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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.

(3)

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_Radix**exponent

where

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• mantissa is a fraction in the number base T'Machine_Radix, the first digit of which is nonzero, and

(5)

• 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.

(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_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_Radix**(k-1) <= |x| < T'Machine_Radix**k.

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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_Radix**(Exponent-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_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.

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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.

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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_Radix**(k-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_Radix**(1-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_Radix**(T'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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