Am 02.09.21 um 15:51 schrieb Hope Rouselle:
Just sharing a case of floating-point numbers. Nothing needed to be
solved or to be figured out. Just bringing up conversation.
(*) An introduction to me
I don't understand floating-point numbers from the inside out, but I do
know how to work with base 2 and scientific notation. So the idea of
expressing a number as
mantissa * base^{power}
is not foreign to me. (If that helps you to perhaps instruct me on
what's going on here.)
(*) A presentation of the behavior
import sys
sys.version
'3.8.10 (tags/v3.8.10:3d8993a, May 3 2021, 11:48:03) [MSC v.1928 64 bit
(AMD64)]'
ls = [7.23, 8.41, 6.15, 2.31, 7.73, 7.77]
sum(ls)
39.599999999999994
ls = [8.41, 6.15, 2.31, 7.73, 7.77, 7.23]
sum(ls)
39.60000000000001
All I did was to take the first number, 7.23, and move it to the last
position in the list. (So we have a violation of the commutativity of
addition.)
I believe it is not commutativity, but associativity, that is violated.
Even for floating point, a+b=b+a except for maybe some extreme cases
like denormliazed numbers etc.
But in general (a+b)+c != a+ (b+c)
Consider decimal floating point with 2 digits.
a=1
b=c=0.04
Then you get LHS;
(1 + 0.04) + 0.04 = 1 + 0.04 = 1
RHS:
1 + (0.04 + 0.04) = 1 + 0.08 = 1.1
Your sum is evaluated like (((a + b) + c) + ....) and hence, if you
permute the numbers, it can be unequal. If you need better accuracy,
there is the Kahan summation algorithm and other alternatives:
https://en.wikipedia.org/wiki/Kahan_summation_algorithm
Christian
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