On Wed, 10 Nov 2010, Bruce Evans wrote:
On Tue, 9 Nov 2010, Ulrich [utf-8] Sp??rlein wrote:
FreeBSD -CURRENT:
freebsd64# cc -o testjn -lm testjn.c && ./testjn
2 4.317548e-01
3 1.989999e-01
4 6.474667e-02
5 1.638924e-02
6 3.404818e-03
7 6.006884e-04
8 9.216579e-05
9 1.251727e-05
FreeBSD agrees perfectly with pari. This is not surprising, since only
7 significant decimal digits are printed and almost 15 should be correct
-- any error here would be an error of about 100 million decimal ulps
or a couple of binary gulps.
freebsd64# cc -o testjnf -lm testjnf.c && ./testjnf
2 4.317548e-01
3 1.989999e-01
4 6.474666e-02
5 1.638924e-02
6 3.404818e-03
7 6.006881e-04
8 9.216577e-05
9 1.251727e-05
This is harder to explain. Now we should only expect 6 digits of accuracy,
but we are getting 7. Anyway, these are all perfectly rounded according to
pari.
Actually we are getting 6. The last digit differs from the double precision
case for args 4, 7 and 8. This is as expected.
[pari output]
Single precision with bad decimal field width:
2 4.317548e-1
3 1.989999e-1
4 6.474667e-2
5 1.638924e-2
6 3.404818e-3
7 6.006884e-4
8 9.216579e-5
9 1.251727e-5
Unfortunately, the correctly rounded (in single precision) result gives
(after further rounding to 1 too many decimal digits) values that agree
with the double precision C output and thus disagree with the single
precision C output. Therefore, the C results in single precision cannot
be correctly rounded.
I may be confused or have a bug somewhere -- manually rounding the
above single precision results to 6 decimal digits gives the same
values in all cases, but FLT_DIG = 6 is supposed to give enough digits
for preserving a single precision result if it is printed and scanned
back. When I started replying to this, I thought that extra decimal
digits were needed (why is DECIMAL_DIG = 21 3 more than LDBL_DIG = 18
on i386...?), and printed 3 extra. I left these out to simplify
comparisons.
Bruce
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