[WP34S] DEG and RAD  diffs

06062014, 10:53 PM
Post: #29




RE: [WP34S] DEG and RAD  diffs
(06062014 07:53 PM)Claudio L. Wrote: Also, when x is close to zero, doing sqrt(1+x^2) is really bad for precision. The x^2 "spreads" your useful digits throughout your exponent range, then the square root compresses them back, and you lost about half of them at the end (see what I meant with "small angles are tougher on precision loss"). Huh?? For z near one, \(\sqrt{z}\) is always closer to one than z is. So yes, you will lose some least significant digits but the result is accurate within your working precision. Hence, \(\sqrt{1+x^2}\) for small x has no accuracy problems by itself, the answer will approach unity much faster than x approaches zero. If, on the other hand, you have something along the lines of \(\sqrt{1+x^2}1\), then the story is completely different. Here, there is a cancellation of digits and loss of precision due to the \(1\). However, these situations are rarely insurmountable. In this case, I'd use the transformation (assuming my algebra is good): $$\sqrt{1+x^2}1 = \frac{x^2}{\sqrt{1+x^2}+1}$$ which doesn't involve any cancellation. The actual transformation required will depend on the exact formula being evaluated of course.  Pauli 

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