1. Re: math.e and misc.e
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.
Matt
