Attributes of Floating Point Types
Author: JTC1/SC22/WG9/ARG, by Randall Brukardt, ARG Editor

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# A.5.3 Attributes of Floating Point Types

#### Static Semantics

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The following representation-oriented attributes are defined for every subtype S of a floating point type T.
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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.
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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
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- mantissa is a fraction in the number base T'Machine_Radix, the first digit of which is nonzero, and

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- exponent is an integer.

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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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For every value x of a floating point type T, the normalized exponent of x is defined as follows:
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- the normalized exponent of zero is (by convention) zero;

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- for nonzero x, the normalized exponent of x is the unique integer k such that T'Machine_Radixk–1 ≤ |x| < T'Machine_Radixk.

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The following primitive function attributes are defined for any subtype S of a floating point type T.
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S'Exponent
S'Exponent denotes a function with the following specification:
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function S'Exponent (X : T)
return universal_integer
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The function yields the normalized exponent of X.
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S'Fraction
S'Fraction denotes a function with the following specification:
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function S'Fraction (X : T)
return T
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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.
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S'Compose
S'Compose denotes a function with the following specification:
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function S'Compose (Fraction : T;
Exponent : universal_integer)
return T
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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.
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S'Scaling
S'Scaling denotes a function with the following specification:
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function S'Scaling (X : T;
Adjustment : universal_integer)
return T
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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.
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S'Floor
S'Floor denotes a function with the following specification:
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function S'Floor (X : T)
return T
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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.
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S'Ceiling
S'Ceiling denotes a function with the following specification:
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function S'Ceiling (X : T)
return T
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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.
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S'Rounding
S'Rounding denotes a function with the following specification:
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function S'Rounding (X : T)
return T
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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.
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S'Unbiased_Rounding
S'Unbiased_Rounding denotes a function with the following specification:
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function S'Unbiased_Rounding (X : T)
return T
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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.
41.1/2
S'Machine_Rounding
S'Machine_Rounding denotes a function with the following specification:
41.2/2
function S'Machine_Rounding (X : T)
return T
41.3/2
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.
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S'Truncation
S'Truncation denotes a function with the following specification:
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function S'Truncation (X : T)
return T
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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.
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S'Remainder
S'Remainder denotes a function with the following specification:
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function S'Remainder (X, Y : T)
return T
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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.
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S'Adjacent
S'Adjacent denotes a function with the following specification:
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function S'Adjacent (X, Towards : T)
return T
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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.
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S'Copy_Sign
S'Copy_Sign denotes a function with the following specification:
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function S'Copy_Sign (Value, Sign : T)
return T
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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.
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S'Leading_Part
S'Leading_Part denotes a function with the following specification:
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function S'Leading_Part (X : T;
Radix_Digits : universal_integer)
return T
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Let v be the value T'Machine_Radixk–Radix_Digits, where k is the normalized exponent of X. The function yields the value
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- Floor(X/v) · v, when X is nonnegative and Radix_Digits is positive;

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- Ceiling(X/v) · v, when X is negative and Radix_Digits is positive.

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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.
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S'Machine
S'Machine denotes a function with the following specification:
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function S'Machine (X : T)
return T
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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.
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The following model-oriented attributes are defined for any subtype S of a floating point type T.
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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.
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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.
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S'Model_Epsilon
Yields the value T'Machine_Radix1 – T'Model_Mantissa. The value of this attribute is of the type universal_real.
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S'Model_Small
Yields the value T'Machine_RadixT'Model_Emin – 1. The value of this attribute is of the type universal_real.
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S'Model
S'Model denotes a function with the following specification:
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function S'Model (X : T)
return T
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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.
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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.
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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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