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| 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 numbers: (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: 0*x=0. 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.