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In the first problem, the bug appears to be in the default gcd;

you can get around this by changing gcd routine:

gcd:'spmod$

The second case is a bug in Taylor. To get around it, first

expand the inner expression, then plug it into asin:

inner: taylor(sqrt(1-x^2)+err*x^6,x,0,6);

taylor(asin(inner),x,0,5);

In general, you have to be careful about the orders of

expansion -- you may need more terms in the inner expression

than the whole expression. That is the sort of thing Taylor is

supposed to do for you, but obviously there is a problem in

this case. That is why I have added an explicit error term --

to be able to trace it through.

The second derivative business is, I believe, a red herring --

the absolute value has to do with returning the principal value

of sqrt, but the Taylor expansion doesn't care about principal

values, it cares about analytic continuation.