Python high-precision for trig functions

[RupertPupkin]

Suppose I want to calculate the cosine of 2 radians to 50 decimal places. I tried using the decimal class like this:

>>> import math
>>> from decimal import *
>>> getcontext().prec = 50
>>> a = Decimal(2)
>>> s = Decimal(str(math.cos(a)))
>>> s
Decimal("-0.416146836547")

So I'm not getting 50 decimal places. If I leave out the str() string conversion then I get "TypeError: Cannot convert float to Decimal. First convert the float to a string". I'm sure this is easy to do and I'm missing something obvious (I'm still a Python n00b). What am I missing?

[PantheraLeo]

Not being a big python programmer, I'm going to step out on a limb and say the 'decimal' data type isn't fully defined. If it's like C/C++ that Python is suppose to be interoperable with, it looks like when the compiler is told what a 'Decimal' data type is it just not told how to convert it to or from a 'float' data type.

[aleph]

Conversion from a Decimal to a float is well supported. Just apply the built-in float() callable.

On the other hand, converting a float instance to a Decimal cannot be well-defined without extra information, I think. A Decimal instance is supposed to be "as precise as needed", just like when a human being performs arithmetic with pencils and paper. However, a float is already a lossy representation of a real number, so we don't gain much by converting one to a Decimal. Converting a text string to Decimal makes better sense though, for a string literal (which can be arbitrarily long within certain practical limits) is compatible with our "natural", "human" and variable-precision notation of real numbers.

OK so much for my own buzzing noise Please refer to the PEP for Decimal type for a sketch of what the Python czars thought when the decimal module was conceived.

Python's support for decimal number is still in development, so there is an absence of some methods such as trig and other transcendental functions. However, I'm not sure how practical they are -- Is the precision gained worth the CPU time and memory consumption? For some applications, maybe yes; but I'm afraid it's "no" most of the time.

[Gödel]

Also can use mpmath library from sympy package ('yum install sympy'), which can set the precision in decimal places (dps) or binary bits (prec). Note that the dps is applied correctly when converting to a string

>>> from sympy.mpmath import *
>>> mp.dps=50
>>> cos(2)
mpf('-0.41614683654714238699756822950076218976600077107554512')
>>> s=str(cos(2))
>>> s
'-0.41614683654714238699756822950076218976600077107555'

For precise bounds on calculations you can try the interval arithmetic python package (mpmath also has some support for intervals, but not transcendental functions) (you need the crlibm package, and use './configure CFLAGS=-fPIC')

$ sudo yum install python-setuptools-devel
$ sudo easy_install pyinterval

$ python
Python 2.6 (r26:66714, Jun  8 2009, 16:07:29) 
[GCC 4.4.0 20090506 (Red Hat 4.4.0-4)] on linux2
Type "help", "copyright", "credits" or "license" for more information.
>>> from interval import imath
>>> imath.cos(2)
interval([-0.41614683654714241, -0.41614683654714235])

which means, cos(2) definitely lies in that interval.