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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.