Re: weird math

[Steven Ritter <sr0o+@andrew.cmu.edu>]

Thanks for the answers. Bill St. Clair and Kim Barrett are right. This
has more to do with how comparisons between rationals and floats work
than with floating point accuracy (coercing 7/5 to a float yields the
same answer as reading 1.4, although neither are really equal to 1.4). I
assumed that, in comparing a ratio to a float, the ratio would be
converted to a float, and then the comparison would take place. X3J13
decided that the float gets converted to a ratio, and then the
comparison takes place. So:

? (rational 1.4)
3152519739159347/2251799813685248 ;;because 1.4 isn't really 1.4

And, of course, the above mess is not equal to 7/5.

Now, let me explain why I think X3J13 is wrong.
The nice thing about ratios is that they are accurate to arbitrary
precision. This means that you can do integer arithmetic, including
integer division, with no roundoff error (until, of course, you want to
print the final ratio as a float, in which case you almost never want
full precision anyway).

What X3J13's comparison rule says, in effect, is that LISP believes that
the floating point number is more accurate than the ratio. This, of
course, will never be true (although, sometimes, they might be equally
accurate).

As explained on p. 290, X3J13's comparison rule preserves transitivity
in cases where you depend on numbers being precise to LISP's limits. In
exchange, we lose the ability to compare floats to ratios in any
reasonable way. We can no longer hold the belief that ratios are always
exact, while floats are always approximate. Now, both are approximate
(at least, in some circumstances). I think this is a bad trade.

Steve