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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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