cram-results.tex

Transmutagen is able to compute the CRAM approximation for any degree $n$ to
any number of decimal digits $d$. We have computed the first $60$ degrees to
200 digits. Our digits agree with the coefficients published
in~\cite{carpenter1984extended}, which has the first 30 degrees up to 20
digits. It should be noted that the digits in~\cite{carpenter1984extended}
have been rounded using decimal rounding, which is often different from
floating point, or binary rounding. For example, the constant coefficient in
the numerator ($p_0$) for degree 11 is reported as
\texttt{1.0000000000146631119}, even though the true coefficient,
\texttt{1.000000000014663111949374871\ldots} is more closely approximated as a
binary floating point number by \texttt{1.0000000000146631120}. Our computed
digits agree with those from~\cite{carpenter1984extended} when rounded using
the default half-even strategy of the Python \texttt{decimal} module (SymPy's
\texttt{Float} objects, which are used in the calculations from
Section~\ref{sec:remez-algorithm}, use binary rounding). This is important
because for practical purposes, when using the computed coefficients to
compute a CRAM approximation on a computer, one should round the digits to a
machine floating point number using binary rounding, to get the floating point
number that most closely approximates the true value. With that being said,
the difference is typically no more than a single bit, and the 20 digits
reported by~\cite{carpenter1984extended} are more than what can fit in a
standard \texttt{double} machine float.