Lx=d: SundayResearch

The floating point reproducibility problem On a typical computer, the results of floating point operations can differ from the mathematical result. In general, floating point operations like addition may not satisfy all mathematical properties for that operation. For example, the associativity property of addition may not hold for all numerical values: $(x + y) + z \stackrel{?}{=} x + (y + z)$ A decimal representation of a number cannot always be represented by a finite binary representation in computer hardware. For example, the computed result of $0.1 + 0.01 - 0.1 - 0.01$ is not equal to $0$, but mathematically these values should be equivalent. For example, you can demonstrate this using the Python interpreter (Python 3.8.5): >>> 0.1 + 0.01 - 0.1 - 0.01 -5.204170427930421e-18 But mathematical properties do hold in other cases. For example, the result of $0.1 - 0.1 + 0.01 - 0.01$ is equal to $0$. >>> 0.1 - 0.1 + 0.01 - 0.01 0.0 Another problem results from rounding

Once I walk around this Hafen with one of my friends, we discussed a book (*) we both read recently. The discussion goes far and in many directions. But I found two points were interesting.  Berlin Westhafen One is as in the title, ``You have no responsibility for what was happened, but you have responsibility for that not happens again. (Original: Ihr seid nicht verantwortlich fuer das, was geschah. Aber das es nicht wieder geschiert, dafuer schon)'' If I think about responsibility, I believe that the people who were not born at that time cannot have ``personal'' responsibility. Some extreme idea says someone has responsibility for all the ancestors did. It does not seem fair because the responsibility is passed on to the descendants. They are words about war, but I think it still holds in the case of war. It should be noted that what I put parentheses around "personal" here may not apply if that is not a person. The other part is an argument that the droppin