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A.5.3 Attributes of Floating Point Types
Static Semantics
1
{representationoriented
attributes (of a floating point subtype)} 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
{canonical form} 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.
6.a
Ramification: Values of
a type held in an extended register are, in general, not machine numbers
of the type, since they cannot be expressed in the canonical form with
a sufficiently short mantissa.
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.
8.a
Ramification: Note that
the above definitions do not determine unique values for the representationoriented
attributes of floating point types. The implementation may choose any
set of values that collectively satisfies the definitions.
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
{denormalized number}
The values described by the formula in the definition
of S'Denorm are called
denormalized numbers.
{normalized
number} A nonzero machine number that
is not a denormalized number is a
normalized number.
{represented
in canonical form} {canonicalform
representation} 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.
10.a
Discussion: The intent
is that S'Denorm be True when such denormalized numbers exist and are
generated in the circumstances defined by IEC 559:1989, though the latter
requirement is not formalized here.
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.
11.a
Discussion:
It is difficult to be more precise about what it means to round the
result of a predefined operation. If the implementation does not use
extended registers, so that every arithmetic result is necessarily a
machine number, then rounding seems to imply two things:
11.b
 S'Model_Mantissa = S'Machine_Mantissa, so that operand
preperturbation never occurs;
11.c
 when the exact mathematical result is not a machine number,
the result of a predefined operation must be the nearer of the two adjacent
machine numbers.
11.d
Technically, this attribute should
yield False when extended registers are used, since a few computed results
will cross over the halfway point as a result of double rounding, if
and when a value held in an extended register has to be reduced in precision
to that of the machine numbers. It does not seem desirable to preclude
the use of extended registers when S'Machine_Rounds could otherwise be
True.
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
{normalized
exponent} 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}.
16.a
Ramification: The normalized
exponent of a normalized number x is the value of exponent
in the canonicalform representation of x.
16.b
The normalized exponent of a denormalized
number is less than the value of T'Machine_Emin.
17
{primitive
function} 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.
23.a
Discussion: Informally,
when X is a normalized number, the result is the value obtained
by replacing the exponent by zero in the canonicalform representation
of X.
23.b
Ramification: Except when
X is zero, the magnitude of the result is greater than or equal
to the reciprocal of T'Machine_Radix and less than one; consequently,
the result is always a normalized number, even when X is a denormalized
number.
23.c
Implementation Note: When
X is a denormalized number, the result is the value obtained by
replacing the exponent by zero in the canonicalform representation
of the result of scaling X up sufficiently to normalize it.
24
 S'Compose

S'Compose denotes a function
with the following specification:
25
function S'Compose (Fraction : T;
Exponent : universal_integer)
return T
26
 {Constraint_Error (raised by failure
of runtime check)} 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. {Range_Check
[partial]} {check, languagedefined
(Range_Check)} 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.
26.a
Discussion: Informally,
when Fraction and v are both normalized numbers, the result
is the value obtained by replacing the exponent by Exponent
in the canonicalform representation of Fraction.
26.b
Ramification: If Exponent
is less than T'Machine_Emin and Fraction is nonzero, the
result is either zero, T'Model_Small, or (if T'Denorm is
True) a denormalized number.
27
 S'Scaling

S'Scaling denotes a function
with the following specification:
28
function S'Scaling (X : T;
Adjustment : universal_integer)
return T
29
 {Constraint_Error (raised by failure
of runtime check)} 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. {Range_Check [partial]} {check,
languagedefined (Range_Check)} 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.
29.a
Discussion: Informally,
when X and v are both normalized numbers, the result is
the value obtained by increasing the exponent by Adjustment
in the canonicalform representation of X.
29.b
Ramification: If Adjustment
is sufficiently small (i.e., sufficiently negative), the result is either
zero, T'Model_Small, or (if T'Denorm is True) a denormalized
number.
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
 {Constraint_Error (raised by failure
of runtime check)} 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. {Division_Check
[partial]} {check, languagedefined
(Division_Check)} Constraint_Error is
raised if Y is zero. A zero result has the sign of X when
S'Signed_Zeros is True.
47.a
Ramification: The magnitude
of the result is less than or equal to onehalf the magnitude of Y.
47.b
Discussion: Given machine
numbers X and Y of the type T, v is necessarily
a machine number of the type T, except when Y is in the
neighborhood of zero, X is sufficiently close to a multiple of
Y, and T'Denorm is False.
48
 S'Adjacent

S'Adjacent denotes a function
with the following specification:
49
function S'Adjacent (X, Towards : T)
return T
50
 {Constraint_Error (raised by failure
of runtime check)} 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. {Range_Check
[partial]} {check, languagedefined
(Range_Check)} 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.
50.a
Ramification: The value
of S'Adjacent(0.0, 1.0) is the smallest normalized positive number of
the type T when T'Denorm is False and the smallest denormalized
positive number of the type T when T'Denorm is True.
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
 {Constraint_Error (raised by failure
of runtime check)} 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. {Range_Check [partial]} {check,
languagedefined (Range_Check)} 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.
53.a
Discussion: S'Copy_Sign
is provided for convenience in restoring the sign to a quantity from
which it has been temporarily removed, or to a related quantity. When
S'Signed_Zeros is True, it is also instrumental in determining the sign
of a zero quantity, when required. (Because negative and positive zeros
compare equal in systems conforming to IEC 559:1989, a negative zero
does not appear to be negative when compared to zero.) The sign
determination is accomplished by transferring the sign of the zero quantity
to a nonzero quantity and then testing for a negative result.
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 (raised by failure
of runtime check)} {Range_Check
[partial]} {check, languagedefined
(Range_Check)} 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.
59.a
Discussion: Informally,
if X is nonzero, the result is the value obtained by retaining
only the specified number of (leading) significant digits of X
(in the machine radix), setting all other digits to zero.
59.b
Implementation Note: The
result can be obtained by first scaling X up, if necessary to
normalize it, then masking the mantissa so as to retain only the specified
number of leading digits, then scaling the result back down if X
was scaled up.
60
 S'Machine

S'Machine denotes a function
with the following specification:
61
function S'Machine (X : T)
return T
62
 {Constraint_Error (raised by failure
of runtime check)} 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. {Range_Check
[partial]} {check, languagedefined
(Range_Check)} 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.
62.a
Discussion: All of the
primitive function attributes except Rounding and Machine correspond
to subprograms in the Generic_Primitive_Functions generic package proposed
as a separate ISO standard (ISO/IEC DIS 11729) for Ada 83. The Scaling,
Unbiased_Rounding, and Truncation attributes correspond to the Scale,
Round, and Truncate functions, respectively, in Generic_Primitive_Functions.
The Rounding attribute rounds away from zero; this functionality was
not provided in Generic_Primitive_Functions. The name Round was not available
for either of the primitive function attributes that perform rounding,
since an attribute of that name is used for a different purpose for decimal
fixed point types. Likewise, the name Scale was not available, since
an attribute of that name is also used for a different purpose for decimal
fixed point types. The functionality of the Machine attribute was also
not provided in Generic_Primitive_Functions. The functionality of the
Decompose procedure of Generic_Primitive_Functions is only provided in
the form of the separate attributes Exponent and Fraction. The functionality
of the Successor and Predecessor functions of Generic_Primitive_Functions
is provided by the extension of the existing Succ and Pred attributes.
62.b
Implementation Note: The
primitive function attributes may be implemented either with appropriate
floating point arithmetic operations or with integer and logical operations
that act on parts of the representation directly. The latter is strongly
encouraged when it is more efficient than the former; it is mandatory
when the former cannot deliver the required accuracy due to limitations
of the implementation's arithmetic operations.
63
{modeloriented
attributes (of a floating point subtype)} 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.
66.a
Discussion: In most implementations,
this attribute yields the absolute value of the difference between one and the
smallest machine number of the type T above one which, when added to
one, yields a machine number different from one. Further discussion can be found
in G.2.2.
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.
67.a
Discussion: In most implementations,
this attribute yields the smallest positive normalized number of the type T,
i.e. the number corresponding to the positive underflow threshold. In some implementations
employing a radixcomplement representation for the type T, the positive
underflow threshold is closer to zero than is the negative underflow threshold,
with the consequence that the smallest positive normalized number does not coincide
with the positive underflow threshold (i.e., it exceeds the latter). Further
discussion can be found in G.2.2.
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.
72.a
Discussion: A predefined
floating point arithmetic operation that yields a value in the safe range
of its result type is guaranteed not to overflow.
72.b
To be honest: An exception is made
for exponentiation by a negative exponent in 4.5.6.
72.c
Implementation defined: The
values of the Model_Mantissa, Model_Emin, Model_Epsilon, Model, Safe_First,
and Safe_Last attributes, if the Numerics Annex is not supported.
Incompatibilities With Ada 83
72.d
{incompatibilities with Ada
83} The Epsilon and Mantissa attributes of floating
point types are removed from the language and replaced by Model_Epsilon
and Model_Mantissa, which may have different values (as a result of changes
in the definition of model numbers); the replacement of one set of attributes
by another is intended to convert what would be an inconsistent change
into an incompatible change.
72.e
The Emax, Small, Large, Safe_Emax,
Safe_Small, and Safe_Large attributes of floating point types are removed
from the language. Small and Safe_Small are collectively replaced by
Model_Small, which is functionally equivalent to Safe_Small, though it
may have a slightly different value. The others are collectively replaced
by Safe_First and Safe_Last. Safe_Last is functionally equivalent to
Safe_Large, though it may have a different value; Safe_First is comparable
to the negation of Safe_Large but may differ slightly from it as well
as from the negation of Safe_Last. Emax and Safe_Emax had relatively
few uses in Ada 83; T'Safe_Emax can be computed in the revised language
as Integer'Min(T'Exponent(T'Safe_First), T'Exponent(T'Safe_Last)).
72.f
Implementations are encouraged
to eliminate the incompatibilities discussed here by retaining the old
attributes, during a transition period, in the form of implementationdefined
attributes with their former values.
Extensions to Ada 83
72.g
{extensions to Ada 83} The
Model_Emin attribute is new. It is conceptually similar to the negation of Safe_Emax
attribute of Ada 83, adjusted for the fact that the model numbers now have the
hardware radix. It is a fundamental determinant, along with Model_Mantissa,
of the set of model numbers of a type (see G.2.1).
72.h
The Denorm and Signed_Zeros attributes
are new, as are all of the primitive function attributes.
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