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Re: infinities reformulated
| From: Aubrey Jaffer <agj@xxxxxxxxxxxx>
| Date: Wed, 25 May 2005 23:19:48 -0400 (EDT)
| Transcendental functions can return irrational numbers which cannot
| be distinguished from each other when represented by finite length
| decimal strings. Thus the precision of an inexact (or exact)
| number representation cannot be unlimited.
| But exponent size does not suffer from the same limitation. An
| inexact number representation with big exponents will never
| overflow into an infinity. Infinities will result only from
| operations on infinities or limit points. Thus there would be no
| continuity between the rational flonums and infinities; which bodes
| poorly for LIMIT.
Increasing exponent sizes without increasing precision has some
problems. With IEEE-754 flonums it is already the case that:
(atan 1 0) ==> 1.5707963267948965
(tan 1.5707963267948965) ==> 16.331778728383844e15
(tan 1.5707963267948966) ==> 16.331778728383844e15
(tan 1.5707963267948967) ==> -6.218352966023783e15
The mantissa does not have enough precision so that there would exist
a number which makes TAN return 1/0. This sort of problem gets worse
if exponents alone are expanded.