You're right. I should have said x / (1+|x|). Atan, and also atanh, have similar shapes. But they are linear near 0, and not superexponential for large x, missing a couple of your objectives. Rich ------ Quoting Steve Witham <sw@tiac.net>:
From: rcs@xmission.com
What's wrong with f(x) = x / (1+x^2)? And f(z) = z / (1+|z^2|) ?
Doesn't that look like this?
^ ----\ / \----- v
Do you mean f(x) = x / ( 1 - x^2 )?
That's more like it but I want it to be superexponential as it approaches -1 and +1.
x sb(x) x / ( 1 - x^2 ) 1/2 1 2/3 3/4 2 12/7 7/8 4 56/15 15/16 16 240/31 31/32 65536 992/63 63/64 2^65536 4032/127
Something involving x/(1-x^2) may do what I want. sinh( x/(1-x^2) ) or sinh( z/|1-x^2| ) is more like it but not it.
Also, sb(x) is really flat near zero, e.g. sb(1/64) = 2^-65536, which is also something I want, whereas x/(1-x^2) is linearish there.
My sb() function is only like a tower, and there are all sorts of largeness beyond that, but it's what I was going for.
--Steve
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