index.rst

API Reference

bigfloat

The BigFloat class

The BigFloat class implements multiple-precision binary floating-point 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 * 2e, 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(p-1) <= 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.

Construct a new BigFloat instance from an integer, string, float or another BigFloat instance, using the rounding-mode and output format (precision, exponent bounds and subnormalization) given by the current context. If the context keyword argument is given, its value should be a Context 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 another BigFloat instance, doing an exact conversion where possible. Unlike the usual BigFloat 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 resulting BigFloat 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 the RoundTiesToEven 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 a Context instance and its attributes override those of the current context.

hex(self)

Return a hexadecimal representation of a BigFloat. The advantage of the hexadecimal representation is that it represents the value of the BigFloat exactly.

precision

Precision of a BigFloat instance, in bits.

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 instance x is independent of the current context, and has the property that eval(repr(x)) recovers x 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, mixed-type operations involving a BigFloat and an integer or float are permitted. These behave as though the non BigFloat operand is first converted to a BigFloat with no loss of accuracy.
• The '+' and '-' unary operators and built-in abs function are supported. Note that these all round to the current context; in particular, '+x' is not necessarily equal to 'x' for a BigFloat instance x.
• 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 nonzero BigFloat and True otherwise. None of these conversions is affected by the current context.
• On Python 3, round, math.floor, math.ceil and math.trunc all behave as expected, returning the appropriate integer. They are unaffected by the current context. Note that Python 2 does not provided type-specific support for these four functions; the functions will all work on Python, but only by doing an implicit conversion to the built-in float type first.
• BigFloat instances are hashable. The hash function obeys the rule that objects that compare equal should hash equal; in particular, if x == n for some BigFloat instance x and some Python int or long n then hash(x) == hash(n), and similarly for floats.

• BigFloat instances support str.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 to BigFloat instances. The type may also be omitted, in which case str-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'	Fixed-point 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 a BigFloat instance to a decimal value. The default is to use round-ties-to-even. 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 floating-point format, together with a rounding mode.

Create a new Context object with the given attributes. Not all attributes need to be specified. Note that all attributes of the generated Context are read-only. Attributes that are unset for this Context instance return None.

precision

Precision of the floating-point 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 (1-2**-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 floating-point numbers have a value that's an integer multiple of 2**(emin-1).

rounding

The rounding mode of this Context, for example ROUND_TIES_TO_EVEN. The available rounding modes are described in the rounding modes section. Note that the values RoundTiesToEven, etc. exported by the bigfloat package are Context 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.

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.

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>

half_precision

single_precision

double_precision

quadruple_precision

These Context instances correspond to the binary16, binary32, binary64 and binary128 interchange formats described in IEEE 754-2008 (section 3.6). They're all special cases of the IEEEContext function.

IEEEContext

precision

rounding

RoundTiesToEven

RoundTowardZero

RoundAwayFromZero

RoundTowardPositive

RoundTowardNegative

Context objects corresponding to the five available rounding modes. RoundTiesToEven rounds the result of an operation or function to the nearest representable BigFloat, with ties rounded to the BigFloat whose least significant bit is zero. RoundTowardZero rounds results towards zero. RoundAwayFromZero rounds results away from zero. RoundTowardPositive rounds results towards positive infinity, and RoundTowardsNegative rounds results towards negative infinity.

Constants

PRECISION_MIN

PRECISION_MAX

Minimum and maximum precision that's valid for Context and BigFloat instances. In the current implementation, PRECISION_MIN is 2 and PRECISION_MAX is 2**31-1.

EMIN_MIN

EMIN_MAX

Minimum and maximum allowed values for the Context emin attribute. In the current implementation, EMIN_MIN == -EMIN_MAX == 1-2**30.

EMAX_MIN

EMAX_MAX

Minimum and maximum allowed values for the Context emax attribute. In the current implementation, -EMAX_MIN == EMAX_MAX == 2**30-1.

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 thread-local 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 and subnormalize 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.

getcontext()

setcontext(context)

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:

extra_precision

Rounding modes

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.

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.

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.

ROUND_TOWARD_POSITIVE

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

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

add

sub

mul

div

pow

fmod

floordiv

mod

remainder

dim

pos

neg

abs

fma

fms

sqr

sqrt

rec_sqrt

cbrt

root

hypot

Exponential and logarithmic functions

exp

exp2

exp10

log

log2

log10

expm1

log1p

Trigonometric functions

cos

sin

tan

sec

csc

cot

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.

acos

asin

atan

These functions return a result in radians.

atan2

Hyperbolic trig functions

cosh

sinh

tanh

sech

csch

coth

acosh

asinh

atanh

Special functions

eint

li2

gamma

lngamma

lgamma

zeta

zeta_ui

erf

erfc

j0

j1

jn

y0

y1

yn

agm

factorial

Constants

const_catalan

const_euler

const_log2

const_pi

Miscellaneous functions

max

min

copysign

frac

ceil

floor

round

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 three-way comparisons.

sgn

cmp

cmpabs

The following functions match the functionality of the builtin Python comparison operators.

greater

greaterequal

less

lessequal

equal

notequal

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.

lessgreater

unordered

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.

is_nan

is_inf

is_zero

is_finite

is_negative

is_integer

Miscellaneous functions

next_up

next_down

Flags

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 or BigFloat construction. Let x 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 than 2**(c.emin-1) in absolute value. If c.subnormalize is True, the Underflow flag is set if and only if x is nonzero, finite, and strictly smaller than 2**(c.emin+c.precision-2) in absolute value.

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 to 2**c.emax in absolute value.

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.

Inexact

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

NanFlag

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

clear_flag

set_flag

test_flag

get_flagstate

set_flagstate

MPFR Version information

MPFR_VERSION_STRING

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

MPFR_VERSION

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

MPFR_VERSION_MAJOR

An integer giving the major level of the MPFR version.

MPFR_VERSION_MINOR

An integer giving the minor level of the MPFR version.

MPFR_VERSION_PATCHLEVEL

An integer giving the patch level of the MPFR version.