>>>>> "Henry" == Henry Baker <hbaker1 at pipeline.com> writes:
Henry> Using my range analysis program:
Henry> Given: sqrt(1-sqr(x))
Henry> where sqr(x) is x^2, we conclude that the result is in the
Henry> range [0,1], and x must be in the range [-1,1].
So you're saying if we know that sqrt(1-sqr(x)) is in [0, 1], then x
must be in [-1, 1]? That makes sense to me.
Henry> Given: sqrt(1-x*x)
Henry> we conclude that the result is in the range [0,inf), and x
Henry> must be in the range (minf,inf). [Note that x*x is not the
Henry> same as x^2, because x*x is a binary operation on two
Henry> arguments that happen to be the same ranges, while x^2 is a
Henry> unary operation on a single range.]
This I don't understand. How can sqrt(1-x*x) be in the range [0,
inf)? If x is in the range (minf, inf), then x*x is (minf, inf)
(assuming the x's are different) and 1 - x*x is (minf, inf). The sqrt
would then be some complex result union [0, inf). Is that what you
mean?
Henry> Given: (exp(x)+exp(-x))/2 = 3
Henry> we conclude that x is in the range (-30/17, 30/17), which
Henry> is pretty good, considering we don't know about acosh() !
How do you derive that result?
Ray