Use 'long double' for x,y,z arguments

[dstndstn]

• use cosl,sinl for radec_to_healpixlf
• compute 1-vz in long double

[astrofrog]

Looks good to me - I don't have time to check whether this fully fixes #34, so I'll leave it up to you whether to merge or wait until we can confirm this is fixed.

[cdeil]

There's a bit of a concern with this, in the sense that it probably won't fix the issue for all users: some will end up with only "double" precision still (see https://en.wikipedia.org/wiki/Long_double).

Even worse would be if there's a distribution problem where binaries (e.g. Windows from conda-forge) don't work at all for some users (e.g. some Windows machines); I don't know if that's a real concern.

I would suggest we merge tomorrow, and continue with more edits and checks in the next days.

Unless there is a more stable way to do that computation using only double.
@mreinicke - do you know if there is a trick to get this numerically stable?

[mreineck]

Hmm, as far as I understand, the problematic term is sqrt(1.-vz).

Wouldn't it be possible to rewrite this as sqrt((vx*vx+vy*vy)/(1.+vz)) ?

That should be more accurate.

[mreineck]

Going to long double is almost never a good idea, because it doesn't increase accuracy on many platforms. In most cases there is a way to rewrite the problematic expressions in a more stable fashion.

[dstndstn]

In this case it's the sin(Dec) that drops precision; Dec->90 degrees so sin(Dec)->1

[mreineck]

Yes, that's why I think it should be safer to use cos(Dec) (i.e. sqrt(x*x + y*y)) instead of sin(Dec) (i.e. z) in this situation.

I see similar code in other places in the package... the z2dec function in starutil.inc for example will fail badly near the poles. I'd recommend something like

double xyz2dec(double x, double y, double z)
  {
  return M_PI/2 - atan2(sqrt(x*x+y*y), z);
  }

It's not much slower, and it works extremely reliably.

[dstndstn]

I tried this in #94 - thanks for the tip! - seems to work in our one polar test case

[cdeil]

@mreineck @dstndstn - Thank you!
Merging #94 now and closing this one without merging.