<s>
All	O
of	O
the	O
Machin-like	B-Algorithm
formulas	I-Algorithm
can	O
be	O
derived	O
by	O
repeated	O
application	O
of	O
equation	O
.	O
</s>
<s>
Since	O
the	O
arctangent	O
of	O
one	O
has	O
a	O
very	O
slow	O
convergence	O
rate	O
if	O
we	O
find	O
two	O
complex	O
numbers	O
that	O
when	O
multiplied	O
will	O
result	O
in	O
the	O
same	O
real	O
and	O
imaginary	O
part	O
we	O
will	O
have	O
a	O
Machin-like	B-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
obtain	O
the	O
Lehmer	O
's	O
measure	O
as	O
small	O
as	O
possible	O
,	O
it	O
is	O
necessary	O
to	O
decrease	O
the	O
ratio	O
of	O
positive	O
integers	O
in	O
the	O
arctangent	O
arguments	O
and	O
to	O
minimize	O
the	O
number	O
of	O
the	O
terms	O
in	O
the	O
Machin-like	B-Algorithm
formula	I-Algorithm
.	O
</s>
<s>
Nowadays	O
at	O
the	O
smallest	O
known	O
Lehmer	O
's	O
measure	O
is	O
due	O
to	O
H	O
.	O
Chien-Lih	O
(	O
1997	O
)	O
,	O
whose	O
Machin-like	B-Algorithm
formula	I-Algorithm
is	O
shown	O
below	O
.	O
</s>
<s>
The	O
calculation	O
was	O
performed	O
on	O
a	O
64-node	O
Hitachi	O
supercomputer	B-Architecture
with	O
1	O
terabyte	O
of	O
main	O
memory	O
,	O
performing	O
2	O
trillion	O
operations	O
per	O
second	O
.	O
</s>
<s>
Machin-like	B-Algorithm
formulas	I-Algorithm
for	O
can	O
be	O
constructed	O
by	O
finding	O
a	O
set	O
of	O
numbers	O
where	O
the	O
prime	O
factorisations	O
of	O
together	O
use	O
no	O
more	O
distinct	O
primes	O
than	O
the	O
number	O
of	O
elements	O
in	O
the	O
set	O
,	O
and	O
then	O
using	O
either	O
linear	O
algebra	O
or	O
the	O
LLL	O
basis-reduction	O
algorithm	O
to	O
construct	O
linear	O
combinations	O
of	O
arctangents	O
of	O
reciprocals	O
of	O
integer	O
denominators	O
.	O
</s>
<s>
The	O
most	O
efficient	O
currently	O
known	O
Machin-like	B-Algorithm
formula	I-Algorithm
for	O
computing	O
is	O
:	O
</s>
<s>
There	O
are	O
further	O
methods	O
to	O
derive	O
Machin-like	B-Algorithm
formulas	I-Algorithm
for	O
with	O
reciprocals	O
of	O
integers	O
.	O
</s>
<s>
For	O
large	O
computations	O
of	O
,	O
the	O
binary	B-Algorithm
splitting	I-Algorithm
algorithm	I-Algorithm
can	O
be	O
used	O
to	O
compute	O
the	O
arctangents	O
much	O
,	O
much	O
more	O
quickly	O
than	O
by	O
adding	O
the	O
terms	O
in	O
the	O
Taylor	O
series	O
naively	O
one	O
at	O
a	O
time	O
.	O
</s>
<s>
In	O
practical	O
implementations	O
such	O
as	O
y-cruncher	O
,	O
there	O
is	O
a	O
relatively	O
large	O
constant	O
overhead	O
per	O
term	O
plus	O
a	O
time	O
proportional	O
to	O
,	O
and	O
a	O
point	O
of	O
diminishing	O
returns	O
appears	O
beyond	O
three	O
or	O
four	O
arctangent	O
terms	O
in	O
the	O
sum	O
;	O
this	O
is	O
why	O
the	O
supercomputer	B-Architecture
calculation	O
above	O
used	O
only	O
a	O
four-term	O
version	O
.	O
</s>
