With this small tool, the coefficients for polynomial approximation
used in kernels can be generated.

Usage

Edit gencoefdp.c. In the beginning of the file, specifications of the
parameters for generating coefficients are listed. Enable one of them
by changing #if. Then, run make to compile the source code. Run the
gencoef, and it will show the generated coefficients in a few minutes.


How it works

There are two phases of the program.

The first phase is the regression for minimizing the maximum relative
error. This problem can be reduced to a linear programming problem,
and the Simplex method is used in this implementation. This requires
multi-precision calculation, and the implementation uses the MPFR
library to do this. In this phase, only a small number of values
(specified by S macro, usually 40 or so) of the function to
approximate are sampled within the argument range. The function to
approximate can be given by FRFUNC function. Specifying higher values
for S does not always give better results.

The second phase is to optimize the coefficients so that it gives good
accuracy with double precision calculation. In this phase, it checks
100000 points (specified by Q macro) within the specified argument
range to see if the polynomial gives good error bound. In some cases,
the last few terms have to be calculated in higher precision in order
to achieve 1 ULP overall accuracy, and this implementation can take
care of that. The L parameter specifies the number of high precision
coefficients.

In some cases, it is desirable to fix the last few coefficients to
values like 1. This can be specified if you define FIXCOEF0
macro. This sometimes does not work, however. In this case, you need
to specify the function to approximate as shown in the definition for
cos.

Finding a set of good parameters is not a straightforward process. You
usually need many iterations of trial and error.