1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"
"http://www.w3.org/TR/html4/loose.dtd">
<HTML>
<HEAD>
<TITLE>Ada95 - Model of Fixed Point Arithmetic</TITLE>
<META NAME="Author" CONTENT="JTC1/SC22/WG9/ARG, by Randall Brukardt, ARG Editor">
<META NAME="GENERATOR" CONTENT="Arm_Form.Exe, Ada Reference Manual generator">
<STYLE type="text/css">
DIV.paranum {position: absolute; font-family: Arial, Helvetica, sans-serif; left: 0.5 em; top: auto}
TT {font-family: "Courier New", monospace}
DT {display: compact}
DIV.Normal {font-family: "Times New Roman", Times, serif; margin-bottom: 0.6em}
DIV.Wide {font-family: "Times New Roman", Times, serif; margin-top: 0.6em; margin-bottom: 0.6em}
DIV.Annotations {font-family: "Times New Roman", Times, serif; margin-left: 4.0em; margin-bottom: 0.6em}
DIV.WideAnnotations {font-family: "Times New Roman", Times, serif; margin-left: 4.0em; margin-top: 0.6em; margin-bottom: 0.6em}
DIV.Index {font-family: "Times New Roman", Times, serif}
DIV.SyntaxSummary {font-family: "Times New Roman", Times, serif; margin-left: 2.0em; margin-bottom: 0.4em}
DIV.Notes {font-family: "Times New Roman", Times, serif; margin-left: 2.0em; margin-bottom: 0.6em}
DIV.NotesHeader {font-family: "Times New Roman", Times, serif; margin-left: 2.0em}
DIV.SyntaxIndented {font-family: "Times New Roman", Times, serif; margin-left: 2.0em; margin-bottom: 0.4em}
DIV.Indented {font-family: "Times New Roman", Times, serif; margin-left: 6.0em; margin-bottom: 0.6em}
DIV.CodeIndented {font-family: "Times New Roman", Times, serif; margin-left: 4.0em; margin-bottom: 0.6em}
DIV.SmallIndented {font-family: "Times New Roman", Times, serif; margin-left: 10.0em; margin-bottom: 0.6em}
DIV.SmallCodeIndented {font-family: "Times New Roman", Times, serif; margin-left: 8.0em; margin-bottom: 0.6em}
DIV.Examples {font-family: "Courier New", monospace; margin-left: 2.0em; margin-bottom: 0.6em}
DIV.SmallExamples {font-family: "Courier New", monospace; font-size: 80%; margin-left: 7.5em; margin-bottom: 0.6em}
DIV.IndentedExamples {font-family: "Courier New", monospace; margin-left: 8.0em; margin-bottom: 0.6em}
DIV.SmallIndentedExamples {font-family: "Courier New", monospace; font-size: 80%; margin-left: 15.0em; margin-bottom: 0.6em}
UL.Bulleted {font-family: "Times New Roman", Times, serif; margin-left: 2.0em; margin-right: 2.0em; margin-top: 0em; margin-bottom: 0.5em}
UL.SmallBulleted {font-family: "Times New Roman", Times, serif; margin-left: 6.0em; margin-right: 6.0em; margin-top: 0em; margin-bottom: 0.5em}
UL.NestedBulleted {font-family: "Times New Roman", Times, serif; margin-left: 4.0em; margin-right: 4.0em; margin-top: 0em; margin-bottom: 0.5em}
UL.SmallNestedBulleted {font-family: "Times New Roman", Times, serif; margin-left: 8.0em; margin-right: 8.0em; margin-top: 0em; margin-bottom: 0.5em}
UL.IndentedBulleted {font-family: "Times New Roman", Times, serif; margin-left: 8.0em; margin-right: 8.0em; margin-top: 0em; margin-bottom: 0.5em}
UL.CodeIndentedBulleted {font-family: "Times New Roman", Times, serif; margin-left: 6.0em; margin-right: 6.0em; margin-top: 0em; margin-bottom: 0.5em}
UL.CodeIndentedNestedBulleted {font-family: "Times New Roman", Times, serif; margin-left: 8.0em; margin-right: 8.0em; margin-top: 0em; margin-bottom: 0.5em}
UL.SyntaxIndentedBulleted {font-family: "Times New Roman", Times, serif; margin-left: 4.0em; margin-right: 4.0em; margin-top: 0em; margin-bottom: 0.5em}
UL.NotesBulleted {font-family: "Times New Roman", Times, serif; margin-left: 4.0em; margin-right: 4.0em; margin-top: 0em; margin-bottom: 0.5em}
UL.NotesNestedBulleted {font-family: "Times New Roman", Times, serif; margin-left: 6.0em; margin-right: 6.0em; margin-top: 0em; margin-bottom: 0.5em}
DL.Hanging {font-family: "Times New Roman", Times, serif; margin-top: 0em; margin-bottom: 0.6em}
DD.Hanging {margin-left: 6.0em}
DL.IndentedHanging {font-family: "Times New Roman", Times, serif; margin-left: 4.0em; margin-top: 0em; margin-bottom: 0.6em}
DD.IndentedHanging {margin-left: 2.0em}
DL.HangingInBulleted {font-family: "Times New Roman", Times, serif; margin-left: 2.0em; margin-right: 2.0em; margin-top: 0em; margin-bottom: 0.5em}
DD.HangingInBulleted {margin-left: 4.0em}
DL.SmallHanging {font-family: "Times New Roman", Times, serif; margin-left: 4.0em; margin-top: 0em; margin-bottom: 0.6em}
DD.SmallHanging {margin-left: 7.5em}
DL.SmallIndentedHanging {font-family: "Times New Roman", Times, serif; margin-left: 8.0em; margin-top: 0em; margin-bottom: 0.6em}
DD.SmallIndentedHanging {margin-left: 2.0em}
DL.SmallHangingInBulleted {font-family: "Times New Roman", Times, serif; margin-left: 6.0em; margin-right: 6.0em; margin-top: 0em; margin-bottom: 0.5em}
DD.SmallHangingInBulleted {margin-left: 5.0em}
DL.Enumerated {font-family: "Times New Roman", Times, serif; margin-right: 0.0em; margin-top: 0em; margin-bottom: 0.5em}
DD.Enumerated {margin-left: 2.0em}
DL.SmallEnumerated {font-family: "Times New Roman", Times, serif; margin-left: 4.0em; margin-right: 4.0em; margin-top: 0em; margin-bottom: 0.5em}
DD.SmallEnumerated {margin-left: 2.5em}
DL.NestedEnumerated {font-family: "Times New Roman", Times, serif; margin-left: 2.0em; margin-right: 2.0em; margin-top: 0em; margin-bottom: 0.5em}
DL.SmallNestedEnumerated {font-family: "Times New Roman", Times, serif; margin-left: 6.0em; margin-right: 6.0em; margin-top: 0em; margin-bottom: 0.5em}
</STYLE>
</HEAD>
<BODY TEXT="#000000" BGCOLOR="#FFFFF0" LINK="#0000FF" VLINK="#800080" ALINK="#FF0000">
<P><A HREF="RM-TOC.html">Contents</A> <A HREF="RM-0-29.html">Index</A> <A HREF="RM-G-2-2.html">Previous</A> <A HREF="RM-G-2-4.html">Next</A></P>
<HR>
<H1> G.2.3 Model of Fixed Point Arithmetic</H1>
<DIV Class="Paranum"><FONT SIZE=-2>1</FONT></DIV>
<DIV Class="Normal"> In the strict mode, the predefined arithmetic
operations of a fixed point type shall satisfy the accuracy requirements
specified here and shall avoid or signal overflow in the situations described.
</DIV>
<H4 ALIGN=CENTER>Implementation Requirements</H4>
<DIV Class="Paranum"><FONT SIZE=-2>2</FONT></DIV>
<DIV Class="Normal"> The accuracy requirements for the predefined fixed
point arithmetic operations and conversions, and the results of relations
on fixed point operands, are given below. </DIV>
<DIV Class="Paranum"><FONT SIZE=-2>3</FONT></DIV>
<DIV Class="Normal"> The operands of the fixed point adding operators,
absolute value, and comparisons have the same type. These operations
are required to yield exact results, unless they overflow.</DIV>
<DIV Class="Paranum"><FONT SIZE=-2>4</FONT></DIV>
<DIV Class="Normal"> Multiplications and divisions are allowed between
operands of any two fixed point types; the result has to be (implicitly
or explicitly) converted to some other numeric type. For purposes of
defining the accuracy rules, the multiplication or division and the conversion
are treated as a single operation whose accuracy depends on three types
(those of the operands and the result). For decimal fixed point types,
the attribute T'Round may be used to imply explicit conversion with rounding
(see <A HREF="RM-3-5-10.html">3.5.10</A>).</DIV>
<DIV Class="Paranum"><FONT SIZE=-2>5</FONT></DIV>
<DIV Class="Normal"> When the result type is a floating point type,
the accuracy is as given in <A HREF="RM-G-2-1.html">G.2.1</A>. <A NAME="I6979"></A>For
some combinations of the operand and result types in the remaining cases,
the result is required to belong to a small set of values called the
<I>perfect result set</I>; <A NAME="I6980"></A>for other combinations,
it is required merely to belong to a generally larger and implementation-defined
set of values called the <I>close result set</I>. When the result type
is a decimal fixed point type, the perfect result set contains a single
value; thus, operations on decimal types are always fully specified.
</DIV>
<DIV Class="Paranum"><FONT SIZE=-2>6</FONT></DIV>
<DIV Class="Normal"> When one operand of a fixed-fixed multiplication
or division is of type <I>universal_real</I>, that operand is not implicitly
converted in the usual sense, since the context does not determine a
unique target type, but the accuracy of the result of the multiplication
or division (i.e., whether the result has to belong to the perfect result
set or merely the close result set) depends on the value of the operand
of type <I>universal_real</I> and on the types of the other operand and
of the result. </DIV>
<DIV Class="Paranum"><FONT SIZE=-2>7</FONT></DIV>
<DIV Class="Normal" Style="margin-bottom: 0.4em"> For a fixed point
multiplication or division whose (exact) mathematical result is <I>v</I>,
and for the conversion of a value <I>v</I> to a fixed point type, the
perfect result set and close result set are defined as follows: </DIV>
<DIV Class="Paranum"><FONT SIZE=-2>8</FONT></DIV>
<UL Class="Bulleted" Style="margin-bottom: 0.3em"><LI TYPE=DISC>If the result type is an ordinary fixed point type with
a <I>small</I> of <I>s</I>, </LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>9</FONT></DIV>
<UL Class="NestedBulleted"><LI TYPE=DISC>if <I>v</I> is an integer multiple of <I>s</I>, then the
perfect result set contains only the value <I>v</I>;</LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>10</FONT></DIV>
<UL Class="NestedBulleted"><LI TYPE=DISC>otherwise, it contains the integer multiple of <I>s</I>
just below <I>v</I> and the integer multiple of <I>s</I> just above <I>v</I>.
</LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>11</FONT></DIV>
<UL Class="Bulleted">The close result set is an implementation-defined
set of consecutive integer multiples of <I>s</I> containing the perfect
result set as a subset.</UL>
<DIV Class="Paranum"><FONT SIZE=-2>12</FONT></DIV>
<UL Class="Bulleted" Style="margin-bottom: 0.3em"><LI TYPE=DISC>If the result type is a decimal type with a <I>small</I>
of <I>s</I>, </LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>13</FONT></DIV>
<UL Class="NestedBulleted"><LI TYPE=DISC>if <I>v</I> is an integer multiple of <I>s</I>, then the
perfect result set contains only the value <I>v</I>;</LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>14</FONT></DIV>
<UL Class="NestedBulleted"><LI TYPE=DISC>otherwise, if truncation applies then it contains only
the integer multiple of <I>s</I> in the direction toward zero, whereas
if rounding applies then it contains only the nearest integer multiple
of <I>s</I> (with ties broken by rounding away from zero). </LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>15</FONT></DIV>
<UL Class="Bulleted">The close result set is an implementation-defined
set of consecutive integer multiples of <I>s</I> containing the perfect
result set as a subset. </UL>
<DIV Class="Paranum"><FONT SIZE=-2>16</FONT></DIV>
<UL Class="Bulleted" Style="margin-bottom: 0.3em"><LI TYPE=DISC>If the result type is an integer type, </LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>17</FONT></DIV>
<UL Class="NestedBulleted"><LI TYPE=DISC>if <I>v</I> is an integer, then the perfect result set
contains only the value <I>v</I>;</LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>18</FONT></DIV>
<UL Class="NestedBulleted"><LI TYPE=DISC>otherwise, it contains the integer nearest to the value
<I>v</I> (if <I>v</I> lies equally distant from two consecutive integers,
the perfect result set contains the one that is further from zero). </LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>19</FONT></DIV>
<UL Class="Bulleted">The close result set is an implementation-defined
set of consecutive integers containing the perfect result set as a subset.
</UL>
<DIV Class="Paranum"><FONT SIZE=-2>20</FONT></DIV>
<DIV Class="Normal"> The result of a fixed point multiplication or
division shall belong either to the perfect result set or to the close
result set, as described below, if overflow does not occur. In the following
cases, if the result type is a fixed point type, let <I>s</I> be its
<I>small</I>; otherwise, i.e. when the result type is an integer type,
let <I>s</I> be 1.0. </DIV>
<DIV Class="Paranum"><FONT SIZE=-2>21</FONT></DIV>
<UL Class="Bulleted"><LI TYPE=DISC>For a multiplication or division neither of whose operands
is of type <I>universal_real</I>, let <I>l</I> and <I>r</I> be the <I>smalls</I>
of the left and right operands. For a multiplication, if (<I>l</I> ·
<I>r</I>) / <I>s</I> is an integer or the reciprocal of an integer (the
<I>smalls</I> are said to be ``compatible'' in this case), the result
shall belong to the perfect result set; otherwise, it belongs to the
close result set. For a division, if <I>l</I> / (<I>r</I> · <I>s</I>)
is an integer or the reciprocal of an integer (i.e., the <I>smalls</I>
are compatible), the result shall belong to the perfect result set; otherwise,
it belongs to the close result set. </LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>22</FONT></DIV>
<UL Class="Bulleted"><LI TYPE=DISC>For a multiplication or division having one <I>universal_real</I>
operand with a value of <I>v</I>, note that it is always possible to
factor <I>v</I> as an integer multiple of a ``compatible'' <I>small</I>,
but the integer multiple may be ``too big.'' If there exists a factorization
in which that multiple is less than some implementation-defined limit,
the result shall belong to the perfect result set; otherwise, it belongs
to the close result set. </LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>23</FONT></DIV>
<DIV Class="Normal"> A multiplication P * Q of an operand of a fixed
point type F by an operand of an integer type I, or vice-versa, and a
division P / Q of an operand of a fixed point type F by an operand of
an integer type I, are also allowed. In these cases, the result has a
type of F; explicit conversion of the result is never required. The accuracy
required in these cases is the same as that required for a multiplication
F(P * Q) or a division F(P / Q) obtained by interpreting the operand
of the integer type to have a fixed point type with a <I>small</I> of
1.0.</DIV>
<DIV Class="Paranum"><FONT SIZE=-2>24</FONT></DIV>
<DIV Class="Normal"> The accuracy of the result of a conversion from
an integer or fixed point type to a fixed point type, or from a fixed
point type to an integer type, is the same as that of a fixed point multiplication
of the source value by a fixed point operand having a <I>small</I> of
1.0 and a value of 1.0, as given by the foregoing rules. The result of
a conversion from a floating point type to a fixed point type shall belong
to the close result set. The result of a conversion of a <I>universal_real</I>
operand to a fixed point type shall belong to the perfect result set.</DIV>
<DIV Class="Paranum"><FONT SIZE=-2>25</FONT></DIV>
<DIV Class="Normal"> The possibility of overflow in the result of
a predefined arithmetic operation or conversion yielding a result of
a fixed point type T is analogous to that for floating point types, except
for being related to the base range instead of the safe range. <A NAME="I6981"></A><A NAME="I6982"></A>If
all of the permitted results belong to the base range of T, then the
implementation shall deliver one of the permitted results; otherwise,
</DIV>
<DIV Class="Paranum"><FONT SIZE=-2>26</FONT></DIV>
<UL Class="Bulleted"><LI TYPE=DISC><A NAME="I6983"></A>if T'Machine_Overflows is True, the
implementation shall either deliver one of the permitted results or raise
Constraint_Error;</LI></UL>
<DIV Class="Paranum"><FONT SIZE=-2>27</FONT></DIV>
<UL Class="Bulleted"><LI TYPE=DISC>if T'Machine_Overflows is False, the result is implementation
defined. </LI></UL>
<HR>
<P><A HREF="RM-TOC.html">Contents</A> <A HREF="RM-0-29.html">Index</A> <A HREF="RM-G-2-2.html">Previous</A> <A HREF="RM-G-2-4.html">Next</A> <A HREF="RM-TTL.html">Legal</A></P>
</BODY>
</HTML>
|
