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Re: comparison operators and *typos
> | > 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?
- sorry, the answer was hidden on the last line below: 0.0 == +0.0
(which is the presumption of fp implementations which support -0.0,
which to me is confusing as then there is no value which is defined
which covers both +-0, hence the notion of ~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.
- you may be correct, but can you provide an example of a practical
problem it would introduce, as I can't honestly think of one?
(as to me, it seems that it may be thought of as simply a synonym
for the abstraction of the combined ordered regions +-0, which are
well ordered both between themselves, and all other numbers, and
seems consistent with the notion that #e0 is the center of ~0.0)
> | (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
- personally, I see no reason to restrict scheme to IEEE-754 idiosyncrasies
or failures; and as noted #i~1/0 may be though of as NAN, which is what
IEEE-754 considers both it and 1/0 to be, so it's actually more consistent
than it may first appear.
> 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
- yes, which is why the simultaneous two sided limit == +-1/0 == ~1/0
and correspondingly (/ (tan pi/2)) => ~0 thereby all functions may be
thought of as being evaluated about simultaneously converging points about
their specified values (f x y) == (f (+ x ~0.0) (+ y ~0.0)).
including (tan (+ pi/2 ~0.0))
and (/ 1 #e0) == (/ 1 (+ #e0 ~0.0)) == #i~1/0 == ~inf.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.
- As I presently believe it's important to adopt a uniform convention
which dictates how multi-variable values are evaluated, using the
the convention above, which I believe to be most generally useful:
then #i0/0 == 1, if one considers the signs of the converging difference
in magnitude of a multi-variable function to uniform albeit ambiguous,
thereby, the ratio of: 0/0 == 1 == 100%, thereby 100% of 0/0 == 0, not
some useless error.
or 0/0 == ~1, if the their signs are not necessarily considered uniform.
which is likely most technically correct, and yields: ~100% of 0/0 == ~0,
but I know you're not too enthusiastic about ~1 which I avoided earlier,
by only introducing ~0 which happens to represent a pair of ordered values
with no intervening potential ordering ambiguity to speak of.