I can actually find one a lot longer, using the structure. Start with ou = 1 + ULP => 1 + 2**-52 -- that's the farthest separation between two active bits possible in a double. Then, shift it as far to the right of the decimal point as you can: x = ou * 2**-1022 = 2**-1022 + 2**-1074. The MPFR shows 303 digits. But I don't think that's all the digits. I actually expect 2**-1022 to take 1022 digits (including leading zeros, not including the "0.") after a fixed decimal point. And 2**-1074 should take 1074 digits (including leading zeros) after a fixed decimal point, so the sum of those two should really go to the 10**-1074th place. Since it starts at the 308th digit after the fixed decmial point, I actually expect 765 digits.
`use warnings;
use strict;
use Data::IEEE754::Tools 0.016 qw/:all/;
use Math::MPFR qw/:mpfr/;
use POSIX qw/floor ceil log10 log2/;
Rmpfr_set_default_prec(1000);
# easy to come up with one that takes many digits to express:
my $x = nextUp( POS_NORM_SMALLEST() );
print "POS_NORM_SMALLEST = 2**-1022\n";
print "POS_NORM_SMALLEST+1ULP = (1 + 2**-52)*(2**-1022)\n";
print "\n";
print "( 1 ) \n";
print "(1 + -----)*(2**-1022)\n";
print "( 2**52) \n";
print "\n";
printf "IEEE754 Hex String: '%s'\n", hexstr754_from_double($x);
printf "%-30a %-30s %-30s\n", $x, to_hex_floatingpoint($x), to_dec_flo
+atingpoint($x);
my $s=Math::MPFR->new($x);
printf "[%d digits long]: %s\n", length($s)-6, $s;
my $d = ceil(-log10($x)); # start digit
printf "For _FIXED POINT_ notation,\n";
printf "\tstarts %d digits after the decimal point\n", $d;
my $ulp = ulp($x); # value of last bit (whether or not it is
+set)
printf "long version of ulp: %s\n", Math::MPFR->new($ulp);
my $l2 = log2($ulp);
printf "log2(ulp) = %d, so I expect it to fit within %d digits of the
+fixed decimal point\n", $l2, -$l2;
`
`__OUTPUT__
POS_NORM_SMALLEST = 2**-1022
POS_NORM_SMALLEST+1ULP = (1 + 2**-52)*(2**-1022)
( 1 )
(1 + -----)*(2**-1022)
( 2**52)
IEEE754 Hex String: '0010000000000001'
0x1.0000000000001p-1022 +0x1.0000000000001p-1022 +0d1.000
+0000000000002p-1022
[303 digits long]: 2.2250738585072018771558785585789482407880088486837
+041956131300312119688603996006965297904292212628858639037013670281908
+017171296072711910355127227413175152199055740043138804567803233377539
+881639177387328959246074229270113078053813397081653361296447449529789
+5212189790907838525833659018517896187998851504e-308
For _FIXED POINT_ notation,
starts 308 digits after the decimal point
`
Since I don't know MPFR, I tried to look at it using Math::BigFloat, but I couldn't get it to stop prematurely rounding at about 30 digits after the fixed point, even with accuracy(1000) and precision(-1000). Apparently, I don't know BigFloat very well, either. I'm pretty busy the rest of the weekend, but I know what my coffee breaks at work next week are going to include: writing my own simplistic string-based fixed-point bigs, that can simply add two bigs, or divide a big by two, without all the overhead of modules I don't understand that seem to be rounding prematurely. |
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