4.5.5 Multiplying Operators
Static Semantics
The
multiplying operators * (multiplication), / (division),
mod (modulus),
and
rem (remainder) are predefined for every specific integer
type
T:
function "*" (Left, Right : T) return T
function "/" (Left, Right : T) return T
function "mod"(Left, Right : T) return T
function "rem"(Left, Right : T) return T
Signed integer multiplication has its conventional
meaning.
Signed integer division
and remainder are defined by the relation:
A = (A/B)*B + (A rem B)
where (A rem
B) has the sign of A and an absolute value less than the absolute value
of B. Signed integer division satisfies the identity:
(-A)/B = -(A/B) = A/(-B)
{
AI05-0260-1}
The signed integer modulus operator is defined such that the result of
A
mod B is either zero, or has the sign of B and an absolute value
less than the absolute value of B; in addition, for some signed integer
value N, this result satisfies the relation:
A = B*N + (A mod B)
The multiplying operators on modular types are
defined in terms of the corresponding signed integer operators[, followed
by a reduction modulo the modulus if the result is outside the base range
of the type] [(which is only possible for the "*" operator)].
Ramification: The above identity satisfied
by signed integer division is not satisfied by modular division because
of the difference in effect of negation.
Multiplication and
division operators are predefined for every specific floating point type
T:
function "*"(Left, Right : T) return T
function "/"(Left, Right : T) return T
The following multiplication
and division operators, with an operand of the predefined type Integer,
are predefined for every specific fixed point type T:
function "*"(Left : T; Right : Integer) return T
function "*"(Left : Integer; Right : T) return T
function "/"(Left : T; Right : Integer) return T
[All of the above multiplying
operators are usable with an operand of an appropriate universal numeric
type.] The following additional multiplying operators for
root_real
are predefined[, and are usable when both operands are of an appropriate
universal or root numeric type, and the result is allowed to be of type
root_real, as in a
number_declaration]:
Ramification: These operators are analogous
to the multiplying operators involving fixed or floating point types
where root_real substitutes for the fixed or floating point type,
and root_integer substitutes for Integer. Only values of the corresponding
universal numeric types are implicitly convertible to these root numeric
types, so these operators are really restricted to use with operands
of a universal type, or the specified root numeric types.
function "*"(Left, Right : root_real) return root_real
function "/"(Left, Right : root_real) return root_real
function "*"(Left : root_real; Right : root_integer) return root_real
function "*"(Left : root_integer; Right : root_real) return root_real
function "/"(Left : root_real; Right : root_integer) return root_real
Multiplication and
division between any two fixed point types are provided by the following
two predefined operators:
Ramification: {
AI12-0005-1}
Universal_fixed is the universal type for the class of fixed point
types, meaning that these operators take operands of any fixed point
type (not necessarily the same) and return a result that is implicitly
(or explicitly) convertible to any fixed point type.
function "*"(Left, Right : universal_fixed) return universal_fixed
function "/"(Left, Right : universal_fixed) return universal_fixed
Name Resolution Rules
{
AI95-00364-01}
{
AI95-00420-01}
The above two fixed-fixed multiplying operators shall not be used in
a context where the expected type for the result is itself
universal_fixed
[— the context has to identify some other numeric type to which
the result is to be converted, either explicitly or implicitly]. Unless
the predefined universal operator is identified using an expanded name
with
prefix
denoting the package Standard, an explicit conversion is required on
the result when using the above fixed-fixed multiplication operator if
either operand is of a type having a user-defined primitive multiplication
operator such that:
{
AI05-0020-1}
{
AI05-0209-1}
it is declared immediately within the same declaration list as the type
or any partial or incomplete view thereof; and
both of its formal parameters are of a fixed-point
type.
Discussion: The small of universal_fixed
is infinitesimal; no loss of precision is permitted. However, fixed-fixed
division is impractical to implement when an exact result is required,
and multiplication will sometimes result in unanticipated overflows in
such circumstances, so we require an explicit conversion to be inserted
in expressions like A * B * C if A, B, and C are each of some fixed point
type.
On the other hand, X := A * B; is permitted
by this rule, even if X, A, and B are all of different fixed point types,
since the expected type for the result of the multiplication is the type
of X, which is necessarily not universal_fixed.
{
AI95-00364-01}
{
AI95-00420-01}
We have made these into Name Resolution rules to ensure that user-defined
primitive fixed-fixed operators are not made unusable due to the presence
of these universal fixed-fixed operators. But we do allow these operators
to be used if prefixed by package Standard, so that they can be used
in the definitions of user-defined operators.
Paragraph 20 was
deleted.
Dynamic Semantics
The multiplication and division operators for real
types have their conventional meaning. [For floating point types, the
accuracy of the result is determined by the precision of the result type.
For decimal fixed point types, the result is truncated toward zero if
the mathematical result is between two multiples of the
small
of the specific result type (possibly determined by context); for ordinary
fixed point types, if the mathematical result is between two multiples
of the
small, it is unspecified which of the two is the result.
]
The
exception Constraint_Error is raised by integer division,
rem,
and
mod if the right operand is zero. [Similarly, for a real type
T with
T'Machine_Overflows True, division by zero raises
Constraint_Error.]
NOTE 1 For
positive A and B, A/B is the quotient and A rem B is the remainder
when A is divided by B. The following relations are satisfied by the
rem operator:
A rem (-B) = A rem B
(-A) rem B = -(A rem B)
NOTE 2 For
any signed integer K, the following identity holds:
A mod B = (A + K*B) mod B
The relations between
signed integer division, remainder, and modulus are illustrated by the
following table:
A B A/B A rem B A mod B A B A/B A rem B A mod B
10 5 2 0 0 -10 5 -2 0 0
11 5 2 1 1 -11 5 -2 -1 4
12 5 2 2 2 -12 5 -2 -2 3
13 5 2 3 3 -13 5 -2 -3 2
14 5 2 4 4 -14 5 -2 -4 1
A B A/B A rem B A mod B A B A/B A rem B A mod B
10 -5 -2 0 0 -10 -5 2 0 0
11 -5 -2 1 -4 -11 -5 2 -1 -1
12 -5 -2 2 -3 -12 -5 2 -2 -2
13 -5 -2 3 -2 -13 -5 2 -3 -3
14 -5 -2 4 -1 -14 -5 2 -4 -4
Examples
Examples of expressions
involving multiplying operators:
I : Integer := 1;
J : Integer := 2;
K : Integer := 3;
X : Real := 1.0; --
see 3.5.7
Y : Real := 2.0;
F : Fraction := 0.25; --
see 3.5.9
G : Fraction := 0.5;
Expression Value Result Type
I*J 2 same as I and J, that is, Integer
K/J 1 same as K and J, that is, Integer
K mod J 1 same as K and J, that is, Integer
X/Y 0.5 same as X and Y, that is, Real
F/2 0.125 same as F, that is, Fraction
3*F 0.75 same as F, that is, Fraction
0.75*G 0.375 universal_fixed, implicitly convertible
to any fixed point type
Fraction(F*G) 0.125 Fraction, as stated by the conversion
Real(J)*Y 4.0 Real, the type of both operands after
conversion of J
Incompatibilities With Ada 83
{
AI95-00364-01}
{
AI95-00420-01}
The universal fixed-fixed multiplying operators are
now directly available (see below). Any attempt to use user-defined fixed-fixed
multiplying operators will be ambiguous with the universal ones. The
only way to use the user-defined operators is to fully qualify them in
a prefix call. This problem was not documented during the design of Ada
95, and has been mitigated by Ada 2005.
Extensions to Ada 83
Explicit conversion of the
result of multiplying or dividing two fixed point numbers is no longer
required, provided the context uniquely determines some specific fixed
point result type. This is to improve support for decimal fixed point,
where requiring explicit conversion on every fixed-fixed multiply or
divide was felt to be inappropriate.
The type universal_fixed is covered by
universal_real, so real literals and fixed point operands may
be multiplied or divided directly, without any explicit conversions required.
Wording Changes from Ada 83
We have used the normal syntax for function
definition rather than a tabular format.
Incompatibilities With Ada 95
{
AI95-00364-01}
We have changed the resolution rules for the universal
fixed-fixed multiplying operators to remove the incompatibility with
Ada 83 discussed above. The solution is to hide the universal operators
in some circumstances. As a result, some legal Ada 95 programs will require
the insertion of an explicit conversion around a fixed-fixed multiply
operator. This change is likely to catch as many bugs as it causes, since
it is unlikely that the user wanted to use predefined operators when
they had defined user-defined versions.
Wording Changes from Ada 2005
{
AI05-0020-1}
{
AI05-0209-1}
Correction: Wording was added to clarify that
universal_fixed
"*" and "/" does not apply if an appropriate operator
is declared for a partial (or incomplete) view of the designated type.
Otherwise, adding a partial (or incomplete) view could make some "*"
and "/" operators ambiguous.
{
AI05-0260-1}
Correction: The wording for the
mod operator was corrected
so that a result of 0 does not have to have "the sign of B"
(which is impossible if B is negative).
Ada 2005 and 2012 Editions sponsored in part by Ada-Europe
