can act as a 53-bit signed integer for some purposes.

To be even more pedantic... :-)

...that would be a 53 bit unsigned integer, or a 54-bit sign and magnitude signed integer (using one bit for the sign); equivalent to a 54-bit 1's complement signed integer.

Since 1's complement machines are as rare as rocking horse fumet, one might worry about the -0x80....0 minimum value. Happily that is quite comfortably accommodated in floating point form.

The -0 value is generally treated as zero, so that shouldn't intrude.

So, for practical purposes a 64-bit IEEE-754 double precision floating point value can act as a 54-bit signed integer (with the usual 2's complement -ve values).

So:

  for my $n (53..55) {
    my $max =   2**($n-1) - 1 ;
    my $min = -(2**($n-1)) ;

    printf "%4d: -%s..%s\n", $n, show($min), show($max) ;
  } ;

  sub show {
    my ($x) = @_ ;

    my $ms = int(abs($x)/2**32) ;
    my $ls = abs($x) - ($ms * 2**32) ;

    sprintf '0x%04X_%04X_%04X_%04X', ($ms >> 16), ($ms & 0xFFFF),
                                     ($ls >> 16), ($ls & 0xFFFF) ;
  } ;

[download]
gives:
    53: -0x0010_0000_0000_0000..0x000F_FFFF_FFFF_FFFF
    54: -0x0020_0000_0000_0000..0x001F_FFFF_FFFF_FFFF
    55: -0x0040_0000_0000_0000..0x0040_0000_0000_0000
showing that 53-bit (2's complement) signed integer is on the small side, 55-bit is to big, but 54-bits is just right.

For extra credit, those with 64-bit integers can consider (a) why:

  my $z = 0x0FED_CBA9_8765_4321 ;
  my $d = 9 ;

  printf " a = %20s * %d + i, for i = 0..%d\n", "$z", $d, $d ;
         # 12: 12345678901234567890 : +2345
  print  "  i:                a / $d : Error\n" ;
 
  foreach my $i (0..$d) {
    my $a = ($z * $d) + $i ;
    my $q = $a / $d ;

    printf " %2d: %20s : %+5d\n", $i, "$q", $q - ($z + int($i/$d)) ;
  } ;

[download]
gives:
 a =  1147797409030816545 * 9 + i, for i = 0..9
  i:                a / 9 : Error
  0:  1147797409030816545 :    +0
  1: 1.14779740903082e+18 :  +128
  2: 1.14779740903082e+18 :  +128
  3: 1.14779740903082e+18 :  +128
  4: 1.14779740903082e+18 :  +128
  5: 1.14779740903082e+18 :  +128
  6: 1.14779740903082e+18 :  +128
  7: 1.14779740903082e+18 :  +128
  8: 1.14779740903082e+18 :  +128
  9:  1147797409030816546 :    +0
and (b) whether this could be fixed, and finally (c) whether round-to-even is a Good Thing, with particular reference to this case.

Bug #53784 refers.

In reply to Re^5: 64-bit digest algorithms by gone2015
in thread 64-bit digest algorithms by BrowserUk

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