Geodesic
On the exponential integral computation
Matematičeskoe modelirovanie, Tome 20 (2008) no. 1, pp. 87-91.

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New high-precision algorithm for exponential integral calculation was developed. It is based on the representation of exponential integral in form of convergent series when $x$-argument is not large and in form of asymptotically convergent continued fraction when $x$ is large. It was shown that the optimal bound between these representations is $x=1$. At the same time using of 18 series members and 220 continued fraction members guarantees relative pre-cision lower than $2\cdot 10^{-15}$, that exceeds practical needs. 
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N. N. Kalitkin; I. A. Panin. On the exponential integral computation. Matematičeskoe modelirovanie, Tome 20 (2008) no. 1, pp. 87-91. http://geodesic.mathdoc.fr/item/MM_2008_20_1_a6/

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