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ea1f8f68 | 1 | Long double format |
2 | ================== | |
3 | ||
4 | Each long double is made up of two IEEE doubles. The value of the | |
5 | long double is the sum of the values of the two parts (except for | |
6 | -0.0). The most significant part is required to be the value of the | |
7 | long double rounded to the nearest double, as specified by IEEE. For | |
8 | Inf values, the least significant part is required to be one of +0.0 | |
9 | or -0.0. No other requirements are made; so, for example, 1.0 may be | |
10 | represented as (1.0, +0.0) or (1.0, -0.0), and the low part of a NaN | |
11 | is don't-care. | |
12 | ||
13 | Classification | |
14 | -------------- | |
15 | ||
16 | A long double can represent any value of the form | |
17 | s * 2^e * sum(k=0...105: f_k * 2^(-k)) | |
18 | where 's' is +1 or -1, 'e' is between 1022 and -968 inclusive, f_0 is | |
19 | 1, and f_k for k>0 is 0 or 1. These are the 'normal' long doubles. | |
20 | ||
21 | A long double can also represent any value of the form | |
22 | s * 2^-968 * sum(k=0...105: f_k * 2^(-k)) | |
23 | where 's' is +1 or -1, f_0 is 0, and f_k for k>0 is 0 or 1. These are | |
24 | the 'subnormal' long doubles. | |
25 | ||
26 | There are four long doubles that represent zero, two that represent | |
27 | +0.0 and two that represent -0.0. The sign of the high part is the | |
28 | sign of the long double, and the sign of the low part is ignored. | |
29 | ||
30 | Likewise, there are four long doubles that represent infinities, two | |
31 | for +Inf and two for -Inf. | |
32 | ||
33 | Each NaN, quiet or signalling, that can be represented as a 'double' | |
34 | can be represented as a 'long double'. In fact, there are 2^64 | |
35 | equivalent representations for each one. | |
36 | ||
37 | There are certain other valid long doubles where both parts are | |
38 | nonzero but the low part represents a value which has a bit set below | |
39 | 2^(e-105). These, together with the subnormal long doubles, make up | |
40 | the denormal long doubles. | |
41 | ||
42 | Many possible long double bit patterns are not valid long doubles. | |
43 | These do not represent any value. | |
44 | ||
45 | Limits | |
46 | ------ | |
47 | ||
48 | The maximum representable long double is 2^1024-2^918. The smallest | |
49 | *normal* positive long double is 2^-968. The smallest denormalised | |
50 | positive long double is 2^-1074 (this is the same as for 'double'). | |
51 | ||
52 | Conversions | |
53 | ----------- | |
54 | ||
55 | A double can be converted to a long double by adding a zero low part. | |
56 | ||
57 | A long double can be converted to a double by removing the low part. | |
58 | ||
59 | Comparisons | |
60 | ----------- | |
61 | ||
62 | Two long doubles can be compared by comparing the high parts, and if | |
63 | those compare equal, comparing the low parts. | |
64 | ||
65 | Arithmetic | |
66 | ---------- | |
67 | ||
68 | The unary negate operation operates by negating the low and high parts. | |
69 | ||
70 | An absolute or absolute-negate operation must be done by comparing | |
71 | against zero and negating if necessary. | |
72 | ||
73 | Addition and subtraction are performed using library routines. They | |
74 | are not at present performed perfectly accurately, the result produced | |
75 | will be within 1ulp of the range generated by adding or subtracting | |
76 | 1ulp from the input values, where a 'ulp' is 2^(e-106) given the | |
77 | exponent 'e'. In the presence of cancellation, this may be | |
78 | arbitrarily inaccurate. Subtraction is done by negation and addition. | |
79 | ||
80 | Multiplication is also performed using a library routine. Its result | |
81 | will be within 2ulp of the correct result. | |
82 | ||
83 | Division is also performed using a library routine. Its result will | |
84 | be within 3ulp of the correct result. | |
7dfbd804 | 85 | |
86 | \f | |
f1717362 | 87 | Copyright (C) 2004-2016 Free Software Foundation, Inc. |
7dfbd804 | 88 | |
89 | Copying and distribution of this file, with or without modification, | |
90 | are permitted in any medium without royalty provided the copyright | |
91 | notice and this notice are preserved. |