The limit of the mantissa is the ULP. For a double-precision float typically used in perl, it's 2**-52 times the main power of two for the number:

perl -MData::IEEE754::Tools=:all -le"$,=qq(\t); map { print $_, ulp($_
+) } @ARGV" 0.25 0.4 0.5 0.6 0.9 1 1.9 2

0.25    5.55111512312578e-017
0.4     5.55111512312578e-017
0.5     1.11022302462516e-016
0.6     1.11022302462516e-016
0.9     1.11022302462516e-016
1       2.22044604925031e-016
1.9     2.22044604925031e-016
2       4.44089209850063e-016

[download]

This can also be seen in your 1+1/2**x by printing the underlying IEEE754, or by using a powers-of-two representation of the float, or by always using %.16f

perl -MData::IEEE754::Tools=:all -e"printf qq(%-3d %-32.32s 0x%-16s %.
+16f %s\n), $_, 1+1/2**$_, hexstr754_from_double(1+1/2**$_), 1+1/2**$_
+, to_dec_floatingpoint(1+1/2**$_) for @ARGV" 47 48 49 50 51 52 53 54 
+55 56

47  1.00000000000001                 0x3FF0000000000020 1.000000000000
+0071 +0d1.0000000000000071p+0000
48  1                                0x3FF0000000000010 1.000000000000
+0036 +0d1.0000000000000036p+0000
49  1                                0x3FF0000000000008 1.000000000000
+0018 +0d1.0000000000000018p+0000
50  1                                0x3FF0000000000004 1.000000000000
+0009 +0d1.0000000000000009p+0000
51  1                                0x3FF0000000000002 1.000000000000
+0004 +0d1.0000000000000004p+0000
52  1                                0x3FF0000000000001 1.000000000000
+0002 +0d1.0000000000000002p+0000
53  1                                0x3FF0000000000000 1.000000000000
+0000 +0d1.0000000000000000p+0000
54  1                                0x3FF0000000000000 1.000000000000
+0000 +0d1.0000000000000000p+0000
55  1                                0x3FF0000000000000 1.000000000000
+0000 +0d1.0000000000000000p+0000
56  1                                0x3FF0000000000000 1.000000000000
+0000 +0d1.0000000000000000p+0000

[download]

for numbers from 0.5 to 0.9999..., it's one better:

perl -MData::IEEE754::Tools=:all -e"printf qq(%-3d %-32.32s 0x%-16s %.
+16f %s\n), $_, 0.5+1/2**$_, hexstr754_from_double(0.5+1/2**$_), 0.5+1
+/2**$_, to_dec_floatingpoint(0.5+1/2**$_) for @ARGV" 47 48 49 50 51 5
+2 53 54 55 56
47  0.500000000000007                0x3FE0000000000040 0.500000000000
+0071 +0d1.0000000000000142p-0001
48  0.500000000000004                0x3FE0000000000020 0.500000000000
+0036 +0d1.0000000000000071p-0001
49  0.500000000000002                0x3FE0000000000010 0.500000000000
+0018 +0d1.0000000000000036p-0001
50  0.500000000000001                0x3FE0000000000008 0.500000000000
+0009 +0d1.0000000000000018p-0001
51  0.5                              0x3FE0000000000004 0.500000000000
+0004 +0d1.0000000000000009p-0001
52  0.5                              0x3FE0000000000002 0.500000000000
+0002 +0d1.0000000000000004p-0001
53  0.5                              0x3FE0000000000001 0.500000000000
+0001 +0d1.0000000000000002p-0001
54  0.5                              0x3FE0000000000000 0.500000000000
+0000 +0d1.0000000000000000p-0001
55  0.5                              0x3FE0000000000000 0.500000000000
+0000 +0d1.0000000000000000p-0001
56  0.5                              0x3FE0000000000000 0.500000000000
+0000 +0d1.0000000000000000p-0001

perl -MData::IEEE754::Tools=:all -e"printf qq(%-3d %-32.32s 0x%-16s %.
+16f %s\n), $_, 1.0-1/2**$_, hexstr754_from_double(1-1/2**$_), 1-1/2**
+$_, to_dec_floatingpoint(1-1/2**$_) for @ARGV" 47 48 49 50 51 52 53 5
+4 55 56
47  0.999999999999993                0x3FEFFFFFFFFFFFC0 0.999999999999
+9929 +0d1.9999999999999858p-0001
48  0.999999999999996                0x3FEFFFFFFFFFFFE0 0.999999999999
+9965 +0d1.9999999999999929p-0001
49  0.999999999999998                0x3FEFFFFFFFFFFFF0 0.999999999999
+9982 +0d1.9999999999999964p-0001
50  0.999999999999999                0x3FEFFFFFFFFFFFF8 0.999999999999
+9991 +0d1.9999999999999982p-0001
51  1                                0x3FEFFFFFFFFFFFFC 0.999999999999
+9996 +0d1.9999999999999991p-0001
52  1                                0x3FEFFFFFFFFFFFFE 0.999999999999
+9998 +0d1.9999999999999996p-0001
53  1                                0x3FEFFFFFFFFFFFFF 0.999999999999
+9999 +0d1.9999999999999998p-0001
54  1                                0x3FF0000000000000 1.000000000000
+0000 +0d1.0000000000000000p+0000
55  1                                0x3FF0000000000000 1.000000000000
+0000 +0d1.0000000000000000p+0000
56  1                                0x3FF0000000000000 1.000000000000
+0000 +0d1.0000000000000000p+0000

[download]

In reply to Re^6: inclusive rand by pryrt
in thread inclusive rand by msh210

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