Take a look at POSIX::frexp(). It seems to do the right thing all the way to the 53-bit limit of doubles. It gets a little flakey thereafter:

printf "-1:%2d %30f:%d +1:%2d\n", 
    (frexp( $_ - 1 ))[1], $_, (frexp( $_ ))[1], (frexp( $_ +1 ))[1] 
    for map 2**$_, 1 .. 64;;
-1: 1                       2.000000:2 +1: 2
-1: 2                       4.000000:3 +1: 3
-1: 3                       8.000000:4 +1: 4
-1: 4                      16.000000:5 +1: 5
-1: 5                      32.000000:6 +1: 6
-1: 6                      64.000000:7 +1: 7
-1: 7                     128.000000:8 +1: 8
-1: 8                     256.000000:9 +1: 9
-1: 9                     512.000000:10 +1:10
-1:10                    1024.000000:11 +1:11
-1:11                    2048.000000:12 +1:12
-1:12                    4096.000000:13 +1:13
-1:13                    8192.000000:14 +1:14
-1:14                   16384.000000:15 +1:15
-1:15                   32768.000000:16 +1:16
-1:16                   65536.000000:17 +1:17
-1:17                  131072.000000:18 +1:18
-1:18                  262144.000000:19 +1:19
-1:19                  524288.000000:20 +1:20
-1:20                 1048576.000000:21 +1:21
-1:21                 2097152.000000:22 +1:22
-1:22                 4194304.000000:23 +1:23
-1:23                 8388608.000000:24 +1:24
-1:24                16777216.000000:25 +1:25
-1:25                33554432.000000:26 +1:26
-1:26                67108864.000000:27 +1:27
-1:27               134217728.000000:28 +1:28
-1:28               268435456.000000:29 +1:29
-1:29               536870912.000000:30 +1:30
-1:30              1073741824.000000:31 +1:31
-1:31              2147483648.000000:32 +1:32
-1:32              4294967296.000000:33 +1:33
-1:33              8589934592.000000:34 +1:34
-1:34             17179869184.000000:35 +1:35
-1:35             34359738368.000000:36 +1:36
-1:36             68719476736.000000:37 +1:37
-1:37            137438953472.000000:38 +1:38
-1:38            274877906944.000000:39 +1:39
-1:39            549755813888.000000:40 +1:40
-1:40           1099511627776.000000:41 +1:41
-1:41           2199023255552.000000:42 +1:42
-1:42           4398046511104.000000:43 +1:43
-1:43           8796093022208.000000:44 +1:44
-1:44          17592186044416.000000:45 +1:45
-1:45          35184372088832.000000:46 +1:46
-1:46          70368744177664.000000:47 +1:47
-1:47         140737488355328.000000:48 +1:48
-1:48         281474976710656.000000:49 +1:49
-1:49         562949953421312.000000:50 +1:50
-1:50        1125899906842624.000000:51 +1:51
-1:51        2251799813685248.000000:52 +1:52
-1:52        4503599627370496.000000:53 +1:53
-1:53        9007199254740992.000000:54 +1:54
-1:55       18014398509481984.000000:55 +1:55
-1:56       36028797018963968.000000:56 +1:56
-1:57       72057594037927936.000000:57 +1:57
-1:58      144115188075855870.000000:58 +1:58
-1:59      288230376151711740.000000:59 +1:59
-1:60      576460752303423490.000000:60 +1:60
-1:61     1152921504606847000.000000:61 +1:61
-1:62     2305843009213694000.000000:62 +1:62
-1:63     4611686018427387900.000000:63 +1:63
-1:64     9223372036854775800.000000:64 +1:64
-1:65    18446744073709552000.000000:65 +1:65
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