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Re: comparison operators and *typos
| Date: Thu, 30 Jun 2005 20:11:03 -0400
| From: Paul Schlie <schlie@xxxxxxxxxxx>
| > If we make (/ +1 0.0) ==> #i+1/0, then (/ -1 0.0) ==> #i-1/0.
| > This choice is arbitrary; ...
| - which seems very reasonable.
| > | > Inexact infinities have reciprocals: zero. Their reciprocals
| > | > are not unique, but that is already the case with IEEE-754
| > | > floating-point representations:
| > ...
| > Zero is at the center of 0.0's neighborhood. R5RS division by 0.0
| > is an error; leaving latitude for SRFI-70's response.
| - also seems very reasonable, and provide the opportunity to reconsider
| eliminating IEEE-754's otherwise inconsistent asymmetry, by defining:
| (/ #i-0/1) => #i-1/0 ; -inf.0
| (/ #i+0/1) => #i+1/0 ; +inf.0
| thereby truly consistently symmetric with the above:
| (/ #i-1/0) => #i-0/1 ; -0.0
| (/ #i+1/0) => #i+0/1 ; +0.0
It does not remove the asymmetry -- which neighborhood does (unsigned)
0.0 belong to: -0.0 or +0.0?
| > Most neighborhoods mapping through piecewise-continuous functions
| > project onto adjacent neighborhoods. But / near 0 is not the
| > only function which does not. TAN near pi/2 is another example.
| - and please reconsider this may be consistently symmetrically defined:
| [where ~ denotes a value being simultaneously positive and negative]
| (/ #i~0) => #i~1/0 ; ~inf.0
| (/ #i~1/0) => #i~0 ; ~0.0
#i~0 is not a real number because it cannot be ordered (relative to
0.0). Damaging the total ordering of the real numbers is too high a
price for symmetry.
| (tan pi/2) => #i~1/0 ; ~inf.0
(atan #i+1/0) ==> 1.5707963267948965
The next larger IEEE-754 number is 1.5707963267948967.
But there is no IEEE-754 number whose tangent is infinite:
(tan 1.5707963267948965) ==> 16.331778728383844e15
(tan 1.5707963267948967) ==> -6.218352966023783e15
Note that the one-sided LIMIT gets it right without needing any new
(limit tan 1.5707963267948965 -1.0e-15) ==> +1/0
(limit tan 1.5707963267948965 1.0e-15) ==> -1/0
| (abs ~inf.0) => +inf.0
| (- (abs ~inf.0) => -inf.0
| (abs ~0.0) => +0.0
| (- (abs ~0.0)) -0.0
| (+ +0.0 -0.0) => ~0.0
| Where I believe it's reasonable to redefine the use of IEEE's NAN
| values to encode these values, as arguably ~inf.0 may be thought
| of as being NAN, and ~0.0 as being 1/NAN (leaving 0.0 == +0.0)
For some expressions returning #i0/0, no number has any more claim to
correctness than any other. For example any number x satisfies:
So #i0/0 could be any number (if we forget that division by zero is
undefined). The reciprocal of this #i0/0 potentially maps to any
number; which is represented by #i0/0.