Approximating (x^2+a^2)^(1/4) w/ Error < 1e-6

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The discussion focuses on approximating the function (x^2 + a^2)^(1/4) with an error less than 1e-6 for x in the range [-1, 1] and a in the real numbers. The user considers using Legendre polynomials but finds that convergence requires a high number of polynomials due to the error decreasing at a rate of n^(3/2). They propose using multiple Taylor polynomials over specific intervals to improve accuracy, acknowledging that Taylor polynomials centered at zero yield significant errors away from that point. The conversation highlights the efficiency of rational function approximations, as referenced in Abramowitz and Stegun's work.

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I want to have the following function approximated (in elementary functions) within a error of 1e-6:

[tex](x^2+a^2)^{1/4}\ \forall x\ \in [-1,1],\ a\in \mathbb{R}[/tex]If I use Legendre polynomials I have to use a lot to get convergence (since if I use an estimate of the error the error decreases with n^3/2 with n the number of polynomials). So I suppose that I can use taylor polynomials, right? But taylored around 0 the polynomials tends to have a large error moving away from zero. So my idea is to use several taylor polynomials for certain intervals. Is this an idea or do I have to do something else?
 
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From the point of view of computer programming, the most efficient approximations of commonly encountered functions are approximations by rational functions ( ratios of polynomials). For example, if you look at the famous book by Abramawitz and Stegun, you find that sort of approximation.
 

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