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

[Bruce Evans]

bde         2005-11-30 04:56:49 UTC

  FreeBSD src repository

  Modified files:
    lib/msun/src         e_expf.c 
  Log:
  Fixed the hi+lo approximation to log(2).  The normal 17+24 bit decomposition
  that was used doesn't work normally here, since we want to be able to
  multiply `hi' by the exponent of x _exactly_, and the exponent of x has
  more than 7 significant bits for most denormal x's, so the multiplication
  was not always exact despite a cloned comment claiming that it was.  (The
  comment is correct in the double precision case -- with the normal 33+53
  bit decomposition the exponent can have 20 significant bits and the extra
  bit for denormals is only the 11th.)
  
  Fixing this had little or no effect for denormals (I think because
  more precision is inherently lost for denormals than is lost by roundoff
  errors in the multiplication).
  
  The fix is to reduce the precision of the decomposition to 16+24 bits.
  Due to 2 bugs in the old deomposition and numerical accidents, reducing
  the precision actually increased the precision of hi+lo.  The old hi+lo
  had about 39 bits instead of at least 41 like it should have had.
  There were off-by-1-bit errors in each of hi and lo, apparently due
  to mistranslation from the double precision hi and lo.  The correct
  16 bit hi happens to give about 19 bits of precision, so the correct
  hi+lo gives about 43 bits instead of at least 40.  The end result is
  that expf() is now perfectly rounded (to nearest) except in 52561 cases
  instead of except in 67027 cases, and the maximum error is 0.5013 ulps
  instead of 0.5023 ulps.
  
  Revision  Changes    Path
  1.10      +4 -4      src/lib/msun/src/e_expf.c