In part 2 we showed that the position of the implicit bit i was also the width of the biased exponent. That makes the bias i-1 bits wide. Since all bits of the bias are 1, the value of the bias is 2i-1-1. With one bit for the sign, that leaves n-i-1 bits for the width of the significand. By specifying i as a function of n the storage format becomes fully defined. This works not only for eight-bit boundaries but for the six-bit boundaries of Color My Data eight-bit compressed binary format (CBF8). CBF8 is designed for loss-less data exchange using binary formats. One base-64 digit could represent plus or minus sixteenths between 0 and 0.9375; eighths between 1 and 1.875; fourths between 2 and 3.75, infinity, quiet NaN or NaN with up to two signals. That represents a potential compression of 87.5% over double precision WITH NO LOSS OF DATA.
The graph below shows how i varies with n in comparison to the current standard.
The graph was generated from the following tables. Here are the column definitions.
bytes
bits
sign
biasExp
significand
precision
bias
dynRange
1
8
1
3
4
1.20
3
1.81
2
16
1
5
10
3.01
15
9.03
3
24
1
7
16
4.82
63
37.93
4
32
1
8
23
6.92
127
76.46
5
40
1
9
30
9.03
255
153.53
6
48
1
10
37
11.14
511
307.65
7
56
1
10
45
13.55
511
307.65
8
64
1
11
52
15.65
1023
615.91
9
72
1
11
60
18.06
1023
615.91
10
80
1
12
67
20.17
2047
1232.42
11
88
1
12
75
22.58
2047
1232.42
12
96
1
13
82
24.68
4095
2465.44
13
104
1
13
90
27.09
4095
2465.44
14
112
1
14
97
29.20
8191
4931.47
15
120
1
14
105
31.61
8191
4931.47
16
128
1
15
112
33.72
16383
9863.55
digits
bits
sign
biasExp
significand
precision
bias
dynRange
1
6
1
2
3
0.90
1
0.60
2
12
1
4
7
2.11
7
4.21
3
18
1
5
12
3.61
15
9.03
4
24
1
7
16
4.82
63
37.93
5
30
1
8
21
6.32
127
76.46
6
36
1
8
27
8.13
127
76.46
7
42
1
9
32
9.63
255
153.53
8
48
1
10
37
11.14
511
07.65
9
54
1
10
43
12.94
511
307.65
10
60
1
11
48
14.45
1023
615.91
11
66
1
11
54
16.26
1023
615.91
12
72
1
11
60
18.06
1023
615.91
13
78
1
12
65
19.57
2047
1232.42
14
84
1
12
71
21.37
2047
1232.42
15
90
1
13
76
22.88
4095
2465.44
16
96
1
13
82
24.68
4095
2465.44
17
102
1
13
88
26.49
4095
2465.44
18
108
1
14
93
28.00
8191
4931.47
19
114
1
14
99
29.80
8191
4931.47
20
120
1
14
105
31.61
8191
4931.47
21
126
1
15
110
33.11
16383
9863.55
22
132
1
15
116
34.92
16383
9863.55