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** 7. New Formulas for Pi and Related Constants**
Many readers may be already familiar with the recent paper by the
authors and others which gives a new method for computing individual
hexadecimal digits of , as well as a number of other constants.
A brief review of these results is as follows. First, let us ask
whether satisfy a relation of the form

where are rational numbers? Indeed it does. Such a formula can
be found by separating the right hand side of the above expression
into eight summations, numerically evaluating these to high precision,
appending the numerical value of , and applying PSLQ to the
resulting 9-long vector. When this is done, PSLQ discovers the
following formula:
A similar formula was discovered by Ferguson shortly after the
discovery of the above formula. In fact, there is a two-dimensional
lattice of such formulas, which lattice can be generated by these two
formulas.
The significance of these formulas for the computation of can be
seen as follows. Let be the first of the sums in the above
formula for . Then we can write

The first sum can be rapidly evaluated by means of the binary
algorithm for exponentiation, where each operation is performed modulo
the integer **8k + 1**. These calculations can be done with either
integer or floating-point arithmetic, provided the format being used
has enough accuracy to exactly represent the integer . Once an
individual exponentiation operation is complete, the resulting integer
value is divided by **8 k + 1**, using floating-point arithmetic, and
added to the sum modulo 1. Only a few terms are required of the
second, since the terms rapidly become smaller than the ``machine
epsilon'' of the floating-point arithmetic system being used. The
resulting fractional value, when expressed in base 16 notation, gives
the hexadecimal digits of beginning at position **d + 1**.
Here are a number of other formulas of this type. As before, these
formulas were originally found using PSLQ searches.
Full details of these calculations, as well as formal proofs of the
above formulas, can be found in [5].

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