The continued logarithm of π

Briefly, a continued logarithm is an arbitrarily long bit string approximating a real number arbitrarily well, and supports arbitrarily precise bit-at-a-time algorithms for rational functions of these numbers. (See http://www.tweedledum.com/rwg/cfup.htm , p 47+)

A six-bits-per-term π series:

In[]:=
ProductMatrixForm@512k
3
,0,32k
3
(-37+42k),(-1+2k)
3
,{k,∞}MatrixForm[{{ooπ,0},{oo,"?"}}]
Out[]=
∞
∏
k

512k
3
0
32k
3
(42k-37)
(2k-1)
3

ooπ
0
oo
?


Where ∏ means matrix product, not Mathematica product, and oo is some quantity that blows up with the number of product terms, the same way you compute continued fractions with 2×2 matrices. An incorrect form of this series is derived in https://dspace.mit.edu/handle/1721.1/6088 .
We initialize the work matrix m to the first term of the matrix product:

In[]:=
MatrixFormm=512k
3
,0,32k
3
(-37+42k),(-1+2k)
3
/.k1
Out[]//MatrixForm=

512
0
160
1


This represents the function

In[]:=
Divide@@(m.{t,1})
Out[]=
512t
1+160t

where t is the tail of the series, starting with k=2 rather than 1. It should be easy to show that, in general,

In[]:=
3/8/k<t<3/8/(k+1)
Out[]=
3
8k
<t<
3
8(1+k)

giving bounds

m/.{{t3/8/1.},{t3/8/2.}}
Out[]=
{3.14754,3.09677}

Since these both exceed 2, we commence output with

In[]:=
Style[cl@π={1},22]
Out[]=
{1}

and left multiply m by the divide-by-2 matrix:

In[]:=
MatrixForm[m={{1,0},{0,2}}.m]
Out[]//MatrixForm=

512
0
320
2


It costs almost nothing to remove the common power of 2:

In[]:=
MatrixForm[m=m/2]
Out[]//MatrixForm=

256
0
160
1


representing the function

In[]:=
Divide@@(m.{t,1})
Out[]=
256t
1+160t

We still are on input term k=1, and can still use

In[]:=
%/.{{t3/8/1.},{t3/8/2.}}
Out[]=
{1.57377,1.54839}

which is smack between 1 and 2, which we celebrate with

In[]:=
Style[AppendTo[cl@π,0],22]
Out[]=
{1,0}

and left multiply m by the subtract-1-and-reciprocate matrix:

In[]:=
(m={{0,1},{1,-1}}.m)//MatrixForm
Out[]//MatrixForm=

160
1
96
-1


representing

In[]:=
Divide@@(m.{t,1})
Out[]=
1+160t
-1+96t

Again using

In[]:=
%/.{{t3/8/1.},{t3/8/2.}}
Out[]=
{1.74286,1.82353}

which dictates another

In[]:=
Style[AppendTo[cl@π,0],22]
Out[]=
{1,0,0}

and

In[]:=
(m={{0,1},{1,-1}}.m)//MatrixForm
Out[]//MatrixForm=

96
-1
64
2


representing