rounding errors in Python floor division

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As you and khelwood already noticed, 0.4 cannot be exactly represented as a float. Why? It is two fifth (4/10 == 2/5) which does not have a finite binary fraction representation.

Try this:

from fractions import Fraction
Fraction('8.0') // Fraction('0.4')
    # or equivalently
    #     Fraction(8, 1) // Fraction(2, 5)
    # or
    #     Fraction('8/1') // Fraction('2/5')
# 20

However

Fraction('8') // Fraction(0.4)
# 19

Here, 0.4 is interpreted as a float literal (and thus a floating point binary number) which requires (binary) rounding, and only then converted to the rational number Fraction(3602879701896397, 9007199254740992), which is almost but not exactly 4 / 10. Then the floored division is executed, and because 

19 * Fraction(3602879701896397, 9007199254740992) < 8.0

and 

20 * Fraction(3602879701896397, 9007199254740992) > 8.0

the result is 19, not 20.

The same probably happens for

8.0 // 0.4

I.e., it seems floored division is determined atomically (but on the only approximate float values of the interpreted float literals).

So why does

floor(8.0 / 0.4)

give the “right” result? Because there, two rounding errors cancel each other out. First 1) the division is performed, yielding something slightly smaller than 20.0, but not representable as float. It gets rounded to the closest float, which happens to be 20.0. Only then, the floor operation is performed, but now acting on exactly 20.0, thus not changing the number any more.

1) As Kyle Strand points out, that the exact result is determined then rounded isn’t what actually happens low2)-level (CPython’s C code or even CPU instructions). However, it can be a useful model for determining the expected 3) result.

2) On the lowest 4) level, however, this might not be too far off. Some chipsets determine float results by first computing a more precise (but still not exact, simply has some more binary digits) internal floating point result and then rounding to IEEE double precision.

3) “expected” by the Python specification, not necessarily by our intuition.

4) Well, lowest level above logic gates. We don’t have to consider the quantum mechanics that make semiconductors possible to understand this.

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