# The bigfloat package — high precision floating-point arithmetic¶

## Introduction¶

The bigfloat package is a Python wrapper for the MPFR library for arbitrary precision floating-point reliable arithmetic.

The MPFR library is a well-known portable C library for arbitrary-precision arithmetic on floating-point numbers. It provides precise control over precisions and rounding modes and gives correctly-rounded reproducible platform-independent results.

The bigfloat package aims to provide a convenient and friendly Python interface to the operations and functions provided by the MPFR library. The main class, BigFloat, gives an immutable multiple-precision floating-point type that can be freely mixed with Python integers and floats. The Context class, when used in conjunction with Python’s with statement, gives a simple way of controlling precisions and rounding modes. Additional module-level functions provide various standard mathematical operations. There is full support for IEEE 754 signed zeros, nans, infinities and subnormals.

Here’s a quick tour:

```>>> from bigfloat import *
>>> sqrt(2, precision(100))  # compute sqrt(2) with 100 bits of precision
BigFloat.exact('1.4142135623730950488016887242092', precision=100)
>>> with precision(100):     # another way to get the same result
...     sqrt(2)
...
BigFloat.exact('1.4142135623730950488016887242092', precision=100)
>>> my_context = precision(100) + RoundTowardPositive
>>> my_context
Context(precision=100, rounding='RoundTowardPositive')
>>> sqrt(2, my_context)      # and another, this time rounding up
BigFloat.exact('1.4142135623730950488016887242108', precision=100)
>>> with RoundTowardNegative: # a lower bound for zeta(2)
...     sum(1/sqr(n) for n in range(1, 10000))
...
BigFloat.exact('1.6448340618469506', precision=53)
>>> zeta(2) # actual value, for comparison
BigFloat.exact('1.6449340668482264', precision=53)
>>> const_pi()**2/6.0  # double check value
BigFloat.exact('1.6449340668482264', precision=53)
>>> quadruple_precision  # context implementing IEEE 754 binary128 format
Context(precision=113, emax=16384, emin=-16493, subnormalize=True)
>>> next_up(0, quadruple_precision)  # smallest subnormal for binary128
BigFloat.exact('6.47517511943802511092443895822764655e-4966', precision=113)
>>> log2(_)
BigFloat.exact('-16494.000000000000', precision=53)
```

## Installation¶

### Where to get it¶

The latest released version of the bigfloat package can be downloaded from its place at the Python Package Index. Development sources can be checked out from the project’s bitbucket page.

### Prerequisites¶

In order to use the bigfloat package you will need to have both the GMP and MPFR libraries already installed on your system. See the MPFR homepage and the GMP homepage for more information about these libraries. Currently, MPFR version 2.3.0 or later is required.

This package requires Python version 2.5 or later. For Python 2.5, you’ll need to do a from __future__ import with_statement if you want to take advantage of all of the features of this package.

### Installation¶

Like most third party Python libraries, the bigfloat package is installed by means of the setup.py script included in the distribution. On most systems, installation should be as simple as doing:

`python setup.py install`

in the top-level directory of the unpacked distribution. You may need superuser privileges to install the library, for example with:

`sudo python setup.py install`

### Locating the MPFR library¶

On import, the bigfloat package attempts to locate an MPFR library on your system; if the library is in the usual library search path (as controlled by the LD_LIBRARY_PATH environment variable on Linux, for example) then it should be found as normal. If the package fails to find the MPFR library, or if you have multiple MPFR libraries installed on your system and want to specify which one to use, you can edit the mpfr_library_location setting in the bigfloat_config.py configuration file to specify the library location. This file should be found in the same location as the other installed source files for the bigfloat package.

### Other configuration¶

The bigfloat_config.py configuration file also allows you to specify some other system-dependent values. On a typical system, with default installs of GMP and MPFR, it’s unlikely that these values will need to be changed. But if you’re getting segmentation faults or crashes with the bigfloat library then you may need to edit the values in this file. In this case it will probably also be useful to have the gmp.h and mpfr.h include files handy to refer to; on Linux systems, these files may be in a different package from the library files (e.g., ‘mpfr-devel’ instead of ‘mpfr’).

## Tutorial¶

Start by importing the contents of the package (assuming that you’ve already installed it and its prerequisites) with:

```>>> from bigfloat import *
```

This import brings a fairly large number of functions into the current namespace, and clobbers some builtin Python functions: abs, max, min and pow. In normal usage you’ll probably only want to import the classes and functions that you actually need.

If you’re using Python 2.5 you’ll also need to do:

```>>> from __future__ import with_statement
```

### BigFloat construction¶

The main type of interest is the BigFloat class. The BigFloat type is an immutable binary floating-point type. A BigFloat instance can be created from an integer, a float or a string:

```>>> BigFloat(123)
BigFloat.exact('123.00000000000000', precision=53)
>>> BigFloat(-4.56)
BigFloat.exact('-4.5599999999999996', precision=53)
```

Each BigFloat instance has both a value and a precision. The precision gives the number of bits used to store the significand of the BigFloat. The value of a finite nonzero BigFloat with precision p is a real number of the form (-1)**s * m * 2**e, where the sign s is either 0 or 1, the significand m is a number in the half-open interval [0.5, 1.0) that can be expressed in the form n/2**p for some integer n, and e is an integer giving the exponent. In addition, zeros (positive and negative), infinities and NaNs are representable. Just like Python floats, the printed form of a BigFloat shows only a decimal approximation to the exact stored value, for the benefit of human readers.

The precision of a newly-constructed BigFloat instance is dictated by the current precision, which defaults to 53. This setting can be overridden by supplying the context keyword argument to the constructor:

```>>> BigFloat(-4.56, context=precision(24))
BigFloat.exact('-4.55999994', precision=24)
```

The first argument to the BigFloat constructor is rounded to the correct precision using the current rounding mode, which defaults to RoundTiesToEven; again, this can be overridden with the context keyword argument:

```>>> BigFloat('3.14')
BigFloat.exact('3.1400000000000001', precision=53)
>>> BigFloat('3.14', context=RoundTowardZero)
BigFloat.exact('3.1399999999999997', precision=53)
>>> BigFloat('3.14', context=RoundTowardPositive + precision(24))
BigFloat.exact('3.14000010', precision=24)
```

More generally, the second argument to the BigFloat constructor can be any instance of the Context class. The various rounding modes are all Context instances, and precision is a function returning a Context:

```>>> RoundTowardNegative
Context(rounding='RoundTowardNegative')
>>> precision(1000)
Context(precision=1000)
```

Context instances can be combined by addition, as seen above.

```>>> precision(1000) + RoundTowardNegative
Context(precision=1000, rounding='RoundTowardNegative')
```

When adding two contexts that both specify values for a particular attribute, the value for the right-hand addend takes precedence:

```>>> c = Context(subnormalize=False, rounding='RoundTowardPositive')
>>> double_precision
Context(precision=53, emax=1024, emin=-1073, subnormalize=True)
>>> double_precision + c
Context(precision=53, emax=1024, emin=-1073, subnormalize=False,
rounding='RoundTowardPositive')
>>> c + double_precision
Context(precision=53, emax=1024, emin=-1073, subnormalize=True,
rounding='RoundTowardPositive')
```

The bigfloat package also defines various constant Context instances. For example, quadruple_precision is a Context that corresponds to the IEEE 754 binary128 interchange format:

```>>> quadruple_precision
Context(precision=113, emax=16384, emin=-16493, subnormalize=True)
BigFloat.exact('1.10000000000000000000000000000000008', precision=113)
```

The current settings for precision and rounding mode given by the current context, accessible via the getcontext() function:

```>>> getcontext()
Context(precision=53, emax=1073741823, emin=-1073741823, subnormalize=False,
rounding='RoundTiesToEven')
```

There’s also a setcontext() function for changing the current context; however, the preferred method for making temporary changes to the current context is to use Python’s with statement. More on this below.

Note that (in contrast to Python’s standard library decimal module), Context instances are immutable.

There’s also a second method for constructing BigFloat instances: BigFloat.exact(). Just like the usual constructor, BigFloat.exact() accepts integers, floats and strings. However, for integers and floats it performs an exact conversion, creating a BigFloat instance with precision large enough to hold the integer or float exactly (regardless of the current precision setting):

```>>> BigFloat.exact(-123)
BigFloat.exact('-123.0', precision=7)
>>> BigFloat.exact(7**30)
BigFloat.exact('22539340290692258087863249.0', precision=85)
>>> BigFloat.exact(-56.7)
BigFloat.exact('-56.700000000000003', precision=53)
```

For strings, BigFloat.exact() accepts a second precision argument, and always rounds using the RoundTiesToEven rounding mode.

```>>> BigFloat.exact('1.1', precision=80)
BigFloat.exact('1.1000000000000000000000003', precision=80)
```

The result of a call to BigFloat.exact is independent of the current context; this is why the repr() of a BigFloat is expressed in terms of BigFloat.exact(). The str() of a BigFloat looks prettier, but doesn’t supply enough information to recover that BigFloat exactly if you don’t know the precision:

```>>> print BigFloat('1e1000', precision(20))
9.9999988e+999
>>> print BigFloat('1e1000', precision(21))
9.9999988e+999
```

### Arithmetic on BigFloat instances¶

All the usual arithmetic operations, with the exception of floor division, apply to BigFloat instances, and those instances can be freely mixed with integers and floats (but not strings!) in those operations:

```>>> BigFloat(1234)/3
BigFloat.exact('411.33333333333331', precision=53)
>>> BigFloat('1e1233')**0.5
BigFloat.exact('3.1622776601683794e+616', precision=53)
```

As with the BigFloat constructor, the precision for the result is taken from the current context, as is the rounding mode used to round the exact mathematical result to the nearest BigFloat.

For mixed-type operations, the integer or float is converted exactly to a BigFloat before the operation (as though the BigFloat.exact constructor had been applied to it). So there’s only a single point where precision might be lost: namely, when the result of the operation is rounded to the nearest value representable as a BigFloat.

Note

The current precision and rounding mode even apply to the unary plus and minus operations. In particular, +x is not necessarily a no-op for a BigFloat instance x:

```>>> BigFloat.exact(7**100)
BigFloat.exact('323447650962475799134464776910021681085720319890462540093389
5331391691459636928060001.0', precision=281)
>>> +BigFloat.exact(7**100)
BigFloat.exact('3.2344765096247579e+84', precision=53)
```

This makes the unary plus operator useful as a way to round a result produced in a different context to the current context.

For each arithmetic operation the bigfloat package exports a corresponding function. For example, the div() function corresponds to usual (true) division:

```>>> 355/BigFloat(113)
BigFloat.exact('3.1415929203539825', precision=53)
>>> div(355, 113)
BigFloat.exact('3.1415929203539825', precision=53)
```

This is useful for a couple of reasons: one reason is that it makes it possible to use div(x, y) in contexts where a BigFloat result is desired but where one or both of x and y might be an integer or float. But a more important reason is that these functions, like the BigFloat constructor, accept an extra context keyword argument giving a context for the operation:

```>>> div(355, 113, context=single_precision)
BigFloat.exact('3.14159298', precision=24)
```

Similarly, the sub function corresponds to Python’s subtraction operation. To fully appreciate some of the subtleties of the ways that binary arithmetic operations might be performed, note the difference in the results of the following:

```>>> x = 10**16+1  # integer, not exactly representable as a float
>>> y = 10**16.   # 10.**16 is exactly representable as a float
>>> x - y
0.0
>>> BigFloat(x) - BigFloat(y)
BigFloat.exact('0', precision=53)
>>> sub(x, y)
BigFloat.exact('1.0000000000000000', precision=53)
```

For the first subtraction, the integer is first converted to a float, losing accuracy, and then the subtraction is performed, giving a result of 0.0. The second case is similar: x and y are both explicitly converted to BigFloat instances, and the conversion of x again loses precision. In the third case, x and y are implicitly converted to BigFloat instances, and that conversion is exact, so the subtraction produces exactly the right answer.

Comparisons between BigFloat instances and integers or floats also behave as you’d expect them to; for these, there’s no need for a corresponding function.

### Mathematical functions¶

The bigfloat package provides a number of standard mathematical functions. These functions follow the same rules as the arithmetic operations above:

• the arguments can be integers, floats or BigFloat instances
• integers and float arguments are converted exactly to BigFloat instances before the function is applied
• the result is a BigFloat instance, with the precision of the result, and the rounding mode used to obtain the result, taken from the current context.
• attributes of the current context can be overridden by providing an additional context keyword argument. Here are some examples:
```>>> sqrt(1729, context=RoundTowardZero)
BigFloat.exact('41.581245772583578', precision=53)
>>> sqrt(1729, context=RoundTowardPositive)
BigFloat.exact('41.581245772583586', precision=53)
>>> atanh(0.5, context=precision(20))
BigFloat.exact('0.54930592', precision=20)
>>> const_catalan(precision(1000))
BigFloat.exact('0.9159655941772190150546035149323841107741493742816721342664
9811962176301977625476947935651292611510624857442261919619957903589880332585
9059431594737481158406995332028773319460519038727478164087865909024706484152
1630002287276409423882599577415088163974702524820115607076448838078733704899
00864775113226027', precision=1000)
>>> 4*exp(-const_pi()/2/agm(1, 1e-100))
BigFloat.exact('9.9999999999998517e-101', precision=53)
```

For a full list of the supported functions, see the reference manual.

### Controlling the precision and rounding mode¶

We’ve seen one way of controlling precision and rounding mode, via the context keyword argument. There’s another way that’s often more convenient, especially when a single context change is supposed to apply to multiple operations: contexts can be used directly in Python with statements. Note: if you’re using Python 2.5, you’ll need to enable with statements with:

```>>> from __future__ import with_statement
```

For example, here we compute high-precision upper and lower-bounds for the thousandth harmonic number:

```>>> with precision(100):
...     with RoundTowardNegative:  # lower bound
...         lower_bound = sum(div(1, n) for n in range(1, 1001))
...     with RoundTowardPositive:  # upper bound
...         upper_bound = sum(div(1, n) for n in range(1, 1001))
...
>>> lower_bound
BigFloat.exact('7.4854708605503449126565182015873', precision=100)
>>> upper_bound
BigFloat.exact('7.4854708605503449126565182077593', precision=100)
```

The effect of the with statement is to change the current context for the duration of the with block; when the block exits, the previous context is restored. With statements can be nested, as seen above. Let’s double-check the above results using the asymptotic formula for the nth harmonic number [1]:

```>>> n = 1000
>>> with precision(100):
...     approx = log(n) + const_euler() + div(1, 2*n) - 1/(12*sqr(n))
...
>>> approx
BigFloat.exact('7.4854708605503365793271531207983', precision=100)
```

The error in this approximation should be approximately -1/(120*n**4). Let’s check it:

```>>> error = approx - lower_bound
>>> error
BigFloat.exact('-8.3333293650807890e-15', precision=53)
>>> -1/(120*pow(n, 4))
BigFloat.exact('-8.3333333333333336e-15', precision=53)
```

A more permanent change to the context can be effected using the setcontext() function, which takes a single argument of type Context:

```>>> setcontext(precision(30))
>>> sqrt(2)
BigFloat.exact('1.4142135624', precision=30)
>>> setcontext(RoundTowardZero)
>>> sqrt(2)
BigFloat.exact('1.4142135605', precision=30)
```

An important point here is that in any place that a context is used, only the attributes specified by that context are changed. For example, the context precision(30) only has the precision attribute, so only that attribute is affected by the setcontext call; the other attributes are not changed. Similarly, the setcontext(RoundTowardZero) line above doesn’t affect the precision.

There’s a DefaultContext constant giving the default context, so you can always restore the original default context as follows:

```>>> setcontext(DefaultContext)
```

Note

If setcontext() is used within a with statement, its effects only last for the duration of the block following the with statement.

### Flags¶

The bigfloat package also provides four global flags: ‘Inexact’, ‘Overflow’, ‘Underflow’, ‘NanFlag’, along with methods to set and test these flags:

```>>> set_flagstate(set())  # clear all flags
>>> get_flagstate()
set([])
>>> exp(10**100)
BigFloat.exact('Infinity', precision=53)
>>> get_flagstate()
set(['Overflow', 'Inexact'])
```

These flags show that overflow occurred, and that the given result (infinity) was inexact. The flags are sticky: none of the standard operations ever clears a flag:

```>>> sqrt(2)
BigFloat.exact('1.4142135623730951', precision=53)
>>> get_flagstate()  # overflow flag still set from the exp call
set(['Overflow', 'Inexact'])
>>> set_flagstate(set())  # clear all flags
>>> sqrt(2)
BigFloat.exact('1.4142135623730951', precision=53)
>>> get_flagstate()   # sqrt only sets the inexact flag
set(['Inexact'])
```

The functions clear_flag(), set_flag() and test_flag() allow clearing, setting and testing of individual flags.

Support for these flags is preliminary, and the API may change in future versions.

## Reference¶

### 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 * 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**(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.

class bigfloat.BigFloat(value, context=None)

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 some 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, and mixed-type operations involving a BigFloat and an integer or float are permitted. For MPFR version >= 2.4.0, the ‘%’ is also supported. Mixed-type operations behave as though the non BigFloat operand is first converted to a BigFloat with no loss of accuracy. 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 first argument, not the second. Floor division is not currently implemented.
• 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 RoundTiesToEven rounding mode. Conversion to bool returns False for a nonzero BigFloat and True otherwise. None of these conversions is affected by the current context.
• BigFloat instances are hashable. For Python 2.6 and later, 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. In Python 2.5, there are some rare cases where x == n does not imply hash(x) == hash(n). For that reason it’s inadvisable to mix integers and BigFloat instances in a set, or to use both integers and BigFloat instances as keys in the same dictionary.

### 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.

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 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]. PRECISION_MIN is usually 2.

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. This should be a string. Valid values are ‘RoundTiesToEven’, ‘RoundTowardZero’, ‘RoundTowardPositive’ and ‘RoundTowardNegative’. Note that the rounding modes RoundTiesToEven, etc. exported by the bigfloat package are Context instances, not strings, 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='RoundTiesToEven')
>>> y = Context(precision=53, subnormalize=True)
>>> x + y
Context(precision=53, subnormalize=True, rounding='RoundTiesToEven')
>>> y + x
Context(precision=200, subnormalize=True, rounding='RoundTiesToEven')
```

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

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.

bigfloat.IEEEContext(bitwidth)

Return IEEE 754-2008 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 754-2008 or the bigfloat source for more details.

bigfloat.precision(prec)

Return context specifying the given precision.

precision(prec) is exactly equivalent to Context(precision=prec).

bigfloat.rounding(rnd)

Return a context giving the specified rounding mode.

rounding(rnd) is exactly equivalent to Context(rounding=rnd).

bigfloat.RoundTiesToEven
bigfloat.RoundTowardZero
bigfloat.RoundTowardPositive
bigfloat.RoundTowardNegative

Contexts corresponding to the four 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. RoundTowardPositive rounds results towards positive infinity, and RoundTowardsNegative rounds results towards negative infinity.

#### Constants¶

bigfloat.PRECISION_MIN
bigfloat.PRECISION_MAX

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

bigfloat.EMIN_MIN
bigfloat.EMIN_MAX

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

bigfloat.EMAX_MIN
bigfloat.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.

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. If context 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 to Context(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)
```

### 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¶

Return x + y.

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

Return x - y.

bigfloat.mul(x, y, context=None)

Return x times y.

bigfloat.div(x, y, context=None)

Return x divided by y.

bigfloat.pow(x, y, context=None)

Return x raised to the power y.

Special values are handled as described in the ISO C99 and IEEE 754-2008 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 non-integer 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.mod(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 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.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:

```>>> 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.

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 base-two logarithm of x.

bigfloat.log10(x, context=None)

Return the base-ten 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:

```>>> 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, then atan2(y, x) is mathematically equivalent to atan(y / x). If x < 0 and y > 0, atan(y, x) is equivalent to π + atan(y, x). If x < 0 and y < 0, the result is + atan(y, x).

Geometrically, atan2(y, x) is the angle (measured counterclockwise, in radians) from the positive x-axis to the line segment joining (0, 0) to (x, y), in the usual representation of the x-y plane.

Special values are handled as described in the ISO C99 and IEEE 754-2008 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 at x.

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 at x.

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 Euler-Mascheroni 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 non-NaN 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 non-NaN 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 the RoundTowardPositive rounding mode:

```>>> with RoundTowardPositive:
...     x = 2**100 + 1
...     ceil(x) >= x
...
True
```

Similar comments apply to the floor(), round() and trunc() functions.

Note

This function corresponds to the MPFR function mpfr_rint_ceil, not to mpfr_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 the ceil() function for more information.

Note

This function corresponds to the MPFR function mpfr_rint_floor, not to mpfr_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 to mpfr_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 to mpfr_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.

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 three-way 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.

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 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.

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 to 2**c.emax in absolute value.

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.

Footnotes