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diff --git a/algorithms.tex b/algorithms.tex index 2ec2d26d0..f1cf2212b 100644 --- a/algorithms.tex +++ b/algorithms.tex @@ -1075,6 +1075,12 @@ that ${\rm erfc} \, x \leq 2^{-n}$, thus ${\rm erf} \, x = 1$ or ${\rm erf} \, x = 1 - 2^{-n}$ according to the rounding mode. +More precisely, \cite[formul{\ae} 7.1.23 and 7.1.24]{AbSt73} give: +\[ \sqrt{\pi} x e^{x^2} {\rm erfc} x \approx + 1 + \sum_{k=1}^n (-1)^k \frac{1 \times 3 \times \cdots \times (2k-1)} + {(2x^2)^k}, \] +with the error bounded in absolute value by the next term and of the same sign. + \subsection{The hyperbolic cosine function} The {\tt mpfr\_cosh} ($\cosh{x}$) function implements the hyperbolic |