the second step of approximating cbrt(x). It turns out to be neither
very magic not nor very good. It is just the (2,2) Pade approximation
to 1/cbrt(r) at r = 1, arranged in a strange way to use fewer operations
at a cost of replacing 4 multiplications by 1 division, which is an
especially bad tradeoff on machines where some of the multiplications
can be done in parallel. A Remez rational approximation would give
at least 2 more bits of accuracy, but the (2,2) Pade approximation
already gives 6 more bits than needed. (Changed the comment which
essentially says that it gives 3 more bits.)
Lower order Pade approximations are not quite accurate enough for
double precision but are plenty for float precision. A lower order
Remez rational approximation might be enough for double precision too.
However, rational approximations inherently require an extra division,
and polynomial approximations work well for 1/cbrt(r) at r = 1, so I
plan to switch to using the latter. There are some technical
complications that tend to cost a division in another way.
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