added section on Notations

fixed notations for roundings


git-svn-id: svn://scm.gforge.inria.fr/svn/mpfr/trunk@4089 280ebfd0-de03-0410-8827-d642c229c3f4

1 files changed, 95 insertions, 87 deletions

diff --git a/algorithms.tex b/algorithms.tex
index 113406c16..400795477 100644
--- a/algorithms.tex
+++ b/algorithms.tex
@@ -43,6 +43,14 @@ Algorithms and Proofs}
 
 \tableofcontents
 
+\section{Notations}
+
+In the whole document, $\N()$ denotes rounding to nearest,
+$\Z()$ rounding towards zero,
+$\pinf()$ rounding towards plus infinity,
+$\minf()$ rounding towards minus infinity,
+and $\circ()$ any of those four rounding modes.
+
 \section{Error calculus}
 
 Let $n$ --- the working precision ---
@@ -120,8 +128,8 @@ Easy: if $a = m_a \cdot 2^{e_a}$, then $2^k a = m_a \cdot 2^{e_a+k}$.
 \end{proof}
 
 \begin{Rule} \label{R6}
-Let $x > 0$, $o(\cdot)$ be any rounding, and $u := o(x)$, then $\frac{1}{2} u
-        < x < 2 u$.
+Let $x > 0$, $\circ(\cdot)$ be any rounding, and $u := o(x)$,
+then $\frac{1}{2} u < x < 2 u$.
 \end{Rule}
 \begin{proof}
 Assume $x \geq 2 u$, then $2u$ is another representable number which is closer 
@@ -138,18 +146,18 @@ and the right one from $|a| \ulp(1) \geq \frac{1}{2} 2^{e_a} 2^{1-n} =\ulp(a)$.
 \end{proof}
 
 \begin{Rule} \label{R8}
-For any $x \neq 0$ and any rounding mode $o(\cdot)$,
-we have $\ulp(x) \leq \ulp(o(x))$, and equality holds when rounding towards
+For any $x \neq 0$ and any rounding mode $\circ(\cdot)$,
+we have $\ulp(x) \leq \ulp(\circ(x))$, and equality holds when rounding towards
 zero, towards $-\infty$ for $x>0$, or towards $+\infty$ for $x<0$.
 \end{Rule}
 \begin{proof}
 Without loss of generality, assume $x > 0$.
 Let $x = m \cdot 2^e$ with $\frac{1}{2} \leq m < 1$.
-As $\frac{1}{2} 2^{e_x}$ is a machine number, necessarily $o(x) \geq
-\frac{1}{2} 2^{e_x}$, thus by Rule~\ref{R2}, then $\ulp(o(x)) \geq
+As $\frac{1}{2} 2^{e_x}$ is a machine number, necessarily $\circ(x) \geq
+\frac{1}{2} 2^{e_x}$, thus by Rule~\ref{R2}, then $\ulp(\circ(x)) \geq
 2^{e_x - n} = \ulp(x)$.
-If we round towards zero, then $o(x) \leq x$ and by Rule~\ref{R2} again,
-$\ulp(o(x)) \leq \ulp(x)$.
+If we round towards zero, then $\circ(x) \leq x$ and by Rule~\ref{R2} again,
+$\ulp(\circ(x)) \leq \ulp(x)$.
 \end{proof}
 
 \begin{Rule} \label{R9}
@@ -159,11 +167,11 @@ $\ulp(o(x)) \leq \ulp(x)$.
 \end{eqnarray}
 
 \begin{eqnarray}\nonumber
-&&\n{For}\;\;  u=o(x),\;\; u.c_u^- \leq x \leq u.c_u^+\\\nonumber
+&&\n{For}\;\;  u=\circ(x),\;\; u.c_u^- \leq x \leq u.c_u^+\\\nonumber
 &&\n{if}\;\;  u=\pinf(x),\n{ then } c_u^+=1\\\nonumber
 &&\n{if}\;\;  u=\minf(x),\n{ then } c_u^-=1\\\nonumber
-&&\n{if}\;\;  \n{for $x<0$ and } u=Z(x),\n{ then } c_u^+=1 \\\nonumber
-&&\n{if}\;\;  \n{for $x>0$ and } u=Z(x),\n{ then } c_u^-=1 \\\nonumber
+&&\n{if}\;\;  \n{for $x<0$ and } u=\Z(x),\n{ then } c_u^+=1 \\\nonumber
+&&\n{if}\;\;  \n{for $x>0$ and } u=\Z(x),\n{ then } c_u^-=1 \\\nonumber
 &&\n{else}\;\;   c_u^{-}=1-2^{1-p} \n{ and } c_u^{+}=1+2^{1-p}
 \end{eqnarray}
 \end{Rule}
@@ -229,7 +237,7 @@ error(w)& \leq &c_w \ulp(w) + k_u \ulp(u) + k_v \ulp(v) \\\nonumber
 error(w)& \leq&(c_w + k_u + k_v) \, \ulp(w)
 \end{eqnarray}
 \begin{eqnarray}\nonumber
-\textnormal{Note:}&&\textnormal{If}\;\; w=N(u+v) \;\;\textnormal{Then}\;\; c_w =\frac{1}{2} \;\;\textnormal{else}\;\; c_w =1\\\nonumber
+\textnormal{Note:}&&\textnormal{If}\;\; w=\N(u+v) \;\;\textnormal{Then}\;\; c_w =\frac{1}{2} \;\;\textnormal{else}\;\; c_w =1\\\nonumber
 \end{eqnarray}
 
 \subsection{Generic error of multiplication}\label{generic:mul}
@@ -239,7 +247,7 @@ We want to compute the generic error of the multiplication.
 We assume here $u, v > 0$.
 
 \begin{eqnarray}\nonumber
-w&=&o(u.v) \\\nonumber
+w&=&\circ(u.v) \\\nonumber
 \textnormal{Note:}&& error(u) \leq k_u \, \ulp(u), \;\; error(v) \leq k_v \, \ulp(v)
 \end{eqnarray}
 \begin{eqnarray}\nonumber
@@ -252,7 +260,7 @@ error(w)& = &|w - x.y| \\\nonumber
 & \leq & [ c_w +  (1+c_u^+) k_v + (1+c_v^+) k_u ] \ulp(w)\U{R8}\\\nonumber
 \end{eqnarray}
 \begin{eqnarray}\nonumber
-\textnormal{Note:}&&\textnormal{If}\;\; w=N(u+v) \;\;\textnormal{Then}\;\; c_w =\frac{1}{2} \;\;\textnormal{else}\;\; c_w =1
+\textnormal{Note:}&&\textnormal{If}\;\; w=\N(u+v) \;\;\textnormal{Then}\;\; c_w =\frac{1}{2} \;\;\textnormal{else}\;\; c_w =1
 \end{eqnarray}
 
 \subsection{Generic error of inverse}\label{generic:inv}
@@ -261,7 +269,7 @@ We want to compute the generic error of the inverse.
 We assume $u > 0$.
 
 \begin{eqnarray}\nonumber
-w&=&o(\frac{1}{u}) \\\nonumber
+w&=&\circ(\frac{1}{u}) \\\nonumber
 \textnormal{Note:}&& error(u) \leq k_u \, \ulp(u)
 \end{eqnarray}
 \begin{eqnarray}\nonumber
@@ -281,7 +289,7 @@ error(w)& \leq & c_w \ulp(w) + c_u\frac{k_u}{u^2} \ulp(u)\\\nonumber
 & \leq & [c_w + 2.c_u.k_u].\ulp(w) \U{R8}
 \end{eqnarray}
 \begin{eqnarray}\nonumber
-\textnormal{Note:}&&\textnormal{If}\;\; w=N(\frac{1}{u}) \;\;\textnormal{Then}\;\; c_w =\frac{1}{2} \;\;\textnormal{else}\;\; c_w =1\\\nonumber\end{eqnarray}
+\textnormal{Note:}&&\textnormal{If}\;\; w=\N(\frac{1}{u}) \;\;\textnormal{Then}\;\; c_w =\frac{1}{2} \;\;\textnormal{else}\;\; c_w =1\\\nonumber\end{eqnarray}
 
 \subsection{Generic error of division}\label{generic:div}
 
@@ -289,7 +297,7 @@ error(w)& \leq & c_w \ulp(w) + c_u\frac{k_u}{u^2} \ulp(u)\\\nonumber
 We want to compute the generic error of the division.
 Without loss of generality, we assume all variables are positive.
 \begin{eqnarray}\nonumber
-w&=&o(\frac{u}{v}) \\\nonumber
+w&=&\circ(\frac{u}{v}) \\\nonumber
 \textnormal{Note:}&& error(u) \leq k_u \, \ulp(u), \;\; error(v) \leq k_v \, \ulp(v)
 \end{eqnarray}
 \begin{eqnarray}\nonumber
@@ -316,7 +324,7 @@ error(w)& \leq & c_w \ulp(w) + 2.k_u  \ulp(w)+ c_u.c_v.\frac{k_v u}{vv} \ulp(v)\
 & \leq & [c_w  + 2.k_u+ 2.c_u.c_v.k_v].\ulp(w) \U{R8}
 \end{eqnarray}
 \begin{eqnarray}\nonumber
-\textnormal{Note:}&&\textnormal{If}\;\; w=N(\frac{u}{v}) \;\;\textnormal{Then}\;\; c_w =\frac{1}{2} \;\;\textnormal{else}\;\; c_w =1
+\textnormal{Note:}&&\textnormal{If}\;\; w=\N(\frac{u}{v}) \;\;\textnormal{Then}\;\; c_w =\frac{1}{2} \;\;\textnormal{else}\;\; c_w =1
 \end{eqnarray}
 Note that we can obtain a slightly different result by writing
 $uy-vx = (uy - uv) + (uv - vx)$ instead of $(uy-xy)+(xy-vx)$.
@@ -335,7 +343,7 @@ this is bounded by $w (|\theta_u|+|\theta_v|)$.
 We want to compute the generic error of the square root of a floating-point
 number $u$, itself an approximation to a real $x$,
 with $|u - x| \leq k_u \ulp(u)$.
-If $v = o(\sqrt{u})$, then:
+If $v = \circ(\sqrt{u})$, then:
 \begin{eqnarray}\nonumber
 error(v) := |v - \sqrt{x}| 
 & \leq &|v - \sqrt{u}| +|\sqrt{u} - \sqrt{x}| \\\nonumber
@@ -362,7 +370,7 @@ $|v-\sqrt{x}| \leq (c_v + k_u)  \ulp(v)$.
 We want to compute the generic error of the exponential.
 
 \begin{eqnarray}\nonumber
-v&=&o(e^{u}) \\\nonumber
+v&=&\circ(e^{u}) \\\nonumber
 \textnormal{Note:}&& error(u) \leq k_u \, \ulp(u)\\\nonumber
 \end{eqnarray}
 \begin{eqnarray}\nonumber
@@ -392,7 +400,7 @@ error(v)& \leq & c_v \ulp(v) +  c_u^* e^u k_u \ulp(u) \\\nonumber
 We want to compute the generic error of the logarithm.
 
 \begin{eqnarray}\nonumber
-v&=&o(\log{u}) \\\nonumber
+v&=&\circ(\log{u}) \\\nonumber
 \textnormal{Note:}&& error(u) \leq k_u \, \ulp(u)\\\nonumber
 \end{eqnarray}
 \begin{eqnarray}\nonumber
@@ -1068,13 +1076,13 @@ cosine as :
 
 \[\cosh x = \frac{1}{2} \left( e^{x} + \frac{1}{e^x} \right).\]
 
-The algorithm used for the calculation of the hyperbolic cosine is as follows\footnote{$o()$ represent the rounding error and $error(u)$ the
+The algorithm used for the calculation of the hyperbolic cosine is as follows\footnote{$\circ()$ represent the rounding error and $error(u)$ the
   error associate with the calculation of $u$}:
 
 \begin{eqnarray}\nonumber
-u&\leftarrow&o(e^x)\\\label{coshalgo1}
-v&\leftarrow&o({u}^{-1})\\\label{coshalgo2}
-w&\leftarrow&o(u+v)\\\label{coshalgo3}
+u&\leftarrow&\circ(e^x)\\\label{coshalgo1}
+v&\leftarrow&\circ({u}^{-1})\\\label{coshalgo2}
+w&\leftarrow&\circ(u+v)\\\label{coshalgo3}
 s&\leftarrow&\frac{1}{2} w\\\label{coshalgo4}
 \end{eqnarray}
 
@@ -1095,7 +1103,7 @@ with $x \geq 0$. We can deduce $e^x \geq 1$ and $0 \leq e^{-x} \leq
 ${\textnormal{error}}(u)$
 
 
-$u \leftarrow o(e^x)$\\
+$u \leftarrow \circ(e^x)$\\
 $-\infty \;\; (\bullet)$
 
 \end{minipage} &
@@ -1113,7 +1121,7 @@ $-\infty \;\; (\bullet)$
 ${\textnormal{error}}(v)$
 
 
-$v \leftarrow o({u}^{-1}) $\\
+$v \leftarrow \circ({u}^{-1}) $\\
 $+\infty \;\; (\bullet\bullet)$ 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -1160,7 +1168,7 @@ $v$ to $+\infty \;\; (\bullet)$\\
 ${\textnormal{error}}(w)$
 
 
-$w \leftarrow o(u+v) $
+$w \leftarrow \circ(u+v) $
 \end{minipage} &
 \begin{minipage}{7.5cm}
 
@@ -1194,7 +1202,7 @@ then $\ulp(u) \leq \ulp(w)$
 \begin{minipage}{2.5cm}
 ${\textnormal{error}}(s)$
 
-$s \leftarrow o(\frac{w}{2}) $
+$s \leftarrow \circ(\frac{w}{2}) $
 \end{minipage} &
 \begin{minipage}{7.5cm}
 
@@ -1260,13 +1268,13 @@ sine as :
 
 \[\sinh x = \frac{1}{2} \left( e^{x} - \frac{1}{e^x} \right).\]
 
-The algorithm used for the calculation of the hyperbolic sine is as follows\footnote{$o()$ represent the rounding error and $error(u)$ the
+The algorithm used for the calculation of the hyperbolic sine is as follows\footnote{$\circ()$ represent the rounding error and $error(u)$ the
   error associate with the calculation of $u$}:
 
 \begin{eqnarray}\nonumber
-u&\leftarrow&o(e^x)\\\nonumber
-v&\leftarrow&o({u}^{-1})\\\nonumber
-w&\leftarrow&o(u-v)\\\nonumber
+u&\leftarrow&\circ(e^x)\\\nonumber
+v&\leftarrow&\circ({u}^{-1})\\\nonumber
+w&\leftarrow&\circ(u-v)\\\nonumber
 s&\leftarrow&\frac{1}{2} w
 \end{eqnarray}
 
@@ -1350,7 +1358,7 @@ it is possible with $v=\pinf(u^{-1})$ $(\bullet\bullet)$
 ${\textnormal{error}}(w)$
 
 
-$w \leftarrow o(u+v) $
+$w \leftarrow \circ(u+v) $
 \end{minipage} &
 \begin{minipage}{7.8cm}
 
@@ -1382,7 +1390,7 @@ see subsection \ref{generic:sous}
 \begin{minipage}{2.5cm}
 ${\textnormal{error}}(s)$
 
-$s \leftarrow o(\frac{w}{2}) $
+$s \leftarrow \circ(\frac{w}{2}) $
 \end{minipage} &
 \begin{minipage}{7.5cm}
 
@@ -1425,11 +1433,11 @@ The {\tt mpfr\_asinh} ($\n{acosh}{x}$) function implements the inverse hyperboli
 The algorithm used for the calculation of the inverse hyperbolic sine is as follows
 
 \begin{eqnarray}\nonumber
-s&\leftarrow&o(x^2)\\\nonumber
-t&\leftarrow&o(s+1)\\\nonumber
-u&\leftarrow&o(\sqrt{t})\\\nonumber
-v&\leftarrow&o(u+x)\\\nonumber
-w&\leftarrow&o(\log  v)
+s&\leftarrow&\circ(x^2)\\\nonumber
+t&\leftarrow&\circ(s+1)\\\nonumber
+u&\leftarrow&\circ(\sqrt{t})\\\nonumber
+v&\leftarrow&\circ(u+x)\\\nonumber
+w&\leftarrow&\circ(\log  v)
 \end{eqnarray}
 
 
@@ -1445,7 +1453,7 @@ with $x \geq 0$.
 ${\textnormal{error}}(s)$
 
 
-$s \leftarrow o(x^2) $
+$s \leftarrow \circ(x^2) $
 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -1490,7 +1498,7 @@ see subsection \ref{generic:sous}
 ${\textnormal{error}}(u)$
 
 
-$u \leftarrow o(\sqrt{t}) $
+$u \leftarrow \circ(\sqrt{t}) $
 
 
 \end{minipage} &
@@ -1516,7 +1524,7 @@ with ($\bullet$)
 ${\textnormal{error}}(v)$
 
 
-$v \leftarrow o(u+x) $
+$v \leftarrow \circ(u+x) $
 
 
 \end{minipage} &
@@ -1540,7 +1548,7 @@ see subsection \ref{generic:sous}
 ${\textnormal{error}}(w)$
 
 
-$w \leftarrow o(\log v) $
+$w \leftarrow \circ(\log v) $
 \end{minipage} &
 \begin{minipage}{7.5cm}
 
@@ -1629,11 +1637,11 @@ The {\tt mpfr\_atanh} ($\n{acosh}{x}$) function implements the inverse hyperboli
 The algorithm used for the calculation of the inverse hyperbolic tangent is as follows
 
 \begin{eqnarray}\nonumber
-s&\leftarrow&o(1+x)\\\nonumber
-t&\leftarrow&o(1-x)\\\nonumber
-u&\leftarrow&o(\frac{s}{t})\\\nonumber
-v&\leftarrow&o(\log u)\\\nonumber
-w&\leftarrow&o(\frac{1}{2} v)
+s&\leftarrow&\circ(1+x)\\\nonumber
+t&\leftarrow&\circ(1-x)\\\nonumber
+u&\leftarrow&\circ(\frac{s}{t})\\\nonumber
+v&\leftarrow&\circ(\log u)\\\nonumber
+w&\leftarrow&\circ(\frac{1}{2} v)
 \end{eqnarray}
 
 
@@ -1696,7 +1704,7 @@ see subsection \ref{generic:sous}
 ${\textnormal{error}}(u)$
 
 
-$u \leftarrow o(\frac{s}{t}) $
+$u \leftarrow \circ(\frac{s}{t}) $
 
 
 \end{minipage} &
@@ -1724,7 +1732,7 @@ with ($\bullet$) and ($\bullet\bullet$)
 ${\textnormal{error}}(v)$
 
 
-$v \leftarrow o(\log(u)) $
+$v \leftarrow \circ(\log(u)) $
 
 
 \end{minipage} &
@@ -1751,7 +1759,7 @@ see subsection \ref{generic:log}
 ${\textnormal{error}}(w)$
 
 
-$w \leftarrow o(\frac{1}{2} v) $
+$w \leftarrow \circ(\frac{1}{2} v) $
 \end{minipage} &
 \begin{minipage}{7.5cm}
 
@@ -2097,16 +2105,16 @@ The \texttt{mpfr\_hypot} function implements the euclidean distance function:
 \]
 The algorithm used is as follows, where all roundings are done towards zero:
 \begin{eqnarray}\nonumber
-u&\leftarrow&o(x^2)\\\nonumber
-v&\leftarrow&o(y^2)\\\nonumber
-w&\leftarrow&o(u+v)\\\nonumber
-s&\leftarrow&o(\sqrt{w})
+u&\leftarrow&\circ(x^2)\\\nonumber
+v&\leftarrow&\circ(y^2)\\\nonumber
+w&\leftarrow&\circ(u+v)\\\nonumber
+s&\leftarrow&\circ(\sqrt{w})
 \end{eqnarray}
 Since the inputs $x$ and $y$ are exact, we have $|u-x^2| \leq \ulp(u)$
 and $|v-y^2| \leq \ulp(v)$;
 then $|w-(x^2+y^2)| \leq \ulp(w) + |u-x^2| + |v-y^2| \leq 3 \ulp(w)$
 since $w$ is greater or equal to $u$ and $v$.
-For the last step $s \leftarrow o(\sqrt{w})$, we use the last formula
+For the last step $s \leftarrow \circ(\sqrt{w})$, we use the last formula
 from~\textsection\ref{generic:sqrt}, which gives
 $|s-\sqrt{x^2+y^2}| \leq 4 \ulp(s)$.
 
@@ -2121,8 +2129,8 @@ The {\tt mpfr\_fma} ($\n{fma}(x,y,z)$) function implements the floating multiply
 The algorithm used for this calculation is as follows:
 
 \begin{eqnarray}\nonumber
-u&\leftarrow&o(x \times y)\\\nonumber
-v&\leftarrow&o(z + u)\\\nonumber
+u&\leftarrow&\circ(x \times y)\\\nonumber
+v&\leftarrow&\circ(z + u)\\\nonumber
 \end{eqnarray}
 
 Now, we have to bound the rounding error for each step of this
@@ -2139,7 +2147,7 @@ algorithm.
 ${\textnormal{error}}(u)$
 
 
-$u \leftarrow o(x \times y)$
+$u \leftarrow \circ(x \times y)$
 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -2156,7 +2164,7 @@ $u \leftarrow o(x \times y)$
 ${\textnormal{error}}(v)$
 
 
-$v \leftarrow o(z+u) $
+$v \leftarrow \circ(z+u) $
 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -2195,8 +2203,8 @@ The {\tt mpfr\_expm1} ($\n{expm1}(x)$) function implements the expm1 function  a
 The algorithm used for this calculation is as follows:
 
 \begin{eqnarray}\nonumber
-u&\leftarrow&o(e^x)\\\nonumber
-v&\leftarrow&o(u-1)\\\nonumber
+u&\leftarrow&\circ(e^x)\\\nonumber
+v&\leftarrow&\circ(u-1)\\\nonumber
 \end{eqnarray}
 
 Now, we have to bound the rounding error for each step of this
@@ -2212,7 +2220,7 @@ algorithm.
 ${\textnormal{error}}(u)$
 
 
-$u \leftarrow o(e^x)$
+$u \leftarrow \circ(e^x)$
 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -2229,7 +2237,7 @@ $u \leftarrow o(e^x)$
 ${\textnormal{error}}(v)$
 
 
-$v \leftarrow o(u-1) $
+$v \leftarrow \circ(u-1) $
 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -2267,9 +2275,9 @@ The {\tt mpfr\_log1p} ($\n{log1p}(x)$) function implements the log1p function  a
 The algorithm used for this calculation is as follows:
 
 \begin{eqnarray}\nonumber
-u&\leftarrow&o(x)\\\nonumber
-v&\leftarrow&o(1+u)\\\nonumber
-w&\leftarrow&o(\log(v))\\\nonumber
+u&\leftarrow&\circ(x)\\\nonumber
+v&\leftarrow&\circ(1+u)\\\nonumber
+w&\leftarrow&\circ(\log(v))\\\nonumber
 \end{eqnarray}
 
 Now, we have to bound the rounding error for each step of this
@@ -2284,7 +2292,7 @@ algorithm.
 ${\textnormal{error}}(u)$
 
 
-$u \leftarrow o(x)$
+$u \leftarrow \circ(x)$
 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -2301,7 +2309,7 @@ $u \leftarrow o(x)$
 ${\textnormal{error}}(v)$
 
 
-$v \leftarrow o(1+u) $
+$v \leftarrow \circ(1+u) $
 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -2320,7 +2328,7 @@ see subsection \ref{generic:sous}
 ${\textnormal{error}}(w)$
 
 
-$w \leftarrow o(\log(v)) $
+$w \leftarrow \circ(\log(v)) $
 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -2363,9 +2371,9 @@ The algorithm used for this calculation is the same for $\n{log2}$ or
 $\n{log10}$ and is described as follows for $t=2 \n{ or } 10$:
 
 \begin{eqnarray}\nonumber
-u&\leftarrow&o(\log(x))\\\nonumber
-v&\leftarrow&o(\log(t))\\\nonumber
-w&\leftarrow&o(\frac{u}{v})\\\nonumber
+u&\leftarrow&\circ(\log(x))\\\nonumber
+v&\leftarrow&\circ(\log(t))\\\nonumber
+w&\leftarrow&\circ(\frac{u}{v})\\\nonumber
 \end{eqnarray}
 
 Now, we have to bound the rounding error for each step of this
@@ -2413,7 +2421,7 @@ $(\bullet\bullet)$
 ${\textnormal{error}}(w)$
 
 
-$w \leftarrow o(\frac{u}{v}) $
+$w \leftarrow \circ(\frac{u}{v}) $
 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -2451,9 +2459,9 @@ The {\tt mpfr\_pow} function implements the power function  as:
 The algorithm used for this calculation is as follows:
 
 \begin{eqnarray}\nonumber
-u&\leftarrow&o(\log(x))\\\nonumber
-v&\leftarrow&o(y u)\\\nonumber
-w&\leftarrow&o(e^v)\\\nonumber
+u&\leftarrow&\circ(\log(x))\\\nonumber
+v&\leftarrow&\circ(y u)\\\nonumber
+w&\leftarrow&\circ(e^v)\\\nonumber
 \end{eqnarray}
 
 Now, we have to bound the rounding error for each step of this
@@ -2468,7 +2476,7 @@ algorithm with $x \geq 0$ and $y$ is a floating number.
 ${\textnormal{error}}(u)$
 
 
-$u \leftarrow o(\log(x))$
+$u \leftarrow \circ(\log(x))$
 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -2514,7 +2522,7 @@ with \U{R8}
 ${\textnormal{error}}(w)$
 
 
-$w \leftarrow o(e^v) $
+$w \leftarrow \circ(e^v) $
 
 \end{minipage} &
 \begin{minipage}{7.5cm}
@@ -2976,13 +2984,13 @@ is an exponent loss in the final subtraction $r = \circ(v_{n+1}-w)$.
 \begin{center}
 \begin{tabular}{|c|c|c|} \hline
    $w$        & $\err(w)/\ulp(w) \le c_w + \ldots$ &special case\\ \hline\hline
-$o(u+v)$ & $k_u 2^{e_u-e_w} + k_v 2^{e_v-e_w}$ & $k_u + k_v$ if $u v \ge 0$\\
-$o(u \cdot v)$ & $(1+c^{+}_u)k_u + (1+c^{+}_v)k_v$ & $2k_u + 2k_v$ if $u \ge x$, $v \ge y$\\
-$o(1/v)$ & $4 k_v$ & $2 k_v$ if $v \le y$ \\
-$o(u/v)$ & $4 k_u + 4 k_v$ & $2 k_u + 2 k_v$ if $v \le y$ \\
-$o(\sqrt{u})$ & $2 k_u/(1+\sqrt{c^{-}_u})$ & $k_u$ if $u \le x$ \\
-$o(e^u)$ & $e^{k_u 2^{e_u-p}} 2^{e_u+1} k_u$ & $2^{e_u+1} k_u$ if $u \ge x$ \\
-$o(\log u)$ & $k_u 2^{1-e_w}$ & \\
+$\circ(u+v)$ & $k_u 2^{e_u-e_w} + k_v 2^{e_v-e_w}$ & $k_u + k_v$ if $u v \ge 0$\\
+$\circ(u \cdot v)$ & $(1+c^{+}_u)k_u + (1+c^{+}_v)k_v$ & $2k_u + 2k_v$ if $u \ge x$, $v \ge y$\\
+$\circ(1/v)$ & $4 k_v$ & $2 k_v$ if $v \le y$ \\
+$\circ(u/v)$ & $4 k_u + 4 k_v$ & $2 k_u + 2 k_v$ if $v \le y$ \\
+$\circ(\sqrt{u})$ & $2 k_u/(1+\sqrt{c^{-}_u})$ & $k_u$ if $u \le x$ \\
+$\circ(e^u)$ & $e^{k_u 2^{e_u-p}} 2^{e_u+1} k_u$ & $2^{e_u+1} k_u$ if $u \ge x$ \\
+$\circ(\log u)$ & $k_u 2^{1-e_w}$ & \\
 \hline
 \end{tabular}
 \end{center}