If we wanted to be fully precised, for values close to 1, we could
actually evaluate sqrt(delta * (2 - delta)) with delta = 1 - something
then
>>> '%.17g' % math.sqrt(1e-7 * (2 - 1e-7))
'0.00044721358431961788'
which matches quite perfectly the full precision result.
Well, revisiting that, the above would be correct if "delta = 1 -
something" would be evaluated with full precision... But in reality, it
doesn't, so:
if something = 1 - 1e-7,
delta = 1 - something = 9.9999999947364415e-08
and
'%.17g' % math.sqrt(9.9999999947364415e-08 * (2 - 9.9999999947364415e-08))
= 0.00044721358420192118
which is not better than using the naive sqrt(1 - x * x)
So my pj_cos_of_asin(x) could simplify be simplified as sqrt(1 - x * x),
possibly with being tolerant for input values in [1, 1 +
EPSILON_FOR_SLIGHTLY_OUT_OF_DOMAIN] or [-1 -
EPSILON_FOR_SLIGHTLY_OUT_OF_DOMAIN, -1] as we do in aasin() (in
src/aasincos.cpp), although I don't know if the use cases where authalic
lat --> geographic lat is involved we can get those slightly out of
range values, due to rounding errors in previous computations.
--
http://www.spatialys.com
My software is free, but my time generally not.
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