cln_mirror/doc/ratseries/paper/binsplit.tex

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%%% * Habe neue Referenzen zu den Karatsuba-Arbeiten hinzugefügt, entnommen
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%%% * Korrigiere die Formel für a(n) bei zeta(3).
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%%% * Drei Literaturverweise für LiDIA statt nur einem.
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%%% Version 7, 1998-01-16a, Bruno
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%%% * Adresse: Praefix F fuer Frankreich
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%%% Version 6, 1998-01-14, Thomas
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%%% * habe meine Adresse ge"andert.
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%%% * Meine Firma schreibt sich mit vier Grossbuchstaben.
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%%% * Das Cohen-Villegas-Zagier brauchen wir nun doch nicht zu zitieren.
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%%% * Figure 2, erste Spalte rechtsbuendig.
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%%% Version 2, 1997-01-06, Thomas
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%%% * Habe Dein Dankeschön an mich entfernt.
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%%% * Habe neue Formel für pi eingefügt. Grund: einfacher,
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%%%   eingefügt. Ein Beispiel über die Wirksamkeit habe ich
%%%   bei der Apery Konstante angegeben. Grund: damit können wir
%%%   das Paper auch bei PASCO '97 einreichen. 
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%%% * Habe Beispiel-C++-Implementierung für abpq Reihen eingefügt.
%%%   Grund: zeigen wie einfach es ist wenn man die Formeln hat ;-)
%%% * Habe Beispiel-Implementierung für abpqcd Reihen eingefügt.
%%%   Grund: dito
%%% * Habe Computational results und Conclusions Abschnitt eingefügt.
%%% * Habe die Namen der Konstanten (C, G, ...) and die entsprechenden
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%%%   Tabellen im Abschnitt Computational results benutzt.
%%% * Habe Verweis an LiDIA eingefügt. Grund: wird bei Computational
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%%% Version 1, 1996-11-30, Bruno
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%%% Bruno Haible, Thomas Papanikolaou.                                     %%%%
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\begin{document}

\title{Fast multiprecision evaluation of series of rational numbers}

\author{
\begin{tabular}{ccc}
{Bruno Haible} & \hspace*{2cm} & {Thomas Papanikolaou}\\
{\normalsize ILOG} && {\normalsize Laboratoire A2X}\\
{\normalsize 9, rue de Verdun} && {\normalsize 351, cours de la Lib\'eration}\\
{\normalsize F -- 94253 Gentilly Cedex} && {\normalsize F -- 33405 Talence Cedex}\\
{\normalsize {\tt haible@ilog.fr}} && {\normalsize {\tt papanik@math.u-bordeaux.fr}}\\
\end{tabular}
}

\maketitle

\begin{abstract}

We describe two techniques for fast multiple-precision evaluation of linearly
convergent series, including power series and Ramanujan series. The computation
time for \(N\) bits is  \( O((\log N)^{2}M(N)) \), where \( M(N) \) is the time
needed to multiply two \(N\)-bit numbers. Applications include fast algorithms
for elementary functions,  \(\pi\), hypergeometric functions at rational points,
$\zeta(3)$, Euler's, Catalan's and Ap{\'e}ry's constant. The algorithms are
suitable for parallel computation.

\end{abstract}

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\section{Introduction}

Multiple-precision evaluation of real numbers has become efficiently possible
since Sch\"onhage and Strassen \cite{71} have showed that the bit complexity of
the multiplication of two \(N\)-bit numbers is
\( M(N)=O(N\:\log N\:\log\log N) \).
This is not only a theoretical result; a C++ implementation \cite{96a} can
exploit this already for  \( N=40000 \) bits. Algorithms for computing
elementary functions (exp, log, sin, cos, tan, asin, acos, atan, sinh, cosh,
tanh, arsinh, arcosh, artanh) have appeared in \cite{76b}, and a remarkable
algorithm for \( \pi \) was found by Brent and Salamin \cite{76c}.

However, all these algorithms suffer from the fact that calculated results
are not reusable, since the computation is done using real arithmetic (using
exact rational arithmetic would be extremely inefficient). Therefore functions
or constants have to be recomputed from the scratch every time higher precision
is required.

In this note, we present algorithms for fast computation of sums of the form

\[S=\sum _{n=0}^{\infty }R(n)F(0)\cdots F(n)\]
where \( R(n) \) and \( F(n) \) are rational functions in \( n \) with rational
coefficients, provided that this sum is linearly convergent, i.e. that the
\( n \)-th term is \( O(c^{-n}) \) with  \( c>1 \). Examples include elementary
and hypergeometric functions at rational points in the {\em interior} of the
circle of convergence, as well as \( \pi  \) and Euler's, Catalan's and
Ap{\'e}ry's constants.

The presented algorithms are {\em easy to implement} and {\em extremely
efficient}, since they take advantage of pure integer arithmetic. The
calculated results are {\em exact}, making {\em checkpointing} and
{\em reuse} of computations possible. Finally,
the computation of our algorithms {\em can be easily parallelised}.

After publishing the present paper, we were informed that the results of
section 2 were already published by E.~Karatsuba in \cite{91,91b,93,95c}.

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\section{Evaluation of linearly convergent series}

The technique presented here applies to all linearly convergent sums of the
form

\[ S=\sum ^{\infty }_{n=0}
\frac{a(n)}{b(n)}\frac{p(0)\cdots p(n)}{q(0)\cdots q(n)}\]
where  \( a(n) \),  \( b(n) \),  \( p(n) \),  \( q(n) \) are integers with
\( O(\log n) \) bits. The most often used case is that  \( a(n) \), \( b(n) \),
\( p(n) \),  \( q(n) \) are polynomials in  \( n \) with integer coefficients.

\begin{description}
\item [Algorithm:]~
\end{description}

Given two index bounds  \( n_{1} \) and  \( n_{2} \), consider the partial sum 

\[
S=\sum _{n_{1}\leq n<n_{2}}
\frac{a(n)}{b(n)}\frac{p(n_{1})\cdots p(n)}{q(n_{1})\cdots q(n)}\]

It is not computed directly. Instead, we compute the integers
\( P={p(n_{1})}\cdots {p(n_{2}-1)} \),  \( Q={q(n_{1})}\cdots {q(n_{2}-1)} \),
\( B={b(n_{1})}\cdots {b(n_{2}-1)} \) and  \( T=BQS \). If  \( n_{2}-n_{1}<5 \),
these are computed directly.  If  \( n_{2}-n_{1}\geq 5 \), they are computed
using {\em binary splitting}: Choose an index  \( n_{m} \) in the middle of
\( n_{1} \) and \( n_{2} \), compute the components \( P_{l} \), \( Q_{l} \),
\( B_{l} \), \( T_{l} \) belonging to the interval \( n_{1}\leq n<n_{m} \),
compute the components  \( P_{r} \),  \( Q_{r} \),  \( B_{r} \),  \( T_{r} \)
belonging to the interval  \( n_{m}\leq n<n_{2} \), and set 
\( P=P_{l}P_{r} \),  \( Q=Q_{l}Q_{r} \),  \( B=B_{l}B_{r} \) and
\( T=B_{r}Q_{r}T_{l}+B_{l}P_{l}T_{r} \).

Finally, this algorithm is applied to  \( n_{1}=0 \) and
\( n_{2}=n_{\max }=O(N) \), and a final floating-point division
\( S=\frac{T}{BQ} \) is performed.

\begin{description}
\item [Complexity:]~
\end{description}

The bit complexity of computing  \( S \) with  \( N \) bits of precision is 
\( O((\log N)^{2}M(N)) \).

\begin{description}
\item [Proof:]~
\end{description}

Since we have assumed the series to be linearly convergent, the  \( n \)-th
term is  \( O(c^{-n}) \) with  \( c>1 \). Hence choosing
\( n_{\max }=N\frac{\log 2}{\log c}+O(1) \) will ensure that the round-off
error is  \( <2^{-N} \). By our assumption that  \( a(n) \),  \( b(n) \),
\( p(n) \),  \( q(n) \) are integers with  \( O(\log n) \) bits, the integers
\( P \),  \( Q \),  \( B \),  \( T \) belonging to the interval
\( n_{1}\leq n<n_{2} \) all have \( O((n_{2}-n_{1})\log n_{2}) \) bits.

The algorithm's recursion depth is  \( d=\frac{\log n_{\max }}{\log 2}+O(1) \).
At recursion depth  \( k \) (\( 1\leq k\leq d \)), integers having each
\( O(\frac{n_{\max }}{2^{k}}\log n_{\max }) \) bits are multiplied. Thus,
the entire computation time \( t \) is
\begin{eqnarray*}
t &=& \sum ^{d}_{k=1}
2^{k-1}O\left( M\left( \frac{n_{\max }}{2^{k}}\log n_{\max }\right)\right)\\
&=& \sum ^{d}_{k=1} O\left( M\left( n_{\max }\log n_{\max }\right) \right)\\
&=& O(\log n_{\max }M(n_{\max }\log n_{\max }))
\end{eqnarray*}
Because of  \( n_{\max }=O( \frac{N}{\log c}) \) and
\begin{eqnarray*}
M\left(\frac{N}{\log c} \log \frac{N}{\log c}\right)
&=& O\left(\frac{1}{\log c} N\, (\log N)^{2}\, \log \log N\right)\\
&=& O\left(\frac{1}{\log c} \log N\, M(N)\right)
\end{eqnarray*}
we have
\[ t = O\left(\frac{1}{\log c} (\log N)^{2}M(N)\right) \]
Considering \(c\) as constant, this is the desired result.

\begin{description}
\item [Checkpointing/Parallelising:]~
\end{description}

A checkpoint can be easily done by storing the (integer) values of
\( n_1 \),  \( n_2 \), \( P \),  \( Q \),  \( B \) and \( T \).
Similarly, if \( m \) processors are available, then the interval 
\( [0,n_{max}] \) can be divided into \( m \) pieces of length 
\( l = \lfloor n_{max}/m \rfloor \). After each processor \( i \) has
computed the sum of its interval \( [il,(i+1)l] \), the partial sums are
combined to the final result using the rules described above.

\begin{description}
\item [Note:]~
\end{description}

For the special case  \( a(n)=b(n)=1 \), the binary splitting algorithm has
already been documented in \cite{76a}, section 6, and \cite{87}, section 10.2.3.

Explicit computation of  \( P \),  \( Q \),  \( B \),  \( T \) is only required
as a recursion base, for  \( n_{2}-n_{1}<2 \), but avoiding recursions for
\( n_{2}-n_{1}<5 \) gains some percent of execution speed.

The binary splitting algorithm is asymptotically faster than step-by-step
evaluation of the sum -- which has binary complexity  \( O(N^{2}) \) -- because
it pushes as much multiplication work as possible to the region where
multiplication becomes efficient. If the multiplication were implemented
as an  \( M(N)=O(N^{2}) \) algorithm, the binary splitting algorithm would
provide no speedup over step-by-step evaluation.

\begin{description}
\item [Implementation:]~
\end{description}

In the following we present a simplified C++ implementation of
the above algorithm\footnote{A complete implementation can be found in 
CLN \cite{96a}. The implementation of the binary-splitting method will
be also available in {\sf LiDIA-1.4}}. The initialisation is done by a
structure {\tt abpq\_series} containing arrays {\tt a}, {\tt b}, {\tt p}
and {\tt q} of multiprecision integers ({\tt bigint}s). The values of
the arrays at the index \( n \) correspond to the values of the functions
\( a \), \( b \), \( p \) and \( q \) at the integer point \( n \).
The (partial) results of the algorithm are stored in the
{\tt abpq\_series\_result} structure.

\begin{verbatim}
// abpq_series is initialised by user
struct { bigint *a, *b, *p, *q; 
       } abpq_series;

// abpq_series_result holds the partial results
struct { bigint P, Q, B, T; 
       } abpq_series_result;

// binary splitting summation for abpq_series
void sum_abpq(abpq_series_result & r, 
              int n1, int n2, 
              const abpq_series & arg)
{
  // check the length of the summation interval
  switch (n2 - n1)
  {
    case 0:
      error_handler("summation device", 
               "sum_abpq:: n2-n1 should be > 0.");
      break;

    case 1: // the result at the point n1
      r.P = arg.p[n1];
      r.Q = arg.q[n1];
      r.B = arg.b[n1];
      r.T = arg.a[n1] * arg.p[n1];
      break;

    // cases 2, 3, 4 left out for simplicity

    default: // the general case

      // the left and the right partial sum
      abpq_series_result L, R;

      // find the middle of the interval
      int nm = (n1 + n2) / 2;

      // sum left side
      sum_abpq(L, n1, nm, arg);

      // sum right side
      sum_abpq(R, nm, n2, arg);

      // put together
      r.P = L.P * R.P;
      r.Q = L.Q * R.Q;
      r.B = L.B * R.B;
      r.T = R.B * R.Q * L.T + L.B * L.P * R.T;
      break;
  }
}
\end{verbatim}

Note that the multiprecision integers could be replaced here by integer
polynomials, or by any other ring providing the operators \( = \) (assignment),
\( + \) (addition) and \( * \) (multiplication). For example, one could regard
a bivariate polynomial over the integers as a series over the second variable,
with polynomials over the first variable as its coefficients. This would result
an accelerated algorithm for summing bivariate (and thus multivariate)
polynomials.

\subsection{Example: The factorial}

This is the most classical example of the binary splitting algorithm and was
probably known long before \cite{87}.

Computation of the factorial is best done using the binary splitting algorithm,
combined with a reduction of the even factors into odd factors and
multiplication with a power of 2, according to the formula

\[
n!=2^{n-\sigma _{2}(n)}\cdot \prod _{k\geq 1}
\left( \prod _{\frac{n}{2^{k}}<2m+1\leq \frac{n}{2^{k-1}}}(2m+1)\right) ^{k}\]
and where the products 
\[
P(n_{1},n_{2})=\prod _{n_{1}<m\leq n_{2}}(2m+1)\]
are evaluated according to the binary splitting algorithm:
\( P(n_{1},n_{2})=P(n_{1},n_{m})P(n_{m},n_{2}) \) with  
\( n_{m}=\left\lfloor \frac{n_{1}+n_{2}}{2}\right\rfloor  \) 
if  \( n_{2}-n_{1}\geq 5 \).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Example: Elementary functions at rational points}

The binary splitting algorithm can be applied to the fast computation of the
elementary functions at rational points  \( x=\frac{u}{v} \), simply 
by using the power series. We present how this can be done for
\( \exp (x) \), \( \ln (x) \), \( \sin (x) \), \( \cos (x) \),
\( \arctan (x) \), \( \sinh (x) \) and \( \cosh (x) \). The calculation
of other elementary functions is similar (or it can be reduced to the
calculation of these functions).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{\( \exp (x) \) for rational \( x \)}

This is a direct application of the above algorithm with  \( a(n)=1 \),
\( b(n)=1 \), \( p(0)=q(0)=1 \), and \( p(n)=u \), \( q(n)=nv \) for \( n>0 \).
Because the series is not only linearly convergent -- \( \exp (x) \) is an
entire function --,  \( n_{\max }=O(\frac{N}{\log N + \log \frac{1}{|x|}}) \),
hence the bit complexity is
\[ O\left(\frac{(\log N)^2}{\log N + \log \frac{1}{|x|}} M(N)\right) \]
Considering \(x\) as constant, this is  \( O(\log N\: M(N)) \).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{\( \exp (x) \) for real \( x \)}

This can be computed using the addition theorem for exp, by a trick due to
Brent \cite{76a} (see also \cite{87}, section 10.2, exercise 8). Write

\[
x=x_{0}+\sum _{k=0}^{\infty }\frac{u_{k}}{v_{k}}\]
with  \( x_{0} \) integer,  \( v_{k}=2^{2^{k}} \) and 
\( |u_{k}|<2^{2^{k-1}} \), and compute 
\[
\exp (x)=
\exp (x_{0})\cdot \prod _{k\geq 0}\exp \left( \frac{u_{k}}{v_{k}}\right) \]

This algorithm has bit complexity
\[ O\left(\sum\limits_{k=0}^{O(\log N)} \frac{(\log N)^2}{\log N + 2^k} M(N)\right)
  = O((\log N)^{2}M(N)) \]


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{ \( \ln (x) \) for rational  \( x \)}

For rational  \( |x-1|<1 \), the binary splitting algorithm can also be applied
directly to the power series for  \( \ln (x) \). Write  \( x-1=\frac{u}{v} \)
and compute the series with  \( a(n)=1 \),  \( b(n)=n+1 \),  \( q(n)=v \),
\( p(0)=u \), and  \( p(n)=-u \) for  \( n>0 \).

This algorithm has bit complexity  \( O((\log N)^{2}M(N)) \).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{\( \ln (x) \) for real \( x \)}

This can be computed using the ``inverse'' Brent trick:

Start with  \( y:=0 \).

As long as  \( x\neq 1 \) within the actual precision, choose  \( k \)
maximal with  \( |x-1|<2^{-k} \). Put \( z=2^{-2k}\left[ 2^{2k}(x-1)\right]  \),
i.e. let  \( z \) contain the first  \( k \) significant bits of  \( x-1 \).
\( z \) is a good approximation for  \( \ln (x) \). Set  \( y:=y+z \) and
\( x:=x\cdot \exp (-z) \).

Since  \( x\cdot \exp (y) \) is an invariant of the algorithm, the final
\( y \) is the desired value  \( \ln (x) \).

This algorithm has bit complexity
\[ O\left(\sum\limits_{k=0}^{O(\log N)} \frac{(\log N)^2}{\log N + 2^k} M(N)\right)
  = O((\log N)^{2}M(N)) \]


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{ \( \sin (x) \),  \( \cos (x) \) for rational  \( x \)}

These are direct applications of the binary splitting algorithm: For
\( \sin (x) \), put  \( a(n)=1 \),  \( b(n)=1 \),  \( p(0)=u \), 
\( q(0)=v \), and  \( p(n)=-u^{2} \),  \( q(n)=(2n)(2n+1)v^{2} \) for
\( n>0 \). For  \( \cos (x) \), put  \( a(n)=1 \),  \( b(n)=1 \),  
\( p(0)=1 \),  \( q(0)=1 \), and  \( p(n)=-u^{2} \),  \( q(n)=(2n-1)(2n)v^{2} \)
for  \( n>0 \). Of course, when both  \( \sin (x) \) and  \( \cos (x) \) are 
needed, one should only compute
 \( \sin (x) \) this way, and then set  
\( \cos (x)=\pm \sqrt{1-\sin (x)^{2}} \). This is a 20\% speedup at least.

The bit complexity of these algorithms is  \( O(\log N\: M(N)) \).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{ \( \sin (x) \),  \( \cos (x) \) for real  \( x \)}

To compute  \( \cos (x)+i\sin (x)=\exp (ix) \) for real  \( x \), again the 
addition theorems and Brent's trick
can be used. The resulting algorithm has bit complexity  
\( O((\log N)^{2}M(N)) \).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{ \( \arctan (x) \) for rational  \( x \)}

For rational  \( |x|<1 \), the fastest way to compute  \( \arctan (x) \) with 
bit complexity  \( O((\log N)^{2}M(N)) \) is
to apply the binary splitting algorithm directly to the power series
for  \( \arctan (x) \). Put  \( a(n)=1 \),  \( b(n)=2n+1 \),  \( q(n)=1 \),  
\( p(0)=x \) and  \( p(n)=-x^{2} \) for  \( n>0 \).


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{ \( \arctan (x) \) for real  \( x \)}

This again can be computed using the ``inverse'' Brent trick:

Start out with  \( z:=\frac{1}{\sqrt{1+x^{2}}}+i\frac{x}{\sqrt{1+x^{2}}} \) 
and  \( \varphi :=0 \). During the algorithm  \( z \) will be a complex number
with  \( |z|=1 \) and  \( \re (z)>0 \).

As long as  \( \im (z)\neq 0 \) within the actual precision, choose  \( k \) 
maximal with  \( |\im (z)|<2^{-k} \).
Put  \( \alpha =2^{-2k}\left[ 2^{2k}\im (z)\right]  \), i.e. let  \( \alpha  \) 
contain the first  \( k \) significant bits of  \( \im (z) \).  \( \alpha  \) 
is a good approximation for  \( \arcsin (\im (z)) \). Set  
\( \varphi :=\varphi +\alpha  \) and  \( z:=z\cdot \exp (-i\alpha ) \).

Since  \( z\cdot \exp (i\varphi ) \) is an invariant of the algorithm, the 
final  \( \varphi  \) is the desired
value  \( \arcsin \frac{x}{\sqrt{1+x^{2}}} \).

This algorithm has bit complexity
\[ O\left(\sum\limits_{k=0}^{O(\log N)} \frac{(\log N)^2}{\log N + 2^k} M(N)\right)
  = O((\log N)^{2}M(N)) \]


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{\( \sinh (x) \), \( \cosh (x) \) for rational and real \( x \)}

These can be computed by similar algorithms as  \( \sin (x) \) and  
\( \cos (x) \) above, with the same asymptotic bit complexity. The
standard computation, using \( \exp (x) \) and its reciprocal (calculated
by the Newton method) results also to the same complexity and works equally
well in practice.

The bit complexity of these algorithms is  \( O(\log N\: M(N)) \) for rational
\( x \) and \( O((\log N)^{2}M(N)) \) for real \( x \).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Example: Hypergeometric functions at rational points}

The binary splitting algorithm is well suited for the evaluation of a
hypergeometric series 

\[
F\left( \begin{array}{ccc}
a_{1}, & \ldots , & a_{r}\\
b_{1}, & \ldots , & b_{s}
\end{array}
\big| x\right) =\sum ^{\infty }_{n=0}
\frac{a_{1}^{\overline{n}}\cdots 
a_{r}^{\overline{n}}}{b_{1}^{\overline{n}}\cdots b_{s}^{\overline{n}}}x^{n}\]
with rational coefficients  \( a_{1} \), ...,  \( a_{r} \),  \( b_{1} \),
...,  \( b_{s} \) at a rational point  \( x \) in the interior of the circle of
convergence. Just put  \( a(n)=1 \),  \( b(n)=1 \),  \( p(0)=q(0)=1 \), and
\( \frac{p(n)}{q(n)}=\frac{(a_{1}+n-1)\cdots 
(a_{r}+n-1)x}{(b_{1}+n-1)\cdots (b_{s}+n-1)} \) for  \( n>0 \). The evaluation 
can thus be done with
bit complexity  \( O((\log N)^{2}M(N)) \) for  
\( r=s \) and  \( O(\log N\: M(N)) \) for  \( r<s \).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Example:  \( \pi  \)}

The Ramanujan series for  \( \pi  \) 
\[
\frac{1}{\pi }=\frac{12}{C^{3/2}}\sum^{\infty }_{n=0}
\frac{(-1)^n(6n)!(A+nB)}{(3n)!n!^{3}C^{3n}}\]
with  \( A=13591409 \),  \( B=545140134 \), \( C=640320 \) found by the
Chudnovsky's \footnote{A special case of \cite{87}, formula (5.5.18),
with N=163.} and which is used by the {\sf LiDIA} \cite{95,97,97b} and the Pari
\cite{95b} system to compute \( \pi \), is usually written as an algorithm
of bit complexity  \( O(N^{2}) \). It is, however, possible to apply
binary splitting to the sum. Put \( a(n)=A+nB \),  \( b(n)=1 \),
\( p(0)=1 \),  \( q(0)=1 \), and \( p(n)=-(6n-5)(2n-1)(6n-1) \),
\( q(n)=n^{3}C^3/24 \) for  \( n>0 \).  This reduces the complexity to 
\( O((\log N)^{2}M(N)) \). Although this is theoretically slower than
Brent-Salamin's quadratically convergent iteration, which has a bit
complexity of  \( O(\log N\: M(N)) \), in practice the binary splitted
Ramanujan sum is three times faster than Brent-Salamin, at least in the
range from  \( N=1000 \) bits to \( N=1000000 \) bits.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% \subsection{Example: Catalan's constant}

\subsection{Example: Catalan's constant \( G \)}

A linearly convergent sum for Catalan's constant 
\[
G:=\sum ^{\infty }_{n=0}\frac{(-1)^{n}}{(2n+1)^{2}}\]
is given in \cite{87}, p. 386:
\[
G = \frac{3}{8}\sum ^{\infty }_{n=0}\frac{1}{{2n \choose n} (2n+1)^{2}}
    +\frac{\pi }{8}\log (2+\sqrt{3})
\]

The series is summed using binary splitting, putting \( a(n)=1 \),  
\( b(n)=2n+1 \),  \( p(0)=1 \),  \( q(0)=1 \), and
\( p(n)=n \),  \( q(n)=2(2n+1) \) for  \( n>0 \). Thus  
\( G \) can be computed with bit complexity  \( O((\log N)^{2}M(N)) \).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Example: The Gamma function at rational points}

For evaluating  \( \Gamma (s) \) for rational  \( s \), we first reduce \( s \)
to the range \( 1\leq s\leq 2 \) by the formula \( \Gamma (s+1)=s\Gamma (s) \).
To compute  \( \Gamma (s) \) with a precision of  \( N \) bits, choose a
positive integer  \( x \) with  \( xe^{-x}<2^{-N} \). Partial integration lets
us write 

\begin{eqnarray*}
\Gamma (s)&=& \int ^{\infty }_{0}e^{-t}t^{s-1}dt\\
          &=& x^{s}e^{-x}\:\sum ^{\infty }_{n=0}
             \frac{x^{n}}{s(s+1)\cdots (s+n)}
             +\int^{\infty }_{x}e^{-t}t^{s-1}dt\\
\end{eqnarray*}
The last integral is  \( <xe^{-x}<2^{-N} \). The series is evaluated as a
hypergeometric function (see above); the number of terms to be summed up is
\( O(N) \), since \( x=O(N) \). Thus the entire computation can be done with
bit complexity  \( O((\log N)^{2}M(N)) \).

\begin{description}
\item [Note:]~
\end{description}

This result is already mentioned in \cite{76b}.

E.~Karatsuba \cite{91} extends this result to \( \Gamma (s) \) for algebraic
\( s \).

For \( \Gamma (s) \) there is no checkpointing possible because of the
dependency on \( x \) in the binary splitting.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Example: The Riemann Zeta value \( \zeta (3) \) \label{zeta}}

Recently, Doron Zeilberger's method of ``creative telescoping'' has been
applied to Riemann's zeta function at  \( s=3 \) (see \cite{96c}), which is
also known as {\em Ap{\'e}ry's constant}: 
\[
\zeta (3)=
\frac{1}{2}\sum ^{\infty }_{n=1}
               \frac{(-1)^{n-1}(205n^{2}-160n+32)}{n^{5}{2n \choose n}^{5}}
\]

This sum consists of three hypergeometric series. Binary splitting can also be
applied directly, by putting  \( a(n)=205n^{2}+250n+77 \), \( b(n)=1 \),
\( p(0)=1 \), \( p(n)=-n^{5} \) for  \( n>0 \), and \( q(n)=32(2n+1)^{5} \).
Thus the bit complexity of computing \( \zeta (3) \) is
\( O((\log N)^{2}M(N)) \).

\begin{description}
\item [Note:]~
\end{description}

Using this the authors were able to establish a new record in the
calculation of \( \zeta (3) \) by computing 1,000,000 decimals \cite{96d}.
The computation took 8 hours on a Hewlett Packard 9000/712 machine. After
distributing on a cluster of 4 HP 9000/712 machines the same computation
required only 2.5 hours. The half hour was necessary for reading the partial
results from disk and for recombining them. Again, we have used binary-splitting
for recombining: the 4 partial result produced 2 results which were combined
to the final 1,000,000 decimals value of \( \zeta (3) \).

This example shows the importance of checkpointing. Even if a machine crashes
through the calculation, the results of the other machines are still usable.
Additionally, being able to parallelise the computation reduced the computing
time dramatically.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Evaluation of linearly convergent series of sums}

The technique presented in the previous section also applies to all linearly
convergent sums of the form

\[
U=\sum ^{\infty }_{n=0}\frac{a(n)}{b(n)}\left( \frac{c(0)}{d(0)}+\cdots 
+\frac{c(n)}{d(n)}\right) \frac{p(0)\cdots p(n)}{q(0)\cdots q(n)}\]
 where  \( a(n) \),  \( b(n) \),  \( c(n) \),  \( d(n) \),  \( p(n) \),  
\( q(n) \) are integers with  \( O(\log n) \) bits. The most often
used case is again that  \( a(n) \),  \( b(n) \),  \( c(n) \),  \( d(n) \),  
\( p(n) \),  \( q(n) \) are polynomials in  \( n \) with
integer coefficients.

\begin{description}
\item [Algorithm:]~
\end{description}

Given two index bounds  \( n_{1} \)and  \( n_{2} \), consider the partial sums 
\[
S=\sum _{n_{1}\leq n<n_{2}}\frac{a(n)}{b(n)}
\frac{p(n_{1})\cdots p(n)}{q(n_{1})\cdots q(n)}\]
and 
\[
U=\sum _{n_{1}\leq n<n_{2}}\frac{a(n)}{b(n)}
\left( \frac{c(n_{1})}{d(n_{1})}+\cdots +\frac{c(n)}{d(n)}\right) 
\frac{p(n_{1})\cdots p(n)}{q(n_{1})\cdots q(n)}\]


As above, we compute the integers  \( P={p(n_{1})}\cdots {p(n_{2}-1)} \),  
\( Q={q(n_{1})}\cdots {q(n_{2}-1)} \),  \( B={b(n_{1})}\cdots {b(n_{2}-1)} \),  
\( T=BQS \),  \( D={d(n_{1})}\cdots {d(n_{2}-1)} \),  
\( C=D\left( \frac{c(n_{1})}{d(n_{1})}+\cdots +
\frac{c(n_{2}-1)}{d(n_{2}-1)}\right)  \) and  \( V=DBQU \). 
If  \( n_{2}-n_{1}<4 \), these
are computed directly. If  \( n_{2}-n_{1}\geq 4 \), they are computed using 
{\em binary splitting}: Choose an index  \( n_{m} \) in the middle of
\( n_{1} \)and  \( n_{2} \), compute the components \( P_{l} \),  \( Q_{l} \),
\( B_{l} \),  \( T_{l} \),  \( D_{l} \),  \( C_{l} \),  \( V_{l} \) belonging
to the interval  \( n_{1}\leq n<n_{m} \), compute the components \( P_{r} \),
\( Q_{r} \),  \( B_{r} \),  \( T_{r} \),  \( D_{r} \),  \( C_{r} \),  
\( V_{r} \) belonging to the interval  \( n_{m}\leq n<n_{2} \), and set  
\( P=P_{l}P_{r} \),  \( Q=Q_{l}Q_{r} \),  \( B=B_{l}B_{r} \), 
\( T=B_{r}Q_{r}T_{l}+B_{l}P_{l}T_{r} \),  \( D=D_{l}D_{r} \),  
\( C=C_{l}D_{r}+C_{r}D_{l} \) and  
\( V=D_{r}B_{r}Q_{r}V_{l}+D_{r}C_{l}B_{l}P_{l}T_{r}+D_{l}B_{l}P_{l}V_{r} \).

Finally, this algorithm is applied to \( n_{1}=0 \) and 
\( n_{2}=n_{\max}=O(N) \), and final floating-point divisions 
\( S=\frac{T}{BQ} \) and  \( U=\frac{V}{DBQ} \) are performed.

\begin{description}
\item [Complexity:]~
\end{description}

The bit complexity of computing  \( S \) and  \( U \) with  \( N \) bits of
precision is \( O((\log N)^{2}M(N)) \).

\begin{description}
\item [Proof:]~
\end{description}

By our assumption that  \( a(n) \),  \( b(n) \),  \( c(n) \),  \( d(n) \),  
\( p(n) \),  \( q(n) \) are integers with  \( O(\log n) \) bits,
the integers  \( P \),  \( Q \),  \( B \),  \( T \),  \( D \),  \( C \),  
\( V \) belonging to the interval  \( n_{1}\leq n<n_{2} \) all have
 \( O((n_{2}-n_{1})\log n_{2}) \) bits. The rest of the proof is as in the 
previous section.

\begin{description}
\item [Checkpointing/Parallelising:]~
\end{description}

A checkpoint can be easily done by storing the (integer) values of
\( n_1 \),  \( n_2 \), \( P \),  \( Q \),  \( B \), \( T \) and additionally
\( D \),  \( C \),  \( V \). Similarly, if \( m \) processors are available,
then the interval \( [0,n_{max}] \) can be divided into \( m \) pieces of
length \( l = \lfloor n_{max}/m \rfloor \). After each processor \( i \) has
computed the sum of its interval \( [il,(i+1)l] \), the partial sums are
combined to the final result using the rules described above.

\begin{description}
\item [Implementation:]~
\end{description}

The C++ implementation of the above algorithm is very similar
to the previous one. The initialisation is done now by a structure 
{\tt abpqcd\_series} containing arrays {\tt a}, {\tt b}, {\tt p},
{\tt q}, {\tt c} and {\tt d} of multiprecision integers. The values of
the arrays at the index \( n \) correspond to the values of the functions
\( a \), \( b \), \( p \), \( q \), \( c \) and \( d \) at the integer point
\( n \). The (partial) results of the algorithm are stored in the
{\tt abpqcd\_series\_result} structure, which now contains 3 new elements
({\tt C}, {\tt D} and {\tt V}).

\begin{verbatim}
// abpqcd_series is initialised by user
struct { bigint *a, *b, *p, *q, *c, *d; 
       } abpqcd_series;

// abpqcd_series_result holds the partial results
struct { bigint P, Q, B, T, C, D, V; 
       } abpqcd_series_result;

void sum_abpqcd(abpqcd_series_result & r, 
                int n1, int n2, 
                const abpqcd_series & arg)
{
  switch (n2 - n1)
  {
    case 0:
      error_handler("summation device", 
            "sum_abpqcd:: n2-n1 should be > 0.");
      break;

    case 1: // the result at the point n1
      r.P = arg.p[n1];
      r.Q = arg.q[n1];
      r.B = arg.b[n1];
      r.T = arg.a[n1] * arg.p[n1];
      r.D = arg.d[n1];
      r.C = arg.c[n1];
      r.V = arg.a[n1] * arg.c[n1] * arg.p[n1];
      break;

    // cases 2, 3, 4 left out for simplicity

    default: // general case
     
      // the left and the right partial sum
      abpqcd_series_result L, R;

      // find the middle of the interval
      int nm = (n1 + n2) / 2;
 
      // sum left side
      sum_abpqcd(L, n1, nm, arg);

      // sum right side
      sum_abpqcd(R, nm, n2, arg);

      // put together
      r.P = L.P * R.P;
      r.Q = R.Q * L.Q;
      r.B = L.B * R.B;
      bigint tmp = L.B * L.P * R.T;
      r.T = R.B * R.Q * L.T + tmp;
      r.D = L.D * R.D;
      r.C = L.C * R.D + R.C * L.D;
      r.V = R.D * (R.B * R.Q * L.V + L.C * tmp) 
            + L.D * L.B * L.P * R.V;
      break;
  }
}
\end{verbatim}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Example: Euler's constant \( C \) \label{eulergamma}}

\begin{description}
\item [Theorem:]~
\end{description}

Let  \( f(x)=\sum ^{\infty }_{n=0}\frac{x^{n}}{n!^{2}} \) and  
\( g(x)=\sum ^{\infty }_{n=0}H_{n}\frac{x^{n}}{n!^{2}} \). Then for  
\( x\rightarrow \infty  \),  
\( \frac{g(x)}{f(x)}=\frac{1}{2}\log x+C+O\left( e^{-4\sqrt{x}}\right)  \).

\begin{description}
\item [Proof:]~
\end{description}

The Laplace method for asymptotic evaluation of exponentially growing
sums and integrals yields 
\[
f(x)=
e^{2\sqrt{x}}x^{-\frac{1}{4}}\frac{1}{2\sqrt{\pi }}(1+O(x^{-\frac{1}{4}}))\]
 and 
\[
g(x)=e^{2\sqrt{x}}x^{-\frac{1}{4}}\frac{1}{2\sqrt{\pi }}
\left(\frac{1}{2}\log x+C+O(\log x\cdot x^{-\frac{1}{4}})\right)\]
On the other hand,  \( h(x):=\frac{g(x)}{f(x)} \) satisfies the
differential equation 
\[
xf(x)\cdot h''(x)+(2xf'(x)+f(x))\cdot h'(x)=f'(x)\]
hence 
\[
h(x)=\frac{1}{2}\log x+C+c_{2}
\int ^{\infty }_{x}\frac{1}{tf(t)^{2}}dt=\frac{1}{2}\log x+C+O(e^{-4\sqrt{x}})\]


\begin{description}
\item [Algorithm:]~
\end{description}

To compute  \( C \) with a precision of  \( N \) bits, set  
\[ x=\left\lceil (N+2)\: \frac{\log 2}{4}\right\rceil ^{2} \]
and evaluate the series for  \( g(x) \) and  \( f(x) \) simultaneously,
using the binary-splitting algorithm,
with  \( a(n)=1 \),  \( b(n)=1 \),  \( c(n)=1 \),  \( d(n)=n+1 \),  
\( p(n)=x \),  \( q(n)=(n+1)^{2} \). Let  \( \alpha =3.591121477\ldots  \) 
be the solution of the equation  \( -\alpha \log \alpha +\alpha +1=0 \). Then
 \( \alpha \sqrt{x}-\frac{1}{4\log \alpha }\log \sqrt{x}+O(1) \) 
terms of the series suffice for the relative error to be bounded
by  \( 2^{-N} \).

\begin{description}
\item [Complexity:]~
\end{description}

The bit complexity of this algorithm is  \( O((\log N)^{2}M(N)) \).

\begin{description}
\item [Note:]~
\end{description}

This algorithm was first mentioned in \cite{80}. It is by far
the fastest known algorithm for computing Euler's constant. 

For Euler's constant there is no checkpointing possible because 
of the dependency on \( x \) in the binary splitting.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational results}

In this section we present some computational results of our CLN and
{\sf LiDIA} implementation of the algorithms presented in this note. We use
the official version (1.3) and an experimental version (1.4a) of {\sf LiDIA}.
We have taken advantage of {\sf LiDIA}'s ability to replace its kernel
(multiprecision arithmetic and memory management) \cite{95,97,97b}, so we were
able to use in both cases CLN's fast integer arithmetic routines.

\subsection{Timings}

The table in Figure \ref{Fig1} shows the running times for the calculation of
\( \exp(1) \), \( \log(2) \), \( \pi \), \( C \), \( G \) and \( \zeta(3) \)
to precision 100, 1000, 10000 and 100000 decimal digits. The timings are given
in seconds and they denote the {\em real} time needed, i.e. system and user
time. The computation was done on an Intel Pentium with 133Hz and 32MB of RAM.

\begin{figure}[htb]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
D      &\( \exp(1) \)&\( \log(2) \)&\( \pi \)&\( C \)  &\( G \)&\( \zeta(3) \)\\
\hline
\hline
\( 10^2 \) &0.0005   & 0.0020      &0.0014   & 0.0309  &0.0179 & 0.0027 \\
\hline
\( 10^3 \) &0.0069   & 0.0474      &0.0141   & 0.8110  &0.3580 & 0.0696 \\
\hline
\( 10^4 \) &0.2566   & 1.9100      &0.6750   & 33.190  &13.370 & 2.5600 \\
\hline
\( 10^5 \) &5.5549   & 45.640      &17.430   & 784.93  &340.33 & 72.970 \\
\hline
\end{tabular}
\caption{{\sf LiDIA-1.4a} timings of computation of constants using
binary-splitting}\label{Fig1}
\end{center}
\end{figure}

The second table (Figure \ref{Fig2}) summarizes the performance of
\( exp(x) \) in various Computer Algebra systems\footnote{We do not list
the timings of {\sf LiDIA-1.4a} since these are comparable to those of CLN.}.
For a fair comparison of the algorithms, both argument and precision are
chosen in such a way, that system--specific optimizations (BCD arithmetic
in Maple, FFT multiplication in CLN, special exact argument handling in 
{\sf LiDIA}) do not work. We use \( x = -\sqrt{2} \) and precision 
\( 10^{(i/3)} \), with \( i \) running from \( 4 \) to \( 15 \).

\begin{figure}[htb]
\begin{center}
\begin{tabular}{|r|l|l|l|l|}
\hline
D           &  Maple & Pari & {\sf LiDIA-1.3} & CLN \\
\hline
\hline
21          & 0.00090   & 0.00047 & 0.00191           & 0.00075 \\
\hline
46          & 0.00250   & 0.00065 & 0.00239           & 0.00109 \\
\hline
100         & 0.01000   & 0.00160 & 0.00389           & 0.00239 \\
\hline
215         & 0.03100   & 0.00530 & 0.00750           & 0.00690 \\
\hline
464         & 0.11000   & 0.02500 & 0.02050           & 0.02991 \\
\hline
1000        & 0.4000    & 0.2940  & 0.0704            & 0.0861 \\
\hline
2154        & 1.7190    & 0.8980  & 0.2990            & 0.2527 \\
\hline
4641        & 8.121     & 5.941   & 1.510             & 0.906 \\
\hline
10000       & 39.340    & 39.776  & 7.360             & 4.059 \\
\hline
21544       & 172.499   & 280.207 & 39.900            & 15.010 \\
\hline
46415       & 868.841   & 1972.184& 129.000           & 39.848 \\
\hline
100000      & 4873.829  & 21369.197& 437.000           & 106.990 \\
\hline
\end{tabular}
\caption{Timings of computation of \( \exp(-\sqrt{2}) \)}\label{Fig2}
\end{center}
\end{figure}

MapleV R3 is the slowest system in this comparison. This is probably due to
the BCD arithmetic it uses. However, Maple seems to have an asymptotically
better algorithm for \( exp (x) \) for numbers having more than 10000 decimals.
In this range it outperforms Pari-1.39.03, which is the fastest system in the
0--200 decimals range.

The comparison indicating the strength of binary-splitting is between
{\sf LiDIA-1.3} and CLN itself. Having the same kernel, the only
difference is here that {\sf LiDIA-1.3} uses Brent's \( O(\sqrt{n}M(n)) \)
for \( \exp(x) \), whereas CLN changes from Brent's method to a
binary-splitting version for large numbers.

As expected in the range of 1000--100000 decimals CLN outperforms
{\sf LiDIA-1.3} by far. The fact that {\sf LiDIA-1.2.1} is faster
in the range of 200--1000 decimals (also in some trig. functions)
is probably due to a better optimized \( O(\sqrt{n}M(n)) \) method
for \( \exp(x) \).

\subsection {Distributed computing of \( \zeta (3) \)}

Using the method described in \ref{zeta} the authors were the first to 
compute 1,000,000 decimals of \( \zeta (3) \) \cite{96d}.
The computation took 8 hours on a Hewlett Packard 9000/712 machine. After
distributing on a cluster of 4 HP 9000/712 machines the same computation
required only 2.5 hours. The half hour was necessary for reading the partial
results from disk and for recombining them. Again, we have used binary-splitting
for recombining: the 4 partial result produced 2 results which were combined
to the final 1,000,000 decimals value of \( \zeta (3) \).

This example shows the importance of checkpointing. Even if a machine crashes
through the calculation, the results of the other machines are still usable.
Additionally, being able to parallelise the computation reduced the computing
time dramatically.

\subsection{Euler's constant \( C \)}

We have implemented a version of Brent's and McMillan's algorithm \cite{80} and
a version accelerated by binary-splitting as shown in \ref{eulergamma}.

The computation of \( C \) was done twice on a SPARC-Ultra machine
with 167 MHz and 256 MB of RAM. The first computation using the non-acellerated
version required 160 hours. The result of this computation was then verified
by the binary splitting version in (only) 14 hours.

The first 475006 partial quotients of the continued fraction of \( C \)
were computed on an Intel Pentium with 133 MHz and 32 MB of RAM in 3 hours
using a programm by H. te Riele based on \cite{96e}, which was translated to
{\sf LiDIA} for efficiency reasons. Computing the 475006th
convergent produced the following improved theorem:

\medskip

\centerline{If \( C \) is a rational number, \(C=p/q\), then \( |q| > 10^{244663} \)}

\medskip

Details of this computation (including statistics on the partial 
quotients) can be found in \cite{98}.


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\section{Conclusions}

Although powerful, the binary splitting method has not been widely used.
Especially, no information existed on the applicability of this method.

In this note we presented a generic binary-splitting summation device for
evaluating two types of linearly convergent series. From this we derived simple
and computationally efficient algorithms for the evaluation of elementary
functions and constants. These algorithms work with {\em exact}
objects, making them suitable for use within Computer Algebra systems.

We have shown that the practical performance of our algorithms is
superior to current system implementations. In addition to existing methods,
our algorithms provide the possibility of checkpointing and parallelising.
These features can be useful for huge calculations, such as those done in
analytic number theory research.

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\section{Thanks}

The authors would like to thank J\"org Arndt, for pointing us to
chapter 10 in \cite{87}. We would also like to thank Richard P. Brent for 
his comments and Hermann te Riele for providing us his program for the
continued fraction computation of Euler's constant.

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