a declaration was not translated to "float" although bit fiddling on
double variables was translated. This resulted in garbage being put
into the low word of one of the doubles instead of non-garbage being
put into the only word of the intended float. This had no effect on
any result because:
- with doubles, the algorithm for calculating -1/(x+y) is unnecessarily
complicated. Just returning -1/((double)x+y) would work, and the
misdeclaration gave something like that except for messing up some
low bits with the bit fiddling.
- doubles have plenty of bits to spare so messing up some of the low
bits is unlikely to matter.
- due to other bugs, the buggy code is reached for a whole 4 args out
of all 2**32 float args. The bug fixed by 1.8 only affects a small
percentage of cases and a small percentage of 4 is 0. The 4 args
happen to cause no problems without 1.8, so they are even less likely
to be affected by the bug in 1.8 than average args; in fact, neither
1.8 nor this commit makes any difference to the result for these 4
args (and thus for all args).
Corrections to the log message in 1.8: the bug only applies to tan()
and not tanf(), not because the float type can't represent numbers
large enough to trigger the problem (e.g., the example in the fdlibm-5.3
readme which is > 1.0e269), but because:
- the float type can't represent small enough numbers. For there to be
a possible problem, the original arg for tanf() must lie very near an
odd multiple of pi/2. Doubles can get nearer in absolute units. In
ulps there should be little difference, but ...
- ... the cutoff for "small" numbers is bogus in k_tanf.c. It is still
the double value (2**-28). Since this is 32 times smaller than
FLT_EPSILON and large float values are not very uniformly distributed,
only 6 args other than ones that are initially below the cutoff give
a reduced arg that passes the cutoff (the 4 problem cases mentioned
above and 2 non-problem cases).
Fixing the cutoff makes the bug affect tanf() and much easier to detect
than for tan(). With a cutoff of 2**-12 on amd64 with -O1, 670102
args pass the cutoff; of these, there are 337604 cases where there
might be an error of >= 1 ulp and 5826 cases where there is such an
error; the maximum error is 1.5382 ulps.
The fix in 1.8 works with the reduced cutoff in all cases despite the
bug in it. It changes the result in 84492 cases altogether to fix the
5826 broken cases. Fixing the fix by translating "double" to "float"
changes the result in 42 cases relative to 1.8. In 24 cases the
(absolute) error is increased and in 18 cases it is reduced, but it
remains less than 1 ulp in all cases.
d568fc134a