How not to write a loop, revisited

[Scott Daniels]

Oh do I have egg all over my face.  Ah well, I will go home and calculate 
fractions for a week.

In article <3637 at teklds.TEK.COM> daniels at teklds.UUCP (Scott Daniels) writes:
--That's me--
>	... [ notation: E/F is F (a binary fraction) * pow(2.0,E) ] ...
>Now notice that in this representation, a number E/F actually represents
>the range:   E / (F - 1/32)  through E / (F + 1/32).  The size of the interval
>depends only on the exponent.
>	1/.1110 = 1.750    represents  1.6875 through 1.8125
>	1/.1111 = 1.875    represents  1.8125 through 1.9375
>	2/.1000 = 2.000    represents  1.8750 through 2.1250 
>	2/.1001 = 2.250    represents  2.1250 through 2.3750 

Assume we treat the last bit as possibly in error.  The numbers are in the 
range: E / (F - 1/16)  through E / (F + 1/16).  Again, the size of the 
interval depends only on the exponent.
	1/.1110 = 1.750    represents  1.625  through 1.875
	1/.1111 = 1.875    represents  1.750  through 2.000
	2/.1000 = 2.000    represents  1.750  through 2.250 
	2/.1001 = 2.250    represents  2.000  through 2.500

>Ok, here we go:
>  To represent 2.0 exactly, we could use 2/.1000, but that represents the
>interval 1.8750:2.1250.  Now, there is a tighter specification which is 
>entirely within that interval: 1/.1111 (which represents 1.8125:1.9375), so
>we should use that tighter interval since no poin inside it is any further
>from the desired value 2.0 than the range that 2/.1000 gives.  Hence the
>besty representation (the tightest) for 2.0 is an interval which does not
>even include 2.0!
>
	1/.1110 = 1.750    represents  1.625  through  1.875
	1/.1111 = 1.875    represents  1.750  through  2.000
	2/.1000 = 2.000    represents  1.750  through  2.250 
	2/.1001 = 2.250    represents  2.000  through  2.500
  To represent 2.0 exactly, we could use 2/.1000, but that represents the
interval 1.750:2.250.  Now, there is a tighter specification which is 
entirely within that interval: 1/.1111 (which represents 1.750:2.000), so
we should use that tighter interval since no poin inside it is any further
from the desired value 2.0 than the range that 2/.1000 gives.  Hence the
best representation (the tightest) for 2.0 is an interval which does not
even include 2.0!

>-Scott Daniels daniels at teklds.TEK.COM (or daniels at teklds.UUCP)

Basically, I remebered the argument as open intervals +/- lsbit, but I
reproduced the argument using +/- 1/2 lsbit (seemed clearer and more 
appropriate), but didn't really check that that also exhibitted the behavior.
I will now sneak off into a corner and cry myself to death.

still:
  -Scott Daniels daniels at teklds.TEK.COM (or daniels at teklds.UUCP)