[Tutor] very odd math problem

[Dave Angel]

On 01/-10/-28163 02:59 PM, Steven D'Aprano wrote:
> Alan Gauld wrote:
>
>> Why would you use a loop when the final value is just
>> the final multiplication. Since you know the final value
>> in advance (you need it to create the loop!) why not
>> just do the final multiplication directly:
>>
>> x = 10*0.1
>>
>> I think I'm missing something?
>
> The context was generating a list of values, not just the final value.
> But the result can be generalized to other situations. Consider some
> function that calculates a result x by an iterative process. You don't
> care about the intermediate results, but you do have to step through
> them on the way to the final result:
>
> x = initial_value
> while condition:
> x += increment(x)
>
>
> When possible, it is better to re-write the formula to avoid repeatedly
> adding an increment to x. The problem is, if each addition has potential
> error of dx, then N additions have potential error N*dx -- you can't
> assume that errors will always cancel. If N is large, so is the expected
> error.
>
>  <snip>
>

I like to think of this as the social security problem, as that was the 
context in which I first saw it.  When figuring social security 
withholding tax (USA), the employer is not allowed to just figure it on 
the basis of the current wage amount.  If he did, the amount deducted 
would be rounded/truncated to the penny, and those partial pennies could 
add up to an enormous amount.  Instead he figures the total soc.sec. tax 
on the pay for the year, and from that subtracts the amount withheld in 
all previous checks.  So the total is always within a fractional penny 
of the correct amount.

Any time you can't store the intermediate amount exactly, you need to 
decide how/if to eliminate accumulating residuals.

DaveA