3.5.9 Fixed Point Types

1
A fixed point type is either an ordinary fixed point type, or a decimal fixed point type. The error bound of a fixed point type is specified as an absolute value, called the delta of the fixed point type.

Syntax

2
fixed_point_definition ::= ordinary_fixed_point_definition | decimal_fixed_point_definition
3
ordinary_fixed_point_definition ::=
delta static_expression  real_range_specification
4
decimal_fixed_point_definition ::=
delta static_expression digits static_expression [real_range_specification]
5
digits_constraint ::=
digits static_expression [range_constraint]

Name Resolution Rules

6
For a type defined by a fixed_point_definition, the delta of the type is specified by the value of the expression given after the reserved word delta; this expression is expected to be of any real type. For a type defined by a decimal_fixed_point_definition (a decimal fixed point type), the number of significant decimal digits for its first subtype (the digits of the first subtype) is specified by the expression given after the reserved word digits; this expression is expected to be of any integer type.

Legality Rules

7
In a fixed_point_definition or digits_constraint, the expressions given after the reserved words delta and digits shall be static; their values shall be positive.
8/2
{AI95-00100-01} The set of values of a fixed point type comprise the integral multiples of a number called the small of the type. The machine numbers of a fixed point type are the values of the type that can be represented exactly in every unconstrained variable of the type. For a type defined by an ordinary_fixed_point_definition (an ordinary fixed point type), the small may be specified by an attribute_definition_clause (see 13.3); if so specified, it shall be no greater than the delta of the type. If not specified, the small of an ordinary fixed point type is an implementation-defined power of two less than or equal to the delta
8.a
Implementation defined: The small of an ordinary fixed point type.
9
For a decimal fixed point type, the small equals the delta; the delta shall be a power of 10. If a real_range_specification is given, both bounds of the range shall be in the range –(10**digits–1)*delta .. +(10**digits–1)*delta.
10
A fixed_point_definition is illegal if the implementation does not support a fixed point type with the given small and specified range or digits
10.a
Implementation defined: What combinations of small, range, and digits are supported for fixed point types.
11
For a subtype_indication with a digits_constraint, the subtype_mark shall denote a decimal fixed point subtype.
11.a
To be honest: Or, as an obsolescent feature, a floating point subtype is permitted — see J.3.

Static Semantics

12
The base range (see 3.5) of a fixed point type is symmetric around zero, except possibly for an extra negative value in some implementations.
13
An ordinary_fixed_point_definition defines an ordinary fixed point type whose base range includes at least all multiples of small that are between the bounds specified in the real_range_specification. The base range of the type does not necessarily include the specified bounds themselves. An ordinary_fixed_point_definition also defines a constrained first subtype of the type, with each bound of its range given by the closer to zero of:
14
the value of the conversion to the fixed point type of the corresponding expression of the real_range_specification;
14.a.1/1
To be honest: The conversion mentioned above is not an implicit subtype conversion (which is something that happens at overload resolution, see 4.6), although it happens implicitly. Therefore, the freezing rules are not invoked on the type (which is important so that representation items can be given for the type).
15
the corresponding bound of the base range.
16
A decimal_fixed_point_definition defines a decimal fixed point type whose base range includes at least the range –(10**digits–1)*delta .. +(10**digits–1)*delta. A decimal_fixed_point_definition also defines a constrained first subtype of the type. If a real_range_specification is given, the bounds of the first subtype are given by a conversion of the values of the expressions of the real_range_specification. Otherwise, the range of the first subtype is –(10**digits–1)*delta .. +(10**digits–1)*delta.
16.a.1/1
To be honest: The conversion mentioned above is not an implicit subtype conversion (which is something that happens at overload resolution, see 4.6), although it happens implicitly. Therefore, the freezing rules are not invoked on the type (which is important so that representation items can be given for the type).

Dynamic Semantics

17
The elaboration of a fixed_point_definition creates the fixed point type and its first subtype.
18
For a digits_constraint on a decimal fixed point subtype with a given delta, if it does not have a range_constraint, then it specifies an implicit range –(10**D–1)*delta .. +(10**D–1)*delta, where D is the value of the expression. A digits_constraint is compatible with a decimal fixed point subtype if the value of the expression is no greater than the digits of the subtype, and if it specifies (explicitly or implicitly) a range that is compatible with the subtype.
18.a
Discussion: Except for the requirement that the digits specified be no greater than the digits of the subtype being constrained, a digits_constraint is essentially equivalent to a range_constraint.
18.b
Consider the following example:
18.c
type D is delta 0.01 digits 7 range -0.00 .. 9999.99;
18.d/1
The compatibility rule implies that the digits_constraint "digits 6" specifies an implicit range of "–9999.99 99.9999 .. 9999.99 99.9999". Thus, "digits 6" is not compatible with the constraint of D, but "digits 6 range 0.00 .. 9999.99" is compatible.
18.e/2
{AI95-00114-01} A value of a scalar type belongs to a constrained subtype of the type if it belongs to the range of the subtype. Attributes like Digits and Delta have no effect affect on this fundamental rule. So the obsolescent forms of digits_constraints and delta_constraints that are called “accuracy constraints” in RM83 don't really represent constraints on the values of the subtype, but rather primarily affect compatibility of the “constraint” with the subtype being “constrained.” In this sense, they might better be called “subtype assertions” rather than “constraints.”
18.f
Note that the digits_constraint on a decimal fixed point subtype is a combination of an assertion about the digits of the subtype being further constrained, and a constraint on the range of the subtype being defined, either explicit or implicit.
19
The elaboration of a digits_constraint consists of the elaboration of the range_constraint, if any. If a range_constraint is given, a check is made that the bounds of the range are both in the range –(10**D–1)*delta .. +(10**D–1)*delta, where D is the value of the (static) expression given after the reserved word digits. If this check fails, Constraint_Error is raised.

Implementation Requirements

20
The implementation shall support at least 24 bits of precision (including the sign bit) for fixed point types.
20.a
Reason: This is sufficient to represent Standard.Duration with a small no more than 50 milliseconds.

Implementation Permissions

21
Implementations are permitted to support only smalls that are a power of two. In particular, all decimal fixed point type declarations can be disallowed. Note however that conformance with the Information Systems Annex requires support for decimal smalls, and decimal fixed point type declarations with digits up to at least 18.
21.a
Implementation Note: The accuracy requirements for multiplication, division, and conversion (see G.2.1, “Model of Floating Point Arithmetic”) are such that support for arbitrary smalls should be practical without undue implementation effort. Therefore, implementations should support fixed point types with arbitrary values for small (within reason). One reasonable limitation would be to limit support to fixed point types that can be converted to the most precise floating point type without loss of precision (so that Fixed_IO is implementable in terms of Float_IO).
NOTES
22
41  The base range of an ordinary fixed point type need not include the specified bounds themselves so that the range specification can be given in a natural way, such as:
23
type Fraction is delta 2.0**(-15) range -1.0 .. 1.0;

24
With 2's complement hardware, such a type could have a signed 16-bit representation, using 1 bit for the sign and 15 bits for fraction, resulting in a base range of –1.0 .. 1.0–2.0**(–15).

Examples

25
Examples of fixed point types and subtypes:
26
type Volt is delta 0.125 range 0.0 .. 255.0;
27
-- A pure fraction which requires all the available
-- space in a word can be declared as the type Fraction:
type Fraction is delta System.Fine_Delta range -1.0 .. 1.0;
-- Fraction'Last = 1.0 – System.Fine_Delta
28
type Money is delta 0.01 digits 15;  -- decimal fixed point
subtype Salary is Money digits 10;
-- Money'Last = 10.0**13 – 0.01, Salary'Last = 10.0**8 – 0.01

28.a
In Ada 95, S'Small always equals S'Base'Small, so if an implementation chooses a small for a fixed point type smaller than required by the delta, the value of S'Small in Ada 95 might not be the same as it was in Ada 83.

28.b/3
{AI05-0005-1} Decimal fixed point types are new, though their capabilities are essentially similar to that available in Ada 83 with a fixed point type whose small equals its delta and both are equals a power of 10. However, in the Information Systems Annex, additional requirements are placed on the support of decimal fixed point types (e.g. a minimum of 18 digits of precision). Ada 2005 and 2012 Editions sponsored in part by Ada-Europe