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I'm trying to develop a architecture hardware to make a implementation of an algorithm that can be descompose in terms of sums, multiplications, subtractions and exponential functions. I'm trying to modelling $\exp(-x)$ through Taylor series. The domain of my function is bounded between $0$ and $1500$, but I want to use a particular Taylor approximation whose domain is bounded between $0$ and $0.5$.

Is there any way to get an approximation using the my tailor series whose domain is bounded between $0$ and $0.5$ to modelling the function whose domain is bounded between $0$ and $1500$?

The function I want to model for bounded domain is $\exp(-x)$. Thank you for your help.

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The computation of the exponential is a well studied problem, I would imagine, and I very much doubt it is done in practice using Taylor series, which tend to be really bad approximations for anything but proving theorems. Have you looked at standard implementations (like the one in the GMP library)? – Mariano Suárez-Alvarez Feb 23 '13 at 22:22

You can certainly find the Taylor series of exp(-x) around 0.25. Wolfram Alpha gives an answer. Then you can plug large numbers into it if you want. It just won't be at all accurate. But I don't think I am understanding what you mean by your boldface question.

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About my boldface question: I mean that if I can use a T.series expression of a function whose domain is bounded between 0 and 0.5 to get an approximation for a function bounded between 0 and 1500. The problem with using the Taylor series approximations is that only are true for a narrow margin around the point of evaluation. – Peterstone Feb 4 '11 at 16:42
How to get a Taylor approximation to a much larger margin than unity, than ten units or even a hundred? .I think TS could be a solution for a margin of 10 units at most, but not for 100 units or more. – Peterstone Feb 4 '11 at 16:49
You can use other series than Taylor, for example a Chebyshev one, which minimizes the error over a domain. It still will need many terms over a wide range. Another approach is to accurately calculate exp(x) over a small range and use the properties of exponents to extend it. And for exp(-x), if x gets very large, you can just return 0. – Ross Millikan Feb 4 '11 at 17:28

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