# Tutorial¶

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

>>> from bigfloat import *


You should be a little bit careful here: this import brings a fairly large number of functions into the current namespace, four of which shadow existing Python builtins, namely abs, max, min and pow. In normal usage you’ll probably only want to import the classes and functions that you actually need.

## 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 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.

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 :

>>> 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 five global flags: ‘Inexact’, ‘Overflow’, ‘ZeroDivision’, ‘Underflow’, and ‘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.

Footnotes