From d9ad9c695dcc0197dcec82930f840caa8d063d31 Mon Sep 17 00:00:00 2001 From: vlefevre Date: Mon, 18 Dec 2017 12:22:26 +0000 Subject: [doc/algorithms.tex] mpfr_tanh: missing absolute value; added a \cdot. git-svn-id: svn://scm.gforge.inria.fr/svn/mpfr/trunk@11999 280ebfd0-de03-0410-8827-d642c229c3f4 --- doc/algorithms.tex | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) (limited to 'doc') diff --git a/doc/algorithms.tex b/doc/algorithms.tex index 4cedf794a..4ab148f40 100644 --- a/doc/algorithms.tex +++ b/doc/algorithms.tex @@ -1865,10 +1865,10 @@ and then $s = \tanh(x) \cdot (1+\theta_4)^{2^k+4}$. \begin{lemma} For $|x| \leq 1/2$, and $|y| \leq |x|^{-1/2}$, we have: -\[ |(1+x)^y-1| \leq 2.5 \cdot |y| \cdot x. \] +\[ |(1+x)^y-1| \leq 2.5 \cdot |y| \cdot |x|. \] \end{lemma} \begin{proof} -We have $(1+x)^y = e^{y \log (1+x)}$, +We have $(1+x)^y = e^{y \cdot \log (1+x)}$, with $|y \cdot \log (1+x)| \leq |x|^{-1/2} \cdot \left|\log (1+x)\right|$. The function $|x|^{-1/2} \cdot \log (1+x)$ is increasing on $[-1/2,1/2]$, and takes as values $\approx -0.980$ in $x=-1/2$ and $\approx 0.573$ in $x=1/2$, -- cgit v1.2.1