cvs commit: src/lib/msun/src e_rem_pio2f.c

[Luigi Rizzo]

wow now this is a commit message!!!

On Tue, Oct 25, 2005 at 12:13:37PM +0000, Bruce Evans wrote:
> bde         2005-10-25 12:13:37 UTC
> 
>   FreeBSD src repository
> 
>   Modified files:
>     lib/msun/src         e_rem_pio2f.c 
>   Log:
>   More fixes for arg reduction near pi/2 on systems with broken assignment
>   to floats (mainly i386's).  All errors of more than 1 ulp for float
>   precision trig functions were supposed to have been fixed; however,
>   compiling with gcc -O2 uncovered 18250 more such errors for cosf(),
>   with a maximum error of 1.409 ulps.
>   
>   Use essentially the same fix as in rev.1.8 of k_rem_pio2f.c (access a
>   non-volatile variable as a volatile).  Here the -O1 case apparently
>   worked because the variable is in a 2-element array and it takes -O2
>   to mess up such a variable by putting it in a register.
>   
>   The maximum error for cosf() on i386 with gcc -O2 is now 0.5467 (it
>   is still 0.5650 with gcc -O1).  This shows that -O2 still causes some
>   extra precision, but the extra precision is now good.
>   
>   Extra precision is harmful mainly for implementing extra precision in
>   software.  We want to represent x+y as w+r where both "+" operations
>   are in infinite precision and r is tiny compared with w.  There is a
>   standard algorithm for this (Knuth (1981) 4.2.2 Theorem C), and fdlibm
>   uses this routinely, but the algorithm requires w and r to have the
>   same precision as x and y.  w is just x+y (calculated in the same
>   finite precision as x and y), and r is a tiny correction term.  The
>   i386 gcc bugs tend to give extra precision in w, and then using this
>   extra precision in the calculation of r results in the correction
>   mostly staying in w and being missing from r.  There still tends to
>   be no problem if the result is a simple expression involving w and r
>   -- modulo spills, w keeps its extra precision and r remains the right
>   correction for this wrong w.  However, here we want to pass w and r
>   to extern functions.  Extra precision is not retained in function args,
>   so w gets fixed up, but the change to the tiny r is tinier, so r almost
>   remains as a wrong correction for the right w.
>   
>   Revision  Changes    Path
>   1.12      +8 -5      src/lib/msun/src/e_rem_pio2f.c