.- Parameters
- Returns
Decimal
instance derived from r- Raises
TypeError – r is not an instance of numbers.Real.
ValueError – exact is True and r can not exactly be converted to a
Decimal
with a precision <= MAX_DEC_PRECISION.
If exact is False and r can not exactly be represented by a Decimal with a precision <= MAX_DEC_PRECISION, the result is rounded to a precision = MAX_DEC_PRECISION.
- quantize(quant: decimalfp.SupportsAsIntegerRatio, rounding: Optional[ROUNDING] = None) numbers.Rational ¶
Return integer multiple of quant closest to self.
- Parameters
quant – quantum to get a multiple from
rounding – rounding mode (default: None)
If no rounding mode is given, the current default rounding mode is used.
- Returns
- Raises
TypeError – quant is not a number or does not support as_integer_ratio
ValueError – quant is not convertable to a Rational
- property denominator: int¶
Return the normalized denominator of ‘self’.
I. e. the smallest positive denominator from the pairs of integers, whose ratio is equal to self.
- property imag: int¶
Return imaginary part of self.
Returns 0 (Real numbers have no imaginary component).
- property magnitude: int¶
Return magnitude of self in terms of power to 10.
I.e. the largest integer exp so that 10 ** exp <= self.
- property numerator: int¶
Return the normalized numerator of self.
I. e. the numerator from the pair of integers with the smallest positive denominator, whose ratio is equal to self.
- property precision: int¶
Return precision of self.
Rounding modes¶
Decimal supports rounding modes equivalent to those defined by the standard library module decimal: ROUND_DOWN, ROUND_UP, ROUND_HALF_DOWN, ROUND_HALF_UP, ROUND_HALF_EVEN, ROUND_CEILING, ROUND_FLOOR and ROUND_05UP.
The rounding modes are wrapped into the Enum ROUNDING
.
- class ROUNDING¶
Enumeration of rounding modes.
- ROUND_05UP = 1¶
Round away from zero if last digit after rounding towards zero would have been 0 or 5; otherwise round towards zero.
- ROUND_CEILING = 2¶
Round towards Infinity.
- ROUND_DOWN = 3¶
Round towards zero.
- ROUND_FLOOR = 4¶
Round towards -Infinity.
- ROUND_HALF_DOWN = 5¶
Round to nearest with ties going towards zero.
- ROUND_HALF_EVEN = 6¶
Round to nearest with ties going to nearest even integer.
- ROUND_HALF_UP = 7¶
Round to nearest with ties going away from zero.
- ROUND_UP = 8¶
Round away from zero.
Unless a rounding mode is explicitely given, the rounding mode set as current default is used. To get or set the default rounding mode, the package decimalfp provides the following two functions: