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author | vlefevre <vlefevre@280ebfd0-de03-0410-8827-d642c229c3f4> | 2017-12-18 12:22:26 +0000 |
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committer | vlefevre <vlefevre@280ebfd0-de03-0410-8827-d642c229c3f4> | 2017-12-18 12:22:26 +0000 |
commit | d9ad9c695dcc0197dcec82930f840caa8d063d31 (patch) | |
tree | e0a0361674ce396f2041b1309774ad8d155c6e01 | |
parent | d5677d5bd42a35b5060396d9d38da00f2cf1650c (diff) | |
download | mpfr-d9ad9c695dcc0197dcec82930f840caa8d063d31.tar.gz |
[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
-rw-r--r-- | doc/algorithms.tex | 4 |
1 files changed, 2 insertions, 2 deletions
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 $x=1/2$, |