A.5.3 Attributes of Floating Point Types
Static Semantics
The
following
representation-oriented attributes are defined for every
subtype S of a floating point type
T.
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.
The values of other representation-oriented
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_Radixexponent
where
mantissa is a fraction in the number base
T'Machine_Radix, the first digit of
which is nonzero, and
exponent is an integer.
S'Machine_Mantissa
Yields the largest value of
p
such that every value expressible in the canonical form (for the type
T), having a
p-digit
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.
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.
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.
S'Denorm
Yields the value True if every
value expressible in the form
±
mantissa
·
T'Machine_RadixT'Machine_Emin
where
mantissa is a nonzero
T'Machine_Mantissa-digit
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.
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
canonical-form representation
of
x.
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.
S'Machine_Overflows
Yields the value True if overflow
and divide-by-zero 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.
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.
For
every value
x of a floating point type
T, the
normalized exponent of
x is defined as follows:
the normalized exponent of zero is (by convention)
zero;
for nonzero x,
the normalized exponent of x is the
unique integer k such that T'Machine_Radixk–1
≤ |x| < T'Machine_Radixk.
The
following
primitive function attributes are defined for any subtype
S of a floating point type
T.
S'Exponent
S'Exponent denotes a function
with the following specification:
function S'Exponent (X : T)
return universal_integer
The function yields the normalized exponent
of X.
S'Fraction
S'Fraction denotes a function
with the following specification:
function S'Fraction (X : T)
return T
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.
S'Compose
S'Compose denotes a function
with the following specification:
function S'Compose (Fraction : T;
Exponent : universal_integer)
return T
Let
v
be the value
Fraction ·
T'Machine_RadixExponent–k,
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.
S'Scaling
S'Scaling denotes a function
with the following specification:
function S'Scaling (X : T;
Adjustment : universal_integer)
return T
Let
v
be the value
X ·
T'Machine_RadixAdjustment.
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.
S'Floor
S'Floor denotes a function with
the following specification:
function S'Floor (X : T)
return T
The function yields the value Floor(X),
that is, 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.
S'Ceiling
S'Ceiling denotes a function
with the following specification:
function S'Ceiling (X : T)
return T
The function yields the value Ceiling(X),
that is, 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.
S'Rounding
S'Rounding denotes a function
with the following specification:
function S'Rounding (X : T)
return T
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.
S'Unbiased_Rounding
S'Unbiased_Rounding denotes a
function with the following specification:
function S'Unbiased_Rounding (X : T)
return T
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.
S'Machine_Rounding
S'Machine_Rounding denotes a
function with the following specification:
function S'Machine_Rounding (X : T)
return T
The function yields the integral value
nearest to
X. If
X
lies exactly halfway between two integers, one of those integers is returned,
but which of them is returned is unspecified. A zero result has the sign
of
X when S'Signed_Zeros is True. This
function provides access to the rounding behavior which is most efficient
on the target processor.
S'Truncation
S'Truncation denotes a function
with the following specification:
function S'Truncation (X : T)
return T
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.
S'Remainder
S'Remainder denotes a function
with the following specification:
function S'Remainder (X, Y : T)
return T
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.
S'Adjacent
S'Adjacent denotes a function
with the following specification:
function S'Adjacent (X, Towards : T)
return T
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.
S'Copy_Sign
S'Copy_Sign denotes a function
with the following specification:
function S'Copy_Sign (Value, Sign : T)
return T
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.
S'Leading_Part
S'Leading_Part denotes a function
with the following specification:
function S'Leading_Part (X : T;
Radix_Digits : universal_integer)
return T
Let v
be the value T'Machine_Radixk–Radix_Digits,
where k is the normalized exponent
of X. The function yields the value
Floor(X/v)
· v, when X
is nonnegative and Radix_Digits is
positive;
Ceiling(X/v)
· v, when X
is negative and Radix_Digits is positive.
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.
S'Machine
S'Machine denotes a function
with the following specification:
function S'Machine (X : T)
return T
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.
The
following
model-oriented attributes are defined for any subtype
S of a floating point type
T.
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.
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.
S'Model_Epsilon
Yields the value
T'Machine_Radix1
– T'Model_Mantissa.
The value of this attribute is of the type
universal_real.
S'Model_Small
Yields the value
T'Machine_RadixT'Model_Emin – 1. The value of this attribute is of the
type
universal_real.
S'Model
S'Model denotes a function with
the following specification:
function S'Model (X : T)
return T
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.
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.
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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