If you want
exp(-r*r/(f*b*b))
to have Full Width at Half Maximum (FWHM) to be b, then you want
exp(-(b/2)*(b/2)/(f*b*b))=1/2
(i.e., the Half width at half max is b/2, so exp(-r*r/(f*b*b))=1/2 at r=b/2). That is, you want
exp(-1/(4*f))=2
or
1/(4*f)=log(2)
or
f=1/(4*log(2))=0.360674.
Here, I use log() as the natural logarithm, as is common among mathematicians (who have no use for log-base-10).
zorkon the magnificent