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
> 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