1. Re: math.e and misc.e
Matt Lewis wrote:
> Juergen Luethje wrote:
> >
> > Thanks for the explanation, Matt!
> > Just to be sure: When the 17th digit is 5 or higher, should then the
> > 18th digit be rounded up, or is it better to keep the original 16th digit?
>
> I'm not sure there's an easy rule like that. Taking a further look,
> it's possible that the seventeenth digit could affect the rounding.
> If you look at how I figure out the fractional part, I look bit by
> bit to see if it should be one or zero.
>
> Basically, I start by putting all of the digits into a sequence, so
>
> 0.1234 => {1,2,3,4} = decimals
> [decimals is the var name in the code]
>
> Then, I make a similar representation for 1 / 2:
>
> 0.5 => {5} = sub
>
> This is the value of (binary) 0.1. If the decimals sequence compares bigger
> than the sub sequence, then I set the bit to 1, and subtract it from
> decimals. Then I divide sub by 2 to get {2,5}, and keep going until
> I've got enough to fill up the entire precision, or decimals has dropped
> to all zeros.
>
> A double has 52 bits of precision, plus one more, if it's a normalized value
> (meaning there's an implicit 1 at the beginning), so really a total of
> 53. But I also consider rounding based on the 54th bit. Watching my
> 'sub' sequence, it takes 3 or four bits (i.e., dividing sub by 2) before
> the first non-zero digit in sub moves farther down (which makes sense
> when you condsider that log10(2) ~ 3.xx). In other words:
>
> bit 1st non-0 sub
> --- --------- ---------
> 1 1 {5}
> 2 1 {2,5}
> 3 1 {1,2,5}
> 4 2 {0,6,2,5}
> ...
> 53 16 {0,.....}
> 54 17 {0,.....}
> 55 17 {0,.....}
> 56 17 {0,.....}
> 57 18 {0,.....}
>
> So best practice is probably to go ahead and use 17 digits for maximum
> precision. But whether the 17th digit gives any added precision
> depends on a lot more than it's own value, because we're converting the
> base-10 fraction to a base-2 fraction. But the 18th digit will have
> absolutely no affect on the value.
>
> Of course, the above all considers that you're using scientific notation,
> so something with a lot of leading fractional zeroes:
>
> 0.000000000000000000123456789
>
> makes a difference. But then you have to consider the exponent. For
> numbers with big or small magnitudes, you have other issues, because the
> representations get sparse in those areas.
Umpf ... This is not exactly easy stuff. 
So I'll use 17 decimal digits for the constants in my next proposal for
"math.e". Thank you so much!
Regards,
Juergen
