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G.2.3 Model of Fixed Point Arithmetic

1
In the strict mode, the predefined arithmetic operations of a fixed point type shall satisfy the accuracy requirements specified here and shall avoid or signal overflow in the situations described.

Implementation Requirements

2
The accuracy requirements for the predefined fixed point arithmetic operations and conversions, and the results of relations on fixed point operands, are given below. 
2.a
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). 
3
The operands of the fixed point adding operators, absolute value, and comparisons have the same type. These operations are required to yield exact results, unless they overflow.
4
Multiplications and divisions are allowed between operands of any two fixed point types; the result has to be (implicitly or explicitly) converted to some other numeric type. For purposes of defining the accuracy rules, the multiplication or division and the conversion are treated as a single operation whose accuracy depends on three types (those of the operands and the result). For decimal fixed point types, the attribute T'Round may be used to imply explicit conversion with rounding (see 3.5.10).
5
When the result type is a floating point type, the accuracy is as given in G.2.1. For some combinations of the operand and result types in the remaining cases, the result is required to belong to a small set of values called the perfect result set; for other combinations, it is required merely to belong to a generally larger and implementation-defined set of values called the close result set. When the result type is a decimal fixed point type, the perfect result set contains a single value; thus, operations on decimal types are always fully specified.
5.a
Implementation defined: The definition of close result set, which determines the accuracy of certain fixed point multiplications and divisions.
6
When one operand of a fixed-fixed multiplication or division is of type universal_real, that operand is not implicitly converted in the usual sense, since the context does not determine a unique target type, but the accuracy of the result of the multiplication or division (i.e., whether the result has to belong to the perfect result set or merely the close result set) depends on the value of the operand of type universal_real and on the types of the other operand and of the result. 
6.a
Discussion: We need not consider here the multiplication or division of two such operands, since in that case either the operation is evaluated exactly (i.e., it is an operation of a static expression all of whose operators are of a root numeric type) or it is considered to be an operation of a floating point type. 
7
For a fixed point multiplication or division whose (exact) mathematical result is v, and for the conversion of a value v to a fixed point type, the perfect result set and close result set are defined as follows: 
8
If the result type is an ordinary fixed point type with a small of s,
9
if v is an integer multiple of s, then the perfect result set contains only the value v;
10
otherwise, it contains the integer multiple of s just below v and the integer multiple of s just above v
11
The close result set is an implementation-defined set of consecutive integer multiples of s containing the perfect result set as a subset.
12
If the result type is a decimal type with a small of s,
13
if v is an integer multiple of s, then the perfect result set contains only the value v;
14/3
{AI05-0264-1} otherwise, if truncation applies, then it contains only the integer multiple of s in the direction toward zero, whereas if rounding applies, then it contains only the nearest integer multiple of s (with ties broken by rounding away from zero). 
15
The close result set is an implementation-defined set of consecutive integer multiples of s containing the perfect result set as a subset. 
15.a
Ramification: As a consequence of subsequent rules, this case does not arise when the operand types are also decimal types. 
16
If the result type is an integer type, 
17
if v is an integer, then the perfect result set contains only the value v;
18
otherwise, it contains the integer nearest to the value v (if v lies equally distant from two consecutive integers, the perfect result set contains the one that is further from zero). 
19
The close result set is an implementation-defined set of consecutive integers containing the perfect result set as a subset.
20
The result of a fixed point multiplication or division shall belong either to the perfect result set or to the close result set, as described below, if overflow does not occur. In the following cases, if the result type is a fixed point type, let s be its small; otherwise, i.e. when the result type is an integer type, let s be 1.0. 
21
For a multiplication or division neither of whose operands is of type universal_real, let l and r be the smalls of the left and right operands. For a multiplication, if (l · r) / s is an integer or the reciprocal of an integer (the smalls are said to be “compatible” in this case), the result shall belong to the perfect result set; otherwise, it belongs to the close result set. For a division, if l / (r · s) is an integer or the reciprocal of an integer (i.e., the smalls are compatible), the result shall belong to the perfect result set; otherwise, it belongs to the close result set. 
21.a
Ramification: When the operand and result types are all decimal types, their smalls are necessarily compatible; the same is true when they are all ordinary fixed point types with binary smalls
22
For a multiplication or division having one universal_real operand with a value of v, note that it is always possible to factor v as an integer multiple of a “compatible” small, but the integer multiple may be “too big.” If there exists a factorization in which that multiple is less than some implementation-defined limit, the result shall belong to the perfect result set; otherwise, it belongs to the close result set. 
22.a
Implementation defined: Conditions on a universal_real operand of a fixed point multiplication or division for which the result shall be in the perfect result set.
23/5
{AI12-0300-1} A multiplication P * Q of an operand of a fixed point type F by an operand of type Integer an integer type I, or vice -versa, and a division P / Q of an operand of a fixed point type F by an operand of type Integer an integer type I, are also allowed. In these cases, the result has the a type of F; explicit conversion of the result is never required. The accuracy required in these cases is the same as that required for a multiplication F(P * Q) or a division F(P / Q) obtained by interpreting the operand of the integer type to have a fixed point type with a small of 1.0.
24
The accuracy of the result of a conversion from an integer or fixed point type to a fixed point type, or from a fixed point type to an integer type, is the same as that of a fixed point multiplication of the source value by a fixed point operand having a small of 1.0 and a value of 1.0, as given by the foregoing rules. The result of a conversion from a floating point type to a fixed point type shall belong to the close result set. The result of a conversion of a universal_real operand to a fixed point type shall belong to the perfect result set.
25
The possibility of overflow in the result of a predefined arithmetic operation or conversion yielding a result of a fixed point type T is analogous to that for floating point types, except for being related to the base range instead of the safe range. If all of the permitted results belong to the base range of T, then the implementation shall deliver one of the permitted results; otherwise,
26
if T'Machine_Overflows is True, the implementation shall either deliver one of the permitted results or raise Constraint_Error;
27
if T'Machine_Overflows is False, the result is implementation defined. 
27.a
Implementation defined: The result of a fixed point arithmetic operation in overflow situations, when the Machine_Overflows attribute of the result type is False.

Inconsistencies With Ada 83

27.b
Since the values of a fixed point type are now just the integer multiples of its small, the possibility of using extra bits available in the chosen representation for extra accuracy rather than for increasing the base range would appear to be removed, raising the possibility that some fixed point expressions will yield less accurate results than in Ada 83. However, this is partially offset by the ability of an implementation to choose a smaller default small than before. Of course, if it does so for a type T then T'Small will have a different value than it previously had.
27.c
The accuracy requirements in the case of incompatible smalls are relaxed to foster wider support for nonbinary smalls. If this relaxation is exploited for a type that was previously supported, lower accuracy could result; however, there is no particular incentive to exploit the relaxation in such a case. 

Wording Changes from Ada 83

27.d
The fixed point accuracy requirements are now expressed without reference to model or safe numbers, largely because the full generality of the former model was never exploited in the case of fixed point types (particularly in regard to operand perturbation). Although the new formulation in terms of perfect result sets and close result sets is still verbose, it can be seen to distill down to two cases: 
27.e
a case where the result must be the exact result, if the exact result is representable, or, if not, then either one of the adjacent values of the type (in some subcases only one of those adjacent values is allowed);
27.f
a case where the accuracy is not specified by the language. 

Wording Changes from Ada 2012

27.g/5
{AI12-0300-1} Correction: Reworded the fixed*integer accuracy requirements to clarify that the only allowed integer type in such operations is Standard.Integer. We make this correction as readers of the Standard have been confused on this point. 

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