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G.2.1 Model of Floating Point Arithmetic

In the strict mode, the predefined operations of a floating point type shall satisfy the accuracy requirements specified here and shall avoid or signal overflow in the situations described. This behavior is presented in terms of a model of floating point arithmetic that builds on the concept of the canonical form (see A.5.3).

Static Semantics

Associated with each floating point type is an infinite set of model numbers. The model numbers of a type are used to define the accuracy requirements that have to be satisfied by certain predefined operations of the type; through certain attributes of the model numbers, they are also used to explain the meaning of a user-declared floating point type declaration. The model numbers of a derived type are those of the parent type; the model numbers of a subtype are those of its type.
{model number} The model numbers of a floating point type T are zero and all the values expressible in the canonical form (for the type T), in which mantissa has T'Model_Mantissa digits and exponent has a value greater than or equal to T'Model_Emin. (These attributes are defined in G.2.2.) 
Discussion: The model is capable of describing the behavior of most existing hardware that has a mantissa-exponent representation. As applied to a type T, it is parameterized by the values of T'Machine_Radix, T'Model_Mantissa, T'Model_Emin, T'Safe_First, and T'Safe_Last. The values of these attributes are determined by how, and how well, the hardware behaves. They in turn determine the set of model numbers and the safe range of the type, which figure in the accuracy and range (overflow avoidance) requirements.
In hardware that is free of arithmetic anomalies, T'Model_Mantissa, T'Model_Emin, T'Safe_First, and T'Safe_Last will yield the same values as T'Machine_Mantissa, T'Machine_Emin, T'Base'First, and T'Base'Last, respectively, and the model numbers in the safe range of the type T will coincide with the machine numbers of the type T. In less perfect hardware, it is not possible for the model-oriented attributes to have these optimal values, since the hardware, by definition, and therefore the implementation, cannot conform to the stringencies of the resulting model; in this case, the values yielded by the model-oriented parameters have to be made more conservative (i.e., have to be penalized), with the result that the model numbers are more widely separated than the machine numbers, and the safe range is a subrange of the base range. The implementation will then be able to conform to the requirements of the weaker model defined by the sparser set of model numbers and the smaller safe range. 
{model interval} A model interval of a floating point type is any interval whose bounds are model numbers of the type. {model interval (associated with a value)} The model interval of a type T associated with a value v is the smallest model interval of T that includes v. (The model interval associated with a model number of a type consists of that number only.) 

Implementation Requirements

The accuracy requirements for the evaluation of certain predefined operations of floating point types are as follows. 
Discussion: This subclause does not cover the accuracy of an operation of a static expression; such operations have to be evaluated exactly (see 4.9). It also does not cover the accuracy of the predefined attributes of a floating point subtype that yield a value of the type; such operations also yield exact results (see 3.5.8 and A.5.3).
{operand interval} An operand interval is the model interval, of the type specified for the operand of an operation, associated with the value of the operand.
For any predefined arithmetic operation that yields a result of a floating point type T, the required bounds on the result are given by a model interval of T (called the result interval) defined in terms of the operand values as follows: 
The result interval of an exponentiation is obtained by applying the above rule to the sequence of multiplications defined by the exponent, assuming arbitrary association of the factors, and to the final division in the case of a negative exponent.
The result interval of a conversion of a numeric value to a floating point type T is the model interval of T associated with the operand value, except when the source expression is of a fixed point type with a small that is not a power of T'Machine_Radix or is a fixed point multiplication or division either of whose operands has a small that is not a power of T'Machine_Radix; in these cases, the result interval is implementation defined. 
Implementation defined: The result interval in certain cases of fixed-to-float conversion.
{Overflow_Check [partial]} {check, language-defined (Overflow_Check)} For any of the foregoing operations, the implementation shall deliver a value that belongs to the result interval when both bounds of the result interval are in the safe range of the result type T, as determined by the values of T'Safe_First and T'Safe_Last; otherwise, 
Implementation defined: The result of a floating point arithmetic operation in overflow situations, when the Machine_Overflows attribute of the result type is False.
For any predefined relation on operands of a floating point type T, the implementation may deliver any value (i.e., either True or False) obtained by applying the (exact) mathematical comparison to values arbitrarily chosen from the respective operand intervals.
The result of a membership test is defined in terms of comparisons of the operand value with the lower and upper bounds of the given range or type mark (the usual rules apply to these comparisons).

Implementation Permissions

If the underlying floating point hardware implements division as multiplication by a reciprocal, the result interval for division (and exponentiation by a negative exponent) is implementation defined.
Implementation defined: The result interval for division (or exponentiation by a negative exponent), when the floating point hardware implements division as multiplication by a reciprocal.

Wording Changes from Ada 83

The Ada 95 model numbers of a floating point type that are in the safe range of the type are comparable to the Ada 83 safe numbers of the type. There is no analog of the Ada 83 model numbers. The Ada 95 model numbers, when not restricted to the safe range, are an infinite set. 

Inconsistencies With Ada 83

{inconsistencies with Ada 83} Giving the model numbers the hardware radix, instead of always a radix of two, allows (in conjunction with other changes) some borderline declared types to be represented with less precision than in Ada 83 (i.e., with single precision, whereas Ada 83 would have used double precision). Because the lower precision satisfies the requirements of the model (and did so in Ada 83 as well), this change is viewed as a desirable correction of an anomaly, rather than a worrisome inconsistency. (Of course, the wider representation chosen in Ada 83 also remains eligible for selection in Ada 95.)
As an example of this phenomenon, assume that Float is represented in single precision and that a double precision type is also available. Also assume hexadecimal hardware with clean properties, for example certain IBM hardware. Then, 
type T is digits Float'Digits range -Float'Last .. Float'Last;
results in T being represented in double precision in Ada 83 and in single precision in Ada 95. The latter is intuitively correct; the former is counterintuitive. The reason why the double precision type is used in Ada 83 is that Float has model and safe numbers (in Ada 83) with 21 binary digits in their mantissas, as is required to model the hypothesized hexadecimal hardware using a binary radix; thus Float'Last, which is not a model number, is slightly outside the range of safe numbers of the single precision type, making that type ineligible for selection as the representation of T even though it provides adequate precision. In Ada 95, Float'Last (the same value as before) is a model number and is in the safe range of Float on the hypothesized hardware, making Float eligible for the representation of T. 

Extensions to Ada 83

{extensions to Ada 83} Giving the model numbers the hardware radix allows for practical implementations on decimal hardware. 

Wording Changes from Ada 83

The wording of the model of floating point arithmetic has been simplified to a large extent. 

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