API Reference¶
The BigFloat
class¶
The BigFloat
class implements multipleprecision binary
floatingpoint numbers. Each BigFloat
instance has both a
value and a precision; the precision is an integer giving the number
of significant bits used to store the value. A finite nonzero
BigFloat
instance with precision p can be thought of as a
(sign, significand, exponent) triple (s, m, e), representing the value
(1)**s * m * 2**e, where m is a value in the range [0.5, 1.0) stored
with p bits of precision. Thus m is of the form n/2**p for some
integer n with 2**(p1) <= n < 2**p.
In addition to nonzero finite numbers, BigFloat
instances can
also represent positive and negative infinity, positive and negative
zero, and NaNs.
BigFloat
instances should be treated as immutable.

class
bigfloat.
BigFloat
(value, context=None)¶ Construct a new
BigFloat
instance from an integer, string, float or anotherBigFloat
instance, using the roundingmode and output format (precision, exponent bounds and subnormalization) given by the current context. If the context keyword argument is given, its value should be aContext
instance and its attributes override those of the current context.value can be an integer, string, float, or another
BigFloat
instance. In all cases the given value is rounded to the format (determined by precision, exponent limits and subnormalization) given by the current context, using the rounding mode specified by the current context. The integer 0 is always converted to positive zero.
as_integer_ratio
(self)¶ Return a pair (n, d) of integers such that n and d are relatively prime, d is positive, and the value of self is exactly n/d.
If self is an infinity or nan then ValueError is raised. Negative and positive zero are both converted to (0, 1).

copy_abs
(self)¶ Return a copy of self with the sign bit unset.
In contrast to
abs(self)
,self.copy_abs()
makes no use of the context, and the result has the same precision as the original.

copy_neg
(self)¶ Return a copy of self with the opposite sign bit.
In constract to
neg(self)
,self.copy_neg()
makes no use of the context, and the result has the same precision as the original.

exact
(cls, value, precision=None)¶ A class method to construct a new
BigFloat
instance from an integer, string, float or anotherBigFloat
instance, doing an exact conversion where possible. Unlike the usualBigFloat
constructor, this alternative constructor makes no use of the current context and will not affect the current flags.If value is an integer, float or
BigFloat
, then the precision keyword must not be given, and the conversion is exact. The resultingBigFloat
has a precision sufficiently large to hold the converted value exactly. If value is a string, then the precision argument must be given. The string is converted using the given precision and theRoundTiesToEven
rounding mode.

fromhex
(cls, value, context=None)¶ Class method that constructs a new
BigFloat
instance from a hexadecimal string. Rounds to the current context using the given precision. If the context keyword argument is given, its value should be aContext
instance and its attributes override those of the current context.

Special methods¶
The BigFloat
type has a full complement of special methods.
Here are some brief notes on those methods, indicating possible
deviations from expected behaviour.
The repr of a
BigFloat
instancex
is independent of the current context, and has the property thateval(repr(x))
recoversx
exactly.The ‘+’ ,’‘, ‘*’, ‘/’, ‘//’, ‘%’ and ‘**’ binary operators are supported. The ‘/’ operator implements true division, regardless of whether
from __future__ import division
is in effect or not. The result of ‘%’ has the same sign as the second argument, so follows the existing Python semantics for ‘%’ on Python floats.For the above operators, mixedtype operations involving a
BigFloat
and an integer or float are permitted. These behave as though the nonBigFloat
operand is first converted to aBigFloat
with no loss of accuracy.The ‘+’ and ‘‘ unary operators and builtin
abs
function are supported. Note that these all round to the current context; in particular, ‘+x’ is not necessarily equal to ‘x’ for aBigFloat
instancex
.The six comparison operators ‘==’, ‘<=’, ‘<’, ‘!=’, ‘>’, ‘>=’ are supported. Comparisons involving NaNs always return False, except in the case of ‘!=’ where they always return True. Again, comparisons with integers or floats are permitted, with the integer or float being converted exactly before the comparison; the context does not affect the result of a comparison.
Conversions to int and long always round towards zero; conversions to float always use the
ROUND_TIES_TO_EVEN
rounding mode. Conversion to bool returns False for a nonzeroBigFloat
and True otherwise. None of these conversions is affected by the current context.On Python 3,
round
,math.floor
,math.ceil
andmath.trunc
all behave as expected, returning the appropriate integer. They are unaffected by the current context. Note that Python 2 does not provided typespecific support for these four functions; the functions will all work on Python, but only by doing an implicit conversion to the builtinfloat
type first.BigFloat
instances are hashable. The hash function obeys the rule that objects that compare equal should hash equal; in particular, ifx == n
for someBigFloat
instancex
and some Python int or longn
thenhash(x) == hash(n)
, and similarly for floats.BigFloat
instances supportstr.format
based formatting, as described in PEP 3101. The format specifier is much as described in the PEP, except that there’s additional support for hexadecimal and binary output, and for specification of a rounding mode. The general form of the format specifier looks like this:[[fill]align][sign][#][0][minimumwidth][.precision][rounding][type]
The
type
field is a single letter, which may be any of the following. The'e'
,'E'
,'f'
,'F'
,'g'
,'G'
and'%'
types behave in the same way as for regular floats. Only the'a'
,'A'
and'b'
formats are particular toBigFloat
instances. The type may also be omitted, in which casestr
style formatting is performed.'a'
Output in hexadecimal format. 'A'
Upper case variant of 'a'
.'b'
Output in binary format. 'e'
Scientific format. 'E'
Upper case variant of 'e'
.'f'
Fixedpoint format. 'F'
Upper case variant of 'f'
.'g'
Friendly format. 'G'
Upper case variant of 'g'
.'%'
Like 'f'
, but formatted as a percentage.The optional
rounding
field consists of a single letter describing the rounding direction to be used when converting aBigFloat
instance to a decimal value. The default is to use roundtiestoeven. Valid values for this field are described in the table below.'U'
Round toward positive 'D'
Round toward negative 'Y'
Round away from zero 'Z'
Round toward zero 'N'
Round ties to even Examples:
>>> from bigfloat import sqrt >>> "{0:.6f}".format(sqrt(2)) '1.414214' >>> "{0:.6Df}".format(sqrt(2)) # round down '1.414213' >>> "{0:.^+20.6e}".format(sqrt(2)) '...+1.414214e+00....' >>> "{0:a}".format(sqrt(2)) '0x1.6a09e667f3bcdp+0' >>> "{0:b}".format(sqrt(2)) '1.0110101000001001111001100110011111110011101111001101p+0'
The Context
class¶
A Context
object is a simple immutable object that packages
together attributes describing a floatingpoint format, together with
a rounding mode.

class
bigfloat.
Context
(precision=None, emin=None, emax=None, subnormalize=None, rounding=None)¶ Create a new
Context
object with the given attributes. Not all attributes need to be specified. Note that all attributes of the generatedContext
are readonly. Attributes that are unset for thisContext
instance returnNone
.
precision
¶ Precision of the floatingpoint format, given in bits. This should be an integer in the range [
PRECISION_MIN
,PRECISION_MAX
].

emax
¶ Maximum exponent allowed for this format. The largest finite number representable in the context self is
(12**self.precision) * 2**self.emax
.

emin
¶ Minimum exponent allowed for this format. The smallest positive number representable in the context self is
0.5 * 2**self.emin
.Note
There’s nothing to stop you defining a context with emin > emax, but don’t expect to get sensible results if you do this.

subnormalize
¶ A boolean value: True if the format has gradual underflow, and False otherwise. With gradual underflow, all finite floatingpoint numbers have a value that’s an integer multiple of 2**(emin1).

rounding
¶ The rounding mode of this
Context
, for exampleROUND_TIES_TO_EVEN
. The available rounding modes are described in the Rounding modes section. Note that the valuesRoundTiesToEven
, etc. exported by thebigfloat
package areContext
instances, not rounding modes, so cannot be used directly here.

Context
instances can be added. If x
and y
are
Context
instances then x + y
is the Context
whose attributes
combine those of x
and y
. In the case that both x
and
y
have a particular attribute set, the value for y
takes
precedence:
>>> x = Context(precision=200, rounding=ROUND_TIES_TO_EVEN)
>>> y = Context(precision=53, subnormalize=True)
>>> x + y
Context(precision=53, subnormalize=True, rounding=ROUND_TIES_TO_EVEN)
>>> y + x
Context(precision=200, subnormalize=True, rounding=ROUND_TIES_TO_EVEN)
Context
instances can be used in with statements to alter
the current context. In effect,
with c:
<block>
behaves roughly like
old_context = getcontext()
setcontext(c)
<block>
setcontext(old_context)
except that nesting of with statements works as you’d expect, and the old context is guaranteed to be restored even if an exception occurs during execution of the block.
Note that for Context
instances x
and y
,
with x + y:
<block>
is exactly equivalent to
with x:
with y:
<block>
The bigfloat package defines a number of predefined Context
instances.

bigfloat.
DefaultContext
¶ The context that’s in use when the bigfloat package is first imported. It has precision of 53, large exponent bounds, no subnormalization, and the RoundTiesToEven rounding mode.

bigfloat.
EmptyContext
¶ Equal to
Context()
. Occasionally useful where a context is syntactically required for a with statement, but no change to the current context is desired. For example:if <want_extra_precision>: c = extra_precision(10) else: c = EmptyContext with c: <do calculation>

bigfloat.
half_precision
¶

bigfloat.
single_precision
¶

bigfloat.
double_precision
¶

bigfloat.
quadruple_precision
¶ These
Context
instances correspond to the binary16, binary32, binary64 and binary128 interchange formats described in IEEE 7542008 (section 3.6). They’re all special cases of theIEEEContext
function.

bigfloat.
IEEEContext
(bitwidth)¶ Return IEEE 7542008 context for a given bit width.
The IEEE 754 standard specifies binary interchange formats with bitwidths 16, 32, 64, 128, and all multiples of 32 greater than 128. This function returns the context corresponding to the interchange format for the given bitwidth.
See section 3.6 of IEEE 7542008 or the bigfloat source for more details.

bigfloat.
precision
(prec)¶ Return context specifying the given precision.
precision(prec)
is exactly equivalent toContext(precision=prec)
.

bigfloat.
rounding
(rnd)¶ Return a context giving the specified rounding mode.
rounding(rnd)
is exactly equivalent toContext(rounding=rnd)
.

bigfloat.
RoundTiesToEven
¶

bigfloat.
RoundTowardZero
¶

bigfloat.
RoundAwayFromZero
¶

bigfloat.
RoundTowardPositive
¶

bigfloat.
RoundTowardNegative
¶ Context
objects corresponding to the five available rounding modes.RoundTiesToEven
rounds the result of an operation or function to the nearest representableBigFloat
, with ties rounded to theBigFloat
whose least significant bit is zero.RoundTowardZero
rounds results towards zero.RoundAwayFromZero
rounds results away from zero.RoundTowardPositive
rounds results towards positive infinity, andRoundTowardsNegative
rounds results towards negative infinity.
Constants¶

bigfloat.
PRECISION_MIN
¶

bigfloat.
PRECISION_MAX
¶ Minimum and maximum precision that’s valid for
Context
andBigFloat
instances. In the current implementation,PRECISION_MIN
is2
andPRECISION_MAX
is2**311
.

bigfloat.
EMIN_MIN
¶

bigfloat.
EMIN_MAX
¶ Minimum and maximum allowed values for the
Context
emin attribute. In the current implementation,EMIN_MIN == EMIN_MAX == 12**30
.

bigfloat.
EMAX_MIN
¶
The current context¶
There can be many Context
objects in existence at one time, but
there’s only ever one current context. The current context is given by a
threadlocal Context
instance. Whenever the BigFloat
constructor is called, or any arithmetic operation or standard function
computation is performed, the current context is consulted to determine:
 The format that the result of the operation or function should take
(as specified by the
precision
,emax
,emin
andsubnormalize
attributes of the context), and  The rounding mode to use when computing the result, as specified by
the
rounding
attribute of the current context.
If an additional context
keyword argument is given to the
operation, function or constructor, then attributes from the context
override the corresponding attributes in the current context.
For example,
sqrt(x, context=my_context)
is equivalent to
with my_context:
sqrt(x)
The current context can be read and written directly using the
getcontext
and setcontext
functions.

bigfloat.
getcontext
()¶ Return the current context.

bigfloat.
setcontext
(context)¶ Set the current context to that given.
Attributes provided by
context
override those in the current context. Ifcontext
doesn’t specify a particular attribute, the attribute from the current context shows through.
It’s usually neater to make a temporary change to the context using a with statement, as described above. There’s also one convenience function that’s often useful in calculations:

bigfloat.
extra_precision
(prec)¶ Return copy of the current context with the precision increased by
prec
. Equivalent toContext(precision=getcontext().precision + p)
.>>> getcontext().precision 53 >>> extra_precision(10).precision 63 >>> with extra_precision(20): ... gamma(1.5) ... BigFloat.exact('0.88622692545275801364912', precision=73)
Rounding modes¶

bigfloat.
ROUND_TIES_TO_EVEN
¶ This is the default rounding mode. The number to be rounded is mapped to the nearest representable value. In the case where that number is exactly midway between the two closest representable values, it is mapped to the value with least significant bit set to zero.

bigfloat.
ROUND_TOWARD_ZERO
¶ The number to be rounded is mapped to the nearest representable value that’s smaller than or equal to the original number in absolute value.

bigfloat.
ROUND_AWAY_FROM_ZERO
¶ The number to be rounded is mapped to the nearest representable value that’s greater than or equal to the original number in absolute value.

bigfloat.
ROUND_TOWARD_POSITIVE
¶ The number to be rounded is mapped to the nearest representable value greater than or equal to the original number.

bigfloat.
ROUND_TOWARD_NEGATIVE
¶ The number to be rounded is mapped to the nearest representable value less than or equal to the original number.
Standard functions¶
All functions in this section follow the same rules:
 Arguments can be
BigFloat
instances, integers or floats, unless otherwise specified.  Integer or float arguments are converted exactly to
BigFloat
instances.  The format of the result and the rounding mode used to obtain that result are taken from the current context.
 Attributes of the current context can be overridden by supplying an
explicit
context
keyword argument.  Results are correctly rounded.
Arithmetic functions¶

bigfloat.
add
(x, y, context=None)¶ Return
x
+y
.

bigfloat.
sub
(x, y, context=None)¶ Return
x
y
.

bigfloat.
mul
(x, y, context=None)¶ Return
x
timesy
.

bigfloat.
div
(x, y, context=None)¶ Return
x
divided byy
.

bigfloat.
pow
(x, y, context=None)¶ Return
x
raised to the powery
.Special values are handled as described in the ISO C99 and IEEE 7542008 standards for the pow function.
 pow(±0, y) returns plus or minus infinity for y a negative odd integer.
 pow(±0, y) returns plus infinity for y negative and not an odd integer.
 pow(±0, y) returns plus or minus zero for y a positive odd integer.
 pow(±0, y) returns plus zero for y positive and not an odd integer.
 pow(1, ±Inf) returns 1.
 pow(+1, y) returns 1 for any y, even a NaN.
 pow(x, ±0) returns 1 for any x, even a NaN.
 pow(x, y) returns NaN for finite negative x and finite noninteger y.
 pow(x, Inf) returns plus infinity for 0 < abs(x) < 1, and plus zero for abs(x) > 1.
 pow(x, +Inf) returns plus zero for 0 < abs(x) < 1, and plus infinity for abs(x) > 1.
 pow(Inf, y) returns minus zero for y a negative odd integer.
 pow(Inf, y) returns plus zero for y negative and not an odd integer.
 pow(Inf, y) returns minus infinity for y a positive odd integer.
 pow(Inf, y) returns plus infinity for y positive and not an odd integer.
 pow(+Inf, y) returns plus zero for y negative, and plus infinity for y positive.

bigfloat.
fmod
(x, y, context=None)¶ Return
x
reduced moduloy
.Returns the value of x  n * y, where n is the integer quotient of x divided by y, rounded toward zero.
Special values are handled as described in Section F.9.7.1 of the ISO C99 standard: If x is infinite or y is zero, the result is NaN. If y is infinite and x is finite, the result is x rounded to the current context. If the result is zero, it has the sign of x.

bigfloat.
floordiv
(x, y, context=None)¶ Return the floor of
x
divided byy
.The result is a
BigFloat
instance, rounded to the context if necessary. Special cases match those of thediv
function.

bigfloat.
mod
(x, y, context=None)¶ Return the remainder of x divided by y, with sign matching that of y.

bigfloat.
remainder
(x, y, context=None)¶ Return x reduced modulo y.
Returns the value of x  n * y, where n is the integer quotient of x divided by y, rounded to the nearest integer (ties rounded to even).
Special values are handled as described in Section F.9.7.1 of the ISO C99 standard: If x is infinite or y is zero, the result is NaN. If y is infinite and x is finite, the result is x (rounded to the current context). If the result is zero, it has the sign of x.

bigfloat.
dim
(x, y, context=None)¶ Return max(x  y, 0).
Return x  y if x > y, +0 if x <= y, and NaN if either x or y is NaN.

bigfloat.
pos
(x, context=None)¶ Return
x
.As usual, the result is rounded to the current context. The
pos
function can be useful for rounding an intermediate result, computed with a temporary increase in precision, back to the current context. For example:>>> from bigfloat import precision >>> pow(3, 20) + 1.234  pow(3, 20) # inaccurate due to precision loss BigFloat.exact('1.2340002059936523', precision=53) >>> with precision(100): # compute result with extra precision ... x = pow(3, 20) + 1.234  pow(3, 20) ... >>> x BigFloat.exact('1.2339999999999999857891452847980', precision=100) >>> pos(x) # round back to original precision BigFloat.exact('1.2340000000000000', precision=53)

bigfloat.
neg
(x, context=None)¶ Return x.

bigfloat.
abs
(x, context=None)¶ Return abs(x).

bigfloat.
fma
(x, y, z, context=None)¶ Return (x * y) + z, with a single rounding according to the current context.

bigfloat.
fms
(x, y, z, context=None)¶ Return (x * y)  z, with a single rounding according to the current context.

bigfloat.
sqr
(x, context=None)¶ Return the square of
x
.

bigfloat.
sqrt
(x, context=None)¶ Return the square root of
x
.Return 0 if x is 0, to be consistent with the IEEE 754 standard. Return NaN if x is negative.

bigfloat.
rec_sqrt
(x, context=None)¶ Return the reciprocal square root of x.
Return +Inf if x is ±0, +0 if x is +Inf, and NaN if x is negative.

bigfloat.
cbrt
(x, context=None)¶ Return the cube root of x.
For x negative, return a negative number. The cube root of 0 is defined to be 0.

bigfloat.
root
(x, k, context=None)¶ Return the kth root of x.
For k odd and x negative (including Inf), return a negative number. For k even and x negative (including Inf), return NaN.
The kth root of 0 is defined to be 0, whatever the parity of k.
This function is only implemented for nonnegative k.

bigfloat.
hypot
(x, y, context=None)¶ Return the Euclidean norm of x and y, i.e., the square root of the sum of the squares of x and y.
Exponential and logarithmic functions¶

bigfloat.
exp
(x, context=None)¶ Return the exponential of x.

bigfloat.
exp2
(x, context=None)¶ Return two raised to the power x.

bigfloat.
exp10
(x, context=None)¶ Return ten raised to the power x.

bigfloat.
log
(x, context=None)¶ Return the natural logarithm of x.

bigfloat.
log2
(x, context=None)¶ Return the basetwo logarithm of x.

bigfloat.
log10
(x, context=None)¶ Return the baseten logarithm of x.

bigfloat.
expm1
(x, context=None)¶ Return one less than the exponential of x.

bigfloat.
log1p
(x, context=None)¶ Return the logarithm of one plus x.
Trigonometric functions¶

bigfloat.
cos
(x, context=None)¶ Return the cosine of
x
.

bigfloat.
sin
(x, context=None)¶ Return the sine of
x
.

bigfloat.
tan
(x, context=None)¶ Return the tangent of
x
.

bigfloat.
sec
(x, context=None)¶ Return the secant of
x
.

bigfloat.
csc
(x, context=None)¶ Return the cosecant of
x
.

bigfloat.
cot
(x, context=None)¶ Return the cotangent of
x
.
The above six trigonometric functions are inefficient for large arguments (for
example, x
larger than BigFloat('1e1000000')
), since reducing x
correctly modulo π requires computing π to high precision. Input
arguments are in radians, not degrees.

bigfloat.
acos
(x, context=None)¶ Return the inverse cosine of
x
.The mathematically exact result lies in the range [0, π]. However, note that as a result of rounding to the current context, it’s possible for the actual value returned to be fractionally larger than π:
>>> from bigfloat import precision >>> with precision(12): ... x = acos(1) ... >>> print(x) 3.1416 >>> x > const_pi() True

bigfloat.
asin
(x, context=None)¶ Return the inverse sine of
x
.The mathematically exact result lies in the range [π/2, π/2]. However, note that as a result of rounding to the current context, it’s possible for the actual value to lie just outside this range.

bigfloat.
atan
(x, context=None)¶ Return the inverse tangent of
x
.The mathematically exact result lies in the range [π/2, π/2]. However, note that as a result of rounding to the current context, it’s possible for the actual value to lie just outside this range.
These functions return a result in radians.

bigfloat.
atan2
(y, x, context=None)¶ Return
atan(y / x)
with the appropriate choice of function branch.If
x > 0
, thenatan2(y, x)
is mathematically equivalent toatan(y / x)
. Ifx < 0
andy > 0
,atan(y, x)
is equivalent toπ + atan(y, x)
. Ifx < 0
andy < 0
, the result isπ + atan(y, x)
.Geometrically,
atan2(y, x)
is the angle (measured counterclockwise, in radians) from the positive xaxis to the line segment joining (0, 0) to (x, y), in the usual representation of the xy plane.Special values are handled as described in the ISO C99 and IEEE 7542008 standards for the atan2 function. The following examples illustrate the rules for positive y; for negative y, apply the symmetry
atan(y, x) == atan(y, x)
.>>> finite = positive = 2.3 >>> negative = 2.3 >>> inf = BigFloat('inf')
>>> print(atan2(+0.0, 0.0)) # pi 3.1415926535897931 >>> print(atan2(+0.0, +0.0)) # 0 0 >>> print(atan2(+0.0, negative)) # pi 3.1415926535897931 >>> print(atan2(+0.0, positive)) # 0 0 >>> print(atan2(positive, 0.0)) # pi / 2 1.5707963267948966 >>> print(atan2(inf, inf)) # 3*pi / 4 2.3561944901923448 >>> print(atan2(inf, inf)) # pi / 4 0.78539816339744828 >>> print(atan2(inf, finite)) # pi / 2 1.5707963267948966 >>> print(atan2(positive, inf)) # pi 3.1415926535897931 >>> print(atan2(positive, +inf)) # 0 0
Hyperbolic trig functions¶

bigfloat.
cosh
(x, context=None)¶ Return the hyperbolic cosine of x.

bigfloat.
sinh
(x, context=None)¶ Return the hyperbolic sine of x.

bigfloat.
tanh
(x, context=None)¶ Return the hyperbolic tangent of x.

bigfloat.
sech
(x, context=None)¶ Return the hyperbolic secant of x.

bigfloat.
csch
(x, context=None)¶ Return the hyperbolic cosecant of x.

bigfloat.
coth
(x, context=None)¶ Return the hyperbolic cotangent of x.

bigfloat.
acosh
(x, context=None)¶ Return the inverse hyperbolic cosine of x.

bigfloat.
asinh
(x, context=None)¶ Return the inverse hyperbolic sine of x.

bigfloat.
atanh
(x, context=None)¶ Return the inverse hyperbolic tangent of x.
Special functions¶

bigfloat.
eint
(x, context=None)¶ Return the exponential integral of x.

bigfloat.
li2
(x, context=None)¶ Return the real part of the dilogarithm of x.

bigfloat.
gamma
(x, context=None)¶ Return the value of the Gamma function of x.

bigfloat.
lngamma
(x, context=None)¶ Return the value of the logarithm of the Gamma function of x.

bigfloat.
lgamma
(x, context=None)¶ Return the logarithm of the absolute value of the Gamma function at x.

bigfloat.
zeta
(x, context=None)¶ Return the value of the Riemann zeta function on x.

bigfloat.
zeta_ui
(x, context=None)¶ Return the value of the Riemann zeta function at the nonnegative integer x.

bigfloat.
erf
(x, context=None)¶ Return the value of the error function at x.

bigfloat.
erfc
(x, context=None)¶ Return the value of the complementary error function at x.

bigfloat.
j0
(x, context=None)¶ Return the value of the first kind Bessel function of order 0 at x.

bigfloat.
j1
(x, context=None)¶ Return the value of the first kind Bessel function of order 1 at x.

bigfloat.
jn
(n, x, context=None)¶ Return the value of the first kind Bessel function of order
n
atx
.n
should be a Python integer.

bigfloat.
y0
(x, context=None)¶ Return the value of the second kind Bessel function of order 0 at x.

bigfloat.
y1
(x, context=None)¶ Return the value of the second kind Bessel function of order 1 at x.

bigfloat.
yn
(n, x, context=None)¶ Return the value of the second kind Bessel function of order
n
atx
.n
should be a Python integer.

bigfloat.
agm
(x, y, context=None)¶ Return the arithmetic geometric mean of x and y.

bigfloat.
factorial
(x, context=None)¶ Return the factorial of the nonnegative integer x.
Constants¶

bigfloat.
const_catalan
(context=None)¶ Return Catalan’s constant.
Returns the value of Catalan’s constant 0.9159655941…, with precision and rounding mode taken from the current context. The Catalan constant is defined as the limit of the series 1  1/3**2 + 1/5**2  1/7**2 + 1/9**2  …

bigfloat.
const_euler
(context=None)¶ Return Euler’s constant.
Returns the value of the EulerMascheroni constant, 0.5772156649…, with precision and rounding mode taken from the current context. The constant is equal to the limit of (1 + 1/2 + 1/3 + … + 1/n)  log(n) as n approaches infinity.

bigfloat.
const_log2
(context=None)¶ Return log(2).
Returns the natural logarithm of 2, 0.6931471805…, with precision and rounding mode taken from the current context.

bigfloat.
const_pi
(context=None)¶ Return π.
Returns π = 3.1415926535…, with precision and rounding mode taken from the current context.
Miscellaneous functions¶

bigfloat.
max
(x, y, context=None)¶ Return the maximum of x and y.
If x and y are both NaN, return NaN. If exactly one of x and y is NaN, return the nonNaN value. If x and y are zeros of different signs, return +0.

bigfloat.
min
(x, y, context=None)¶ Return the minimum of x and y.
If x and y are both NaN, return NaN. If exactly one of x and y is NaN, return the nonNaN value. If x and y are zeros of different signs, return −0.

bigfloat.
copysign
(x, y, context=None)¶ Return a new BigFloat object with the magnitude of x but the sign of y.

bigfloat.
frac
(x, context=None)¶ Return the fractional part of
x
.The result has the same sign as
x
.

bigfloat.
ceil
(x, context=None)¶ Return the next higher or equal integer to x.
If the result is not exactly representable, it will be rounded according to the current context. Note that the rounding step means that it’s possible for the result to be smaller than
x
. For example:>>> x = 2**100 + 1 >>> ceil(2**100 + 1) >= x False
One way to be sure of getting a result that’s greater than or equal to
x
is to use theRoundTowardPositive
rounding mode:>>> with RoundTowardPositive: ... x = 2**100 + 1 ... ceil(x) >= x ... True
Similar comments apply to the
floor
,round
andtrunc
functions.Note
This function corresponds to the MPFR function
mpfr_rint_ceil
, not tompfr_ceil
.

bigfloat.
floor
(x, context=None)¶ Return the next lower or equal integer to x.
If the result is not exactly representable, it will be rounded according to the current context.
Note that it’s possible for the result to be larger than
x
. See the documentation of theceil
function for more information.Note
This function corresponds to the MPFR function
mpfr_rint_floor
, not tompfr_floor
.

bigfloat.
round
(x, context=None)¶ Return the nearest integer to x, rounding halfway cases away from zero.
If the result is not exactly representable, it will be rounded according to the current context.
Note
This function corresponds to the MPFR function
mpfr_rint_round
, not tompfr_round
.

bigfloat.
trunc
(x, context=None)¶ Return the next integer towards zero.
If the result is not exactly representable, it will be rounded according to the current context.
Note
This function corresponds to the MPFR function
mpfr_rint_trunc
, not tompfr_trunc
.
Other Functions¶
These are the functions exported by the bigfloat
package that
don’t fit into the above section, for one reason or another.
Comparisons¶
These functions provide threeway comparisons.

bigfloat.
sgn
(x)¶ Return the sign of x.
Return a positive integer if x > 0, 0 if x == 0, and a negative integer if x < 0. Raise ValueError if x is a NaN.
This function is equivalent to cmp(x, 0), but more efficient.

bigfloat.
cmp
(op1, op2)¶ Perform a threeway comparison of op1 and op2.
Return a positive value if op1 > op2, zero if op1 = op2, and a negative value if op1 < op2. Both op1 and op2 are considered to their full own precision, which may differ. If one of the operands is NaN, raise ValueError.
Note: This function may be useful to distinguish the three possible cases. If you need to distinguish two cases only, it is recommended to use the predicate functions like ‘greaterequal’; they behave like the IEEE 754 comparisons, in particular when one or both arguments are NaN.

bigfloat.
cmpabs
(op1, op2)¶ Compare the absolute values of op1 and op2.
Return a positive value if op1 > op2, zero if op1 = op2, and a negative value if op1 < op2. Both op1 and op2 are considered to their full own precision, which may differ. If one of the operands is NaN, raise ValueError.
Note: This function may be useful to distinguish the three possible cases. If you need to distinguish two cases only, it is recommended to use the predicate functions like ‘greaterequal’; they behave like the IEEE 754 comparisons, in particular when one or both arguments are NaN.
The following functions match the functionality of the builtin Python comparison operators.

bigfloat.
greater
(x, y)¶ Return True if x > y and False otherwise.
This function returns False whenever x and/or y is a NaN.

bigfloat.
greaterequal
(x, y)¶ Return True if x >= y and False otherwise.
This function returns False whenever x and/or y is a NaN.

bigfloat.
less
(x, y)¶ Return True if x < y and False otherwise.
This function returns False whenever x and/or y is a NaN.

bigfloat.
lessequal
(x, y)¶ Return True if x <= y and False otherwise.
This function returns False whenever x and/or y is a NaN.

bigfloat.
equal
(x, y)¶ Return True if x == y and False otherwise.
This function returns False whenever x and/or y is a NaN.

bigfloat.
notequal
(x, y)¶ Return True if x != y and False otherwise.
This function returns True whenever x and/or y is a NaN.
New in version 0.4: The notequal
function was added in version 0.4.
There are two additional comparison functions that don’t correspond to any of the Python comparison operators.

bigfloat.
lessgreater
(x, y)¶ Return True if x < y or x > y and False otherwise.
This function returns False whenever x and/or y is a NaN.

bigfloat.
unordered
(x, y)¶ Return True if x or y is a NaN and False otherwise.
Number classification functions¶
The following functions all accept a single BigFloat
instance (or a
float, or an integer) and return a boolean value. They make no
use of the current context, and do not affect the state of the flags.

bigfloat.
is_nan
(x)¶ Return True if x is a NaN, else False.

bigfloat.
is_inf
(x)¶ Return True if x is an infinity, else False.

bigfloat.
is_zero
(x)¶ Return True if x is a zero, else False.

bigfloat.
is_finite
(x)¶ Return True if x is finite, else False.
A BigFloat instance is considered to be finite if it is neither an infinity or a NaN.

bigfloat.
is_negative
(x)¶ Return True if x has its sign bit set, else False.
Note that this function returns True for negative zeros.

bigfloat.
is_integer
(x)¶ Return True if x is an exact integer, else False.
Miscellaneous functions¶

bigfloat.
next_up
(x, context=None)¶ next_up(x): return the least representable float that’s strictly greater than x.
This operation is quiet: flags are not affected.

bigfloat.
next_down
(x, context=None)¶ next_down(x): return the greatest representable float that’s strictly less than x.
This operation is quiet: flags are not affected.
Flags¶

bigfloat.
Underflow
¶ Underflow flag. Set whenever the result of an operation underflows. The meaning of this flag differs depending on whether the subnormalize attribute is true for the operation context. In the language of IEEE 754, we use the after rounding semantics. The Underflow flag is set on underflow even when the result of an operation is exact.
In detail: let
c
be the context that’s in effect for an operation, function orBigFloat
construction. Letx
be the result of the operation, rounded to the context precision with the context rounding mode, but as though the exponent were unbounded.If c.subnormalize is False, the Underflow flag is set if and only if
x
is nonzero, finite, and strictly smaller than2**(c.emin1)
in absolute value. If c.subnormalize is True, the Underflow flag is set if and only ifx
is nonzero, finite, and strictly smaller than2**(c.emin+c.precision2)
in absolute value.

bigfloat.
Overflow
¶ Set whenever the result of an operation overflows. An operation performed in a context
c
overflows if the result computed as if with unbounded exponent range is finite and greater than or equal to2**c.emax
in absolute value.

bigfloat.
ZeroDivision
¶ Set whenever an exact infinite result is obtained from finite inputs. Despite the name, this flag is not just set for divisions by zero. For example,
log(0)
will set the ZeroDivision flag.

bigfloat.
Inexact
¶ Inexact flag. Set whenever the result of an operation is not exactly equal to the true mathematical result.

bigfloat.
NanFlag
¶ NaN flag. Set whever the result of an operation gives a NaN result.

bigfloat.
clear_flag
(f)¶ Clear the given flag.

bigfloat.
set_flag
(f)¶ Set the given flag.

bigfloat.
test_flag
(f)¶ Return True if the given flag is set, and False otherwise.

bigfloat.
get_flagstate
()¶ Return a set containing the flags that are currently set.

bigfloat.
set_flagstate
(flagset)¶ Set all flags in
flagset
, and clear all other flags.
MPFR Version information¶

bigfloat.
MPFR_VERSION_STRING
¶ The version of the MPFR library being used, as a string.

bigfloat.
MPFR_VERSION
¶ The version of the MPFR library being used, as an integer.

bigfloat.
MPFR_VERSION_MAJOR
¶ An integer giving the major level of the MPFR version.

bigfloat.
MPFR_VERSION_MINOR
¶ An integer giving the minor level of the MPFR version.

bigfloat.
MPFR_VERSION_PATCHLEVEL
¶ An integer giving the patch level of the MPFR version.