The bigfloat module 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 module 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)
The latest released version of the bigfloat module can be downloaded from its place at the Python Package Index. Development sources can be checked out from the project’s bitbucket page.
In order to use the bigfloat module 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 2.4.0 or later is required.
This module 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 module.
Like most third party Python libraries, the bigfloat module 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
The setup.py installation script attempts to locate the MPFR library on your system. If it fails, or if you have multiple MPFR libraries installed on your system and want to specify which one to use, you should edit the mpfr_library_location setting in the bigfloat/bigfloat_config.py.in file to specify the library location.
The bigfloat/bigfloat_config.py.in 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’).
Start by importing the contents of the module (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
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 module 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('1.1', quadruple_precision)
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
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 module 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.
The bigfloat module 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.
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.
The bigfloat module 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.
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.
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.
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).
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.
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.
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.
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.
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 module defines a number of predefined Context instances.
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>
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:
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.
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:
Return a copy of the current context with the precision increased by p. 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)
All functions in this section follow the same rules:
Return e**x - 1. Useful for values of x close to 0, when the expression exp(x)-1 would lose significant accuracy.
>>> exp(1e-10)-1
BigFloat.exact('1.0000000827403710e-10', precision=53)
>>> exp(1e-10, precision(100))-1
BigFloat.exact('1.0000000000500000e-10', precision=53)
>>> expm1(1e-10)
BigFloat.exact('1.0000000000500000e-10', precision=53)
Return the floor of x. Note that since the result is rounded to the current context, it’s quite possible for the result to be larger than x:
>>> with DefaultContext:
... floor(2**100-1) <= 2**100-1
...
False
If you want to be sure of getting a result that’s no larger than x, use the RoundTowardNegative rounding mode. Alternatively, if you want the exact floor you may want to clear the Inexact flag before the call and test it afterwards. Similar comments apply to the ceil(), round() and trunc() functions.
These are the functions exported by the bigfloat module that don’t fit into the above section, for one reason or another.
There are two additional comparison functions that don’t correspond to any of the Python comparison operators.
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.
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.
Footnotes
[1] | See http://mathworld.wolfram.com/HarmonicNumber.html |