Dear Pythonistas, How many times have we seen posts recently along the lines of "why is it that 0.1 appears as 0.10000000000000001 in python?" that lead to posters being sent to the definition of the IEEE 754 standard and the decimal.py module? I am teaching an introductory numerical analysis class this fall, and I realized that the best way to teach this stuff is to be able to play with the representations directly, in particular to be able to see it in action on a simpler system than full 64-bit precision, especially when str(f) or repr(f) won't show *all* of the significant digits stored in a float. The decimal class deliberately avoids binary representation issues, and I can't find what I want online. Consequently, I have written a module to simulate the machine representation of binary floating point numbers and their arithmetic. Values can be of arbitrary fixed precision or infinite precision, along the same lines as python's in-built decimal class. The code is here: http://www2.gsu.edu/~matrhc/binary.html The design is loosely based on that decimal module, although it doesn't get in to threads, for instance. You can play with different IEEE 754 representations with different precisions and rounding modes, and compare with infinite precision Binary numbers. For instance, it is easy to learn about machine epsilon, representation/rounding error using a much simpler format such as a 4-bit exponent and 6-bit mantissa. Such a format is easily defined in the new module and can be manipulated easily:

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context = define_context(4, 6, ROUND_DOWN)

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zero = context(0) Binary("0", (4, 6, ROUND_DOWN))

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print zero # sign, characteristic, significand bits 0 0000 000000

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zero.next() Binary("0.001E-9", (4, 6, ROUND_DOWN))

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print zero.next() 0 0000 000001

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largest_denormalized = context('0 0000 111111') # direct spec of the sign, characteristic, and significand bits largest_denormalized Binary("0.111111E-6", (4, 6, ROUND_DOWN))

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largest_denormalized.as_decimal() Decimal("0.015380859375")

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n01 = context(0.1) # nearest representable is actually stored

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print n01, " rounded to ", n01.as_decimal() 0 0011 100110 rounded to 0.099609375

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Binary('-10111.0000001').as_decimal() Decimal("-23.0078125")

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Binary('-10111.0000001', context).as_decimal() # not enough precision in this context Decimal("-23.0000")

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diff = abs(Binary('-10111.0000001') - Binary('-10111.0000001', context)) diff Binary("0.1E-6", (4, 6, ROUND_DOWN))

The usual arithmetic operations are permitted on these objects, as well as representations of their values in decimal or binary form. Default contexts for half, single, double, and quadruple IEEE 754 precision floats are provided. Binary integer classes are also provided, and some other utility functions for converting between decimal and binary string representations. The module is compatible with the numpy float classes and requires numpy to be installed. The source code is released under the BSD license, but I am amenable to other licensing ideas if there is interest in adapting the code for some other purpose. Full details of the functionality and known issues are in the module's docstring, and many examples of usage are in the accompanying file binary_tests.py (which also acts to validate the common representations against the built-in floating point types). I look forward to hearing feedback, especially in case of bugs or suggestions for improvements. -Rob -- Robert H. Clewley, Ph. D. Assistant Professor Department of Mathematics and Statistics Georgia State University 720 COE, 30 Pryor St Atlanta, GA 30303, USA tel: 404-413-6420 fax: 404-413-6403 http://www2.gsu.edu/~matrhc http://brainsbehavior.gsu.edu/