[Bug math/13942] x86 acos inaccurate near 1

["bugdal at aerifal dot cx" <sourceware-bugzilla at sourceware dot org>]

http://sourceware.org/bugzilla/show_bug.cgi?id=13942

--- Comment #3 from Rich Felker <bugdal at aerifal dot cx> 2012-04-04 14:13:00 UTC ---
Hi Joseph,

I did some preliminary tests, and that formula does not seem to fix it.

Given the following, which has been written in C to use extended precision for
intermediate results:

  double x = 0x0.ffffffffffffp0;
  printf("%.17g\n", acos(x));
  printf("%.17g\n", (double)atanl(sqrtl(1-(long double)x*x)));
  printf("%.17g\n", (double)atanl(sqrtl((1-(long double)x)*(1+(long
double)x))));

I'm getting:

8.4293697021788083e-08
8.4293697021787858e-08
8.4293697021787791e-08

The first result (our current implementation in musl) agrees with Wolfram
Alpha's value for acos(0.999999999999996447286321199499070644378662109375).
Both of the latter results have relatively huge error. So unless I screwed up
something in doing the intermediate results at extended precision, I don't
think your solution works.

Note that our algorithm avoids performing any multiplication or division that
would lose precision. The only loss of precision takes place at the sqrt
operations (which will be correctly rounded for extended precision) and the
atan operation.

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