Regarding Mike Croucher Mathematica says that the numbers are equal with default adaptive precision:. Thanks, Grisha. Notify me of follow-up comments by email. Notify me of new posts by email. Walking Randomly Because it's more fun than getting there in a straight line. Floating point addition is not associative. February 28th, Categories: general math , matlab , Numerics , programming , python Tags: Nume.
Leave a comment Trackback. Chris Nahr. Reply Quote 1. Imre Polik. Floating-point cannot return a result with this many bits, so it has to be rounded to the nearest representable number. In this case, there are two numbers that are equally near, 0x1. When there is a tie, the number with a zero in the lowest bit of the significand is used. The final result of the addition is 0x1. Next, add the number for. Again, we scale to make the exponents match, so 0x1. Then add that to 0x1.
Rounding that to 53 bits gives 0x1. Now consider. Recall the latter is scaled to 0x. Then the exact sum is 0x1. Then add the number for. The exact sum is 0x2. Floating-point significands are always scaled to start with 1 except for special cases: zero, infinity, and very small numbers at the bottom of the representable range , so we adjust this to 0x1.
Finally, we round to 53 bits, giving 0x1. Thus, because of errors that occur when rounding,. Neither addition nor multiplication is associative with IEEE double precision bit numbers. Here are examples for each evaluated with Python 3. Stack Overflow for Teams — Collaborate and share knowledge with a private group.
It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. The example was taken from Mike Croucher's blog entry. As rasher and the documentation both say, Equal has a certain level of fuzziness. The same is true of SameQ , though it has a more stringent tolerance. The following computations are all done with machine precision numbers. Similar things should hold with arbitrary precision numbers but the analysis might be trickier.
It might also be worth mentioning that more traditional floating point comparisons can be easily emulated. For example, since the "fuzziness" is based on Precision , we can check if the difference is equal to zero. See the documentation for Equal. There is a tolerance for inexact numbers. The order of operations combined with precision of targets can affect whether things fall "in" or "out" of tolerance. See specifically the "Possible Issues" section in the documents for Equal. However, this is not true in all cases.
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It is evident from the above-given output that the floating-point arithmetic may not follow the law of associativity in every case. This is due to the format in which the floating-point numbers are stored and represented, it rounds off the numbers during calculations, hence, the associative laws of algebra do not necessarily hold for floating-point numbers.
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