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On 2018-08-28, Frank Millman <frank at chagford.com> wrote: >>>> x = 1.1 + 2.2 >>>> x > 3.3000000000000003 > > According to the docs, the reason is that "numbers like 1.1 and 2.2 do not > have exact representations in binary floating point." Right. > So when I do this - > >>>> y = 3.3 >>>> y > 3.3 > > what exactly is happening? By default, Python shows only a certain number of significant digits (17?), and the decimal value of y rounded to 17 places is 3.3. > What is 'y' at this point? If you want to see the exact value: >>> y = 3.3 >>> y.hex() '0x1.a666666666666p+1' Or in decimal, you can request more significant digits than default: >>> '%0.60f' % y '3.299999999999999822364316059974953532218933105468750000000000' > Or if I do this - > >>>> z = (1.1 + 2.2) * 10 / 10 >>>> z > 3.3 >>> z = (1.1 + 2.2) * 10 / 10 >>> '%0.60f' % y '3.300000000000000266453525910037569701671600341796875000000000' >>> z.hex() '0x1.a666666666666p+1' >>> y = 1.1 + 2.2 >>> '%0.60f' % z '3.299999999999999822364316059974953532218933105468750000000000' >>> y.hex() '0x1.a666666666667p+1' As you can see from the hex values, they differ by one (least significant) bit in the significand. > What makes it different from the first example? The additional multiplication and division operations are not exact, so "* 10 / 10" produces a result that's slightly different than the one you started with. Two things to remember: 1. The actual numbers in the computer are in base-2. The mapping between what you see in base-10 and the real values in base-2 _is_always_exact_. 2. The _operations_ (multiplicaton/division/addition/subtraction) _are_not_always_exact_. Even if you start with values that map exactly from base-10 to base-2 (some do) doing operations on them may not produce an exact result. -- Grant Edwards grant.b.edwards Yow! Spreading peanut at butter reminds me of gmail.com opera!! I wonder why?

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