<s>
The	O
Bailey	B-Algorithm
–	I-Algorithm
Borwein	I-Algorithm
–	I-Algorithm
Plouffe	I-Algorithm
formula	I-Algorithm
(	O
BBP	B-Algorithm
formula	I-Algorithm
)	O
is	O
a	O
formula	O
for	O
.	O
</s>
<s>
The	O
BBP	B-Algorithm
formula	I-Algorithm
gives	O
rise	O
to	O
a	O
spigot	B-Algorithm
algorithm	I-Algorithm
for	O
computing	O
the	O
nth	O
base-16	O
(	O
hexadecimal	O
)	O
digit	O
of	O
(	O
and	O
therefore	O
also	O
the	O
4nth	O
binary	O
digit	O
of	O
)	O
without	O
computing	O
the	O
preceding	O
digits	O
.	O
</s>
<s>
Formulas	O
of	O
this	O
form	O
are	O
known	O
as	O
BBP-type	B-Algorithm
formulas	I-Algorithm
.	O
</s>
<s>
A	O
few	O
manipulations	O
are	O
required	O
to	O
implement	O
a	O
spigot	B-Algorithm
algorithm	I-Algorithm
using	O
this	O
formula	O
.	O
</s>
<s>
Now	O
,	O
for	O
a	O
particular	O
value	O
of	O
n	O
and	O
taking	O
the	O
first	O
sum	O
,	O
we	O
split	O
the	O
sum	O
to	O
infinity	B-Application
across	O
the	O
nth	O
term	O
:	O
</s>
<s>
While	O
there	O
are	O
some	O
carries	B-Algorithm
that	O
are	O
not	O
counted	O
,	O
computers	O
usually	O
perform	O
arithmetic	O
for	O
many	O
bits	O
(	O
32	O
or	O
64	O
)	O
and	O
round	O
,	O
and	O
we	O
are	O
only	O
interested	O
in	O
the	O
most	O
significant	O
digit(s )	O
.	O
</s>
<s>
Though	O
the	O
BBP	B-Algorithm
formula	I-Algorithm
can	O
directly	O
calculate	O
the	O
value	O
of	O
any	O
given	O
digit	O
of	O
with	O
less	O
computational	O
effort	O
than	O
formulas	O
that	O
must	O
calculate	O
all	O
intervening	O
digits	O
,	O
BBP	O
remains	O
linearithmic	O
(	O
)	O
,	O
whereby	O
successively	O
larger	O
values	O
of	O
n	O
require	O
increasingly	O
more	O
time	O
to	O
calculate	O
;	O
that	O
is	O
,	O
the	O
"	O
further	O
out	O
"	O
a	O
digit	O
is	O
,	O
the	O
longer	O
it	O
takes	O
BBP	O
to	O
calculate	O
it	O
,	O
just	O
like	O
the	O
standard	O
-computing	O
algorithms	O
.	O
</s>
<s>
D	O
.	O
J	O
.	O
Broadhurst	O
provides	O
a	O
generalization	O
of	O
the	O
BBP	B-Algorithm
algorithm	I-Algorithm
that	O
may	O
be	O
used	O
to	O
compute	O
a	O
number	O
of	O
other	O
constants	O
in	O
nearly	O
linear	O
time	O
and	O
logarithmic	O
space	O
.	O
</s>
