cvs commit: src/lib/msun/src s_log1pf.c

[Bruce Evans]

bde         2005-12-04 12:30:44 UTC

  FreeBSD src repository

  Modified files:
    lib/msun/src         s_log1pf.c 
  Log:
  For log1pf(), fixed the approximations to sqrt(2), sqrt(2)-1 and
  sqrt(2)/2-1.  For log1p(), fixed the approximation to sqrt(2)/2-1.
  
  The end result is to fix an error of 1.293 ulps in
      log1pf(0.41421395540 (hex 0x3ed413da))
  and an error of 1.783 ulps in
      log1p(-0.292893409729003961761) (hex 0x12bec4 00000001)).
  The former was the only error of > 1 ulp for log1pf() and the latter
  is the only such error that I know of for log1p().
  
  The approximations don't need to be very accurate, but the last 2 need
  to be related to the first and be rounded up a little (even more than
  1 ulp for sqrt(2)/2-1) for the following implementation-detail reason:
  when the arg (x) is not between (the approximations to) sqrt(2)/2-1
  and sqrt(2)-1, we commit to using a correction term, but we only
  actually use it if 1+x is between sqrt(2)/2 and sqrt(2) according to
  the first approximation. Thus we must ensure that
  !(sqrt(2)/2-1 < x < sqrt(2)-1) implies !(sqrt(2)/2 < x+1 < sqrt(2)),
  where all the sqrt(2)'s are really slightly different approximations
  to sqrt(2) and some of the "<"'s are really "<="'s.  This was not done.
  
  In log1pf(), the last 2 approximations were rounded up by about 6 ulps
  more than needed relative to a good approximation to sqrt(2), but the
  actual approximation to sqrt(2) was off by 3 ulps.  The approximation
  to sqrt(2)-1 ended up being 4 ulps too small, so the algoritm was
  broken in 4 cases.  The result happened to be broken in 1 case.  This
  is fixed by using a natural approximation to sqrt(2) and derived
  approximations for the others.
  
  In logf(), all the approximations made sense, but the approximation
  to sqrt(2)/2-1 was 2 ulps too small (a tiny amount, since we compare
  with a granularity of 2**32 ulps), so the algorithm was broken in 2
  cases.  The result was broken in 1 case.  This is fixed by rounding
  up the approximation to sqrt(2)/2-1 by 2**32 ulps, so 2**32 cases are
  now handled a little differently (still correctly according to my
  assertion that the approximations don't need to be very accurate, but
  this has not been checked).
  
  Revision  Changes    Path
  1.9       +11 -4     src/lib/msun/src/s_log1pf.c