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.
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