- 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).
- 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.
- 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.)
- A model interval of a floating point type is any interval whose bounds
are model numbers of the type. 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
- The accuracy requirements for the evaluation of certain predefined
operations of floating point types are as follows.
- 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 is the smallest model interval of T that
includes the minimum and the maximum of all the values obtained
by applying the (exact) mathematical operation to values
arbitrarily selected from the respective operand intervals.
- 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
- 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,
- if T'Machine_Overflows is True, the implementation shall either
deliver a value that belongs to the result interval or raise
- if T'Machine_Overflows is False, the result is implementation
- 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).
- 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.
-- Email comments, additions, corrections, gripes, kudos, etc. to:
Magnus Kempe -- Magnus.Kempe@di.epfl.ch
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