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A.5.3 Attributes of Floating Point Types
Static Semantics
1
The
following
representationoriented attributes are defined for every
subtype S of a floating point type
T.
2
 S'Machine_Radix

Yields the radix of the hardware
representation of the type T. The value of this attribute is of
the type universal_integer.
3
The values of other representationoriented
attributes of a floating point subtype, and of the ``primitive function''
attributes of a floating point subtype described later, are defined in
terms of a particular representation of nonzero values called the
canonical
form. The canonical form (for the type
T) is the form
±
mantissa ·
T'Machine_Radix
^{exponent}
where
4
 mantissa is a fraction in the
number base T'Machine_Radix, the first digit of which is nonzero,
and
5
6
 S'Machine_Mantissa

Yields the largest value of p
such that every value expressible in the canonical form (for the type
T), having a pdigit mantissa and an exponent
between T'Machine_Emin and T'Machine_Emax, is a machine
number (see 3.5.7) of the type T.
This attribute yields a value of the type universal_integer.
7
 S'Machine_Emin

Yields the smallest (most negative)
value of exponent such that every value expressible in the canonical
form (for the type T), having a mantissa of T'Machine_Mantissa
digits, is a machine number (see 3.5.7) of
the type T. This attribute yields a value of the type universal_integer.
8
 S'Machine_Emax

Yields the largest (most positive)
value of exponent such that every value expressible in the canonical
form (for the type T), having a mantissa of T'Machine_Mantissa
digits, is a machine number (see 3.5.7) of
the type T. This attribute yields a value of the type universal_integer.
9
 S'Denorm

Yields the value True if every
value expressible in the form
± mantissa · T'Machine_Radix^{T'Machine_Emin}
where mantissa is a nonzero T'Machine_Mantissadigit fraction
in the number base T'Machine_Radix, the first digit of which is
zero, is a machine number (see 3.5.7) of
the type T; yields the value False otherwise. The value of this
attribute is of the predefined type Boolean.
10
The values described by the
formula in the definition of S'Denorm are called
denormalized numbers.
A nonzero machine number that is not a denormalized
number is a
normalized number.
A
normalized number
x of a given type
T is said to be
represented
in canonical form when it is expressed in the canonical form (for
the type
T) with a
mantissa having
T'Machine_Mantissa
digits; the resulting form is the
canonicalform representation
of
x.
11
 S'Machine_Rounds

Yields the value True if rounding
is performed on inexact results of every predefined operation that yields
a result of the type T; yields the value False otherwise. The
value of this attribute is of the predefined type Boolean.
12
 S'Machine_Overflows

Yields the value True if overflow
and dividebyzero are detected and reported by raising Constraint_Error
for every predefined operation that yields a result of the type T;
yields the value False otherwise. The value of this attribute is of the
predefined type Boolean.
13
 S'Signed_Zeros

Yields the value True if the
hardware representation for the type T has the capability of representing
both positively and negatively signed zeros, these being generated and
used by the predefined operations of the type T as specified in
IEC 559:1989; yields the value False otherwise. The value of this attribute
is of the predefined type Boolean.
14
For
every value
x of a floating point type
T, the
normalized
exponent of
x is defined as follows:
15
 the normalized exponent of zero is
(by convention) zero;
16
 for nonzero x, the normalized
exponent of x is the unique integer k such that T'Machine_Radix^{k1}
<= x < T'Machine_Radix^{k}.
17
The
following
primitive function attributes are defined for any subtype
S of a floating point type
T.
18
 S'Exponent

S'Exponent denotes a function
with the following specification:
19
function S'Exponent (X : T)
return universal_integer
20
 The function yields the normalized
exponent of X.
21
 S'Fraction

S'Fraction denotes a function
with the following specification:
22
function S'Fraction (X : T)
return T
23
 The function yields the value
X · T'Machine_Radix^{k},
where k is the normalized exponent of X. A zero result,
which can only occur when X is zero, has the sign of X.
24
 S'Compose

S'Compose denotes a function
with the following specification:
25
function S'Compose (Fraction : T;
Exponent : universal_integer)
return T
26
 Let v
be the value Fraction · T'Machine_Radix^{Exponentk},
where k is the normalized exponent of Fraction. If v
is a machine number of the type T, or if v >= T'Model_Small,
the function yields v; otherwise, it yields either one of the
machine numbers of the type T adjacent to v. Constraint_Error
is optionally raised if v is outside the base range of S. A zero
result has the sign of Fraction when S'Signed_Zeros is True.
27
 S'Scaling

S'Scaling denotes a function
with the following specification:
28
function S'Scaling (X : T;
Adjustment : universal_integer)
return T
29
 Let v
be the value X · T'Machine_Radix^{Adjustment}.
If v is a machine number of the type T, or if v
>= T'Model_Small, the function yields v; otherwise, it
yields either one of the machine numbers of the type T adjacent
to v. Constraint_Error
is optionally raised if v is outside the base range of S. A zero
result has the sign of X when S'Signed_Zeros is True.
30
 S'Floor

S'Floor denotes a function with
the following specification:
31
function S'Floor (X : T)
return T
32
 The function yields the value
Floor(X), i.e., the largest (most positive) integral value
less than or equal to X. When X is zero, the result has
the sign of X; a zero result otherwise has a positive sign.
33
 S'Ceiling

S'Ceiling denotes a function
with the following specification:
34
function S'Ceiling (X : T)
return T
35
 The function yields the value
Ceiling(X), i.e., the smallest (most negative) integral
value greater than or equal to X. When X is zero, the result
has the sign of X; a zero result otherwise has a negative sign
when S'Signed_Zeros is True.
36
 S'Rounding

S'Rounding denotes a function
with the following specification:
37
function S'Rounding (X : T)
return T
38
 The function yields the integral
value nearest to X, rounding away from zero if X lies exactly
halfway between two integers. A zero result has the sign of X
when S'Signed_Zeros is True.
39
 S'Unbiased_Rounding

S'Unbiased_Rounding denotes a
function with the following specification:
40
function S'Unbiased_Rounding (X : T)
return T
41
 The function yields the integral
value nearest to X, rounding toward the even integer if X
lies exactly halfway between two integers. A zero result has the sign
of X when S'Signed_Zeros is True.
42
 S'Truncation

S'Truncation denotes a function
with the following specification:
43
function S'Truncation (X : T)
return T
44
 The function yields the value
Ceiling(X) when X is negative, and Floor(X)
otherwise. A zero result has the sign of X when S'Signed_Zeros
is True.
45
 S'Remainder

S'Remainder denotes a function
with the following specification:
46
function S'Remainder (X, Y : T)
return T
47
 For nonzero
Y, let v be the value X  n · Y,
where n is the integer nearest to the exact value of X/Y;
if n  X/Y = 1/2, then n is chosen
to be even. If v is a machine number of the type T, the
function yields v; otherwise, it yields zero. Constraint_Error
is raised if Y is zero. A zero result has the sign of X
when S'Signed_Zeros is True.
48
 S'Adjacent

S'Adjacent denotes a function
with the following specification:
49
function S'Adjacent (X, Towards : T)
return T
50
 If Towards
= X, the function yields X; otherwise, it yields the
machine number of the type T adjacent to X in the direction
of Towards, if that machine number exists. If
the result would be outside the base range of S, Constraint_Error is
raised. When T'Signed_Zeros is True, a zero result has the sign
of X. When Towards is zero, its sign has no bearing on
the result.
51
 S'Copy_Sign

S'Copy_Sign denotes a function
with the following specification:
52
function S'Copy_Sign (Value, Sign : T)
return T
53
 If the value
of Value is nonzero, the function yields a result whose magnitude
is that of Value and whose sign is that of Sign; otherwise,
it yields the value zero. Constraint_Error
is optionally raised if the result is outside the base range of S. A
zero result has the sign of Sign when S'Signed_Zeros is True.
54
 S'Leading_Part

S'Leading_Part denotes a function
with the following specification:
55
function S'Leading_Part (X : T;
Radix_Digits : universal_integer)
return T
56
 Let v be the value T'Machine_Radix^{kRadix_Digits},
where k is the normalized exponent of X. The function yields
the value
57
 Floor(X/v)
· v, when X is nonnegative and Radix_Digits
is positive;
58
 Ceiling(X/v)
· v, when X is negative and Radix_Digits
is positive.
59
 Constraint_Error
is raised when Radix_Digits is zero or negative. A zero result,
which can only occur when X is zero, has the sign of X.
60
 S'Machine

S'Machine denotes a function
with the following specification:
61
function S'Machine (X : T)
return T
62
 If X
is a machine number of the type T, the function yields X;
otherwise, it yields the value obtained by rounding or truncating X
to either one of the adjacent machine numbers of the type T. Constraint_Error
is raised if rounding or truncating X to the precision of the
machine numbers results in a value outside the base range of S. A zero
result has the sign of X when S'Signed_Zeros is True.
63
The
following
modeloriented attributes are defined for any subtype
S of a floating point type
T.
64
 S'Model_Mantissa

If the Numerics Annex is not
supported, this attribute yields an implementation defined value that
is greater than or equal to Ceiling(d · log(10)
/ log(T'Machine_Radix)) + 1, where d is the requested decimal
precision of T, and less than or equal to the value of T'Machine_Mantissa.
See G.2.2 for further requirements that apply
to implementations supporting the Numerics Annex. The value of this attribute
is of the type universal_integer.
65
 S'Model_Emin

If the Numerics Annex is not
supported, this attribute yields an implementation defined value that
is greater than or equal to the value of T'Machine_Emin. See G.2.2
for further requirements that apply to implementations supporting the
Numerics Annex. The value of this attribute is of the type universal_integer.
66
 S'Model_Epsilon

Yields the value T'Machine_Radix^{1
 T'Model_Mantissa}. The value of this attribute
is of the type universal_real.
67
 S'Model_Small

Yields the value T'Machine_Radix^{T'Model_Emin
 1}. The value of this attribute is of the type universal_real.
68
 S'Model

S'Model denotes a function with
the following specification:
69
function S'Model (X : T)
return T
70
 If the Numerics Annex is not
supported, the meaning of this attribute is implementation defined; see
G.2.2 for the definition that applies to
implementations supporting the Numerics Annex.
71
 S'Safe_First

Yields the lower bound of the
safe range (see 3.5.7) of the type T.
If the Numerics Annex is not supported, the value of this attribute is
implementation defined; see G.2.2 for the
definition that applies to implementations supporting the Numerics Annex.
The value of this attribute is of the type universal_real.
72
 S'Safe_Last

Yields the upper bound of the
safe range (see 3.5.7) of the type T.
If the Numerics Annex is not supported, the value of this attribute is
implementation defined; see G.2.2 for the
definition that applies to implementations supporting the Numerics Annex.
The value of this attribute is of the type universal_real.
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