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This thread arose on srfi-73@xxxxxxxxxxxxxxxxx, but wandered into inexact infinities. | Date: Tue, 28 Jun 2005 02:38:14 -0400 | From: Paul Schlie <schlie@xxxxxxxxxxx> | | > From: Aubrey Jaffer <agj@xxxxxxxxxxxx> | > | Date: Mon, 27 Jun 2005 18:09:04 -0400 | > | From: Paul Schlie <schlie@xxxxxxxxxxx> | > | ... | > ... | > My point is that (expt 0 0) is unlikely to occur when EXPT is being | > used as a continuous function; its occurrences will be exponentiating | > integers. In the integer context, arguments about limits of | > continuous functions are irrelevant. | | - Typically it seem more broadly accepted that (expt 0 0) == 1 | particularly for integers,... Hmm... "Concrete Mathematics" p.162 (R. Graham, D. Knuth, O. Patashnik) argues that 0^0=1 for the sake of the binomial theorem. I will change SRFI-70 to return 1 for 0^0. | ... | > | | > | - only if it's not considered important that inexact | > | infinities have corresponding reciprocals; | > | > Inexact infinities have reciprocals: zero. Their reciprocals are | > not unique, but that is already the case with IEEE-754 | > floating-point representations: | | - yes, among other idiosyncrasies. | | > 179.76931348623151e306 ==> 179.76931348623151e306 | > 179.76931348623157e306 ==> 179.76931348623157e306 | > (/ 179.76931348623151e306) ==> 5.562684646268003e-309 | > (/ 179.76931348623157e306) ==> 5.562684646268003e-309 | > | > | which seems clearly desirable as otherwise any expression | > | which may overflow the dynamic range of the number system | > | can't preserve the sign of it's corresponding infinitesimal | > | value, which if not considered important, there's no reason to | > | have signed infinities, either, etc. ? | > | > #i+1/0 is the half-line beyond the largest floating-point value. | > The projection of that interval through / is a small open | > interval bordering 0.0. That interval overlaps the interval of | > floating-point numbers closer to 0.0 than to any other. Thus the | > reciprocal of #i+1/0 is 0.0. | | - but the problem seems to be the reciprocal of #i-1/0? #i-1/0 is the half-line beyond the most negative floating-point value. The projection of that interval through / is a small open interval bordering 0.0. That interval overlaps the interval of floating-point numbers closer to 0.0 than to any other. Thus the reciprocal of #i-1/0 is 0.0. | And it's reciprocal?, which should be where one began? In IEEE-754, the reciprocal of the reciprocal is not always equal to the original: (/ 179.76931348623149e306) ==> 5.56268464626801e-309 (/ (/ 179.76931348623149e306)) ==> 179.76931348623143e306 (/ 179.76931348623151e306) ==> 5.562684646268003e-309 (/ (/ 179.76931348623151e306)) ==> #i+1/0 This can be seen as the consequence of the inexact intervals projected through functions not aligning with the inexact intervals on the real line. | (where it one introduces -0.0 then 0.0 is implied as being +0.0 | leaving one with ether a + or - 0, but nothing which is either? | unless one introduces yet another 0, 0 (or ~0 hypothetically, | which then implies ~inf)? 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. If we make (/ +1 0.0) ==> #i+1/0, then (/ -1 0.0) ==> #i-1/0. This choice is arbitrary; but the other way is confusing. Most neighborhoods mapping through functions project onto adjacent neighborhoods. But / near 0 is not the only function which does not. TAN near pi/2 is another example.